Plasma Dynamics for Aerospace Engineering 9781108292566

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Plasma Dynamics for Aerospace Engineering This valuable resource summarizes the past fifty years’ basic research accom­ plishments in plasma dynamics for aerospace engineering, presenting these results in a comprehensive volume that will be an asset to any professional in the field. It offers a comprehensive review of the foundation of plasma dynamics while inte­ grating the most recently developed modelling and simulation techniques with the theoretic physics, including the state-of-the-art numerical algorithms. Several first-ever demonstrations for innovations and incisive explanations for previously unexplained observations are included. All the necessary formulations for technical evaluation to engineering applications are derived from the first principle by statistic and quantum mechanics, and led to physics-based computational simulations for practical applications. The computer-aided procedures directly engage the reader to duplicate findings that are nearly impossible by using ground-based experimental facilities. Plasma Dynamics for Aerospace Engineering will allow readers to reach an incisive understanding of plasma physics. Dr. Joseph J. S. Shang is Emeritus Research Professor at Wright State University, Ohio, and Emeritus Senior Scientist of the Air Force Research Laboratory. His research focuses on computational fluid dynamics and computational electro­ dynamics and plasma dynamics. He is a Fellow of the American Institute of Aeronautics and Astronautics (AIAA). Professor Sergey T.  Surzhikov is Director of the Institute for Problems in Mechanics, Russian Academy of Sciences. He is also Professor and Head of the Physical-Chemical Mechanics Department of the Moscow Institute for Physics and Technology. He is a Fellow of the American Institute of Aeronautics and Astronautics (AIAA).

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Cambridge Aerospace Series Editors: Wei Shyy and Vigor Yang 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

J. M. Rolfe and K. J. Staples (eds.): Flight Simulation P. Berlin: The Geostationary Applications Satellite M. J. T. Smith: Aircraft Noise N. X. Vinh: Flight Mechanics of High-​Performance Aircraft W. A. Mair and D. L. Birdsall: Aircraft Performance M. J. Abzug and E. E. Larrabee: Airplane Stability and Control M. J. Sidi: Spacecraft Dynamics and Control J. D. Anderson: A History of Aerodynamics A. M. Cruise, J. A. Bowles, C. V. Goodall, and T. J. Patrick:  Principles of Space Instrument Design G. A. Khoury (ed.): Airship Technology, Second Edition J. P. Fielding: Introduction to Aircraft Design J. G. Leishman: Principles of Helicopter Aerodynamics, Second Edition J. Katz and A. Plotkin: Low-​Speed Aerodynamics, Second Edition M. J. Abzug and E. E. Larrabee:  Airplane Stability and Control:  A History of the Technologies that Made Aviation Possible, Second Edition D. H. Hodges and G. A. Pierce: Introduction to Structural Dynamics and Aeroelasticity, Second Edition W. Fehse: Automatic Rendezvous and Docking of Spacecraft R. D. Flack: Fundamentals of Jet Propulsion with Applications E. A. Baskharone: Principles of Turbomachinery in Air-​Breathing Engines D. D. Knight: Numerical Methods for High-​Speed Flows C. A. Wagner, T. Hüttl, and P. Sagaut (eds.): Large-​Eddy Simulation for Acoustics D. D. Joseph, T. Funada, and J. Wang: Potential Flows of Viscous and Viscoelastic Fluids W. Shyy, Y. Lian, H. Liu, J. Tang, and D. Viieru: Aerodynamics of Low Reynolds Number Flyers J. H. Saleh: Analyses for Durability and System Design Lifetime B. K. Donaldson: Analysis of Aircraft Structures, Second Edition C. Segal: The Scramjet Engine: Processes and Characteristics J. F. Doyle: Guided Explorations of the Mechanics of Solids and Structures A. K. Kundu: Aircraft Design M. I. Friswell, J. E. T. Penny, S. D. Garvey, and A. W. Lees: Dynamics of Rotating Machines B. A. Conway (ed): Spacecraft Trajectory Optimization R. J. Adrian and J. Westerweel: Particle Image Velocimetry G. A. Flandro, H. M. McMahon, and R. L. Roach: Basic Aerodynamics H. Babinsky and J. K. Harvey: Shock Wave–​Boundary-​Layer Interactions C. K. W. Tam: Computational Aeroacoustics: A Wave Number Approach A. Filippone: Advanced Aircraft Flight Performance I. Chopra and J. Sirohi: Smart Structures Theory W. Johnson: Rotorcraft Aeromechanics vol. 3 W. Shyy, H. Aono, C. K. Kang, and H. Liu: An Introduction to Flapping Wing Aerodynamics T. C. Lieuwen and V. Yang: Gas Turbine Emissions P. Kabamba and A. Girard: Fundamentals of Aerospace Navigation and Guidance R. M. Cummings, W. H. Mason, S. A. Morton, and D. R. McDaniel:  Applied Computational Aerodynamic P. G. Tucker: Advanced Computational Fluid and Aerodynamics Iain D. Boyd and Thomas E. Schwartzentruber:  Nonequilibrium Gas Dynamics and Molecular Simulation Joseph J. S. Shang and Sergey T. Surzhikov: Plasma Dynamics for Aerospace Engineering Bijay K. Sultanian: Gas Turbines: Internal Flow Systems Modeling

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Plasma Dynamics for Aerospace Engineering J O S EP H J .   S.   SH AN G Wright State University

S E R G EY T.   SURZH IKO V Russian Academy of Sciences

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06-​04/​06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108418973 DOI: 10.1017/9781108292566 © Joseph J. S. Shang and Sergey T. Surzhikov 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United States of America by Sheridan Books, Inc. A catalogue record for this publication is available from the British Library. ISBN 978-​1-​108-​41897-​3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-​party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Contents

Preface page xi 1

Plasma Physics Fundamentals

1

Introduction 1 1.1 Intrinsic Electromagnetic Forces 4 1.2 Charged Particle Motion 7 1.3 Debye Shielding Length 11 1.4 Plasma Sheath 14 1.5 Plasma Frequency 16 1.6 Magnetohydrodynamic Waves in Plasma 18 1.7 Landau Damping 22 1.8 Joule Heating 23 1.9 Plasma Kinetics Formulations 25 1.10 Electric Conductivity 27 1.11 Electric Conductivity in a Magnetic Field 29 1.12 Ambipolar Diffusion 31 References 35 2

Plasma in a Magnetic Field

36

Introduction 36 2.1 Hall Current and Parameter 39 2.2 Transverse Waves 41 2.3 Polarization of Electromagnetic Waves 44 2.4 Microwave Propagation in Plasma 49 2.5 Drift Diffusion in Transverse Magnetic Fields 56 2.6 Magnetic Mirrors 61 2.7 Plasma Pinch and Instability 65 References 69 3

Maxwell Equations

71

Introduction 71 3.1 Faraday, Generalized Ampere, and Gauss Laws 73 3.2 Maxwell Equations in the Time Domain 76

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Contents

3.3 Poisson Equation of Plasma Dynamics 81 3.4 Interface Boundary Conditions 82 3.5 Eigenvalues and Characteristic Variables 87 3.6 Characteristic Formulation 91 3.7 Far-​Field Boundary Conditions 96 3.8 High-​Resolution Numerical Algorithms 100 References 106 4

Plasma Dynamics Formulation

108

Introduction 108 4.1 Boltzmann-​Maxwell Equation 111 4.2 Fokker-​Plank Equation and Lorentz Approximation 115 4.3 Vlasov Equations for Collisionless Plasma 117 4.4 Multi-​temperature and Multi-​fluid Models 118 4.5 Low Magnetic Reynolds Number Formulation 127 4.6 Transport Properties via Kinetic Theory 131 4.7 Solving Procedures 137 References 144 5

Magnetohydrodynamics Equations

146

Introduction 146 5.1 Basic Assumptions of MHD 148 5.2 Generalized Ohm’s Law 150 5.3 Ideal MHD Equations 153 5.4 Eigenvalues and Electromagnetic Waves 160 5.5 Full MHD Equations 166 5.6 Similarity Parameters of MHD 171 5.7 Modified Rankine-​Hugoniot Shock Conditions 174 178 5.8 Classic Solutions of MHD Equations References 183 6

Ionization Processes in Gas

185

Introduction 185 6.1 Basic Ionization Mechanisms 188 6.2 Lighthill and Saha Equations 194 6.3 Electron Impact Ionization 197 6.4 Thermal Ionization by Chemical Kinetics 202 6.5 Inelastic Collision Ionization Models 210 6.6 Database of Chemical Kinetics 219 References 222

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Contents

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Plasma and Magnetic Field Generation

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225

Introduction 225 7.1 Direct Current Discharge 227 7.2 Dielectric Barrier Discharge 234 7.3 Shock Tubes 240 7.4 MHD Electric Generators 243 7.5 Arc Plasmatron 246 7.6 Induction Plasma Generators 248 7.7 Microwave Plasmatron 252 7.8 Plasma by Radiation 255 7.9 Magnetic Field Generations 258 References 263 8

Plasma Diagnostics

266

Introduction 266 8.1 Electrode Arrangements of Langmuir Probe 268 8.2 Data Reduction for Langmuir Probes 272 8.3 Emission Spectroscopy 276 8.4 Microwave Attenuation in Plasma 285 8.5 Microwave Dispersion in Plasma 289 8.6 Microwave Probing Simulations 292 8.7 Retarding Potential Analyzer 300 References 302 9

Radiative Energy Transfer

304

Introduction 304 9.1 Fundamental of Thermal Radiation 306 9.2 Integro-​differential Radiation Transfer Equation 309 313 9.3 Half-​Moment Method 9.4 Spherical Harmonic (PN) Method 316 9.5 Method of Discrete Ordinates 321 9.6 Governing Equations of Gas Dynamics Radiation 325 9.7 Ray-​Tracing Procedure 328 9.8 Monte Carlo Method 336 References 339 10 Applications

342

Introduction 342 10.1 Ion Thrusters 345 10.2 Reentry Thermal Protection 351 10.3 Plasma Actuators for Flow Control 361

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Contents

10.4 Remote Energy Deposition 369 10.5 Scramjet MHD Energy Bypass 373 10.6 Plasma-​Assisted Ignition and Combustion 375 References 378 Appendix: Physical Constants and Dimensions Index

383 385

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Preface

Infusing plasma physics into engineering applications was initiated as early as the 1940s. At the beginning it was tentatively used to investigate nuclear fusion and fission problems, and then it was applied to astrophysics and geophysics to better understand the earth’s magnetic field and Van Allen belts, as well as solar wind phenomena. In the later 1950s scientists and engineers pioneered space exploration; plasma science was used to answer the formidable engineering challenges presented by reentry thermal protection and telecommunication blackout issues. At the same time, the ion thruster was initialized for satellites’ station keeping and interplane­ tary flights. However, the plasma dynamics research for engineering applications has experienced a rise and ebb for the past fifty years. Scientists and engineers rejuvenated plasma research interest in the early 1990s by advocating the concept of magnetohydrodynamic-​bypass scramjet engines. Since then an extremely wide range of plasma applications was generated on remote energy deposition for flight vehicle drag reduction, flow control by plasma actuator, radiation-​driven hyper­ sonic wind tunnels, sonic boom mediation, and enhanced ignition and combus­ tion stability. The plasma applications in traditional sterilization and pasteurization now have also been extended to wound healing and tissue regeneration. Today plasma dynamic research for engineering applications has been sustained worldwide; any sizable technical symposia will have many sessions on this vigorous research arena. More than a thousand articles have been presented and published at international conferences and in professional journals. However, nearly all plasma only exists in a high-​enthalpy, nonequilibrium, thermodynamic state that is rarely supportable by ground-​based experimental facilities. Modeling and sim­ ulation capability for plasma dynamic research becomes mandatory for a scien­ tific discipline that must be studied in extremely demanding environments. Scholars have published excellent books and superb treatises on classic plasma dynamics, but very few neither bridge the gap between theoretical physics and computational simulations, nor summarize the most recent progress. Some of the current research efforts are still conducted in isolation to reveal the need for guidance in a productive research direction based on accumulated knowledge. More important, it is crucial to share and to disseminate basic knowledge for mutual support in science and engi­ neering. The present manuscript intends to carry out that humble service by sharing research accomplishments and converting the research results into knowledge for

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Preface

future advancement. For this reason, this text is prepared as a professional refer­ ence for researchers and, we hope, as a textbook for advanced engineering graduate study in plasma dynamics. For these purposes, a variety of multi-​physics models that couple quantum physical-​chemical and radiative processes with gas dynamics is assembled. In this aspect, the engineering approaches based on continuum mechanics for describing high enthalpy ionized gas include thermodynamics, statistical quantum physics, nonequilibrium chemical kinetics, and transport property via gas kinetic theory with computational fluid dynamics techniques. All these approaches are based on the Born-​Oppenheimer approximation involving a wide range of the reso­ nant energy exchange between quantum states. For flow control using electron impact ionization, the recent innovation is presented in detail by combining the classic drift-​diffusion theory with inelastic collision quantum-​chemistry models for ionized species generation and depletion through quantum transitions. The rigorous boundary conditions at the interfaces across electrodes and plasma are derived from the Maxwell equations, which explain fully the self-​limiting feature of the direct barrier charge that prevents transition from discharge to spark. In view of the diversity of all methodologies, the uncertainty in our knowledge of chemical kinetic modeling inviolably leads to many unresolved issues. At the same time, the reliable computational methods still have not yet been completely resolved, despite the significant progress researchers and scientists have made in recent decades. For radiative energy transfer, a systematic presentation from the classic multi-​ flux methods that build on the Milne-​Eddington approximation is highlighted. For the important interaction between radiation and gas dynamics, the direct compu­ tational simulation that interweaves the radiation rate equation with gas dynamics and quantum optic physics is included. The directly usable methods for volumetric emission to analyze optical thin and dense gas, as well as the influence by the fine structure of atomic spectral line to radiation transfer utilizing ray-​tracing and Monte Carlo techniques, are included as a state-​of-​the-​art assessment. This manuscript is organized in ten chapters; the first three chapters evolve around the basic electromagnetic phenomena and fundamental laws of plasma dynamics and electromagnetics. The linking of intrinsic characteristics of the Debye length and plasma frequency to the electrostatic force and the prolate cycloid charged particle motions in electromagnetic fields will be illustrated. The drastically altered electrical conductive property from a scalar to a tensor in an electromagnetic field is also derived. The unique and peculiar behavior of plasma under the influence of an external applied magnetic field that leads to the Hall current, magnetic mirror, and plasma confinement will also be highlighted. At this connection, a discussion of plasma instability is offered to illustrate the self-​constricted pinch effect by the interaction of the static pressure gradient and the magnetic force. In Chapter  3, the Maxwell equations and boundary conditions for plasma are articulated and emphasized for analytic studies and computational simulations. Furthermore, the different hydro-​electromagnetic wave speeds are shown by eigenvalue analysis for the system of hyperbolic partial differential equations. At the same time,

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Preface

xiii

the characteristic formulation is instituted to alleviate the inherent limitation of imposing far-​field boundary conditions for a boundless initial value and boundary condition problem. The fundamental plasma formulation for analytic and computational simulations is detailed in Chapter 4 from the first principle, and the best current practices are included. A  detailed and progressive evolution from the Boltzmann-​ Maxwell, Fokker-​ Plank, and Vlasov equations to the most frequently adopted multi-​ temperature and multi-​fluid plasma models will be systematically introduced. The important transport property of ionized gas will be described from the collision kinetics through the collision integrals with a screened Coulomb potential and verified by experimental observations. The inter-​nuclear interaction by inelastic collision kinetics is utilized to develop the drift-​diffusion theory as a viable plasma modeling for the electron impact ionization. The classic formulation of magnetohydrodynamics (MHD), which is built on the generalized Ohm’s law, is presented next to link the physical observations and approximated numerical simulations. One of the underlying assumptions is that neglecting the effect of displacement electric current actually limits the simulated plasma phenomena beneath the microwave frequency. Equally important, plasma is treated as a multicomponent fluid medium; the electric conductivity is simplified to become a scalar constant. Therefore, the formulation when applying for prac­ tical engineering applications is restricted to the low magnetic Reynolds number environment. Chapter 5 also points out the elegance of analytical clarity, and the limitations of computational simulations using magnetohydrodynamic equations and defining the valid scope for the classic approximations. In the next two chapters, the ionizations of gas and the plasma generation process are articulated. The collision transfer and the energy states are quantum restricted, but the individual mechanism still can be distinctively described. Chapter  6 delineates the basic mechanisms of ionization, which include elastic and inelastic collisions, electronic impact, radiative interaction, charge exchange, dissociation recombination, dielectric recombination, and electron attachment. The intercon­ nection between the microscopic particle dynamics and the macroscopic properties of the ionized gas is first illustrated by the statistic thermodynamics in equilibrium condition, then by nonequilibrium collisional kinetic theory, and it is finally treated by the general principle of computational quantum chemistry. The plasma genera­ tion processes for engineering applications are presented in Chapter 7, in which the plasma generations from electron impact, shock tube, arc plasmatron, electromag­ netic induction, microwave excitation, to ionization by radiation are included with descriptions of existent capabilities. In Chapter  8, all practical plasma diagnostic techniques are presented with supporting experimental observations and accuracy assessments. The plasma diagnostics techniques are systematically presented from the classic Langmuir probe to the nonintrusive approaches such as spectrometry and the microwave attenuation principle. The concept of the retarding potential analyzer is included for detecting the ion energy distribution in ion thrusters at high temperature

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Preface

conditions. Based on these scientific disciplines, the necessary quantification and qualification of plasma parameters for applications can be determined. Chapter 9 addresses the very important and the most complex issues of plasma emission, absorption, and scattering spectroscopic phenomenon by radiation. In theory, the fundamental approach must be built on the perturbation theory of quantum mechanics and a thorough understanding of atomic and molecular spectral structures and collision kinetics. Therefore, the discussion starts from the Boltzmann equation for photon dynamics, and then leads to the phenomenological approach and finally to the general practices of solving radiative energy transfer equations for engineering applications. The traditional methods of multi-​ flux approach  –​the half-​moment, spherical harmonics, and method of discrete ordi­ nate –​are included. The detailed and directly usable methods based on computa­ tional gas dynamic radiation by ray-​tracing and Monte Carlo methods are carefully assessed for their physical fidelity. Some of the cornerstone validating computa­ tional investigations and computational results are also presented and discussed. Chapter  10 summarizes the most viable applications of plasma dynamics to engineering; the current progress and unique features in electrostatic and Hall ion thrusters are incorporated and evaluated for their capability of and potential for interplanetary explorations. Only a brief, but critically important summary of radiation heat transfer for reentry thermal protection is presented. The plasma actuators for flow control either by Joule heating or by electrostatic and Lorentz forces are highlighted for best practices. Using plasma for enhancing ignition and maintaining combustion stability, together with the remote energy deposition by microwave, are brought out as the current innovation in plasma applications. Other possible applications, such as the magnetohydrodynamics scramjet bypass and pulsed plasma thrusters, are also included. Finally, last but not least, we would like to express our sincere appreciation to our colleagues in the scientific research community and our gratitude for the lifetime benefit of learning from each other.

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1

Plasma Physics Fundamentals

Introduction The accepted definition of plasma is that in an electrically conducting medium, all paired and closely positioned positive and negative charges shield each other from an externally applied electromagnetic field (Langmuir 1929). Therefore, plasma is the matter in a particular state that has the fundamental property of global electrical neutrality. The characteristic charge separation distance is the Debye shielding length and is the smallest length scale compared to all other macroscopic physical dimensions in plasma. Within this shielding sphere, the paired charged particles are not free from each other, but interact permanently, which leads to many intrinsic properties of the medium. Charged particles in plasma interact with each other primarily by Coulomb force, magnetic induction, and collisions with neutral particles. The probability of collision can be expressed in terms of an effective momentum transfer cross-​section area and the mean free path between collisions. The collision mechanism leads to the transport properties such as diffusion, mobility, and resistivity in the plasma. Plasma exists not only in gas and liquid but also in a solid conductor, except the electrons in a solid are closely bound but still can move within atomic or molecular structures between collisions. Although mass motion of charges does not generally take place, when an external electromagnetic field is applied, the dynamic effects can always be observed in electric conduction, the Hall effect, and polarization. Plasma is electrically global neutral, and is dominated by interactions of charged particles of opposite polarity within the Debye shielding length. A strong electrostatic force exists between the paired charges; any small perturbation to the equilibrium separation distance will trigger a high-​frequency oscillation by the restoring force. This oscillatory motion is referred to as the plasma frequency, which is distinguished from the lower-​frequency oscillation involving mass motion. In addition, plasma does not naturally conform to its surroundings and will alter its domain according to local and distant conditions by the Coulomb force and the Lorentz acceleration. The other unique feature from the quasi-​neutrality of plasma is that it stores inductive energy, and it contributes to resistance and inductance when the current circuit forms a closed loop. This characteristic is exemplified by the drastic change in electric conductivity σ through a strong response to electromagnetic fields.

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Plasma Physics Fundamentals

Figure 1.1  Classification for types of plasma.

Finally, the disturbance communication in plasma is not only by means of collision processes but also conveyed by transverse wave with phase velocities equal to the speed of light. Plasma in the gas phase has a negligible shear stress and therefore does not have a definite shape or volume. The electric-​conducting medium responds to electromagnetic fields, so plasma can be confined either by a solid container or by a magnetic field. Plasma appears in a wide range of formations such as electric arcs, filaments, micro-​discharges, multiple layers in the presence of shock waves, and cellular structure in outer space. The spontaneous formations of unique spatial features on a wide range of length scales manifest the complexity of plasma. In fact, plasma is the most common state of matter by volume and is the fourth most common state of matter in rarefied and intergalactic environments. Description of plasma is therefore usually by its energy state and degree of ionization. The classification of plasma is commonly recognized by the electron number density and temperature in electron volts or the static temperature as displaced in Figure 1.1. The topics of degenerated quantum plasma and relativistic plasma are beyond the scope of plasma dynamics for aerospace engineering. Therefore, present discussions are focused on plasma that exists around atmospheric pressure, and on mixed thermal and nonthermal conditions. According to convention, the thermal condition of plasma is based on the relative temperatures between electrons, ions, and neutral components. The nonthermal or cold plasma means that the ions and neutral particles have a much lower temperature than the electrons. For most gas discharges generated by electron impact, the electron temperature is at most 3 eV

3

Introduction

3

Figure 1.2  Range of electric conductivity and magnetic Reynolds numbers for common

engineering applications.

(around 3 × 105 105 K), while the ions and neutrals are at near room temperature (Raizer 1991). When ionization is achieved by a strong shock compression in high pressure, the temperatures of the plasma composition are on the same order, around ten thousand degrees Kelvin or higher, but are not always in thermodynamic nonequilibrium. Therefore, the plasma of interest for aerospace applications is generally limited to an electron number density up to 1020/​cm3 and an overall temperature lower than 106 K (Surzhikov 2013). In engineering applications, the relative magnitude of the electromagnetic force and the inertia of gas motion have a strong bearing on the characteristics and structure of plasma. A classification of plasma by the magnetic Reynolds number becomes very important for this purpose, because the magnetic induction is generated by electric current, which is described by electric conductivity σ, charge particle velocity u, and the charge number density. The magnetic Reynolds number for some characteristic scale l is defined as Rm = ul µσ in which σµ is often considered as the magnetic diffusivity and µ is the magnetic permeability of the medium. The magnetic Reynolds number has often been interpreted as a measure of the ratio of the induced and total magnetic field of the plasma. For flows with a low magnetic Reynolds number, the convection by the magnetic field lines is negligible, and the induced magnetic field by electric current is also negligible. The relationship is presented in Figure  1.2 between the electrical conductive and magnetic Reynolds numbers for most engineering applications. In general, all the assembled

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4

Plasma Physics Fundamentals

engineering applications presented here have the same order of magnitude in characteristic lengths and velocity scales, and the ionized gas is generated mostly by an electrical field. Thus, the magnetic Reynolds number is often much less than unity. The condition of ionosphere is included as a comparative reference, because all other astrophysical applications always have a huge physical dimension that differs from the common aerospace engineering applications. As an example, the solar atmosphere has a similar value of electric conductivity in an arc heater, but its length scale is ten million times greater and leads to a proportional greater magnetic Reynolds number (Mitchner and Kruger 1973; Shang 2016). All physical phenomena of plasma are governed by the Maxwell equations, but once the plasma is treated as a continuum for engineering applications, it becomes an interdisciplinary subject at macroscopic scales. The global thermodynamic and kinematic properties of plasma must be described through a distribution function between microscopic and macroscopic dynamics. This approach is accurate when the microscopic structure can be linked approximately to macroscopic motion by the Maxwell-​Boltzmann distribution. However, by this approach the detailed wave-​ particle interaction will be unresolved.

1.1

Intrinsic Electromagnetic Forces Plasma always exists in electric and magnetic fields. The electric charged particles interact and respond to externally applied fields and always create induced field components; the total electromagnetic field is therefore a sum of the applied and the induced components. One of two elementary electromagnetic forces arises from the attraction and repulsion of charged particles of the opposite and same polarities known as the electrostatic force. A free charged particle motion will produce a magnetic field, and when it is moving within an applied magnetic field will be compelled by the Lorentz force to accelerate in the direction perpendicular to both the charged particle motion and the magnetic field. The electrostatic force between two singly charged particles is described by Coulomb’s law. The force between charged particles is collinear along the unit space vector between two charges separated by a distance rij as:



Fij =

1 qi q j rij 2 πε o rij2

(1.1)

In Equation (1.1), the symbol rij designates the unit vector between the two charged particles. The magnitude of the force is directly proportional to the product of the two electric charges and inversely proportional to the square of the distance between them. The symbol ε is designated as the electric permittivity and in the international system of units (SI) it has a value of 8.85 × 10–​12 Farad/​m (Krause 1953). The charges qi and q j are measured in Coulombs, which attract each other if they are of the opposite polarity, but repulse each other if they are the same. The resulting force per unit charge is usually defined as the electric field intensity E, and

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1.1  Intrinsic Electromagnetic Forces

5

the source of the electric intensity is derived from the electric charge. However, a conductor and an isolator will respond to the electric field differently. Through the definition of electric intensity; the electrostatic force can be given as: Fi = E qi



(1.2)

The SI unit of the electric field intensity is the Newton per Coulomb, however, in practical application, it is often given in Volt per meter. For multiple point charges, the principle of superposition applies so the force is the sum of all other particle charges. The entire electric field of space charges is the vector sum of the entire field from all the individual source charges, in which the symbol rˆij denotes the unit vector between the charges:

Ei =

1 q Σ 2i rij 4 πε o rij

(1.3a)

In an electrically conducting medium, the free charge particle movement will produce electric current and exert additional force on each other. In an isolator, the molecules of a dielectric material will polarize to reduce the net local field intensity. The different effects on two kinds of media are described by a dielectric constant κ and Coulomb’s law becomes:

Fij =

1 qi q j rij 4 πε o κ rij2

(1.3b)

The dielectric constant κ has a value of unity in vacuum and for most gas species of air. However, for isolators such as polystyrene, glass, and rubber, the constant has the values of 2.7, 4.7, and 6.7, respectively. The free electric charge in motion produces a conductive electric current. In metallic conductors the charge is carried by electrons with an elementary negative charge of 1.61 × 10–​19 Coulomb. In liquid conductors, such as electrolytes, the charge is carried by both positive and negative ions. The electric field compels the free charges into continuous motion and results in an electric current that can be defined in terms of the electric flux vector per unit area. The conductive current density is different from the convective and displacement current densities. The convective current does not involve an electrically conducting medium and consequently does not satisfy Ohm’s law. The convective current flows through an isolating medium such as a liquid, rarified gas, or vacuum, and the best example is an electron beam within a vacuum tube (Jahn 1968). The displacement current arises from the time-​varying electric field and is introduced by Maxwell to account for the generation of a magnetic field when the conductive current is zero. Without the concept of the displacement current, the electromagnetic wave propagation would be impossible (Stratton 1953). The conductive electric current density J has a physical dimension of coulomb per second or ampere. It is a vector for which its orientation is dictated by the vector sum of all electric particle velocities:

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Plasma Physics Fundamentals

J = Σni qi ui = Σρi ui (1.4) where ni denotes the charge particle number density and ui is the averaged velocity of the charged particles. The current charge density ρi is defined as the sum of the total electric charges per unit volume of species i. A magnetic field generated by an electric current and the orientation of the induced magnetic field is defined by the right-​hand rule. The current may be due to an externally applied electromagnetic field, or an electron beam, or a conductive current in current-​carrying wire. Analogous to the electrostatic force, the magnetic field intensity H is defined by the field intensity at a distance r from the source. In fact, the differential magnetic field intensity is governed by the Biot-​Savart law for magnetostatics, which is the counterpart of Coulomb’s law for electrostatics. The field intensity is proportional to the product of a differential current element J d l , direction sine of the angle between the points of interest, and inversely proportional to the square of the mutual distance. The induced magnetic field line is continuous. In vector form the Biot-​Savart law gives (Jackson 1999):



dH =

J × dl 4 πr 2

(1.5a)

A current-​carrying wire produces a magnetic field perpendicular to the electric current, and the magnetic field strength H has the physical unit of Amp/​m2. The magnetic field orientation is defined by the right-​hand rule with respect to the generating electric current. Similar to electrostatics, the intermediary magnetic field variable is the magnetic flux density B or is also called the magnetic induction, which has the physic unit of Weber/​m2 or Amp/​m2. The connection between the magnetic field strength H and the magnetic flux density B in an isotropic medium is the constitute relation; B = µ H and µ is referred to as the magnetic permeability. In free space, it has the dimensional value of µ = 4 π × 10 −1 Hennery/​meter. From Equation (1.5a), the magnetic flux density B therefore can be given as:

B=

µ J × dl dv ∫∫∫ r2 4π

(1.5b)

Like that of electrostatics, the force on a current element becomes:

d Fij = J i d li × d Bij

(1.5c)

Note that the induced forces components Fij and F ji are symmetric, thus opposite each in directions. However, the electric current–​induced forces still can contradict Newton’s third law by the free charge movement and by the transient electric current in an incomplete circuit (Jahn 1968). If two current-​carrying wires are brought into the vicinity of each other, each wire is surrounded by two individual magnetic fields, leading to a force that acts on the wires. When the wires are carrying current in the same direction, the wires are attracted to each other; when the wires are carrying current in opposite directions, the wires are repelled. The interaction of two electric elements is depicted in

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1.2  Charged Particle Motion

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Figure 1.3  Induced magnetic force by two electric current elements.

Figure 1.3. The incremental force existing between the elements of the current path Ji and Jj is:

d Fij =

µ Ji J j [ d li × ( d l j × rij )] 4 π rij2

(1.5d)

Basically, there are two modes of the electromagnetic body force in an electrically conducting medium that can exert on the charged particles. The interaction of the electric field with the free charge density of the medium Fe = ρe E is known as the electrostatic force. The interaction with an externally applied magnetic field by an electric current that is driven by a force within the medium is Fm = J × B, known as the Lorentz force, both Fe and Fm have the physical unit of Newton per meter square. From Equation (1.4), the current density is directly related to charge particle motion then Fm = J × B and it is the well-​known Lorentz force. It is also often considered that the electromagnetic or Lorentz force is F = q( E + u × B ).

1.2

Charged Particle Motion The moving charged particles in an electrostatic field will induce a magnetic force perpendicular to the orientations of the applied electric field; therefore these are always associated with an electromagnetic field. According to Coulomb’s law and the Biot-​Savart law, the motion of an electrically charged particle always consists

8

Plasma Physics Fundamentals

of a rectilinear and rotational component. The rectilinear acceleration is aligned with the externally applied electric field and a gyration intrinsically revolves around an induced and applied magnetic field vector. In the absence of an electric field, the angular acceleration is restricted in the plane perpendicular to the magnetic field. A moving charge q with a velocity u in a steady and uniform electromagnetic field with an applied magnetic flux density B will be pushed by a force normal to B, and also directly accelerated by the electric field intensity E. The force that is perpendicular to the directions of the charged particle velocity and the magnetic field is the Lorentz force or acceleration, u × B . As a consequence the equation of motion for a charged particle with a velocity of u in an electromagnetic field is:

F = m ( d u dt ) = q( E + u × B ) (1.6a) If the electric field is negligible, the magnetic flux density is applied along the z-​coordinate. Under this circumstance, the motion of a charged particle along the z-​coordinate is unaltered, thus the normal force component of the charge particles is always restricted in the plane perpendicular to a constant and steady magnetic B. The velocity components in three-​dimensional space can be combined into the component that is parallel u and perpendicular u⊥ to the magnetic field. The velocity component u that is parallel to B will not be affected by the magnetic field, but the velocity component u⊥ normal to B will gyrate around it. The scalar components for the equation of a charge motion in the x-​y plane perpendicular to an applied magnetic field and aligned with the z-​coordinate becomes:



dux dt = ( qB m )uy , duy dt = − ( qB m )ux , duz dt = 0

(1.6b)

Here and further: B = B , ux = i ⋅ u ⊥ , uy = j ⋅ u ⊥ , uz = k ⋅ u ,...; i, j , k are the unit vectors of the Cartezian reference system. The general solution to the first-order partial differential equation in time; Equation (1.6b) is:



ux = u⊥ exp(i ω b t ) uy = u⊥ exp(i ω b t ) uz = c

(1.6c)

In Equation (1.6c), u⊥ = u⊥ = ux2 + uy2 the symbol ω b is denoted as the electron cyclotron frequency or the Larmor frequency as (for electron e=|q|):

ω b = eB m = 1.76 × 1011 B ( rad/s ) (1.6d) Using Equation (1.6d) for calculating the Larmor frequency of a single charge, the magnetic flux density B needs to be in the SI unit of Weber/​m2. Equation (1.6c) actually describes the single charged particle motion by a simple harmonic oscillator with the cyclotron or Lamor frequency. The trajectory of the charged particle in a pure magnetic field can be obtained by integrating the equation of motion, Equation (1.6c), and taking the real component of the results to get the location of the particle in the x-​ and y-​coordinates:

9

1.2  Charged Particle Motion

9

B || E

E=0

E=0

E=0

E=0

Figure 1.4  Trajectory of electron motion in electromagnetic field.



x − xo = ( u⊥ ω b ) cos( ω b t ) y − yo = ( u⊥ ω b )sin( ω b t )

(1.6e)

As before by defining a gyro radius as rb = u⊥ ω b , the charge particle motion in the normal plane to the magnetic field must follow a circular path and the charged particles will execute a spiral motion. The gyro radius can be determined easily by the balance of the centrifugal acceleration and the electromagnetic force exerted on a singly charged particle, eu⊥ B = mu⊥2 rb . The radius of the circular motion of a group of charged particles in the plane perpendicular the magnetic field is rb = mu⊥ eB , with an angular gyro velocity ω b = eB m . In short, the gyro radius is often referred to as the Larmor or cyclotron radius and can be evaluated based on a single electron, rb = mu⊥ eB . Similarly, the rate of the angular gyro velocity has also been widely referred to as the Larmor or cyclotron frequency. The charge particle trajectory in a three-​dimensional electromagnetic field is then a helix with its axis parallel to B and a pitch of 2 πu  ω B . When the electric field vector is applied in addition to the magnetic field, the charged particles will execute a persisted spiral trajectory as shown in Figure 1.4. The orientation of the resultant helical trajectory is dictated by the direction of the applied electric field intensity. In fact, the direction of gyration is opposite to the applied magnetic field and tends to reduce the applied magnetic field strength, a process known as diamagnetism. In a steady state the electromagnetic force components are balanced; thus from Equation (1.6a) we shall have E = − u × B. By taking the cross-​product with respect to B of the equation, using a vector identity, ( u × B ) × B ≡ ( B ⋅ u ) B − ( B ⋅ B )u and recognizing that B ⋅ u ≡ 0, it yields:

E × B = uB 2 (1.6f)

10

Plasma Physics Fundamentals

The resultant velocity u = ( E × B ) B 2 is the well-​known drift velocity of E × B, which moves in a direction perpendicular to both the electric and magnetic fields. The drift velocity leads to the well-​known Hall effect (Hall 1897). Meanwhile, in a magnetic field that is increasing in strength, the kinetic energy parallel to the field converts into rotational energy of the particle and increases its Larmor radius. However, the energy of the system is invariant because the magnetic field does no work to change the total kinetic energy of the charged particle. When the magnetic field increases to a point, the velocity parallel to the magnetic field vanishes; it leads to a unique phenomenon of the magnetic mirror for plasma by the Lorentz force (Goldston and Rutherford 1995; Goebel and Katz 2008). In a large group of charged particles, the charged particles will encounter numerous collisions with each other and neutrals in partially ionized plasma. The charged particles’ dynamics are untraceable, but the global behavior for the system of particle has been thoroughly studied by the field of statistic thermodynamics. The global effect of collisions can be determined by the velocity distribution function for each species in the microscopic motions to the macroscopic behavior of plasma. In the absence of the other forces, these particles can be characterized by a speed solely as a function of the group temperature and mass of the species. The most probable distribution of velocity in thermal equilibrium is the Maxwell distribution:

f ( u ) = ( m 2 πkT )3 2 exp[ − mu 2 2 kT ] (1.7a) Where k is the Boltzmann constant, in SI units k = 1.3806 × 10 −23 m2 kg / s 2 K , and in plasma dynamic applications the value of k = 8.61733 × 10 −8 eV / K is frequently used. In Boltzmann distribution Equation (1.7a), the notation T is the system temperature in a thermal equilibrium state. From this distribution function, the average velocity of the system can be found as: ∞



uavg = ∫ 4 πu( m 2 πkT )3 2 exp[ − mu 2 2 kT ] du 0



(1.7b)

The classic result of the average velocity of a particle in a thermodynamic equilibrium state is obtained by integrating Equation (1.7b), which is proportional to the square root value of the temperature and inversely proportional to the unit mass of the particle. The result reveals the huge difference between the average velocities of electrons and ions due to the difference in masses of electrons and ions.

uavg = (8πT π m )1 2

(1.7c)

It becomes clear that the stationary charges generate an electrostatic field; in fact, the remotely acting Coulomb force is the genesis of the intrinsic characteristics of plasma for the Debye shielding length and the plasma frequency. While the direct current produces a magnetostatic field, the dynamics of the electromagnetic field must more often be accompanied by a time-​varying electric current density. In short, within a static electromagnetic field, the electric and magnetic

1

1.3  Debye Shielding Length

11

fields are independent from each other, whereas in dynamic states the two fields are interdependent.

1.3

Debye Shielding Length Two fundamental parameters associated with the electric properties of plasma are the Debye length and the plasma frequency; both characterize the macroscopic behavior for a collection of charged particles. The basic property of plasma is its tendency to maintain electric neutrality (Langmuir 1929). This particular state requires an enormously large electrostatic force between an electron and ion. A quantitative estimate of this dimension over which a deviation from charge neutrality may occur can be obtained by the Gauss law for electric field, ∇ ⋅ D = ρe. The physical meaning of this law simply states that the property of the electric displacement is determined by the local electric charge density. The electric displacement is related to the electric field intensity by a constitutive relationship, D = ε E , where ε is the electric permittivity. In a homogeneous field and isotropic medium, the electric field intensity is inversely proportional to the gradient of the electric potential, E = −∇ϕ . The resultant equation is known as the Poisson equation of plasma dynamics:

ε∇2 ϕ( r ) = −ρe = −e( ni − ne ) (1.8a)



In a thermodynamic equilibrium state, the number densities of electrons and ions are given by the Boltzmann distributions, and by invoking the fact the plasma is electrically neutral; namely, ni = ne.

ne = ne eϕ ( r ) κT and ni = ne − eϕ ( r ) κT (1.8b) Substituting these expressions into the Poisson equation of plasma dynamics and reducing the equation to a one-​dimensional problem:

ε



d 2ϕ = en[e( eϕ κTe ) − 1] dx 2

(1.8c)

Expanding the exponential function of the right-​hand side of the equation by Taylor series in ascending order of eϕ κT  gives:

ε

 eϕ 1  eϕ  2  d 2ϕ = en + + ⋅⋅⋅   ∞   dx 2  κTe 2  κTe  

(1.8d)

Although the magnitude of eϕ κT may not be necessarily negligible over an entire domain between charged particles, the potential is known to drop rapidly with respect to the separation distance between the charged particles. Therefore, it is permissible to keep only the leading term, then:

ε

d 2 ϕ e2 n ≈ ϕ dx 2 κTe

(1.8e)

12

Plasma Physics Fundamentals

It is recognized that Equation (1.8e) is a simple harmonic equation; the general solution is: 12



  εkT   ϕ = exp  −  2 e  x    e n  

(1.9a)

The electric potential of Equation (1.9a) reaffirms an exponential decay behavior as a function of distance between charged particles, and can be given as:

ϕ = ϕ o exp( − | x | λ d ) (1.9b) Equation (1.9b) leads naturally to the definition of the Debye shielding length. This is the minimum and defining characteristic length scale for plasma, and within this space a pair of electron and ion isolates each other from the external electric field. 1



 εκT  2 λd =  2 e   e n 

(1.10a)

In practical applications, the Debye length can be calculated easily by:

λ d = 69(Te n )1 2 or λ d = 7430(κ Te n )1 2 (1.10b) In Equation (1.10b), the Debye shielding length is given in meters. The input electron number density and temperature are required to be number per cubic meter and in electron volts only. The temperature of the free-​electron component of the gas Te is not necessarily equilibrated with the ions or neutrals of the plasma. The Debye length has always been adopted as an index of the typical charge separation distance in plasmas that are sustained by the random thermal energy of the electrons. These simplified results for the Debye length by Equation (1.10b) are evaluated by the following constants: electron volt (1 eV = 1.1604 × 104 K); Boltzmann constant of 1.3807 × 10–16 erg/​K; elementary charge of 4.8032 × 10–10 Coulomb; and electric permittivity of free space 8.8542 × 10–12 Farad/​m. The typical value of the Debye length in a magnetohydrodynamics (MHD) generator is in the same order of magnitude of the electron mean free path O(10–6 – 10–7 m) at a temperature of 2,500 K and charge number density of 1020/​m3. In plasma generated by electronic impact at standard atmospheric conditions, the Debye length is 1.14 × 10–7 m. A most illustrative phenomenon of Debye shielding length is the plasma sheath. The charge neutrality of plasma does not prevail in the immediately adjacent region to a solid surface, especially over electrodes of an externally applied electric field. The boundaries of the plasma represent a physical interface through which energy and charged particles enter and leave the plasma; the plasma will establish the potential and density variations at the interface to satisfy charged particle balance or by the imposed electrical boundary. The narrow domain surrounding the media interface is the so-​called plasma sheath. The Debye length gives the order of magnitude for the thickness of the sheath that separates the neutral plasma from an interface. Under this circumstance, by a first-​order analysis for quasi-​neutrality, the ratio of the electron to the ion current density at the interface is:

13

1.3  Debye Shielding Length

13

Figure 1.5  Cathode layer of a DC parallel discharge ϕ = 439.0 V, I = 5.2 mA, p = 5.0 Torr.



J e J i = ne eue ni e ui = ue ui (1.11a) The ratio of current density on the interface can be further developed by the conservation law of energy for the charged components of the plasma, me ue2 2 = kTe e and mi ui2 2 = kTi e, to yield:



J e J i = ue ui = ( mi Te me Ti )1 2 (1.11b) Recall that the mass ratio between ion and electron is at a value greater than one thousand to one. In general, electrons left the plasma volume faster than ions because of a greater mobility; a charge imbalance would generate a positive electric potential at the interface. This physical characteristic is easily identifiable in the cathode and anode layers of a glow discharge that is generated by a strong electric field. Another contributing factor to this behavior is the recombination mechanisms of chemical kinetics by depleting the charges of ionized gas when striking the surface. The electron absorption and emission properties of the solid surface also play an important role. This phenomenon is closely related to the catalytic of the surface material. The computational simulated direct-​current discharge duplicating the experimental observation is depicted side by side in Figure  1.5. The discharge is generated between parallel metallic electrodes with a gap distance of 1.0 cm and the external electric potential is applied by a voltage of 400 volts under an air ambient pressure of 10 Torr. The total current of the charge is measured to be 10 mA. The detailed discharge structure from the experiment is overwhelmed by the emission or the glow in the photography record, but the cathode layer is clearly captured by the computational simulation. The numerical result displays the positive discharge column over the cathode layer; the plasma sheath is well formed by the cathode fall. The computational result is presented by twenty uniformly increment electron number density contours with a maximum value of 1.0 × 1010 cm −3.

14

Plasma Physics Fundamentals

1.4 Plasma Sheath The plasma sheath is another unique feature of plasma at the interface of different media, which is directly connected to the vastly different unit mass between the electron and ion. For a hydrogen atom the ratio of an ion to electron is 1,836 to one. Away from the media interface the bonded electron and ion must move at the same speed to maintain the global neutrality. Near the boundary, charge separation occurs and the charged particles tend to move at different random motions. The average electron and ion velocities at thermodynamic equilibrium condition according to the Maxwellian distribution are:

ue = (8kTe π me )

(1.12a)



ui = (8kTi π mi )

(1.12b)

The electron and ion mass flux vectors, Γ = ∫∫∫ nu( m 2 πkT )3 2 exp( − mu 2 2 kT ) d 3v , are given as:

Γ e = ne e kTe 2 π me

(1.12c)



Γ i = ni e kTi 2 π mi

(1.12d)

Therefore if the temperatures of electrons and ions are equal, the average velocity of an electron is at least forty-​two times greater than that of ion. However, in an electron-​impact plasma generation process, the electron temperature is much higher than that of the ion to make an even greater disparity. In free space and without the presence of an electric field, the electron and ion pair in plasma is restrained from a separation distance of the Debye shielding length to each other by the electrostatic force. Near the interface boundary, charged particles separate; the higher collision rate of electrons with a surface is much greater than ions to make the surface acquire a net negative potential (Riemann 1991). The ions will recombine at the surface and return the plasma to neutral particles; the electrons either recombine or enter the conduction band for an electric conducting material. Most important, the plasma loses the globally neutral property in this region and the electric potential increases monotonically toward a negative value from the unperturbed neutral state. When the collision process reaches an equilibrium state, the net electric current at the interface surface vanishes. A plasma boundary layer is formed over the interface known as the plasma sheath. It may be anticipated that the plasma sheath often has the same order of magnitude of the Debye shielding length. The thickness of a plasma sheath is dependent greatly on the interface surface recombination rate and the curvature of the surface, thus the plasma thickness is problem dependent and it is difficult to provide a general solution. However, on a flat interface surface the electric potential and the inner structure can be estimated by neglecting the drift velocities of the charged particles and assuming the electrons and ions have the same temperature and under thermal equilibrium

15

1.4  Plasma Sheath

15

condition. Under the stated condition, the number densities of charged particles are Maxwellian:

ne = no exp[ eϕ( r ) kT ] (1.13a)



ni = no exp[ −eϕ( r ) kT ] (1.13b) At the considered condition, there is no net charge building up on the surface:



J e ( 0 ) = J i ( 0 ) (1.13c) The zero surface charge condition for an electrically conducting flat surface is implemented by substituting the mass flux densities of electron and ion Γ e and Γ i, Equations (1.12c) and (1.12d), into Equation (1.13c) to get:



(1 me )1 2 exp( eϕw kT ) = (1 mi )1 2 exp( − eϕw kT ) (1.13d) The estimated electric potential in the plasma sheath becomes:



ϕw = −( kT 4e ) ln( mi me ) (1.14) The estimated plasma sheath potential (Bittencourt 1986) surprisingly reaches a qualitative affinity with more accurate analysis. At the present, the plasma sheath and electric potential immediately next to the interface can be routinely calculated by numerical simulation utilizing the classical drift-​diffusion theory. In the case for simulating the direct current discharge, the secondary emission from the cathode surface has also been taken into consideration. As an example, the electric field strength distributions in the positive column between the anode and cathode of a direct current discharge are displayed in Figure  1.6. The typical cathode sheath is clearly displayed; in a direct current discharge, the electrons carrying most of the current in the positive column now are prevented from reaching the cathode, and the massive ions are incapable of carrying the total current. The discharge is maintained by the secondary emission of electrons by the bombardment of incoming ions, leading to the exponential growth of electron density and flux from the cathode. The value of the exponent is known as the first Townsend coefficient (Lieberman and Lichtenberg 2005). In Figure 1.6, the numerical results are obtained for three different ambient pressures. Three discharges are generated at a fixed electrodes gape distance of L = 2.00 cm between metallic electrodes from ambient pressures from 3.0, 5.0 to 10.0 Torr. The electric potentials are required to increase from 417.6, 534.0 to 826.1 Volts to maintain the discharge for the elevated pressure. On the other hand, the discharge currents decrease from 5.27, 4.88 to 3.9 mA as the ambient pressure is elevated. The computational simulations reveal that the electric field intensity is nearly constant across the positive column but increases rapidly in both the anode and cathode layers. The electric intensity drops drastically from the cathode surface toward the positive column for all three ambient pressures simulated to display the distinctive behavior of cathode fall. The electric field strengths at three different ambient pressures show a monotonic drop by two orders of magnitude from the cathode

16

Plasma Physics Fundamentals

1

0.8 p = 3 Torr p = 5 Torr p = 10 Torr

y/L

0.6

0.4

0.2

0

102

103 Ey (v/cm)

104

Figure 1.6  Electric field structure in direct current discharge.

surface. The plasma sheath of the discharge is also more noticeable in the cathode layer than in that over the anode. The pressure or density of the discharges exerts little influence on the thickness of the plasma sheath, because the sheath thickness is dependent strongly on the charged number density. For the direct current discharge, the electron number density is sustainable at a nearly constant and maximum value of 1010/​cm3. Under the higher-​density environment, the plasma sheath is slightly thinner, but the steep electric potential gradient remains.

1.5

Plasma Frequency Plasma has a tendency to be macroscopically neutral; the relative position of paired charges will always return to its neural equilibrium state after a perturbation. The inertia of electrons will unavoidably lead to overshoot and oscillate around the equilibrium condition by a characteristic rate known as the plasma frequency. The plasma frequency is therefore a measure of a time scale of restoring forces that exerts on a displaced electron returning within Debye length to an ion and keep plasma to an electrically neutral state. In the absence of a magnetic field and without externally applied thermal perturbation, the coupled ion can be considered in a fixed location in uniform plasma that extends to infinite (Mitchner and Kruger 1973). The mass and momentum conservation equations of electron motion in one dimensional case are:



∂ne d ( ne ue ) + =0 ∂t dx ∂u   ∂u mne  e + ue e  = −ene E ∂x   ∂t

(1.15)

17

1.5  Plasma Frequency

17

The compatible electric field E can be determined by the classic Poisson equation of plasma dynamics in one-​dimensional formulation: ∂E e = ( ni − ne ) ∂x ε



(1.16)

The system of equations can be solved easily with a linearized perturbation approach by setting the known initial conditions and considering the displaced electron as a small disturbance:

n = no + εn1 ,

ue = ue + εu1 ,

E = Eo + εE1 ,

ε 0. In fact, Landau damping prevents instability from developing and is also a manifestation of the mathematical property of electron and electromagnetic wave interaction when the electrons move near the phase velocity of the wave.

1.8

Joule Heating Joule heating is also known as Ohmic and resistive heating. The phenomenon arises from interactions between moving charged particles through collisions. The resulting scattering motion from elastic and inelastic collisions is completely random to become thermal energy. It is the process by which the passage of an electric current through a medium releases heat. In other words, the electromagnetic field does work to overcome the resistive force that equals to the dissipated energy due to resistance. The amount of energy released is proportional to the square of the current. This relationship is known as Joule’s first law and the IS unit of the energy was subsequently named after Joule by the symbol J (J = 1 Newton-​meter, or J = 107 erg in SGS). Joule heating is the consequence of the interaction of charged particles in an electric circuit accelerated by an electric field, but electrons must give up some of their kinetic energy every time they collide with ions of the conductor. The increase in kinetic or vibrational energy of the ions manifests itself as heat and an elevation in the temperature of the conductor. In steady state, the electron momentum equation including the electron–​ion and electron–​neutral collision can be approximated as:



ne( E + ue × B ) + ∇ ⋅ pe = mn[ νei ( ue − ui ) + νen ( ue − un )]

(1.24a)

24

Plasma Physics Fundamentals

where νei and νen characterize the collisional frequencies of electrons with ions and with neutral particles, respectively. Since the electron velocity is so much greater than that of neutral, the foregoing equation can be approximated as:

ne( E + ∇pe en ) + enu e × B = mn( νei + νen )ue − nmνei ui (1.24b) By neglecting the collision frequency of electrons and neutrals in comparison to the collision between electrons and ions, and by the negligible partial pressure gradient of electrons, Equation (1.24b) is simplified. Recall the total electric current is defined as J = ne( ui − ue ), and J e = neue is the electric current of electrons; we have then:



( mνei ne 2 )J = E − J e × B en (1.24c) Equation (1.24c) is known as the approximate Ohm’s law for partially ionized plasma, and the group of parameters ( mνei ne 2 ) is the electric resistivity of partially ionized plasma. It is the reciprocate of electric conductivity σ. In the absence of an externally applied magnetic field, the most general and fundamental formula for Joule heating is:



q = E ⋅ J = I 2 σ = σ E 2 (1.24d) The electric current in a circuit is the volumetric integral of the electric current density:



I = ∫∫∫ J dv (1.24e) The time-​averaged Ohmic power per unit area has been given as (Lieberman and Lichtenberg 2005):



P ≈ ( I 2 2 )∫ mνd ξ [ e 2 n( ξ )] (1.24f) Recall that at low ambient pressure, the Ohmic heating is small in comparison to the stochastic heating. However, the Ohmic heating is dominant at high pressure due to the high collision frequency. The total energy dissipated in the conductor in time is then W = ∫ E ⋅ I dt and known as Joule’s law. An identical formula can also be used for alternating current (AC) power, except the parameters are averaged over one or more cycles. In these cases of a phase difference between I and E, a multiplier cos φ must be included. As an example of Joule heating in a direct current discharge, a side-​by-​side electrode configuration is presented in Figure 1.8. The composite graph includes the computational simulation and the photograph observation under identical conditions; the gap between the cathode (on the left) and the anode is 1.5  cm. Each electrode has a length of 0.5 cm and a width of 0.25 cm. The discharge is maintained by an electric potential of 439.0 V, a discharge current of 5.2 mA at an ambient air of 5.0 Torr. The experimental observation shows intensive glows over the closest edges of electrodes. As can be seen from the numerical simulation, the electric field intensifies at the sharp edge of the electrodes. The electron number

25

1.9  Plasma Kinetics Formulations

25

Figure 1.8  Direct current discharge over a side-​by-​side electrode arrangement ϕ = 439.0V , I = 5.2 mA, p = 5.0 torr.

density has a maximum value of 1.7 × 1010/​cm3 locally, and this value is 3.5 times greater than the infinite parallel electrode counterpart. At the inner edge of the anode, the anode layer is completely distorted by the stronger local electric field. From the contour presentation, the cathode layer can be clearly discerned even through the discharge structure over the cathode and is no longer uniform. The plasma sheath is clearly displayed in the cathode layer, and the current density is highly concentrated over the cathode. Thus, the Joule heating is concentrated along the inner edge of the cathode layer closest to the anode, at a total applied direct current electric power of 21.52 j/​cm2s, the Joule heating of the glow discharge at the pressure of 5 Torr is estimated to be 1.30 j/​cm2s (Shang et al. 2009). This rate of Joule heating amounts to 6.04 percent of the total plasma generation power input.

1.9

Plasma Kinetics Formulations The charged particles of partially ionized plasma involve a large number of collisions between electrons, ions, and neutral species. The net effect of collisions is obtainable by applying a distribution function of the velocities for each species. In the absence of other forces and on the average, each particle will move with a speed that is a unique function of the macroscopic temperature and mass of the species. The statistical behavior of charged particles thus can usually be described by different velocity functions; the random motions are usually evaluated by taking the moments of the distribution functions. As introduced by Equation (1.7a), the most probable three-​dimensional velocity distribution function for a group of particles in thermal equilibrium is the Maxwell equation:



f ( u ) = ( m 2πkT )3 2 exp[ − m( u 2 + v 2 + w 2 ) 2 kT ] (1.25) The average speed per particle is uavg = (8kT π m )1 2 , which is the well-​known result from the kinetic theory of gas. Any flux variables are obtainable by taking the

26

Plasma Physics Fundamentals

higher-​order moment of the distribution function. For example, the conservation momentum equation is acquired by taking the first-​order moment with the distribution function, and includes the Lorentz force: F = m ( d u dt ) = e( E + u × B )



(1.26a)

mnu = ∫∫∫ mnu f ( u )dv



The resultant momentum conservation equation is: mn[ ∂u ∂t + ( u ⋅ ∇ )u ] = en( E + u × B ) − ∇p − mnu( u − ui )



(1.26b)

The first term of the right-​hand side of Equation (1.26b) is the externally applied Lorentz force and the second term is partial pressure gradient. The pressure gradient in general should be a stress tensor, but in an anisotropic medium it reduces to a normal vector. The pressure is then given as a scalar partial pressure of a species, p = nkT . The resultant pressure force is given as Fp = −( ∇n n )kT . The momentum transfer by binary collisions between different charges and the collisions with neutral particles generates a force due to collisions that can be expressed as Fc = − m∑ νij ( ui − u j ). The collision frequency is designated as νi , j; the characteri, j

istic velocities of species i and j are denoted as ui  and u j. The conservation of energy equation is obtained by taking the second moment with the distribution function. e = ∫∫∫ uu f ( u )dv



(1.27a)

The general form of the energy equation for charge species by neglecting viscous heating of the species has been provided by Goebel and Katz (2008):

∂ [ nm u 2 2 + 3 p 2 ] + ∇ ⋅ [ nm u 2 2 + 5 p 2 ]u + ∇ ⋅ q = en[ E + R en ] ⋅ u + Q − Ψ (1.27b) ∂t where the transport energy includes the work done by the pressure, the macroscopic energy flux, and heat transfer by conduction, q = k ∇T . The coefficient of thermal conductivity of a species given in SI units is k = 3.2 τ e ne 2T m , and the temperature needs to be specified in electron volts. The mean energy change by collision is R = − ∑ nmνin ( u − un ), and Q and Ψ are energy exchange from elastic collisions and n

energy lost by inelastic collisions. Technically, the plasma kinetic formulation has assumed that the plasma can be treated as an isotropic electrically conducting medium with Maxwellian distribution. Only the interactions and motions of all the fluid elements are considered, and the link between the macroscopic and microscopic properties is based on the most probable states.

27

1.10  Electric Conductivity

27

Table 1.1  Electric conductivity of common materials.

1.10

Substance

Type

σ( mho m )

quartz

insulator

10–​17

air

insulator

10–​15

glass

insulator

10–​12

water

insulator

10–​4

carbon

conductor

3.0 × 104

copper

conductor

5.7 × 107

silver

conductor

6.1 × 107

Electric Conductivity The electric and magnetic fields in plasma always produce an electric current that is related to the drift or diffusion velocities of the various charged particle species. Therefore, it is necessary to evaluate the electrical conductivity as microscopic properties together with the thermodynamic state of the plasma. Between collisions, the individual particles move in accordance to the E × B drift and diffusion velocities that are the driving mechanisms for the charged particle motions. The electrical resistivity is the resultant phenomenon of the collision process. By introducing a mean or drift motion of the electrons in the gas, the electric current is:



J = neu = ( ne 2 mνc ) E (1.28) where m is the mass of the electron and νc is the average electron collision frequency, which is related to the atomic momentum transfer cross-​section and the random thermal motions of the individual particles. The current density is related empirically to the applied electric and magnetic fields by a bulk electric conductivity J = σ( E + u × B ). The proportional constant is known as the electrical conductivity:



σ = ne 2 mνc (1.29) In fact, it is the reciprocal of the electric resistance that has the SI unit of Ohm-​ meter and is a function of the medium temperature. The electrical conductivity is expressed in the reciprocal of the ohm-​meter, or mho. Note that Ohm’s law at a point is J = σ E , that J and E have the same direction in an isotropic medium and in the absence of a magnetic field. Under this special circumstance, electrical conductivity is a scalar quantity. The typical values of electric conductivity of the common substances at the standard condition are displayed in Table 1.1. For electromagnetic wave propagation, the definition of conductor and dielectric must include the wave frequency ν and the electric permittivity ε of the transmitting

28

Plasma Physics Fundamentals

medium. A  rule of thumb is that the electric conductivity must be one hundred times greater than the product of the wave frequency and the electric permittivity of the medium, σ > 100 εν, and for the semiconductor these properties shall have the same magnitude, σ  εν. Whereas for the dielectrics the magnitudes of these properties are just reversed, σ < 0.01εν (Stratton 1953). A fundamental relationship between electric current and electric field intensity that has been widely used in classic magnetohydrodynamics is the generalized Ohm’s law. The law is rigorous for plasmas in the absence of the displacement of electric current and has the distinct advantage in compact formulation to bypass the complex and detailed description of plasma composition. The derivation and approximations of the law are based on the two-​fluid, quasi-​neutral plasma constituted by electrons and singly charged ions. First the mass density, averaged mass velocity, and current density are defined as follows:

ρ = mi ni + me ne ≈ ne ( mi + ne ) 1 m u + me ue u = ( ni mi ui + ne me ue ) ≈ i i ρ mi + me J = e( ni ui + ne ue ) ≈ ene ( ui + ue )



(1.30a)

where symbols mi and me denote the unit mass of ions and electrons, respectively, and n is the number density of charged species. The equation of charged particles’ motion in an electromagnetic field has been given by Equation (1.26b). Additional forces such as the gravitational or any non-​electromagnetic body force exerting on the plasma can also be included. But, first, the convective terms ( u ⋅ ∇ )u for both electrons and ions are neglected, because the velocities of the mass-​averaged organized motion for electrons and ions are assumed to be small. The shear stress is also neglected for simplicity and the error is limited as long as the Larmor radius is much smaller than the other characteristic length scales of charged particles’ motion, the equation of charged particles’ motion can be given as: ∂ui = en( E + ui × B ) − ∇pi + nmi g + Pie ∂t ∂u me n e = −en( E + ue × B ) − ∇pe + me ng + Pei ∂t

nmi

(1.30b)

These two equations, Equation (1.30b), describe the motion of the ions and electrons, respectively. In this equation, pe and pi are the partial pressures of the electrons and ions, respectively. The total momentum transferred to ions or electrons by the collision process is Pie and Pei. The detailed derivation for the generalized Ohm’s law will be deferred to Chapter  5.2. By manipulating the momentum conservation equations, Equation (1.30b), and by invoking the definitions of number density, mass-​averaged velocity, and current density, the result is:

E + ue × B − J σ =

1  mi me n ∂  ( J n ) + ( mi − me )J × B + me ∇pi − mi ∇pe   ρe  e ∂t 

(1.30c)

29

1.11  Electric Conductivity in a Magnetic Field

29

Further simplifications are carried out by considering the temporal variation of the organized motion to be negligible and the relative lower unit mass of electrons in comparison to ions and neutrals. The resultant approximation is known as the generalized Ohm’s law, which describes the electric properties in an electrically conducting medium:

σ( E + ue × B ) = J + ( J × B − ∇pe ) en (1.30d) In the classic formulation of Ohm’s law, the number density of the charged particles is considered to be overwhelmingly large; the following expression is frequently referred to as Ohm’s law for electric conductivity.



J = σ( E + ue × B ) (1.31a) The physical interpretation of this equation is that the current density is driven by an effective electric field that is composed of the applied field E and an induced field component ue × B . Ohm’s law has also been modified to include the Hall current as (Mitchner and Kruger 1973):



J = σ[ E + u × B − β( J × B )]. (1.31b) where σ is the scalar electric conductivity, and hence the Hall current and frequency have been defined previously as J × B and β = 1 en = B ω b n . Again the cyclotron or Larmor frequency is given as  ω b = eB m .

1.11

Electric Conductivity in a Magnetic Field In the presence of a magnetic field, collisions play an increasingly critical role in determining the drift velocity of a charged particle. The magnetic field through the Lorentz acceleration generates a gyro motion for the charged particle with a gyro radius proportional to the velocity component perpendicular to and inversely proportional to the magnetic flux density B. The angular velocity of the gyro motion has been known as the gyro frequency. It may be clear that the helical trajectory of the charged particles has a wide and varying range of orientations in collision; under this condition the electrical conductivity is no longer a scalar quantity but is a tensor of rank two. The trajectory of the drift motion of a charged particle in an electromagnetic field is fascinating and complex. In the presence of both electric and magnetic fields, the charged particle in any inertial frame will execute a helical motion. The electric force usually is decomposed to components parallel and perpendicular to the magnetic field. The relationship between the transformed (reference) frame moving with the mean velocity of plasma and the laboratory (fixed) frame is Etr = E + u × B . In the nonrelativistic approximation, the transformed magnetic flux density is Btr = B − u × E c 2 . For the case where the coordinates system is moving with a velocity perpendicular to both the electric and magnetic fields or on a laboratory

30

Plasma Physics Fundamentals

frame, the helix trajectory becomes a prolate cycloid with loops or a curtate cycloid (Jahn 1968). The ratio of the gyro frequency to the collision frequency of the charged particle is known as the Hall parameter ω b νc . This parameter characterizes the charged particle’s motion as it responds to the applied E and B fields. The E × B component of the current is the so-​called Hall current, which defines the gyro motion. When ω b νc  1, the charged particle rarely can execute one cycle of its motion between collisions, and hence cannot move in the helical trajectory. Thus the primary component of the electric current is parallel to E. On the opposite limit, ω b νc  1, the charged particles complete multiple gyro motions between collisions. As a consequence, the major component of the electric current will be in the direction of E × B, which flows in a direction perpendicular to both the electric and magnetic fields. Finally under the condition ω b νc  1, the current will have comparable components parallel and perpendicular to E. The Hall parameter depends on the charge to mass ratio and therefore has different values for various charged species. A singly charged ion and an electron have substantially different masses by a ratio of around 1,836 to one in a hydrogen atom. This disparity complicates the formulation for the overall electrical conductivity. In practical applications and in the presence of an applied magnetic field, the electrical conductivity has been formulated directly by using the plasma frequency, or by describing the modified electric field in parallel and normal component with respect to an applied magnetic flux density Bo. The derivation for the non-​scalar electric conductivity is tedious. For the purpose of illustrating that the electric conductivity is really a tensor, an abbreviated derivation is presented for the electric conductivity in a steady magnetic field. Three distinct current components have been identified by the component lying along the electric field: E follows the drift E × Bo , and the third component aligns with Bo but in proportion to the component of E parallel to Bo. 2



 ω  [( ν ω ) + i ] E + ( ω b ω )( E × B ) + ( ω b ω )2 [( ν ω ) + i ]−1 ( E ⋅ B ) B J = εω  b  (1.32a)  ω [(( ν ω ) + i ]2 + ( ω b ω )2 If Bo is aligned in the z-​coordinate of a Cartesian system, the conductivity appears as a tensor of rank two (Jahn 1968):



 νe ( ω b2 + ν2 ) ω b ( ω b2 + ν2 ) 0  σ = ω 2b  − ω b ( ω b2 + ν2 ) νe ( ω b2 + ν2 ) 0  ε  0 0 1 νc 

(1.32b)

The conductivity tensor describes a swarm of negative charged particles, where ω b = e B m and the collision frequency ν denotes all possible encounters of charged particles. The gyro frequency is a rotational vector that requires a sign to distinguish the clockwise and counterclockwise rotations. Again, because of the substantial difference in the mass of an electron and an ion, the electrical conductivity tensors are qualitatively different from each other.

31

1.12  Ambipolar Diffusion

31

In partially ionized plasma, the electric conductivity is complicated by the huge disparity in mass of the electrons and ions. Therefore, a systematic approximation must be applied in the derivation. From the basic characterization of an electromagnetic field, it can be recognized that the B field plays no role in determining the component of the motion of particles parallel to it. Therefore, if one lets the magnetic field be aligned with the z-​coordinate, the particle motion can be decomposed into components parallel and perpendicular to the magnetic field. The effective electron conductivity may be written accordingly as:

σ e ,⊥ = σ e (1 + β2e ), σ e , = βe σ e (1 + βe2 ) (1.33a)



By introducing the slip factor s = βe βi = µ e µi B2 for weakly ionized plasma, the components of the electrical conductivity can be approximated as (Mitchner and Kruger 1973):

σ⊥ ≈



βe σ σ(1 + s ) , σ ≈ (1 + s )2 + βe2 (1 + s )2 + βe2

(1.32b)

The electric conductivity in Cartesian tensor form appears as: σ ⊥  σ =  σ  0



− σ σ⊥ 0

0 0  σ ⊥ 

(1.32c)

The presence of a magnetic field in plasma means that the electric conductivity is no longer a scalar quantity, but rather a tensor of rank two.

1.12

Ambipolar Diffusion Diffusion is one of the major complications in the analysis of plasma dynamics. The diffusive mass flux is generated by three driving mechanisms due to the gradients of species concentration and temperature, as well as external forces exerted on the electrically charged species. From the kinetic theory of dilute gases, the description of diffusion does not extend to the presence of internal freedoms in molecules. In an electromagnetic field, additional random motions by the electrostatic force and by Lorentz acceleration are recognized as new mechanisms for the forced diffusion. In a diffusing mixture, the velocities of individual species can be significantly different from each. However, the peculiarity of the Debye length of plasma creates a unique constraint known as ambipolar diffusion (Howatson 1975). In an ionized gas, the random or thermal motion of a charged particle is the result of mutual collisions plus forces acting on the particle by an electromagnetic field. The acceleration of an electron has a magnitude of eE m in the negative direction of E. After numerous collisions, the average kinetic energy reaches a constant value and the force diffusion has an average velocity. This motion is the well-​known drift

32

Plasma Physics Fundamentals

velocity. A similar drift velocity also is attained by the positively charged ion but in the positive E direction, and, due to the greater mass of the ions its drift velocity is much slower than that of the electron. The averaged energy gain between collisions is dependent on the ratio of E p, which is often referred to as the reduced electrical field. The proportionality constant between the drift velocity and the electric field ue = −µ e E is called mobility:

µ e = ue E = e me ν (1.33a)



In this equation, ν is the averaged collision frequency for momentum transfer, and the mobility of the drift velocity of ion can also be given as:

µi = e mi ν (1.33b)



The self-​diffusion or ordinary diffusion is proportional to the concentration gradient of the number particle density ∇n. The diffusion coefficient d determined by the elementary kinetic theory of gases is proportional to the mean random velocity and the mean free path between collisions. The rate of flow particles per unit area is: Γ = − d ∇n; d  λu (1.34a)



From the viewpoint of the conservation of number particle density, the mass flux density is Fick’s second law of diffusion. This relationship is valid for both electrons and ions. Since the mean free path of electrons is greater than that of ions, the electrons’ random velocity is much greater; it follows that d e > di . The diffusion of charged particles is related to mobility; both arise from the random motion and unbalanced collision force. In one-​dimensional motion of an ion: ui =



Γi 1 ∂n 1 ∂p = di i = di i ni ni ∂x pi ∂x

(1.34b)

The gradient of the partial pressure ∂pi ∂x must be balanced by the total force acting on the ion. The force is exerted by the electric field onto charged particles, thus e E ni = ∂pi ∂x . By the definition of mobility, we have:

ui 1 ∂pi 1 ui pi =E= = µi eni ∂x eni di

(1.34c)

di p κT = i = µi eni e

(1.34d)

or

This relationship between mobility and diffusion or µi = edi kT and is known as the Einstein relation (Surzhikov and Shang 2004). The often encountered charge separation is the result of the disparity in the diffusion of electrons and ions in a strong electric field. The electromagnetic field augments the drift velocity of the ions and retards the electrons. From the charge conservation equation, the net charged number density of electrons and ions must

3

1.12  Ambipolar Diffusion

33

be identical. The plasma will establish the required electric field in the system to slow the more mobile electrons so that the electron moving rate is the same as that of the slower ions. When this process reaches a state of local equilibrium for the drift velocity, the resultant process is called ambipolar diffusion. The simple charge conservation principle requires the charged number flex density of both ions and electrons must be equal. From this requirement, the limiting and reasonable estimated value of the ambipolar diffusion coefficient can be found. Since the flux of diffusing electrons and ions must be balanced at all times, the following equality holds:

−d e ∇ne − ne µ e E = −di ∇ni + ni µi E (1.35a) For the globally neutral plasma and singly charged ions, the number of electrons and ions must be equal: ni = ne = n. The electric field intensity that satisfies equality condition is then:



E = ( di − d e ) ( µi + µ e )[ ∇n n ] (1.35b) The equation of ambipolar diffusion can be written as:



d a ∇n = −d e ∇n − nµ e E = −di ∇n + nµi E (1.35c) Eliminating the identical electric intensity E from Equation (1.35c) to yield the ambipolar diffusion coefficient: da =



d e µ i + di µ e µ e + µi

(1.35d)

The ambipolar diffusion reflects the impact of the unique global neutrality plasma behavior and of the different charged particle mobilities. As we have mentioned, the mobility of electrons is much greater than that of ions, µ e  µi , and the ambipolar diffusion coefficient can be approximated by means of the Einstein relationship between the mobility and diffusion as:

da =

 T  d e µ i + di µ e = di  1 + e  µe  Ti 

(1.35e)

Equation (1.35e) is the general definition of the ambipolar diffusion in plasma in the absence of a magnetic field. Two conditions are commonly encountered in practical applications. First, at the same temperature for electrons and ions, and it is the common condition of ionization by thermal ionization:

d e µ e = di µi (1.36a) Under this circumstance, we have the ambipolar coefficient to be twice the value of the ion diffusion.



d a = 2.0di (1.36b) Second, when the plasma is generated by electron impact ionization, the electron temperature of electrons is much higher than that of ions. The later frequently

34

Plasma Physics Fundamentals

retains the value of the ambient condition Te  Ti thus d e µ e  di µi . Again through the Einstein relationship, the ambipolar diffusion coefficient can then be approximated as:

d a = d e ( µi µ e ) = (κTe e )µi (1.36c) The ambipolar diffusion in a magnetic field is more complicated because the mobility and diffusion coefficient become anisotropic in the presence of a magnetic field. The condition of quasi-​neutrality or charge balance must still be satisfied; the ambipolar diffusion requires that the sum of net change for both the ions and electrons be identical across and parallel to the field. The rigorous approach to the ambipolar diffusion therefore needs to build on the frame of kinetics of charged particles. The equation of electron motion in a multiple-​ species (electron, ion, and neutral species) formulation the same as Equation (1.26c) is given as:



mn[ ∂ue ∂t + ( ue ⋅ ∇ )ue ] = −en( E + ue × B ) + ∇pe − mnνen ( ue − u n ) − mnνei (u e − ui )

(1.37)

However, only the result of a special case in the transverse magnetic field is known in the open literature (Goebel and Katz 2008). For practical applications, the result needs to be further deduced by a simplified basic equation of electron motion in steady state from Equation (1.37); a simple analysis is presented as the following:

ne mu e = ne u e × B − ∇pe (1.38a) The effects of the magnetic field on the diffusion are twofold: by the peculiarity of the biased orientation of an externally applied magnetic field, and by the interaction of the Hall current eue × B . The force exerted on electrons is quite different in the direction perpendicular and parallel to the magnetic field and is described in the following (Howatson 1975). The partial pressure gradient transverse to B corresponds to a diffusion coefficient perpendicular to the magnetic field and can be given as:



de ⊥ =

de 1 + ( ω b vc )2

(1.38b)

The diffusion component parallel to B is altered by both the partial pressure gradients and the magnetic field to yield:

de = de

ω b νc 1 + ( ω b νc )2

(1.38c)

Recall ω b = eB m is the Larmor frequency and νc is the averaged collision frequency for the momentum transfer. However, bear in mind that a more elaborate analysis is required for accurate determination of diffusion for particles across and parallel to a magnetic field.

35

References

35

References Alfven, H., Cosmical electrodynamics, Clarendon Press, Oxford, 1950. Bittencourt, J.A., Fundamentals of plasma physics, Pergamon Press, Oxford, 1986. Friedrichs, K.O. and Kranzer, H., Nonlinear wave motion, Notes on magnetohydrodynamics, VIII, N.Y. University Rept. NYO 6486, July 31, 1958. Goebel, D.M. and Katz, I., Fundamentals of electric propulsion: Ion and Hall thrusters, JPL Space Science and Technology Series, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, 2008, pp. 37–​89. Goldston, R.J. and Rutherford, P.H., Introduction to plasma physics, Institute of Physics Publishing, London, 1995. Hall, E., On a new action of the magnet on electric current, Am. J. Math., Vol. 2, No. 3, 1879, pp. 287–​292. Howatson, A.M., An introduction to gas discharge, Pergamon Press, Oxford, 1975. Jackson, J.D., Classic electrodynamics (3rd edn.), John Wiley & Sons, New York, 1999. Jahn, R.G., Physics of electric propulsion, McGraw-​Hill, New York, 1968. Krause, J.D., Electromagnetics (1st edn.), McGraw-​Hill, New York, 1953. Landau, L., On the vibration of the electronic plasma, USSR J. Physics, Vol. 10, 1946, p. 25. Landau, L.D. and Lifshitz, E.M., Electrodynamics of continuous media, Pergamon Press, New York, 1960. Langmuir, L., The interaction of electron and positive ion space charge in cathode sheath, Phys. Rev., Vol. 33, No. 6, 1929, p. 954. Lieberman, M.A. and Lichtenberg, A.J., Principle of plasma discharges and materials processing, John Wiley & Sons, New York, 2005. Mitchner, M. and Kruger, C.H., Partially ionized gases, John Wiley & Sons, New York, 1973. Raizer, Yu.P., Gas discharge physics, Springer-​Verlag, Berlin, 1991. Riemann, K.U., The Bohm criterion and sheath formation, J. Phys. D., Vol. 24, 2991, pp. 493–​518. Shang, J.S., Computational electromagnetic-​aerodynamics, IEEE Press Series on RF and Microwave Technology, John Wiley & Sons, Hoboken, New Jersey, 2016. Shang, J.S., Huang, P.G., Yan, H., and Surzhikov, S.T., Computational simulation of direct current discharge, J. Appl. Phys., Vol. 105, 2009, pp. 023303-​1-​14. Spitzer, L., Physics of fully ionized gases, Interscience Publishers, New York, 1956. Stratton, J.A., Electromagnetic theory, McGraw-​Hill, New York, 1953. Surzhikov, S.T., Computational physics of electric discharges in gas flows, De Gruyter, Berlin, 2013. Surzhikov, S.T. and Shang, J.S., Two-​component plasma model for two-​dimensional glow discharge in magnetic field, J. Comp. Phys., Vol. 199, 2, Sept. 2004, pp. 437–​464. Sutton, G.W. and Sherman, A., Engineering magnetohydrodynamics, McGraw-​Hill, New York, 1965.

2

Plasma in a Magnetic Field

Introduction The interaction of neutral particles is associated only with a short-​range force, which is effective only when the distance between particles is approaching their mass centers for collision, and the interaction ceases as soon as the particles move away from each other. Plasma has a unique property in contrasting to the electrically neutral medium due to the long-​range electromagnetic forces, the Coulomb force and the Lorentz acceleration. The paired charged particles interact simultaneously with each other as well as their surrounding electric and magnetic fields. The magnitude of the Coulomb force is inversely proportional to the separation distance, and the force is collinear along the centers of charged particles. The electrostatic force contributes to the most outstanding characteristics of plasma to isolate all paired charged particles from interacting with the electromagnetic field, and the Debye shielding length is the measure of a distance for which the external electric field cannot exert strong influence on the bounded charges. The magnetic field is often generated by an electric current and creates acceleration to charged particles perpendicular to the direction of its motion. The difference in mass of electrons and ions also leads to a unique transport property of plasma. By virtue of the high electron mobility, it tends to diffuse faster than ions; the polarized electric field enhances the diffusion of ions and retards the diffusion of electrons to make the pair disperse at the same rate, known as the ambipolar diffusion. In the presence of a magnetic field, the plasma accentuates a large group of remarkable novel phenomena. First, the motion of charged particles is restrained, which leads to plasma confinement in a strong magnetic field. Another important characteristic of magnetized plasma is the transverse electromagnetic waves. In the low-​frequency spectrum, the Alfven and magnetoacoustic wave dominate. These wave motions can be described by a dispersion relationship between wave frequency and wave number; in turn, this relationship yields the important information of the plasma for the purpose of plasma diagnostics. The dissipation process by inter-​particle collisions damps the amplitude of waves through which the energy is transferred from wave motion to charged particles. However, the plasma also has a non-​collisional mechanism for wave attenuation by trapping charged particles under some condition at a certain particle velocity to be known as Landau damping. The unusual energy transfer mode exists between electromagnetic waves

37

Introduction

37

with particles moving closely with the phase velocity and has a reversible mode to increase the amplitude of the wave. The growth in wave amplitude increases the possibility of wave instability, which complicates engineering applications. An inherent property of plasma is its instability, and this covers a wide range of physical situations from the dynamics of ionization and depletion processes. The plasma instability, especially involving electric current in diagnostic procedure always generating excessive data fluctuations, renders serious measurement inaccuracy and unreliability. Sometimes the plasma instability even makes experimental observations and certain measurement procedures invalid (Matsuda et  al. 2008; Gulhan et al. 2009). For aerodynamic applications, the spectrum of instability is mostly beneath the threshold of effective flow control mechanisms –​except in the case of remote laser energy deposition for aerodynamic drag reduction, where the pulsations are generated by the intermediate microwave transmission, leading to undesirable consequences. In applications for hot plasma confinement in thermonuclear fusion research, understanding and controlling instability is the foremost critical concern. The research of plasma instability is an important subject for high-​ temperature plasma applications. One often overlooked but especially important aspect of plasma dynamics is the radiative transmission; it is the fundamental phenomenon of electromagnetism. The radiative energy transfer behaves as waves and yet as motions of energy-​ carried photons. The radiative energy transfer is equally critical as conductive and convective heat transfer mechanisms on aerospace vehicles for thermal protection. The radiative energy transfer is originated from the emission and absorption in the ionizing and recombining processes, as well as in processes of vibrational and electronic excitation transition between quantum states of atoms, molecules, and their ions. The radiation is made evident by the line spectra of atomic and molecular structures of the highly excited charged particles. The dominant collisional interaction is the Bremsstrahlung radiation when a charged particle is decelerated by colliding with other particles. This mechanism has been further classified as the free-​ free and free-​bound Bremsstrahlung radiation depending on whether the charged particles remain bound or unbound from the collisional encounters. Sometimes the radiation of atoms’ and molecules’ line structures is also significant, because the spectral optical properties of line strength and half-​widths are important to impact radiative energy transfer (Surzhikov 2005; Surzhikov and Shang 2015). The cyclotron radiation is also generated in magnetized plasma by the spiral motion of charged particles around the magnetic field line. From the aforementioned discussions, the behavior of plasma in a magnetic field impacts significantly its applicability in engineering. In fact, the presence of a magnetic field is often considered as a relativistic distortion to the electric field. Using plasma for flow control, the control effectiveness by amplifying the perturbation to flow field can be drastically enhanced by applying a transverse magnetic field. The presence of a magnetic field in plasma injects more challenges for analyzing plasma dynamics, but it also is necessary for viable applications. An illustration of its significance is easily demonstrable. The rate of change of magnetic flux density with

39

38

Plasma in a Magnetic Field

respect to time yields another aspect of plasma characteristics. Based on the generalized Ohm’s law J = σ( E + u × B ) the rate equation can be shown as:

∂B ∂t = ∇ × ( u × B ) − (1 µσ )∇2 B (2.1a) where B is the magnetic flux density, µ and σ are denoted the magnetic permittivity and electric conductivity of the plasma, u is the characteristic velocity of the charged particles’ motion. The second term on the right-​hand side of Equation (2.1a) is identified as the magnetic diffusion. In perfect electrically conducting plasma, the electric conductivity approaches an asymptotic value of infinity. The diffusion effect under a strong magnetic field diminishes and the convection dictates the plasma motion by locking it along the magnetic field line. In a weak applied and induced magnetic field, the magnetic diffusion will dominate over the forced or convective term, ∇ × ( u × B ). The rate of change for magnetic induction is similar to fluid motion by convective and diffusive terms; therefore, the ratio of the convective term to the diffusion term of Equation (2.1a) is designated as the magnetic Reynolds number:



Rm = uL (1 µσ ) = uLµσ (2.1b) In Equation (2.1b) the symbol L denotes the characteristic length scale of the problem. The dimensionless parameter Rm plays an important role for studying plasma dynamics, especially when the magnetohydrodynamics approximations are adopted. The complexity of magnetized plasma behavior is caused by the orientation of its resultant electromagnetic force that exerts perpendicular to the applied magnetic field on charged particles. For this reason, the fundamental property of electric conductivity in a pure electric field that can be described by a scalar quantity is negated, but in an electromagnetic field it must be treated as a tensor. This change also alters the basic drift and diffusion properties of plasma by the Hall effect and makes the physics much more complex for comprehension. Another peculiar characteristic of an electromagnetic field is that the electric and magnetic components of an electromagnetic wave are propagating in synchronization but perpendicular from each other. Therefore, even for a simple plane wave, a distinct characterization is needed to classify them into electric and magnetic transverse waves passing through a waveguide. In addition, the magnetic field introduces the Maxwell stress tensor to plasma and the normal stress component is recognized as the magnetic pressure. These global magnetized plasma behaviors are observed to increase the stand-​off distance of supersonic bow shock waves over a blunt body and to reduce heat transfer rate at the stagnation point. The mitigation of high heating is derived from the decreasing gradients of temperature and velocity profile normal to the magnetic pole that is embedded onto the blunt body surface. At the same time, the phenomena of magnetic confinement by a magnetic field such as the pinch effect, magnetic mirror, and atmospheric whistlers waves are very intriguing and offer tremendous possibility for technical innovations. In order to receive fully the benefits for engineering

39

2.1  Hall Current and Parameter

39

applications utilizing plasma, the following discussion of the present chapter will be highlighted to better explain plasma in a magnetic field.

2.1

Hall Current and Parameter The magnetic field produces the Hall effect (Hall 1879) by the gyrating acceleration of charged particles in plasma; the circulatory motion is revolving around and in a plane perpendicular to the axis of the magnetic field. The charged particles’ motion parallel to the magnetic field is completely independent from its influence, but accelerates by the electric field intensity E. Due to the disparity of unit mass of plasma composition, a transverse current is generated by the different curvature of the electrons’ trajectory from that of the heavier ions. The Hall current therefore will be shown to be dependent on the ratio of the electron cyclotron frequency to the electron collision frequency. In uniform electrostatic and magnetic fields, the equation of a charged particle’s steady motion can be given by components parallel and perpendicular to the magnetic flux density B as:



m d u dt = q E (2.2) m d u⊥ dt = q( E⊥ + u⊥ × B ) On a moving frame of reference with a constant speed, the velocity component in the plane normal to the magnetic field is:



u⊥ = ( E⊥ × B ) B 2 (2.3a) In other words, in the constant velocity frame of reference, the particle in the transverse plane to the magnetic field has a circular motion with a cyclotron frequency and moving with cyclotron or Larmor radius:



u⊥ = É b × rb (2.3b) Therefore, the particles’ motion in steady, uniform electrostatic and magnetic fields is a superposition of a circular motion in the plane normal to the magnetic field and a constant velocity perpendicular to both the magnetic field B and the electric field component. The particle velocity can be represented in vector form independent from the coordinate system as:



u = É b × rb + E⊥ × B + ( q m ) Et + uo (2.3c) The motion of a charged particle consists of cyclotron circular motion, which is the drift velocity of mass center, the constant velocity by the parallel electric field, and the initial particle velocity uo parallel to the magnetic field. The instantaneous center of gyration is commonly referred to as the guiding center of magnetized plasma. A vector component of the drift velocity is perpendicular both to the magnetic field and to the perpendicular component of the electric field to the magnetic induction. The drift velocity is defined as the rate of change of electrons’ movement along the electric field between collisions. One notices immediately that the drift

41

40

Plasma in a Magnetic Field

velocity is independent of the mass and the polarity of the charged particles. This velocity component is referred to as the plasma drift velocity in the presence of an electromagnetic field:

ud = E × B (2.4a) From Equation (2.3c), the particle motion in the plane normal to the magnetic field is a cycloid. Since the electric force qE⊥ accelerates or decelerates the particle based on the polarity of charge carried, the radius of curvature of the particle’s trajectory will change according to the normal electric field component. Accentuating the vastly different masses of electrons and ions, the Larmor radius of ions will be greater than that of electrons and the Larmor frequency of ions will be lower than that of electrons. Because the drift velocity must be identical to the paired ions and electrons, the curvature of the ions’ trajectory is greater than that of the electrons. Thus the collision frequency of ion-​neutral is greater than the electron-​neutral collision, and the ion motion will be retarded by the more frequent collisions with neutral species. Now the drift velocity of ions and electrons will be differed and an electric current is produced normal to both the electric and magnetic fields and flows in the opposite direction of the drift velocity to be identified as the Hall current (Mitchner and Kruger 1973). In essence, the curvature of the electron trajectory in a magnetic field is responsible for a transverse current; this phenomenon is also called the Hall effect. In other words, the Hall effect leads to a voltage difference across plasma due to the interaction of charged particles’ motion with an external applied magnetic field that transverses both the electric and magnetic fields. From the definition of a conductive electric current, the Hall current is a component of the current driven by the drift velocity:



J h = ne e( E × B ) (2.4b) The Hall current occurs in solid, liquid, and ionized gas; it depends on the ratio of electron cyclotron frequency to electron collision frequency. The Hall parameter is, then:



be = ω b νec = eB mν  λ e rb (2.5) where λ e denotes the mean free path of the electrons and rb is the Larmor radius of the electrons. The ratio is actually the number of gyrations by an electron between collisions and increases with increasing magnetic flux density B with a linear relationship. For this reason, the Hall effect is often adopted as a magnetic field detector to measure the magnetic field intensity, but due to the low signal level, its applications are limited to laboratory conditions. In essence, the Hall current is significant at high electron cyclotron frequency or in a low Larmor radius environment. The Hall effect in solid substances is significantly different from that in gas mediums, and reflects the much lower value of Hall parameters in solids, which generally are much less than unity. In particular, the voltage difference of the Hall effect in semiconductors always exists only on the outer edge of a semiconductor (Lieberman and Lichtenberg 2005). This peculiar property can lead to many

41

2.2  Transverse Waves

41

innovative practical applications for material surface treatments. From a pure physics viewpoint, the Hall effect offers the first concrete proof that electric currents in metal are carried by moving electrons. On the other hand, the Hall current in ionized gas is an important mechanism for aerospace engineering applications in ion thrusters for space propulsion and magnetohydrodynamic (MHD) electric generators, as well as MHD accelerators.

2.2 Transverse Waves The magnetic field introduces a distinctive different wave propagation mode to ionized gas mediums; in its absence, the information is transmitted through the mediums only by particle collisions in the form of a longitudinal wave or an acoustic wave. Electromagnetic waves are usually further classified as electron waves and ion waves. The former is considered to consist of electron oscillation such as light and Whistler waves. The ion waves are constituted by the acoustic wave, the ion cyclotron wave, and Alfven waves. The wave characteristics have been thoroughly investigated at low plasma density where the thermal kinetic energy is negligible for analytic studies. The key findings and classification of waves have been presented in excellent work in the Clemmow-​Mullaly-​Allis diagram (Allis 1959) to show the range of the magnetic field versus the electron number density. In addition, the diagram displays all the resonance and reflections as a function of the plasma frequency and electron cyclotron frequency ω ec = eB me . This knowledge is important for plasma diagnostics to determine the property of plasma in application and can be found in most books of plasma dynamics (Chen 1986; Bittencourt 1986). In most studies of high-​ frequency electromagnetic waves, the ion cyclotron frequency becomes an important criterion, because when wave frequency is greater than the ion cyclotron spectrum ωic = zeB mi , the motion of ions can be neglected. Whistler waves occupy the opposite frequency spectrum of electromagnetic waves; its waves packet is very rich in low frequencies from 100 Hz to 10 kHz. These waves occur in nature and are propagated through the ionosphere nearly along the earth’s magnetic field, and thus they will not be elaborated further in our discussion. The electromagnetic waves usually constitute the longitudinal electron plasma mode and longitudinal ion plasma mode. These two longitudinal wave modes are electrostatic in nature and contain all the charge accumulation but not the magnetic component. The ideal wave propagation velocity in gas can be derived from the adiabatic process of gas, pρ− γ = constant, and γ is the ratio of specific heats at constant pressure and volume processes. The wave propagated in the medium can be treated as a perfect gas (p ρ = γ RT ) and is given as:

us = ( ∂p ∂ρ)1s 2 = ( γ RT )1 2 (2.6a) The other mode is the transverse electromagnetic mode driven by the magnetic field to appear as a transverse wave. In a magnetic field the electrically conducting medium is subjected to magnetic stress by a tension along the magnetic lines like

43

42

Plasma in a Magnetic Field

a hydrostatic pressure of B 2 2µ . Therefore the motion of a small perturbation is subjected to an additional pressure normal to the magnetic field as p + B 2 2µ , and the magnetic field lines act effectively as elastic bands. Whenever the conducting medium is perturbed from its equilibrium state, the magnetic field lines will restore the perturbed medium back to its equilibrium state. The restoring force in turn generates a transverse vibration and the propagation velocity of the transverse wave motion with a velocity known as the Alfven velocity, which is equal to the square-​root value of the tension over the medium density, as given by Equation (1.22a) ua = B 2 µρm . The combined modes of electromagnetic wave propagation velocity are given as:

us = [( ∂p ∂ρ) + ∂( B 2 2µ ) ∂ρ] (2.6b) The existence of a transverse wave mode in the presence of a magnetic field becomes one of the unique features of electromagnetic wave propagation. In a uniform plasma motion, the adiabatic flow field is isentropic; the conservation energy equation describes the invariant entropy of the entire field. Under this condition the total derivatives of (p ργ ) vanish: D( p ργ ) Dt = ∂( p ργ ) ∂t + u ⋅ ( p ργ ) = 0. It is important to develop the dispersion relationship to understand the amplitude and phase angle of a transverse wave motion; the traditional approach is based on a small perturbation to the governing equations including the law of mass and momentum conservation and Faraday’s induction laws. ∂ρ ∂t + ρ∇ ⋅ u = 0



ρ∂u ∂t + ∇ ⋅ ( pI ) + B × ( ∇ × B ) µ = 0 (2.7a) ∂B ∂t − ∇ × ( u × B ) = 0 The dependent variables are decomposed into the original components that satisfy the governing equations of an undisturbed state and the perturbations to the induced magnetic flux and plasma densities. The perturbation to the charged particle velocity u is considered negligible:



B p = B + B' ; ρp = ρ + ρ' ; u p = u (2.7b) Substitute the expansions of dependent variables into the governing equations and eliminate the second-​order terms to obtain the following linearized equation system as:



∂ρ′ ∂t + ρ∇ ⋅ u = 0 ρ ∂u ∂t + ( γ p ρ)∇ρ′ + B × ( ∇ × B ′ ) µ = 0 (2.7c) ∂B ′ ∂t − ∇ × ( u × B ) = 0 By combining these equations, Equation (2.7c) rearranges and differentiates once more with respect to time to achieve a single second-​order partial differential; Equation (2.7d). The resultant partial differential equation in time and space is recognized immediately as the classic wave equation (Bittencourt 1986):

43

2.2  Transverse Waves



∂ 2 u ∂t 2 − ( γ p ρ)∇( ∇ ⋅ u ) + ( B 2 µρ) × {∇ × [ u × ( B

43

µρ )]} = 0 (2.7d)

Taking a step further by retaining only the wave motion parallel to the magnetic field, a simplified equation of wave motion becomes (Bittencourt 1986):

[ k 2 ( B 2 µρ) − ω 2 ]u + [( γµ p B 2 ) − 1]k 2 [ u ⋅ ( B

µρ )]( B

µρ ) = 0 (2.7e)

For a transverse wave, the velocity motion is perpendicular to the undisturbed magnetic field B and the unit vector of the wave number k is thus also perpendicular to the unperturbed magnetic field: u ⋅ B = 0. Thus, the inner product of u and the wave vector vanishes: u ⋅ ( B µρ ) = 0. Equation (2.7e) reduces to:

[ k 2 ( B 2 µρ) − ω 2 ]u = 0 (2.7f) Since in general, the particle velocity u has a nonzero value, the phase velocity of the transverse wave is obtained and recognized immediately as that of the Alfven wave:



up = (ω k ) = B

µρ (2.7g)

It is clearly shown that the phase velocity of the Alfven wave is independent from the wave frequency, thus the transverse wave is non-​dispersive. Again, the Alfven wave only involves plasma motion perpendicular to the magnetic field; it can be expected that the phase velocity of the Alfven wave holds for both the collisionless and collisional plasma. The perturbed electromagnetic field by the transverse wave can be found from the equation of charged particles’ motion from the general solution to the wave in Equation (2.7d):

u( r, t ) = u exp(ikr − i ωt ) (2.8a) After substituting Equation (2.8a) into the system of Equation (2.7c) and using a vector identity, the relationships between velocity, wave number k, and phase angle ω  are:



−ωρ′ + ρk ⋅ u = 0 −ωρu + kp − ( k × B ′ ) × B µ = 0 (2.8b) ω B′ + k × ( u × B ) = 0 The transverse electromagnetic wave reveals a perturbed density as well as magnetic and electric field components:



ρ′ = ρ( k ⋅ u ω ) (2.8c)



B ′ = − u B ( ω k ) (2.8d)



E ′ = − u × B (2.8e) However, note that the perturbed magnetic component is normal to the originally undisturbed and steady magnetic field. The magnetic pressure generated by the perturbed field actually forces a sinusoidal oscillation to appear as the observed transverse wave. In addition, the transverse wave does not involve the

45

44

Plasma in a Magnetic Field

fluctuations of either the medium density or the static pressure, but only oscillates laterally with the perturbed magnetic field line. According to the principle of equipartition of energy for the magnetic field (Mitchner and Kruger 1973), the magnetic energy density of the Alfven motion equals the kinetic energy density of fluid medium.

B ′ 2 2µ = B 2 u 2 [ 2µ( ω k )2 ] (2.8f) In summary, a transverse or Alfven wave will appear when a magnetic field is presented and is generally coupled to the acoustic wave. The transverse wave speed is directly related to the magnitude and direction of the magnetic induction; this is a unique phenomenon of plasma in a magnetic field. The phase and group velocities of the Alfven wave are equal in magnitude but differ in direction; the phase velocity is given as u p = B ⋅ k k ρµ = ω k, because ω = B ⋅ k ρµ and the group velocity is ug = ∂ω ∂ k = B ρµ . Thus the group velocity of the Alfven wave is in the direction of the applied magnetic field at an angle θ to the phase velocity (Bittencourt 1986). A unique characteristic of the Alfven wave is that the phase velocity of the transverse wave vanishes parallel to the magnetic field line.

2.3

Polarization of Electromagnetic Waves All three-​dimensional electromagnetic waves propagate in transverse wave mode; the electric and magnetic components are synchronized but propagate perpendicularly with respect to each other. The interdependence of the electric and magnetic fields is a striking characteristic of an electromagnetic wave when propagating through free space or a lossless dielectric medium. When the conducting current density does not exist in a medium, Ampere’s law reduces to ∂D ∂t = ∇ × H , which is the definition of the displacement current and the key ingredient for electromagnetic wave propagation in a vacuum. Another governing equation for the wave motion is Faraday’s law: ∂B ∂t = −∇ × E . For a plane electromagnetic wave traveling in an x-​coordinate direction of the Cartesian coordinate, the only contributing electric and magnetic components are: ∂H z ∂x = − ε ∂E y ∂t



and

(2.9a)

∂E y ∂x = − µ ∂H z ∂t Two distinctive types of linearly polarized waves are identified as the transverse magnetic (TM) mode where the magnetic field is entirely transverse while the electric field has a longitudinal component, or the modes are vice versa. When the electric field is transverse to the direction of wave motion, this is designated as the transverse electric (TE) mode. The classic equation of wave motion can be derived by differentiating Equation (2.9a) with respect to time and using mutual identities to achieve the D’Alembert wave equations (Krause 1953):

45

2.3  Polarization of Electromagnetic Waves



45

∂ 2 E y ∂t 2 = (1 µε ) ∂ 2 E y ∂x 2 (2.9b) ∂ 2 H z ∂t 2 = (1 µε ) ∂ 2 H z ∂x 2 The waves described by Equation (2.9b) are called the linearized polarized, and the general solution to the foregoing equation is simply:



E y = E0 exp[i ( ωt ± βx )] (2.9c) H z = H 0 exp[i ( ωt ± βx )] The exponent ( ωt ± βx ) of the electromagnetic wave components is referred to as the wave number, which has a complex value. The wave number describes the phase relationship and wave amplitude of all harmonic components of electromagnetic waves. Since the phase velocity is the product of wavelength and frequency u p = λ f , then βu p = ( 2 π λ ) f λ = 2 π f = ω. The symbol ω is referred to as the angular frequency, and the key parameters for a propagating wave can be found from the elementary solution. By a constant value of ωt − βx, the phase velocity of the wave is u p = ω β = λ f = 1 µε . In free space the phase velocity 1 εµ , which is the speed of light (3.998 × 108 m/​s), and the group velocity is given as u = d ω d β = β ( du p d β ) = u p − λ du p d λ . Transmitting electromagnetic energy is a very important function of microwaves for telecommunication and plasma diagnostic applications. The transverse microwave is frequently guided to minimize loss from expanding into a three-​dimensional wave, which is always subject to the diminishing wave amplitude by the geometric inverse square law. The waveguide transmits energy in transverse magnetic or transverse electric high-​order wave modes, and practical waveguides usually take the form of metallic rectangular or circular cylinders. The electromagnetic field structure within the waveguide is obtainable by solving the Maxwell equation by imposing null values to either the normal E or H components at the surface of the conductor. Assuming that temporal harmonic variations of a microwave are given by ei ωt , the variations in the z direction may be expressed as e − γ z , where γ = α + iβ is known as the wave propagation constant. The Maxwell equations for Ez and Hz become:



∂2 E z ∂2 E z + + γ 2 E z = −ω 2 µεE z ∂x 2 ∂y 2 (2.10) ∂2 H z ∂2 H z 2 2 + + γ Z = − ω µε H z z ∂x 2 ∂y 2 These types of wave are called transverse electromagnetic or TEM waves, which simultaneously consist of mutually perpendicular electric and magnetic components. However, the TEM waves may not always be able to propagate through a waveguide, because in an enclosed domain the electric and magnetic components cannot always be entirely perpendicular to each other in the direction of propagation. Since all the transverse electromagnetic waves are governed by the Maxwell equations, the electric and magnetic components must satisfy the boundary condition of the governing equations on the enclosing surfaces. The limited waveguide dimension

47

46

Plasma in a Magnetic Field

with respect to the wavelength can prevent the electric and magnetic fields from being mutually perpendicular in an electromagnetic field, thus not all electric and magnetic components of a transverse electromagnetic wave by a given wavelength can propagate through a waveguide. The polarization of a three-​dimensional electromagnetic wave is often referred to as an elliptically polarized wave, because at a point in space, the electromagnetic field rotates as a function of time and the vector follows an ellipse trajectory. It is usual to designate the possible field structures into the transverse magnetic (TM) waves for which the magnetic field strength H is identical to zero along the direction of wave propagation. On the other hand, the transverse electric (TE) waves for the electric field intensity E vanish. The transverse waves are designated as TEm,n or TMm,n to be the transverse electric and magnetic waves of order mn, and the lowest cut-​off frequency will occur for m and n equal to unity or zero. In fact, the lowest TE wave frequency in a rectangular guide is the TE1,0 wave, which has the lowest cut-​off frequency and is called the dominant wave. The higher-​order waves (larger values of m and n) are generated at higher frequencies in order to be propagated within a waveguide of given physical dimensions. Figure  2.1 presents the linearly polarized electromagnetic waves in free space and propagating through a waveguide in transverse electric mode TE11. The electric field strength E and magnetic field strength H are traveling in phase but perpendicular to each other. To demonstrate this peculiar wave structure in a waveguide, the computational simulated magnetic field is projected on the top surface of a rectangular cross-​section waveguide and the electric field is displayed on the bottom and side surfaces of the waveguide. The numerical results are projected onto surfaces perpendicular to each other, so the distinct polarized structures can be clearly discerned. The polarized and mutually perpendicular electric and magnetic components of a traveling wave are the unique feature of the transverse electromagnetic waves. The analytic solution for a lossless dielectric medium in a rectangular waveguide is well known, and the electromagnetic wave can propagate unimpeded (Stratton 1953). As has been discussed previously, there is a low frequency limit for which a linear polarized transverse magnetic and electric wave can be transmitted through a waveguide with a given cross-​section dimension. An electromagnetic wave also rapidly attenuates in a conducting medium in the collision process. In a perfect electrically conducting solid conductor, the microwave attenuation is so rapid that the wave may penetrate the conductor by only a very shallow depth. In order to validate numerical simulations’ capability for the performance of waveguides, a theoretical reference is invaluable. In practical application or in analytic analysis, the polarized wave is generated by a time-​harmonic mode by the known initial conditions of both the electric field intensity and the magnetic field strength (Bo or Ho).The generalized transverse electric (TE) wave traveling through a lossy medium or a semiconductor along the z-​coordinate has been derived as (Krause 1953):

47

2.3  Polarization of Electromagnetic Waves

47

Figure 2.1  Polarization of electromagnetic waves TE11 in free space and waveguide.



H x = ( H o κ 2 )( nπ a )e − α z [ A cos( ωt − βz ) − B sin( ωt − βz )]sin( nπx ) cos( mπ y ) H y = ( H o κ 2 )( mπ b )e − α z [ A cos( ωt − βz ) − B sin( ωt − βz )] cos( nπx )sin( mπ y ) H z = H o e − α z cos( nπx ) cos( mπ y ) (2.11a) E x = −( H o ων κ 2 )( nπ b )e − α z sin( ωt − βz ) cos( nπx )sin( mπ y ) E y = ( H o ων κ 2 )( nπ a )e − α z sin( ωt − βz )s in( nπx ) cos( mπ y ) Ez = 0 where the parameter k 2 is tied directly to the width and height of the rectangular waveguide (a,b), and the order of the transverse wave m and n: k 2 = ( mπ a )2 + ( nπ b )2 . The interrelated coefficients A and B of the solution are: A = ( ω c )2 ( σ ωε ) / 2 B



B = β2o + βo4 − [( ω c )2 ( σ ωε )]2 / 2 (2.11b) β2o = ( ω c )2 − k 2 Equation (2.11a) indicates the wave can attenuate exponentially in lossy mediums, and the phase shift in an electrically conducting medium is linear. Most important, it is also clearly shown that in the transverse electric mode, the electric field intensity is suppressed in the propagation direction along the z-​axis but not the z-​component of the magnetic intensity. For the transverse magnetic wave, only the electric field intensity along the wave-​traveling direction is permitted. For this reason, the transverse magnetic transverse waves are referred to as the E mode. The propagation constant or the wave number of the transverse electromagnetic waves, γ = α + iβ , in Equation (2.10) is: 2



2

 mπ   n π  γ=  + − ω 2 µε (2.11c)  a   b 

49

48

Plasma in a Magnetic Field

Figure 2.2  Cut-​off frequency of TM1,0 wave in a square waveguide.

From the ordinary transmission line theory, γ is a complex number γ = α + iβ . At a sufficiently high wave frequency; ω > [( mπ a )2 + ( nπ b )2 ] µε ; the value of the propagation constant γ is imaginary, and the microwave will propagate without impedance. At a low frequency spectrum ω < [( mπ a )2 + ( nπ b )2 ] µε the condition is reversed and the propagation constant γ is real; thus the wave amplitude will be attenuated and the wave motion eventually ceased. Again when γ is real, β must be zero, and there cannot be a phase shift along the guide, which means there is no wave motion at low frequency. Therefore, at a frequency below this value, wave propagation will not occur in the waveguide. This gives rise to the definition of the cut-​off frequency:

fc =

2

2

 mπ   n π    +   (2.11d) 2 π µε  a   b  1

A numerical simulation by solving Equation (2.11a) for the cut-​off frequency is depicted in Figure 2.2. A total of five wave numbers is calculated for the value of angular frequency up to 3 π for a TE1,1 wave on a ( 24 × 24 × 128) grid by a finite-​ volume scheme. It is seen that the decreasing frequency of a microwave corresponds to a reciprocally increasing wavelength. Until the value of ω is equal to the cut-​off frequency for the square waveguide of a unity cross-​section dimension, then the wave motion ceases at the cut-​off frequency f = π 2 . It is an important property of the transverse electromagnetic wave mode because any transverse wave will not be transmitted through the waveguide unless the wavelength is sufficiently short in comparison with the cross-​sectional dimensions of a waveguide, or the wave frequency must be above the cut-​off frequency of a hollow rectangular waveguide. The numerical solution agrees perfectly with the theoretical result for the cut-​off frequency or wavelength, Equation (2.11c).

49

2.4  Microwave Propagation in Plasma

49

Another peculiarity of an electromagnetic wave traveling in a waveguide is that the velocity of wave propagation in a guide can be greater than the phase velocity in free space.

up = ω β = w

ω 2 µε − ( mπ a )2 − ( nπ b )2 (2.11d)

The electromagnetic wave is an essential element for transmitting energy for plasma applications, and its dispersive behavior and unique characteristics at the medium interface lead to applications in microwave communication and plasma diagnostics. The energy transfer by a traveling electromagnetic plane wave can be evaluated by the electric and magnetic energy density De = εE 2 2 and Dm = µH 2 2; both have the physical unit of Joules per cubic meter. From the Poynting theorem, the rate of energy transfer per unit volume within a control space must be equal to the rate of work done on the charged particles plus the energy flux across the control surface (Poynting 1884). Therefore, the energy per unit area passing per unit time at any location is required to satisfy a divergent condition over an elementary control volume, which is the physical interpretation of the Poynting theorem.

∫ S ⋅ d s = −∂(1 2εE

2

+ 1 2µH 2 ) ∂t ∆V (2.12a)

By means of the Stokes theorem, the divergence of the Poynting vector becomes:

∇ ⋅ S = − ∂ ∂t[ 1 2 ( εE 2 ) + 1 2 ( µH 2 )] (2.12b) Perform the inner product of H with Faraday’s law, and conduct a similar operation to Ampere’s law with E; we have, then:



H ⋅ ( ∇ × E ) − E ⋅ ( ∇ × H ) = −( E ⋅∂D ∂t + H ⋅ ∂B ∂t ) (2.12c) From a vector identity, H ⋅ ( ∇ × E ) − E ⋅ ( ∇ × H ) ≡ ∇ ⋅ ( E × H ), we have:



( E ⋅∂D ∂t + H ⋅ ∂B ∂t ) = [ ∂ ∂t( εE 2 2 + ∂ ∂t( µH 2 2 )] (2.12d) Comparing Equation (2.12b) with Equation (2.12d), it follows immediately that:



S = E × H (2.12e) The symbol S is the Poynting vector, which gives the rate of energy flow per unit area in a wave that is perpendicular to the electric and magnetic fields. Therefore, it is a fundamental statement of conservation of energy of the electromagnetic field relating the energy stored and the work done by the charged particles. The Poynting vector can also be regarded as a surface power density of an electromagnetic wave.

2.4

Microwave Propagation in Plasma One of the most dramatic characteristics of plasma is the vastly different behavior of an electromagnetic (TEM) wave propagating through an electrically conducting

51

50

Plasma in a Magnetic Field

medium. As has been discussed previously for microwave transmission in a waveguide, the wave always attenuates in a weakly ionized gas or partially ionized plasma, because electromagnetic perturbations in a semi-​conductive medium or lossy material will always encounter resistance to external disturbance. The wave attenuation is a direct consequence of collision and interaction between charged particles and the electromagnetic field associated with the incident microwave. The relative magnitude of incident wave frequency and plasma frequency becomes a critical time parameter, which controls the reacting time for shielding paired charges from the external electromagnetic disturbance. The complete reflection of an incident EM wave at the interface of media can also occur when the frequency of the incident wave is lower than the plasma frequency to behave similarly as the cut-​off frequency. All the direction-​dependent disturbance propagations lead to a very complex wave system and introduce fascinating physics in resonance and energy damping. Especially, the magnetic field always introduces a charged particle motion that is perpendicular to both the electric and magnetic fields. In the presence of an applied magnetic field, the electromagnetic wave will always possess the longitudinal and transverse wave components and simultaneously incur a vast range of phenomena of resonances and degenerations for traveling waves. Some of these bifurcations of wave dispersive properties unfortunately are beyond the scope of our discussion. The equation of electromagnetic wave propagation in an electrically conducting medium and in an open space is (Krause 1953; Mitchner and Kruger 1973):

εµ

∂2 E ∂J −µ = ∇2 E − ∇( ∇ ⋅ E ) (2.13a) 2 ∂t ∂t

The general solutions of a transverse plane wave for Equation (2.13a) are described by harmonic functions and depend on the initial conditions of the electromagnetic field:

E = Eo exp[i ( ωt ± γ r )] (2.13b) J = J o exp[i ( ωt ± γ r )] where γ = ( εµω 2 − i µσω )1 2 is a complex wave number where the electric conductivity was defined previously as σ = ne e 2 me νc , and νc characterizes the averaged electron collision frequency of the partially ionized plasma. In the absence of an external applied magnetic field, the magnitude of the induced magnetic flux density is on the order of | E | c, which is usually negligible. If the effect of the magnetic flux density B is neglected in the generalized Ohm’s law, the temporal behavior of the electric current density in stationery plasma can be approximated as (Mitchner and Kruger 1973):



1 ∂J ≈ σ E − J (2.13c) νc ∂t By satisfying the plasma wave equation, Equations (2.13a), and Equation (2.13c) for describing the rate equation for electrical current density, the result establishes

51

2.4  Microwave Propagation in Plasma

51

the relation between the electric field strength and current density with the wave number and angular frequency of the electromagnetic wave:

γ 2 E = i ωµ J + εµω 2 E (2.13d) [1 − i ( ω ν)]J = σ E Solve the foregoing two equations by eliminating the current density to achieve a dispersion relation for the electromagnetic wave motion in an electrically conducting medium, including the wave attenuation by the collision process, as:



γ 2 = εµω 2 + i µωσ [1 − i ( ω ν)] (2.13e) Substitute the definition of electric conductivity σ = ne e 2 me ν and recognize that the magnetic permeability and the electric permittivity are related by the speed of light in free space, as (c 2 = 1 εµ ), a dispersive relationship for a transverse plane wave, becomes (Mitchner and Kruger 1973):



2  ( ε − ε o ) ( ω p ω )2   ( ω p ω )2 ( νc ω )   cγ  − (2.14a)   = 1 + +i  ω ε 1 + ( νc ω )2   1 + ( νc ω )2  

In this equation, ω p = ( ne e 2 ε me )1 2 is the plasma frequency given by Equation (1.18). In general, the small difference between the electric permittivity in free space and plasma can be ignored: ε − ε o ≈ 0. The complex wave number can be expressed as γ = α + iβ ; then, the electric field intensity, Equation (2.13b), becomes:

E = Eo exp( −α x ) exp[i (βx − ωt )] (2.14b) The real number α of the complex wave number is the attenuation constant associated with the wave amplitude, and β is the phase constant and related to the wavelength β = 2 π λ . The phase velocity of the electromagnetic wave as it has been given is u p = ω β. At the limiting condition that the incident wave frequency is much greater than the average collision frequency of plasma mediums ω >> νc , the effect of collisions is negligible and the dispersion relation reduces to:



(c γ ω )2 = 1 − ( ω p ω )2 (2.14c) Therefore, when the incident wave frequency is higher than the plasma frequency ω > ω p, γ is a real number and the wave propagates through the plasma without attenuation. However, if the condition reverses, ω < ω p, the wave number γ becomes an imaginary number and the solution E = Eo exp[i ( ωt ± γ r )] can no longer describe a propagating wave. It is referred to as an evanescent wave, and on average it does not transport energy, and will be reflected from the media interface. This particular behavior of electromagnetic wave reflection at the media interface is similar to the cut-​off frequency of the waveguide, and is the direct consequence of the Debye shielding phenomenon of plasma. From the simplified analysis, the electromagnetic waves with frequencies lower than the plasma frequency, the incident wave will be reflected at the interface of plasma. The reflected wave from the interface and rapidly attenuated wave amplitude produces the communication blackout phenomenon

53

52

Plasma in a Magnetic Field

Figure 2.3  Electromagnetic wave attenuation in plasma as a function of collision versus

propagating frequency.

that occurs in the earth reentry phase of returning space vehicles (Fedrick, Blevins, and Coleman 1995; Smoot and Underwood 1966). The attenuation of an incident electromagnetic wave in plasma occurs at a wave frequency lower than the plasma frequency ω p = ( ne e 2 ε me )1 2 . The dissipation of the wave amplitude is increased with the higher plasma collision frequency that is associated with a higher degree of ionization for the plasma. The plasma penetration depth in the intermediate frequencies domain, νc < ω < ω p, can be given as (Howatson 1975):

δ ≈ (c ω p )(1 + 3νc 8ω 2 + ω 2 2ω 2p ) (2.14d) From these approximations, the penetration depth of the evanescent wave is practically constant in the interior of the frequency region. The physical dimension of the depth is comparable to the wavelength of the plasma frequency to have a value at most a few millimeters in electric impact-​generated plasma, and is not much greater than that by thermally excited ionized air of the space reentering environment. The attenuation of propagating electromagnetic waves by collisions in plasma is depicted by computational results in Figure  2.3, at a propagating microwave frequency lower than the plasma frequency ω p = ( ne e 2 ε me )1 2. A  total of six different rates of the collision-​to-​incident wave frequency has been carried out to show the modification of the wave amplitude by the collision process. The numerical results describe the rapid attenuation of wave amplitude in terms of the penetration distance versus the ratio of frequencies in decibels (Shang 2002). The phase

53

2.4  Microwave Propagation in Plasma

53

shift can also be determined by the dispersion relationship; the phase constant β exhibits similarly a drastic dependence on the ratio between plasma collision and incident wave frequency. From these electromagnetic wave properties, the attenuation measurements of the incident of microwave by means of the electronic number density and collision frequency of a stationery plasma region are achievable. In fact, the dissipation of the wave amplitude is the consequence of the collision process of charged particles and the Debye shielding of charges of plasma. While collisions tend to modify the attenuation of the electromagnetic wave, the general behavior of the cut-​off frequency is still similar to electromagnetic wave propagation in a waveguide. The wave reflection at the media interface is established by the critical electron number density.

nec = ε me ω 2 e 2 (2.14e) In essence, when the electron number density of plasma is lower than the critical number ne < nec (ω > ω p), the electromagnetic waves can propagate through a nearly collision-​free plasma, but the wave amplitude will be attenuated by collisions. If the electron number density is greater than the critical value ne > nec(ω < ω p), the electromagnetic waves may still be able to propagate through the plasma, but are now strongly attenuated. The 1/​e depth of penetration of a microwave into a medium is inversely proportional to the square root of the product by frequency of incident wave, the electric permeability, and the conductivity of the medium 1 πωµσ . Both the wavelength and the wave attenuation rate are implicit functions of the free electron density and the collision frequency. These properties determine the ability of an electromagnetic wave to penetrate the plasma by describing the phase change and reflection coefficients at the interface of the medium. Once the real and imaginary parts of the wave numbers are known, the electron number density and average plasma collision frequency can be found. In short, the peculiar behavior of electromagnetic wave attenuation in plasma establishes the base for plasma diagnostics and is also the cause of communication blackout in earth reentry. In the intermediate frequency spectrum, ν < ω < ω p, the condition of an electromagnetic wave that will not propagate through the plasma is similar to that of the cut-​off frequency of waveguides. In essence, the wave will not be able to propagate through the electrically conducting medium when an incident microwave has a lower frequency than the cut-​off frequency. This behavior can be understood readily because the plasma now has sufficient time to rearrange the electromagnetic field and to shield the paired charged particles from the external disturbance. It is the manifestation of the basic characteristic of the Debye shield length in action. In the high-​frequency domain, the incident microwave of a greater frequency than the plasma frequency ω > ω p will be able to propagate freely. Under this circumstance, the plasma behaves as a dielectric in which the electromagnetic wave travels through with a very low attenuation similar to that of a relatively low-​ loss medium. The electromagnetic wave propagation in a magnetic field is much more complex than without it because of the transverse wave component. The complication,

5

54

Plasma in a Magnetic Field

in fact, creates significant difficulty for plasma diagnostics using the nonintrusive technique based on spectral measurements (Matsuda et  al. 2008; Gulhan et  al 2009). The general dispersion relationship for a longitudinal wave can be given by the kinetic integral formulation:

I = ω 2e ( no ω )∫∫∫ [ u ( ku − ω )] ∂f ( u ) ∂u dV (2.15) When the distribution function is Maxwellian, the collision frequency of electrons is much lower than the propagating wave frequency (νc ui >> un; un ≈ 0. Although the velocities of electrons and ions are substantially different, by comparison, the charged particle velocities can be considered in the same order O( ue ) ≅ O( ui ), again by the fact that in a partially ionized gas, the collision frequency between charged particles and neutrals is orders of magnitude greater than the collision frequencies between charged particles:  νen >> νe + , νe −, thus Equation (2.20a) reduces further to appear as:  kTe   e   e  ∇ne −  ne E −  ne ( ue × B ) (2.21a) ne ue = −     me νen   me νen   me νen 



It is recognized that the first term on the right-​hand side of Equation (2.21a) defines the electron diffusion coefficient, d e = kTe me νen for the ordinary diffusion mechanism due to species concentration. The random velocity of electrons is also contributed by the force diffusion mechanisms by the electrostatic force and Lorentz force. The group of parameters in front of the second and third terms ( e me νen ) is the well-​known electron mobility µ e , as previously introduced by Equation (1.33b). By the same arguments that the collision frequencies between charged species are much lower than those between neutral particles, ν+ n >> ν+ e , ν+ − , and even the electron drift velocity is much greater than the ion drift velocity, but it still makes the velocity of organized motion of neutral particles negligible: un  0. By comparison, we have, then:  kT+   e   e  ∇ne −  ne E −  n+ ( u+ × B ) (2.21b) n+ u+ = −     m+ ν+ n   m+ ν+ n   m+ ν+ n 

and

 kT−   e   e  ∇ne −  ne E −  n− ( u− × B ) (2.21c) n− u− = −     m− ν− n   m− ν− n   m− ν− n  Similarly, the ion mobility and diffusion coefficient for positively and negatively charged ions is recognized as µ ± = e m± ν± n and d ± = kT± m± ν± n . For all practical purposes, the masses of the positively and negatively charged particles are considered identical. The individual charged species velocities from Equations (2.21a), (2.21c), and (2.21d) can be summarized as:

59

2.5  Drift Diffusion in Transverse Magnetic Fields



59

ne ue = −d e ∇ne − µ e ne E − µ e ne ( ue × B )



n+ u + = −d + ∇n+ + µ + n+ E − µ + n+ ( u+ × B ) (2.22a)



n− u− = −d − ∇n− − µ − n− E − µ − n− ( u− × B ) Equation (2.22a) actually defines the charge number density flux vectors for each species and in most drift-​diffusion formulations is designated as:



Γ e = −d e ∇ne − µ e ne [ E + ( ue × B )] Γ + = −d + ∇n+ + µ + n+ [ E + ( u+ × B )] Γ − = −d − ∇n− − µ − n− [E + ( u− × B )] (2.22b) These flux vectors are contributions by species drift and diffusion velocities for an inhomogeneous, electrically conducting gas mixture by the multi-​fluid model for plasma in steady state. The leading terms describe the ordinary diffusion due to local concentration of electrons, positively charged ions, and negatively charged ions. The second terms are the force diffusion by drift motion of an electrically charged species in an externally applied electric field. The last terms are the force diffusion driven by the Lorentz acceleration. The system of equations Equation (2.22a) is the result of the well-​known classic drift-​diffusion theory (Raizer 1991). From applications of plasma to aerospace engineering, the transverse magnetic field has been found the most effective for flow control and electric propulsion. By applying the magnetic field along the z-​coordinate of the Cartesian frame, the effect of the magnetic field must be laid in the x-​y plane. The interaction of an externally applied magnetic field modifies charge particle velocities driven only by the Coulomb force. A component of charged velocities is generated both parallel and perpendicular to the electric current and the magnetic field. According to Ohm’s law, Equation (1.31b), the electric current by induced drift velocity component is accurately aligned with the direction of E × B. This cross-​product involves all x and y components of electric field intensity and the magnetic flux density is along the z-​coordinate. By the intrinsic characteristic of magnetic induction, the magnetic field plays no role in determining the motion of particles parallel to it. Therefore, when the magnetic field is aligned with the z-​coordinate, the particle motion under its influence can be decomposed into x and y components perpendicular to the magnetic field. The effective electric field components are displayed in Figure 2.4, thus the electric field may be written according to the electric conductivity by Equation (1.33a) as the components perpendicular and parallel to the electric field intensity:



E⊥ = E (1 + be2 ) (2.23) E = be E (1 + be2 ) where the symbol be is recognized as the electron Hall parameter be = µ e Bz c as introduced by Equation (2.5). The electron velocity under the influence of a transverse magnetic field thus has two components, ue = ue ( ue ,x , ue , y ) . The detailed description of charged particles’ velocities is provided by collision dynamics; the

61

60

Plasma in a Magnetic Field

Figure 2.4  Electromagnetic field vector components.

velocities of charged particles are similarly described by Equations (2.21a) through (2.21c).



ne ue ,x =

1  ∂n b  −d e e − µ e ne E x  − e 2 2   1 + be 1 + be  dx

  ∂ne  −d e dy − µ e ne E y 

ne ue , y =

1 1 + be2

  be ∂ne  −d e dy − µ e ne E y  + 1 + b2 e

∂ne   − µ e ne E x   − d e  dx

(2.24a)

The effective electric field strength can be introduced to include all components as the electric field intensity perpendicular to the applied transverse magnetic field along the x-​and y-​coordinates. By using the electron Hall parameter be = µ e Bz c = ω e νe , the expressions of the effective electric field components are simplified as:

Ee ,x = ( be E y − E x ) (1 + be2 ) (2.24b) Ee , y = −( be E x + E y ) (1 + be2 ) The resultant components of the electron number flux density are similar to that in the absence of an externally applied magnetic field, Equation (2.22b):



Γ e ,x = −µ e ne Ee ,x −

1 ∂n b ∂n d e e + e 2 d e e (2.24c) 1 + be2 ∂x 1 + be ∂y



Γ e , y = −µ e ne Ee , y −

1 ∂n b ∂n d e e − e 2 d e e (2.24d) 1 + be2 ∂y 1 + be ∂x

For the positively charged ions, the effective electric field strength, including the ion Hall parameter, b+ = µ + Bz c = ω + νe + , is defined as:

E + ,x = ( b+ E y − E x ) (1 + b+2 ) (2.25a) E + , y = −( b+ E x + E y ) (1 + b+2 ) By introducing the effective electric field, the components of positively charged ion mass flux density acquire the following forms:



Γ + ,x = µ + n+ E + ,x −

1 ∂n+ b ∂n d − + 2 d + + (2.25b) 2 + 1 + b+ ∂x 1 + b+ ∂y

61

2.6  Magnetic Mirrors



Γ + , y = µ + n+ E + , y −

61

1 ∂n+ b ∂n d + + 2 d + + (2.25c) 2 + 1 + b+ ∂y 1 + b+ ∂x

Similarly, by setting the effective electric field strength, including the contribution by the transverse magnetic field B, as E −,x and E −, y: E −,x = ( b− E y − E x ) (1 + b−2 ) (2.26a) E −, y = −( b− E x + E y ) (1 + b−2 )



The two components of the mass flux density of the negatively charged ions are:

Γ −,x = −µ − n− E −,x −

∂n ∂n 1 b d − − + − 2 d − − (2.26b) ∂x 1 + b− ∂y 1 + b−2



Γ −, y = −µ − n− E − , y −

1 ∂n b ∂n d − − − − 2 d − − (2.26c) 1 + b−2 ∂y 1 + b− ∂x

Again in these equations, b+ , and b− are the Hall parameters for the positively charged and negatively charged ions, which are directly related to the Larmor frequencies of the ions, b+ = µ + Bz c = ω + ν+ , and b− = µ − Bz c = ω − ν− . Here, ν− , and ν+ are the averaged electron and ion collision frequencies. The foregoing formulations, Equations (2.24c), (2.24d), (2.25b), (2.25c), (2.26b), and (2.26c), establish the charged species diffusion velocities by the multi-​fluid model for plasma in an electromagnetic field with a uniform externally applied transverse magnetic field. Namely, the orientations of the electric field intensity can be arbitrarily aligned without restriction, but need only to be specified in the x-​y plane that is perpendicular to the externally applied magnetic field along the z-​coordinate. The classic drift-​diffusion theory has been developed and verified over more than half a century, but has only been applied for modeling and simulation recently (Surzhikov and Shang 2004). The much more detailed formulations for the mass flux density, based on the concept of an equivalent induced electric field due to the Lorentz force exerting on charged particles, are limited to the circumstance that the externally applied magnetic field applies only in the transverse orientation. The drift diffusion formulation has proven to be a very effective plasma actuator modeling for flow control by electron impact ionization (Surzhikov and Shang 2004; Shang 2016). These equations not only replace the continuity equation for charged species of plasma, but also generate quantified electric field data when combined with the inelastic collisional models for ionization. These formulations are needed for evaluating the Joule heating and electrostatic force for direct and alternating current discharges for flow control by plasma actuators.

2.6

Magnetic Mirrors The magnetized plasma offers some unique plasma confinement procedures for flowing ionized gas just by imposing a magnetic force on moving charged particles.

63

62

Plasma in a Magnetic Field

Figure 2.5  Equivalent diamagnetic dipole moment by current loops of an applied

magnetic field.

These mechanisms are impossible to realize for aerodynamics with electrically neutral media. In an open-​end or endless configuration such as a tube or a loop, the initial condition of normal and parallel velocity components together with externally applied magnetic field under certain conditions can reflect or confine the charged particles’ motion. This phenomenon is known as the plasma mirror. The basic mechanism is based upon the observation that the induced circular motion of charged particles by an applied magnetic field can be considered as a group of electric current loops, as is illustrated in Figure 2.5 (Mitchner and Kruger 1973). Directions of the current loop relative to the magnetic field in the z-​coordinate are obtainable by the particle trajectory with respect to the guiding center of the charged particle drift motion. The location for the guiding center of plasma is defined as the instantaneous center of gyration of a charged particle.

x = xg + r cos( ωt + ϕ ) (2.27a) y = yg ± r sin( ωt + ϕ ) where ϕ is the phase angle determined by the initial condition. These current loops at the same time will generate an induced magnetic field by a dipole in the opposite direction of the applied magnetic field inside the loop. Because plasma is a collection of charged particles and possesses the diamagnetic property, the magnetic field generated by the current rings is located at distances much greater than the Larmor radius. The dipole is frequency referred to as the diamagnetic dipole:



d m = − ( mu⊥2 ) B (2.27b) For all practical purposes, the diamagnetic moment dm is nearly a constant. This behavior is referred to as the first adiabatic invariant (Mitchner and Kruger 1973). In general, the intrinsic phenomenon of plasma is associated with the energy of

63

2.6  Magnetic Mirrors

63

Figure 2.6  Schematic of magnetic mirror.

either an electric or a magnetic field that does not change and is called the adiabatic invariant. The total kinetic energy of charged particles is the simplest example, because the force of a magnetic field is always perpendicular to the charged particle velocity and will not contribute to work to the system. Therefore, in the absence of an electric field, the net change in the kinetic energy of the particles must be zero by virtue of Faraday’s induction law (Sutton and Sherman 1965):

∫ E ⋅ d l = ∫∫ (∇ × E ) ⋅ d s = d

dt ( B ⋅ d s ) = 0 (2.27c)

In other words, for a slow-​varying magnetic field, the surface normal to magnetic flux density can be approximated as πrb2, and then the quantity πrb2B is also a constant, together with the kinetic energy of the particle m( u⊥2 + u2 ) 2. In the absence of an electric field, they are the well-​known adiabatic invariants. An important consequence of the adiabatic invariants is that a charged particle that moves into an increasing magnetic field will be decelerated, even reflected. An increase in magnetic flux density must be accomplished by an increase in the rotational energy of the particle in the magnetic field. From the property of the adiabatic invariant, the kinetic energy of the charged particle must be constant. Thus, velocity parallel to the magnetic field with increasing strength must decrease until it vanishes. This sequence of events is known as the magnetic mirror. A schematic of the magnetic mirror arrangement and the magnetic flux density distribution in the mirror region is presented in Figure 2.6. A magnetic mirror is formed by applying a suitable electric current to the two coils located at the outer edge of the mirror region. When the electric current flows through these coils, a maximum magnetic flux density will be produced and the magnetic force will decelerate the reflect charged particles parallel to the increasing magnetic field strength until the motion ceased. The detailed condition for the magnetic mirror is given as the following: in the absence of an applied electric field and when the magnetic field is aligned with the

65

64

Plasma in a Magnetic Field

z-​coordinate, the equation of a charged particle moving in a helical trajectory can be given as:

m Du Dt = −µ dB dz (2.28a) where Du Dt = d u dt + u ⋅ ∇u is the substantial derivative, including the temporal and convective rate of change. In the steady state, the equation of charged particles motion becomes:



mu d u dz = −µ dB dz (2.28b) When the magnetic field flux density B increases in the z direction, a force exerts on the particle in the opposite direction to the gradient of the magnetic field. Integrate Equation (2.28b) with respect to the z-​coordinate to get:



m( u2 2 − u20 2 ) = −µ( B − Bo ) (2.28c) where uo and Bo are the initial value of the axial velocity and magnetic flux density along the z-​coordinate. It is clear that when the charged particle moves along the z-​coordinate, if the magnetic flux density is increasing, it will require the parallel velocity component to decrease based on the property of the first adiabatic invariant. At the point where B − Bo = ( mu2o 2µ ), the particle motion along the z-​axis must vanish. At that point, the gradient of the magnetic field is positive, and then the particles will be reflected. However, the condition for a particle trapped by the magnetic mirror can be avoided if:



mu2o 2 > ( mu⊥2 o 2 )( B − Bo ) Bo (2.29a) The particle will move continuously through the magnetic mirror. The specific condition for the magnetic mirror also arises by the relative magnitude of the initial and the maximum applied magnetic field intensity, which is closely linked to the ratio of the normal velocity component of the particle velocity. The charged particle moving into the mirror region has a pitching angle and a limiting value is defined by:



α p = sin −1 Bo Bmax = sin −1 ( u⊥ u ) o (2.29b) If the pitch angle of the charge particle is greater than α p, the particle will be reflected. On the other hand, if the pitch angle of the charged particle is less than α p, its pitch angle will never reach the value of π 2. This means that the parallel velocity component will not approach the vanishing point, and hence the charged particle can escape the trap of the magnetic mirror. This type of plasma confinement by a magnetic field is best suited for applications that the plasma is without ends or the magnetic field lines close by themselves such as the ion propulsion engine or a toroidal magnetic field such as in the Tokomak (Jahn 1968; Jackson 1975). The magnetic mirror is also important to nuclear fusion and the phenomenon also explains the observation of the Van Allen belt

65

2.7  Plasma Pinch and Instability

65

with respect to the earth’s magnetic field through either collision or radiative decay. However, the major problem of this magnetic confinement scheme is the issue of instability and small fluctuations from the designed configuration. Instability is a fundamental characteristic of plasma; it would occur in all possible plasma confinement schemes.

2.7

Plasma Pinch and Instability The confinement of an ionized gas or plasma is achieved by the induced magnetic field with an electric current. From experimental observations, the plasma with electric current flow always leads to instability and hysteresis. The phenomenon is particularly noticeable in a high-​temperature arc, which consists of a strong radiating mixture of electrons, ions, and neutral particles at nearly the same temperature from one to ten eVs (104 to 105 K). The electric discharge is also distinguished by these features at relatively high electric current but relatively low voltage away from the cathode or anode. In the high-​temperature positive column, the self-​constrictive magnetic force of high electric current amplifies and squeezes the electric conducting channel to a small cross-​ section. The self-​ constriction of the discharge column is the well-​known plasma pinch, which is the result of electrically conducting filament by magnetic compression. The simplest configuration for studying the plasma pinch is considering the plasma motion in a cylindrical column on the cylindrical polar coordinate, and by applying an axial current along the z-​coordinate. An induced magnetic flux density will be generated around the axis as a function along the radius line B( r ) . The J × B force exerts on the plasma to constrict plasma laterally to generate the plasma pinch effect for plasma confinement. The compressed plasma increases the charge number density and the temperature of the plasma column. Realistically an equilibrium-​pinched plasma column seldom exists, because the pinched effect is a dynamic balanced between the electric current-​induced magnetic pressure and the hydrodynamic pressure perpendicular to the current flow. The lateral magnetic pressure compressed against the plasma column and the kinetic pressure will counteract it by a series of reflecting pressure waves. The kinetic pressure reflects from the center of the plasma column to reach the outer interface between the plasma and the ambient atmosphere, and bounces inward continuously toward the downstream. These dynamic events always lead to instability of the plasma column. In general the instability is associated either with magnetohydrodynamics by the aforementioned interaction between induced magnetic pressure, or Maxwell stress and the particle motion. The instability from the induced magnetic pressure can be illustrated by an idealized cylindrical plasma column with a radius of r. By neglecting the viscous effects and in steady state with a nearly constant velocity so the plasma convection is negligible, the equation of particle motion on

67

66

Plasma in a Magnetic Field

cylindrical polar coordinates in a dynamic equilibrium state becomes the balancing act of the pressure gradient and the Lorentz force:

∇p( r ) = − J ( r ) × B( r ) (2.30) The total electric current of the flowing ionized gas along the axis is obtained by integrating the electric current density over the cylindrical cross-​section area, from the center of the cylindrical column to the outer radius of the cylinder R:



R

I ( r ) = ∫ 2 πrJ ( r )dr (2.31a) 0

Thus, the rate of change for the total electric current in the ionized gas column is:

dI ( r ) dr = 2 πrJ ( r ) (2.31b) From Ampere’s law for electric circuit, the relationship between the induced magnetic flux density and the plasma’s electric density can be found:



B ( r ) = ( µ 2 πr )I ( r ) = ( µ r )∫ J ( r )rdr (2.31c) Substitute these relationships, Equations (2.31b) and (2.31c), into an equation of steady motion for plasma column, Equation (2.30), to get:



2

dp( r ) dr = −( µ 2 π r 2 )I ( r ) dI ( r ) dr (2.32a) Rearrange Equation (2.32a) and perform the integration from the center to the outer edge of the cylindrical plasma column to get:



R

[ 4 π 2 r 2 p( r )]R0 − 4 π ∫ 2 πrp( r )dr = −µ( I 2 2 ) (2.32b) 0

It is also realized that beyond the outer radius of the plasma column the pressure will be vanished. However within the cylindrical domain 0 ≤ r < Rp(r), the pressure will be finite. Based on the physical meaningful boundary condition at the center of the plasma cylindrical column, again from Equation (2.31a), at the center of the cylindrical plasma column, the local current has the null value, I(0) = 0; thus, the first term of Equation (2.32b) drops out. The total electrical current of the plasma column along the axis is exclusively dependent upon the pressure integration across the plasma column.

r

I 2 = (8π µ )∫ 2 πrp( r )dr (2.32c) 0

The composition of the pressure distribution across the plasma column can be also obtained from Equation (2.30) by letting p(r) serve as a function of B(r). From the steady Ampere’s circuit equation, we have ∇ × B = µ J , and on a cylindrical coordinate system with the axial symmetric assumption, Ampere’s law appears as:

1 r d [ rB ( r )] dr = µJ ( r ) (2.33a)

67

2.7  Plasma Pinch and Instability

67

or

J ( r ) = (1 µ ) dB ( r ) dr + B ( r ) µr (2.33b) Substituting Equation (2.33b) into Equation (2.30) and performing integration with the knowledge that beyond the outer edge of the plasma column or the pressure beyond the plasma column must vanish: p(R) = 0. r



d [ r 2 B 2 ( r )] dr (2.33c) dr 0

p( r ) = (1 2µ ) ∫

For a constant electrical current density in the plasma column, the induced magnetic flux density will be linearly dependent on the total current in the column and inversely proportional to the square value of the plasma column radius. The induced magnetic flux density increases linearly toward the outer boundary of the plasma column and decays gradually away from the cylindrical plasma column. The kinetic pressure will then yield a parabolic distribution across the pinched plasma column, by Ampere’s circuit law of slow varying electric displacement; the electric current density is directly related to the curl of the magnetic field strength: J = ∇ × H . The magnetic field strength is therefore an azimuthal function and linearly proportional to the product of the current density and radius of the plasma column: H ( r )  rJ ( r ). The magnetic field will then decay rapidly proportional to the inverse distance from the center of the plasma column. Therefore, the induced force by the lateral magnetic field is:

(1 r ) ∂p ∂r + µJ 2 2 = 0 (2.34a) In the equilibrium condition of a uniform plasma column, the pressure distribution across the plasma column is parabolic:



p( r ) = ( µJ 2 4 )( rp2 − r 2 ) + p p (2.34b) As we see in the following discussion, the axial pressure within the column could be doubled to the average value. However, the electric current density is highly concentrated within a very thin layer near the outer edge of the plasma column as one would expect for a highly conducting medium. Meanwhile, the electric current density must vanish immediately beyond the plasma column. As a consequence, the pressure will be a constant within the plasma column to maintain a uniform average pressure level. The average pressure inside the cylindrical plasma column has been shown previously to relate to the total current I: Equation (2.32a). The average values is defined as:



R

p = 1 rR 2 ∫ 2 πrp( r )dr (2.35a) 0

Substituting Equation (2.33c) into the finite integral over the plasma column and perform the integration by part, we have:

p = µI 2 (8π 2 R 2 ) = B 2 ( R ) 2µ (2.36)

69

68

Plasma in a Magnetic Field

Figure 2.7  Typical pinched plasma columns, kink and sausage instabilities.

Quite a few observations can be made from the averaged pressure over the pinch effect of the plasma column. First, the average kinetic pressure in an equilibrium plasma column is balanced by the magnetic pressure at the outer edge boundary of the plasma column, B 2 ( R ) 2µ . The magnetic pressure is in fact the normal component of the Maxwell stress tensor and is elaborated in later discussions. Second, the same magnetic pressure is also presented in the shock layer over magnetized plasma flowing over a blunt body in a supersonic flow. The induced magnetic pressure within the shock layer actually pushes the enveloping bow shock farther away from the body than the neutral gas flow (Ziemer 1959). The self-​constricted magnetic mechanism is intrinsically unstable. The two typical commonly observed plasma pinches in an arch discharge are depicted in Figure 2.7. The magnetic pinching force acts stronger on the smaller radius of the column than the thicker portion of the column; the sausage shape of the arc column is formed. The kink in the plasma occurs due to an unbalanced magnetic force exerted on the concave curvature of the positive column. The constricted conformations actually are enhancing the pinching effect until the column is severed. Therefore, the plasma pinch as a nonequilibrium dynamic event seldom existed as depicted, but only in some unusual circumstances. These phenomena will take place independent of the current density distribution of the discharged column, and the self-​restriction mechanism is inherently unstable.

69

References

69

For the pinch effect, the average pressure within the equilibrium plasma column is dependent upon exclusively the square power of the total electric current density and inversely proportional to the radius of the plasma column. In a typical high-​temperature thermonuclear application, the pressure is maintained at around fourteen atmospheric pressures (1.4 × 107 dynes cm2 ). The plasma under this condition often has a high charged number density of 1015 cm3 and a temperature of 10 keV (108 K). At the magnetic induction of 19 kilogauss, the plasma confinement by the pinch effect requires an electric current of 9 × 10 4 amperes (Jackson 1975). Therefore, the plasma pinch effect generally does not occur in aerospace engineering applications, but it is a unique phenomenon of plasma in a magnetic field. References Allis, W.P., Waves in plasma, Sherwood confined controlled fusion, Gatlinburg, MD, TID-​ 7582, 1957, p. 32. Bittencourt, J.A., Fundamentals of plasma physics, Pergamon Press, Oxford, 1986. Chapman, S. and Cowling, T.G., The mathematical theory of non-​uniform gases (2nd edn.), Cambridge University Press, Cambridge, 1964. Chen, F.F., Introduction to plasma physics and controlled fusion, Plenum Press, New York, 1984. Fedrick, R.A., Blevins, J.A., and Coleman, H.W., Investigation of microwave attenuation measurements in a laboratory-​scale motor plume, J. Spacecraft and Rockets, Vol. 32, No. 5, 1995, pp. 923–​925. Gulhan, A. B., Koch, U., Siebe, F., and Riehmer, J., Experimental verification of heat-​flux mitigation by electromagnetic fields in partially-​ionized-​argon flows, J. Spacecr. Rockets, Vol. 46, No. 2, 2009, pp. 274–​282. Hall, E., On a new action of the magnet on electric current, Am. J. Math. Vol. 2, No. 3, 1879, pp. 287–​292. Howatson, A.M., An introduction to gas discharges (2nd edn.), Pergamon Press, Oxford, 1975. Jackson, J.D., Classical Electrodynamics (2nd edn.), John Wiley & Sons, New York, 1975. Jahn, R.G, Physics of electric propulsion, McGraw-​Hill, New York, 1968. Krause, J.D., Electromagnetics, McGraw-​Hill, New York, 1953. Landau, L., On the vibration of the electronic plasma, USSR J. Phys., Vol. 10, 1946, p. 25. Lieberman, M.A. and Lichtenberg, A.J., Principle of plasma discharges and materials processing, John Wiley & Sons, New York, 2005. Matsuda, A., Otsu, H., Kawamura, M., Konigorski, D., Takizawa, Y., and Abe, T., Model surface conductivity effect for the electromagnetic heat shield in re-​entry flight. Phys. Fluids, Vol. 20, 127103, 2008, pp. 1–​10. Mitchner, M. and Kruger, C.H., Partially ionized gases, John Wiley & Sons, New York, 1973. Poynting, T.L., On the transfer of energy in the electromagnetic field, Phil. Trans. R. Soc. Lond., Vol. 175, 1884, pp. 343–​361. Raizer, Yu. P., Gas discharge physics, Springer-​Verlag, Berlin, 1991. Raizer, Yu. P. and Surzhikov, S.T., Diffusion of charges along current and effective numerical methods of eliminating of numerical dissipation at calculations of glow discharge, High Temp., Vol. 28, No. 3, 1990, pp. 324–​328. Shang, J.S., Computational electromagnetic-​aerodynamics, IEEE Press Series on RF and Microwave Technology, John Wiley & Sons, Hoboken, NJ, 2016.

BP

70

Plasma in a Magnetic Field

  Shared knowledge in computational fluid dynamics, electromagnetics, and magneto-​aerodynamics, Prog. Aerosp. Sci., Vol. 38, No. 6–​7, 2002, pp. 449–​467. Smoot, L.D. and Underwood, D.L., Prediction of microwave attenuation characteristics of rocket exhausts, J. Space Rockets, Vol. 3, No. 3, 1966, pp. 302–​309. Stratton, J.A., Electromagnetic theory, McGraw-​Hill, New York, 1953. Surzhikov, S.T. Radiation modeling and spectral data. Lecture series 2002-​ 07:  Physic-​ Chemical Models for High Enthalpy and Plasma Flows. Von Karman Institute for Fluid Dynamics. Surzhikov, S.T. and Shang, J.S., Two-​component plasma model for two-​dimensional glow discharge in magnetic fields, J. Comp. Phys., Vol. 199, No, 2, September 2004, pp. 437–​464. Surzhikov, S.T. and Shang, J.S., Fire-​II flight test data simulations with different physical-​ chemical kinetics data and radiation models, Front. Aerosp. Eng., Vol. 4, No. 2, 2015, pp. 70–​92. Sutton, G.W. and Sherman, A., Engineering magnetohydrodynamics, McGraw-​Hill, New York 1965. Vlasov, A.A., On vibration properties of electron gas, J. Exp. Theor. Phys., Vol. 8, No. 3, 1938, p. 291. Ziemer, R.W., Experimental investigation in magneto-​aerodynamics, J. Am. Rocket Soc., Vol. 29, 1959, pp. 642–​647.

71

3

Maxwell Equations

Introduction Three-​ dimensional, time-​ dependent Maxwell equations consist of four fundamental laws of electromagnetics: Faraday’s induction law, the generalized Ampere’s electric circuit law, and the two Gauss laws for electric displacement and magnetic flux density. The original formulations are based on experimental observations, but all are developed from the phenomenon that electromagnetic waves interact with transmitting mediums on molecular/​atomic scales. In fact, all comply with the rigorous theory to become the most basic law of physics. These equations govern all electromagnetic phenomena and are equally applicable to all electromagnetic fields in free space and in all other media. These fundamental equations were established by James Clerk Maxwell in 1873 and verified experimentally by Heinrich Hertz in 1888. Albert Einstein’s special theory of relativity further affirmed the rigorousness of the Maxwell equations in 1905 (Kong 1986). For these reasons the Maxwell equations occupy the widest range of applicability and a unique position in plasma physics. It is worthy of note that all these laws follow a single physical concept. An understanding of these concepts is invaluable in order to apply these equations. Faraday’s law of induction simply states that a changing magnetic flux density will induce electric field intensity in the path surrounding it. The generalized Ampere’s law on a varying time frame defines the displacement electric current and is also a partial definition of magnetic intensity and magnetic force. Gauss’s law for magnetic flux describes that it has no source; the lines of magnetic flux have no beginning or end in an electromagnetic field. In contrast, Gauss’s law for electric displacement is that an electric field must be originated and terminated on an electric charge, and it is a partial definition of the electric flux density. The electromagnetic field equations are expressed by six dependent variables: the electric field intensity E, the magnetic field intensity H, the electric flux density D, the magnetic flux density B, the electric current density J, and the electric charge density ρe . The system of equations is not necessarily linearly independent under certain circumstances, but definitely does not form a closed system; namely, the numbers of unknowns and equations are unbalanced. The closure of the equation system is achieved by constitutive relationships that specify the specific features for a medium in an electromagnetic field. The most widely known constitutive relationships are for the interconnections of electric flux,

73

72

Maxwell Equations

magnetic flux, and electric current density to the overall electromagnetic field. Free space as usually meant is a vacuum or any other medium that has the same property as a vacuum, such as air. In free space, the constitutive relationships assume a particular simple form and the electric current density vanishes. The system of linear, first-​order partial differential equations in the time domain constitutes the hyperbolic system and represents the most general and versatile formulation for analyzing electromagnetic phenomena. In analyzing plasma with a multidisciplinary approach, the equation system is compatible with magnetohydrodynamics or plasma dynamics by a multi-​fluid plasma model. The hyperbolic differential system is an initial-​value and boundary-​condition problem that has a finite domain of dependence, and these domains are partitioned by characteristics extending to infinity (Courant and Hilbert 1965). An appropriate treatment of the far-​field boundary becomes an analytic issue for analysis, because solutions to the governing equation exist in the boundless space domain and any reflected wave from an artificial boundary will distort the wave motion. Another fundamental resolution limitation also emerges from describing wave motion via the numerical solutions in a vastly extended discrete space. A  high-​frequency electromagnetic wave usually has to travel over numerous wavelengths in an electromagnetic field from its source. This requirement is regulated by the basic and absolute numerical accuracy constraint by the Nyquist criterion for simulating wave motion. In short, a minimum of two discretized points per wavelength is necessary to achieve a physical meaningful simulation. This theoretical limit actually defines the upper bound for practical high-​frequency wave applications that can be accurately simulated. The computational resource requirement for three-​dimensional simulations over an extensive distance measured in wavelength can be truly daunting. Meanwhile, when an incident electromagnetic wave impinges on the interface across two different media, the wave will be partially reflected and transmitted through. The boundary conditions for the electric and magnetic vector field are derived from theorems of Gauss and Stokes that connect volume integrals to surface integrals on the media interface. However, the properties of electromagnetic waves are not necessarily continuous functions across the media interface. As a consequence, there are required continuity and jump conditions for the electric and magnetic fields across the interface. These boundary conditions give rise to some unique behaviors of the electromagnetic field. The random micro discharge or streamer structure of dielectric barrier discharge exemplifies the critical importance of imposing physically meaningful interface boundary conditions to capture the self-​limiting transition phenomenon from discharge by the impact ionization to arc. The detailed and appropriate interface boundary conditions together with the circuit law are presented in our following discussion. The dynamic nature of electromagnetic wave propagation and the interactions with fluid motion at the media interface require accurate time integration of the governing equations. The accuracy of numerical solutions to the Maxwell equations depends on wave frequency and propagating distance. The governing equations are classified as the hyperbolic partial differential system; the general

73

3.1  Faraday, Generalized Ampere, and Gauss Laws

73

solutions to this system are unbounded in space. Thus the detailed formulation must include an appropriate and physically meaningful initial/​boundary condition at its source and media interface; especially for the no-​reflection artificial far field, treatments will be included for numerical solutions only obtainable in a finite domain. The most widely used approach to alleviate this fundamental dilemma is derived from this unique feature by the zone of dependence for the hyperbolic partial differential equation. The partition between the different domains is known as the line or surface of the characteristics, which is an invariant (Courant and Hilbert 1965). Through this intrinsic feature, the characteristic formulation of the Maxwell equation in the time domain has overcome this serious limitation. For this reason, the detailed analyses based on the method of characteristics are included, which must start from the eigenvalue and eigenvector evaluations for the Maxwell equations in the time domain. From these unique properties of the governing equations, viable numerical algorithms are also included to be of direct use for readers. In addition, interesting applications to the electric and magnetic energy distributions and transmissions in electrically conducting fields will be discussed. The study of electromagnetic wave propagation in radiation and waveguides with and without an embedded plasma region will be included in our discussion. The antenna for plasma diagnostics through the dispersion relationship of microwave in an electrically conducting medium will also be highlighted.

3.1

Faraday, Generalized Ampere, and Gauss Laws Faraday’s law describes a time-​varying magnetic flux density B (Weber/​m2) that can generate a spatial distribution of the surrounding electric field intensity E (Volt/​m). In essence, it states that time rate of change in magnetic flux density across a control surface always induces a change in electric field intensity over a line that encircles the control surface. In a control volume, this relationship is given as:



∂B

∫∫ ∂t ⋅ ds (3.1a) ∫ E ⋅ dl = −  Apply the Stokes divergent theorem for the line and surface integrals to obtain:



∂B

∫∫ (∇ × E ) ⋅ d s = −  ∫∫∫ ∂t dv (3.1b)  The order of partial time-​derivative and volumetric integration is exchangeable, and the integration over the control volume is a function of time only. The partial derivative within the control volume becomes the total derivative in time; we have:



∂

∫∫ (∇ × E ) ⋅ d s = − ∂t  ∫∫∫ Bdv (3.1c) 

75

74

Maxwell Equations

The relationship between the induced electric component and the time-​varying magnetic component must hold at any point within the electromagnetic field. By shrinking the control volume to an infinitesimal dimension, it becomes a point in space. And assuming the field vectors B and E to be finite but possibly discontinuous across boundaries, Faraday’s induction law transforms to a partial differential equation.

∂B + ∇ × E = 0 (3.1d) ∂t The original steady state Ampere’s law for electric circuit only defines a relationship between magnetic field strength and electric current in an electric circuit, but is unable to describe the plasma wave propagation in a vacuum. The generalized Ampere’s equation by Maxwell remedies this shortcoming, giving the definition of displacement current ∂D ∂t. Now the time-​dependent Ampere’s law states that the conductive current and the displacement current of an electric circuit are generated by a magnetic field line around the control surface:



∫∫ ( J + ∫ H ⋅ d l = 

∂D ) ⋅ d s (3.2a) ∂t

Again by applying the Stokes divergence theorem, the line and surface integrals in Equation (3.2a) can be expressed as volume integrals over the control volume. This integral relationship holds only if the first derivative of the electric displacement D and the magnetic strength B are continuous throughout the control volume:

∂D

∫∫ (∇ × H ) ⋅ d s −  ∫∫∫ ∂t dv =  ∫∫∫ J dv (3.2b)  By exchanging the order of temporal and spatial derivatives and recognizing that the integration of the control volume and the results are functions of time only, the partial derivative becomes the total derivative in time.



∂

∫∫ (∇ × H ) ⋅ d s − ∂t  ∫∫∫ Ddv =  ∫∫∫ J dv (3.2c)  The relationship within the control volume holds even when the control volume contracts to a simple point in space; we have then:



∂D − ∇ × H = − J (3.2d) ∂t Equation (3.2d) is often referred to as the generalized Ampere’s law, and it introduced the new concept of the displacement electric current for electromagnetic wave propagation even within a dielectric medium such as a vacuum. The two Gauss laws for electric displacement and magnetic flux density of an electromagnetic field are key for maintaining consistent and compatible electric and magnetic fields. Gauss’s law for electric displacement can be derived simply by

75

3.1  Faraday, Generalized Ampere, and Gauss Laws

75

recognizing that electrons attain the Boltzmann distribution, ne = ne eϕ ( r ) κT in thermodynamic equilibrium condition. The potential energy of an electron at position r is −eϕ( r ). Similarly, under the same equilibrium condition, the ions also retain the Boltzmann distribution, ni = ne − eϕ ( r ) κT . The charged number density is defined as the sum of individual charges within a unit control volume. By this definition, the individual charged particles are no longer recognized but are described by the charge number density ρe , which has a physical unit of Coulomb per cubic meter. The sum of charges in an elementary volume can be given as:

∑ ze = e ∑ n z (3.3a) ρe = lim i i i ∆ν→ ε ∆v i where ni is the number density of the charge species, and zi denotes the number of elementary charges carried by the particles. The total electric field generated by a group of charges can be evaluated by the principle of superposition. The integrated surface electric field over control surface must be balanced by the total number of charges within the control volume. Thus:



∫∫ ( ε E )ds = ∫∫∫ ρ dv (3.3b) e

By means of the Stokes divergence theorem, both terms of Equation (3.3b) can be brought into the control volume; we have:

∫∫∫ ∇ ⋅ ( ε E )dv = ∫∫∫ ρ dv (3.3c) e

Again, this equality will hold at any point within the control volume; in fact, it is one of the two Gauss laws of electromagnetics in differential form.

∇ ⋅ D = ρe (3.3d) Gauss’s law relating the electric displacement and electric charge density is the foundation for the most widely applied Poisson equation of electrodynamics. This equation is built on the condition that the electric field intensity can be derived from an electric field potential. The potential energy distribution for plasma is governed, in general, by the Poisson equation of plasma dynamics. Therefore, this equation of plasma dynamics is a fundamental formula in describing the collective behavior of plasma in a continuum. Gauss’s law for magnetic flux density is straightforward by requiring the magnetic flux density to be an invariant over any control surface within any electromagnetic field, or the magnetic flux must have a vanished gradient across a control surface. In other words, Gauss’s law for magnetic flux density imposes additional constraint on the magnetic field.



∫∫ B ⋅ ds = 0 (3.4a) 

76

Maxwell Equations

Again this condition must hold at any point in space and time, shrinking the control volume to an infinitesimal value to obtain:

∇ ⋅ B = 0 (3.4b) Gauss’s divergence law for magnetic flux density in fact negates the existence of a magnetic dipole in direct contrast to the electric dipole. The two Gauss laws are not independent equations, but can be derived from the independent Faraday induction law and the Ampere electric circuit law. For most computational simulations, the Gauss law for magnetic flux density is always imposed implicitly or explicitly. The most widely applied conservation law for charge and electric current can be derived by taking the divergence of Ampere’s law and substituting Gauss’s law for electric displacement. The resultant equation becomes the most widely applied conservation law for charge and electric current.



∂( ∇ ⋅ D ) ∂t − ∇ ⋅ ( ∇ × H ) + ∇ ⋅ J = 0 (3.5a) From Gauss’s divergence law for electric displacement and by means of an elementary vector identity ∇ ⋅ ( ∇ × H ) ≡ 0, we get:



∂ρe + ∇ ⋅ J = 0 (3.5b) ∂t The charge conservation equation, Equation (3.5b), has been demonstrated to be invaluable for verifying the global behavior for computational electro-​aerodynamics and plasma dynamics simulations. For this reason, the conservation law for electric charge and current density has also been regarded as a fundamental equation for plasma dynamics.

3.2

Maxwell Equations in the Time Domain The Maxwell equations are accepted and established as the cornerstone of electromagnetic theory and are developed on a very elaborate and solid theoretical foundation. Their position as the intrinsic rules of nature is beyond any doubt and links many other scientific disciplines especially with radiation, plasma dynamics, and magnetohydrodynamics. The Maxwell equations in the time domain consist of a first-​order divergence-​curl system that is classified as hyperbolic partial differential equations. In essence, Maxwell equations are the foundation for describing all electromagnetic phenomena, including electromagnetic wave propagation, reflection, refraction, and interaction with reflective surfaces. In practical aerospace engineering applications, this system of equations has been widely used for plasma generation, microwave energy transfer, scattering, absorption, and emission, as well as antenna and integrated circuit design. From the previous derivations, the system of linear, partial differential equations can be presented as:



∂B + ∇ × E = 0 (3.6a) ∂t

7

3.2  Maxwell Equations in the Time Domain

77



∂D − ∇ × H + J = 0 (3.6b) ∂t



∇ ⋅ B = 0 (3.6c)



∇ ⋅ D = ρe (3.6d)



The first equation, Faraday’s law, relates the time rate of change of the magnetic flux density B (Weber/​m2) to the curl of the electric field intensity E (Volt/​m). Physically, it means a changing magnetic flux density in time will induce a gradient of electric field intensity in space. The second equation is the generalized Ampere’s circuit law relating the time rate of the electric displacement D (coulomb/​m2) to the curl of the magnetic field strength H (Ampere/​m) and electric current density J (Ampere/​ m2). The most significant fact is that the generalized Ampere’s law introduces the displacement current ∂D ∂t, so there is no ambiguity that electromagnetic waves can propagate in a vacuum. The last two equations are the Gauss laws for magnetic and electric fields. Equation (3.6c) simply requires that the divergence of B must vanish in any electromagnetic field to preclude the existence of a magnetic dipole. The second Gauss law defines the direct relationship between the divergence of the electric displacement D and the electric charge density ρe (Coulomb/​m3). It is known that the two Gauss laws, Equations (3.6c) and (3.6d), are not independent equations, therefore they are not always included explicitly in plasma dynamics analysis. However, these two laws are always imposed implicitly and automatically in the formulation. From a pure mathematical point of view, Maxwell’s equations have a closure issue, namely, the number of dependent variables is unbalanced by the number of unknowns. Meanwhile for problem solving, the material media also need more definite characterization to achieve physical fidelity. This dilemma is removed by the constitutive relations. The most general form is based on the relativistic consideration described by four tensors, P, L, M, and Q, in four dimensional spaces, and they are Lorentz-​covariant (Kong 1986). D = P ⋅ E + L ⋅ cB (3.6e) H = M ⋅ E + Q ⋅ cB In Equation (3.6e), the notations P, L, M, and Q are designated as tensors and all described by 3 × 3 matrices. The constitutive matrices L and M relate the electric and magnetic fields. The coupling between electric and magnetic vanishes when matrices L and M equal zero and the medium is anisotropic. In an anisotropic medium, the parallel property between electric intensity and its flux, as well as the alignment of magnetic intensity and magnetic flux, does not exist. However, the medium does not always have to be both electrically and magnetically anisotropic. For an anisotropic medium, if the matrices P = cεI ,Q = (1 cµ )I , and I is the identity 3 × 3 matrix, then the medium is isotropic. In an isotropic media, the required constitutive properties are simply:



D = εE (3.6f) B = µH

78

Maxwell Equations

where ε denotes the scalar electric permittivity in free space, which has a value of 4 π × 10-​7 Henry/​m, and the symbol µ is the scalar magnetic permeability in free space, which has a value of approximately 8.85 × 10-​12 Farad/​m. From Equation (3.6f), it follows immediately that the electric field intensity E and flux density D are parallel. The magnetic field intensity H and flux density B also have the same orientation. A dielectric medium under an externally applied electric field, polarization takes place by an induced electric dipole moment to align the permanent moments of the medium. As a consequence, a perturbation will appear in the alignment of the electric flux with the electric field intensity. Similarly, the diamagnetism occurs by an induced magnetic moment but tends to oppose the externally applied magnetically moments. There are more complications in the constitutive relationship for ferromagnetic anti-​ferromagnetic substances. For ferromagnetic material such as permanent magnets, spontaneous magnetization is also a function of the Curie temperature, which is beyond the scope of the present discussions. The solutions to Maxwell equations need not be analytic functions (Courant and Hilbert 1965). More importantly, the initial values together with any possible discontinuities are continued along a time–​space trajectory, which is commonly referred to as the characteristic (Sommerfeld 1949). This unique feature is derived from the eigenvalues and eigenvectors of the hyperbolic partial differential equation system. In a way, it reflects the physical characteristics of wave motions, which are directional dependent in that a signal from a source has a preferred orientation for information propagation to and from distinct and different zones of dependence. This particular feature of wave motion is easily demonstrated by applying the derivative of the Faraday and Ampere equations with respect to time; the second-​order curl-​curl form of the governing equations is immediately recognized as the wave equations. For a time-​harmonic wave motion, these equations are foundational to studying the wave in phase space (Harrington 1961).

1 c2 1 ∇×∇× E + 2 c ∇×∇× B+

∂2 B = ∇ × (µJ ) ∂t 2 (3.7a) ∂2 E ∂( µ J ) =− ∂t 2 ∂t

where c is the phase velocity of the electromagnetic wave. In an unbounded medium, it is a function of the magnetic permeability and electric permittivity, and is the speed of light in a vacuum (Krause 1953):

c =1

εµ (3.7b)

A three-​dimensional Maxwell equation in the time domain has been applied to investigate all aspects of electromagnetic phenomena, from the layout of complex integrated electric circuits, electric discharges, antenna design, telecommunication, and microwave interference, to radar cross-​sections evaluations. Its applicability to computational simulations spans the complete spectra of phenomena from acoustic to optical regimes and is limited only by computational capability.

79

3.2  Maxwell Equations in the Time Domain

79

Figure 3.1  Wave reflection from an electromagnetic wave impinging on an electrically perfect

conductor, far-​field energy distribution, and the near field of electric dipole.

Figure  3.1 displays the computational solutions of electromagnetic waves emitting from a dipole antenna and reflecting on an electrically perfect conducting sphere by the three-​dimensional, time-​dependent Maxwell equations. The orthogonal perpendicular electric and magnetic near-​ field structure is completely captured by the numerical simulation. The contour of the electromagnetic field intensities is depicted to highlight the reflected wave propagation from an incoming microwave. Meanwhile, the far-​field energy distributions are also included for verification with theory. The complex wave reflection pattern fully captures the scattering phenomena at three different wave number k values from 5.3 to 10.0 on a grid system of (73 × 48 × 96)  via a scattered field variable formulation (Shang 1999). The far-​field boundary condition is placed at a distance of 2.53 spherical radius away from the center of the scatter, and the numerical simulations reveal some degree of sensitivity to numerical accuracy on the placement of the far-​field boundary. The calculated electromagnetic energy of the horizontal polarity over the entire sphere at the far field reaches excellent agreement with classic results of the Mie series (Bowman, Senior, and Uslenghi 1987).

80

Maxwell Equations

For some practical applications, the Maxwell equations are required to investigate the problem on a nonstationary frame of reference. On the inertial coordinates with a steady velocity u, the Maxwell equations, after applying the Galilean coordinate transformation, have been shown to produce a different form; Maxwell equations are not invariant under the Galilean transformation. The result is in direct contrast to Newton’s law of motion, which shall be an invariant under such coordinate transformation. To resolve this paradox, Einstein postulates the speed of light in both sets of coordinates is a constant, and then applies the Lorentz transformation to the Maxwell equations. First, the origins of both coordinates are set on the same origins, and then the governing equations of both coordinates are transformed as:

x 2 + y2 + z 2 − c 2t 2 = 0 (3.8a) ξ 2 + η2 + ζ 2 − c 2 t 2 = 0 If all independent variables of Equation (3.8a) are regarded as components of a vector in a four-​dimensional space, the coordinate transformation can be written as a matrix operation. For any orthogonal coordinate, the products of direction cosine must be zero, and the length of the vector shall not be changed under the transformation. The coordinate transformation becomes:



 ε   1 1 − ( u c )2  η  0  =  0 ζ   ict  2  − i (u c ) 1 − (u c )

1 − ( u c )2   x     0   y  =  z  (3.8b)    1 1− ( u c )2   ict

0 0 i (u c ) 1 0 0 1 0 0

where u is the constant velocity of the moving frame. Under this condition, if the velocity is much slower than the speed of light, u c fi ′ f j′ or fi f j < fi ′ f j′. Thus the value of H can never increase; it is known as Boltzmann’s H-​theorem. In a uniform and steady state, the rate of change for the distribution function must vanish, thus the conclusion is also applied to the rate of the H function:

∫∫ ( f

i

′

f j′ − fi f j )dxi dx j = ∫∫  f (ci′ ) f (c ′j ) − f (ci ) f (c j ) d 3ci d 3c j (4.4c)

Therefore f (ci′ ) f (c ′j ) = f (ci ) f (c j ) is the necessary and sufficient condition for equilibrium, and the subscripts i and j are designated the different velocities for the colliding particles. It is found the most general solution to the Boltzmann equation under the condition by Equation (4.4c) is the Maxwell distribution function for the translational degree of freedom:

f ( xi , ci , t ) = ( m 2 πkT )1 2 exp( − mu 2 2kT ) (4.4d) The Maxwell distribution function for energy is:



f ( xi , ci , t ) = 2[ e π( kT )3 ]1 2 exp( − e kT ) (4.4e) Under thermodynamic equilibrium conditions, the energy of other possible internal degrees of freedom for rotational, vibrational, and electronic excitation can be described by the partition functions from statistic thermodynamics (Clarke and McChesney 1964). For each quantized internal state, the energy distribution functions are given as:



f ( xi , ci , t ) = 1 πεkT exp( − ε kT ) (4.4f) where the symbol ε represents the characteristic energy of each individual mode of internal quantized excitation. The Boltzmann constant has a value of k = 1.381 × 10 −16 ergs o c. The Boltzmann-​Maxwell equation is difficult to solve,

15

4.2  Fokker-Plank Equation and Lorentz Approximation

115

and the only known solution is Maxwell’s analytic solution under equilibrium conditions. It is interesting to note that the Maxwell equilibrium distribution function (1860) was found before the Boltzmann equation was derived (1872). The classic method for solving the Boltzmann-​Maxwell equation is by Enskog’s infinite series expansion (Chapman and Cowling 1964)

f = f ( 0 ) + f (1) + f ( 2 ) + ......... (4.5a) By an operator ξ in such a manner, the result can be expressed as:



ξ( f ) = ξ( 0 ) [ f ( 0 ) ] + ξ(1) [ f ( 0 ) , f (1) ] + ξ( 2 ) [ f ( 0 ) , f (1) , f ( 3 ) ] + ..... (4.5b) All function of the series is subjected to the condition that each term in the series must be vanished when ξ( f ) = 0. The individual term of series must satisfy this condition to yield:

ξ( 0 ) [ f ( 0 ) ] = 0 ξ (1 ) [ f ( 0 ) , f (1 ) ] = 0 (4.5c) ξ( 2 ) [ f ( 0 ) , f (1) , f ( 2 ) ] = 0 ................



The result of f ( 0 ) , f ( 0 ) + f (1) , f ( 0 ) + f (1) + f ( 2 ) ,..... will be the successive approximations to f . In most practical applications, the zeroth order approximation to the Boltzmann-​Maxwell distribution is the Maxwell equilibrium distribution given by Equation (4.4d).

4.2

Fokker-​Plank Equation and Lorentz Approximation In an elastic collisions environment, the predominant small angular deflection features by the electric charged particles give rise to the Fokker-​Plank collision term for the Boltzmann equation (Bololyubov and Krylov 1939). The underlying consideration is derived from the understanding that the rate of change in the distribution function is the consequence of a series of consecutive shallow deflection angle binary collisions. The scattering field shall be a good representation for the simultaneous Coulomb interactions among charged particles, namely, when a charge enters a shield electromagnetic force field or a number of charged particles are encountered within the realm of the Debye shielding length. The characteristic of a slowly diminishing field strength between charged particles and only high field intensity at a close distance between particles is well known. Therefore, a large velocity change by the collision process is an unlikely event. The modified collision term suggested by Fokker and Plank is widely adopted for binary charged particles collision as:



[ f ( xi , ci t ) ∂t ]c = ∫

∞

∫

4π

−∞ 0

[ f (ci′ ) f (c ′j ) − f (ci ) f (c j )] g σd Ωd 3c (4.6a)

In Equation (4.6a), the collision cross-​section σ is the Rutherford cross-​section within the Debye shielding length (Mitchner and Kruger 1973)

17

116



Plasma Dynamics Formulation

σ = ( ei e j 4 πε mij )[1 g 4 sin 4 ( χ 2 )] (4.6b) and the reduced mass is given as mij = mi m j ( mi + m j ). The deflection angle is related to the Debye shielding length λ D :



sin2 ( χ min 2 ) = 1 {1 + [ λ D ( ei e j 4 πε mi g 2 )]} (4.6c) To determine the particle number density, the distribution functions f (ci’ ) and f (c ’j ) are expanded with respect to a small change in a relative velocity ∇g during a binary collision. Perform the integration by part and after rearranging to yield:



[ ∂f ( xi , ci , t ) ∂t ]c = Γ ij {− ∂( fi ∂H ∂ci ) ∂ci + (1 2 )[ ∂ 2 fi ( ∂ 2G ∂ci ∂c j ) ∂ci ∂c j ]} (4.6d) 4π

Where Γ ij = ( mij2 mi2 ) g 4 Qij, and the function is defined as Qij = ∫ 2 sin2 ( χ 2 )σd Ω. 0 The symbols H and G are known as the Rosen Blatt potential in the Fokker-​Plank collision term (Mitchner and Kruger 1973):

∞

∞

−∞

−∞

H = ( mi mij )∫ [ f (ci ) g ]d 3c; G = ∫ gf (c j )d 3c (4.6e) The approximate Boltzmann-​Maxwell equation now acquires the following form:

∂f ∂t + ci ∂f ∂xi + Fi ∂f ∂ci = Γ ij {− ∂( fi ∂H ∂ci ) ∂ci + (1 2 )[ ∂ fi ( ∂ G ∂ci ∂c j ) ∂ci ∂c j ]} (4.6f) 2

2

In summary, the approximate Fokker-​Plank collision term transforms the integro-​ differential Boltzmann-​Maxwell equation into a partial differential equation. The first term on the right-​hand side of the Fokker-​Plank equation is frequently referred to as the dynamical friction and the second term is called the diffusion or dispersion due to the collision process. For weakly ionized plasma the degree of ionization is low, thus collisions between electrons and other heavy particles may be neglected. In addition, the thermal motion of heavy particles is also slow in comparison to that of electrons. Under this circumstance, only the kinetics of electrons is needed to consider for the distribution function and the probability distribution function is assumed to be Maxwellian. For a self-​consistent approximation, the charged particle number density shall be also very low to permit the electric current density to be negligibly small and remains nearly a constant. In a steady and uniform plasma field, the approximated Boltzmann-​Maxwell equation for electrons reduces to:

− ( e me )( E + ce × B ) ⋅ ∇f (ce ) = ∫∫∫ [ f (ci’ ) f (c ’j ) − f (ci ) f (c j )] gbdbd ζd 3c (4.7a) The resultant equation is still a nonlinear integro-​differential equation. In order to make the derivation simpler, the electric field intensity is considered weak and aligned with a principal coordinate. For an elastic sphere in binary collision, the impact parameter is simply b = r cos θ and db = −r sin θd θ where the r is the radius of the elastic sphere. The angle θ of the electron trajectory during the collision is the bisected angle that has formed by

17

4.3  Vlasov Equations for Collisionless Plasma

117

the apse vector on the electron trajectory; see Figure 4.2. Thus, it gives the classic collision cross-​section as:

σ = ∫∫ (1 − cos χ )bdb = πr 2 (4.7b) The Lorentz approximation leads to an electron distribution function for electrons including explicitly the electron and collision frequencies (Sutton and Sherman 1965):

f (ce ) = f (c0 ){| 1 − [ eνe E ⋅ ue + eci B × E ⋅ ue B ] {kTe [( me 3kTe )( eE me ) + ω e + νe ]}|} (4.7c) 2

2

2

where the electron frequency and collision frequency with the neutral are characterized by the symbols νe and ω e , respectively. It is observed that the electron distribution function exhibits dependence on the electromagnetic fields, thermodynamic and kinematic states of the electron, and the influence from the electron collision mechanisms.

4.3

Vlasov Equations for Collisionless Plasma Vlasov treats the Boltzmann-​Maxwell equation by neglecting the right-​hand side collision integral term, but explicitly includes the electromagnetic force in the rate equation (Vlasov 1938). For this reason, the Vlasov equation is often called the probability distribution equation for collisionless plasma. By this approach, the distribution function is evolved in time only by the contribution from the macroscopic smoothed electromagnetic field, but the formulation will not include the effects of short-​range collision. In his view, the short-​range effects of the Coulomb force are intensive, but the long-​range effect becomes the most unique and dominant mechanism for plasma. Nevertheless, the basic approach is based on the theory of kinetics with long-​range Coulomb interaction. Similar to the Fokker-​Plank formulation, the probability distribution function is analyzed as a nonlinear partial differential equation instead of the integro-​differential system. The drastically simplified formulation for the distribution function becomes a significant contribution to many analyses and simulation approaches. The governing equations consist of the rate equation for the distribution functions and the full set of Maxwell equations; the latter are included to ensure that the evolution is supported by a self-​consistent electromagnetic field. The formulation is simply given as:



∂f ( ui , xi ) ∂t + ui ∇ ⋅ f ( ui , xi ) + ( e m )( E + ui × B )∇ui f ( ui , xi ) = 0 (4.8a) The description of the distribution function has been further split into the electrons and positively charged ions to show the effects of different polarities:



∂fe ∂t + u ⋅ ∇c fe + ( e me )( − E + u × B ) ⋅ ∇u fe = 0 (4.8b) ∂fi ∂t + u ⋅ ∇c fi + ( Ze mi )( E + u × B ) ⋅ ∇u fi = 0

19

118

Plasma Dynamics Formulation

Because the short-​range effect by collisions is neglected in comparison with the long-​ range electromagnetic forces, the Maxwell equations are included in the Vlasov equations system:

∂E ∂t + ∇ × B = µ J ∂B ∂t + ∇ × E = 0 (4.8c) ∇⋅ E = ρ ε, ∇⋅ B = 0 where Z is the number of possible elementary charges that may be carried by the positive ions. The foregoing equations must be solved simultaneously to ensure that the probability distribution functions for electrons and ions fe ( xi , ui , t ) and fi ( xi , ui , t ) are self-​consistent with the immersed electromagnetic field. When distribution functions for electrons and positively charged ions are known, the macroscopic properties of the charged number density and electrical current density are obtainable by the following integrated results:



ρ( xi , t ) = ∫∫∫ [ Zfi ( xi , ui , t ) − fe ( xi , ui , t )]d 3 ui

J ( xi , t ) = ∫∫∫ [ Zfi ( xi , ui , t ) − fe ( xi , ui , t )]ui d 3 ui

(4.8d)

More important, the approximate distribution function by partial differential equation provides a significant advantage for description of the macroscopic properties of plasma by the method of moments based on the probability theory. The concept of the Vlasov approximation actually opens a new avenue to analyze plasma as a multi-​fluid medium to impact greatly the understanding and application of plasma dynamics for engineering. The detailed procedure using the method of moments (Chapman and Cowling 1964) to derive plasma dynamics conservation laws will be presented in the following section. The Vlasov equations simplify significantly the solving procedure for the distribution functions from a set of integro-​differential systems to a group of partial differential equations. The omission of the collision integral from the Boltzmann equation is a clearly valid approximation for collisionless plasma. However, there are not known any general systematic evaluations for the validation of the Vlasov approximation to physical fidelity, but the approach impacts profoundly the approach linking the microscopic and macroscopic phenomena in plasma dynamics. Nevertheless, some comparisons of MHD and collisionless models are presented in works dedicated to high-​altitude space experiments (Surzhikov and Raizer 1995b); the Vlasov approximation has exhibited a substantial valid range in theoretic plasma physics.

4.4

Multi-​temperature and Multi-​fluid Models The multi-​fluid model of plasma describes the coupling between the outstanding electromagnetic properties of plasma and fluid dynamics on the macroscopic scale. In concept, the multi-​fluid approach is similar to that of Vlasov’s formulation (1938), in which the electromagnetic force is explicitly implemented in the

19

4.4  Multi-temperature and Multi-fluid Models

119

conservation equations. The difference is that instead of solving for the distribution function, the macroscopic properties of plasma will be solved for the dependent variables in the continuum regime. The multi-​fluid model neglects the irreversible close-​range intermolecular interaction but focuses on the equilibrium solution to Boltzmann-​Maxwell equations with constant coefficients (Birdsall and Langdon 1985). By this assumption, the plasma is composed of multiple different fluids with different velocities and partial pressures as a heterogeneous medium. In most analytic works, this multi-​fluid plasma model is further simplified to only the distinctive two-​fluid composition with electrons and positively charged ions (Shumlak and Loverich 2003). For aerospace engineering applications, the ions are split further to include the positively and negatively charged ions and to become a three-​fluid partially ionized plasma model (Shang 2016). The common starting point of deriving the multi-​fluid model for plasma dynamics is through kinetic theory by a combination of the Boltzmann-​Maxwell distribution and Maxwell equations in the time domain: ∂fi ∂f q ∂f  ∂f  + ui i + i ( E + ui × B ) i =  i  (4.9a) ∂t ∂c i mi ∂u  ∂t  c



For a class of problems, the detailed knowledge of ionized compositions of the plasma is not essential, but only the global behavior of plasma dynamics is sought after. An attractive attribute of the multi-​fluid model is its ability to treat the dynamics of electrons and positively and negatively charged ions without the complication from the chemical kinetic model (Surzhikov and Shang 2004). The identical procedure follows the process for deriving the Euler equations from the Boltzmann-​Maxwell equation by taking the moments of the distribution equation and by performing the integration over the entire velocity space. The process is also known as the transformation of various terms of variable φ by method of moments of order n (Chapman and Cowling 1964):

∫φ



n

f [t, ci , xi ,( E + u × B )]d 3c = 0 (4.9b)

Such that:

∫ φ(∂f ∂t )d c = ∂( ∫ φ fd c ) ∂t − ∫ f (∂φ ∂t )d c = ∂( nφ) ∂t − n (∂φ ∂t ) ∫ φu(∂f ∂x )d c = ∂( ∫ φufd c ) ∂x − ∫ fu (∂φ ∂x )d c = ∂( nφu ) ∂x − nu (∂φ ∂x ) 3



i

3

3

3

3

i

i

3

i

i

and

∫ φ(∂f

∂ci )d 3c = ∫∫ [ φ f ]∞−∞ dvdw − ∫ f ( ∂φ ∂u )d 3c = − n ∂φ ∂ci (4.9c)

The last equation of Equation (4.9c) gives the integrated results over the entire range of the phase space. By hypothesis, the values of φ f must be vanished on the integral limits at infinity. The general integral transformation can be given as:

∫ φ(∂f

∂t + c ⋅ ∂f ∂xi + F ⋅ ∂f ∂ci ]d 3c

(4.9d) = ∂( nφ ) ∂t + Σ ∂( nφu ) ∂xi − n{∂ϕ ∂t + Σu( ∂φ ∂xi ) + ΣFi ( ∂φ ∂u )

12

120

Plasma Dynamics Formulation

The equation of change, Equation (4.9d), is a generalization by Enskog expansion to the Boltzmann-​Maxwell equation. The equation of distribution function is referred to only as the function of the peculiar velocity of particles; thus the terms ∂φ ∂t and ∂φ ∂r do not appear on the left-​hand side of Equation (4.9d). The dependent variables for the resultant equations are then yielded the macroscopic property for mass, momentum, and energy of plasma. The zero-​order moment of the Boltzmann equation is simply by integrating the distribution function with a scalar variable, numbers of particle densities, over the entire velocity space. The integral result by Equation (4.9c) yields essentially the species continuity equation:

n( x, t ) = ∫ n( x, c, t ) f ( x, c, t )d 3c (4.10a)



∂ne ∂we + ∇ ⋅ ( ne ue ) = (4.10b) ∂t ∂t



∂n+ ∂w+ + ∇ ⋅ ( n+ u+ ) = (4.10c) ∂t ∂t



∂n− ∂w− + ∇ ⋅ ( n− u− ) = (4.10d) ∂t ∂t The right-​hand term is the production or depletion rates of the specific species by chemical reaction for ionization. The source term of the ionizing process can be adopted either by the nonequilibrium chemical kinetics or by the inelastic collision modeling. Take the first moment of the Boltzmann equations for each species, which is identical to the integral integrational transformation by Equation (4.9d):

ρi ui ( x, t ) = ∫ ρu( x, t ) f ( x, c, t )d 3c (4.11a)



Perform the indefinite integral by Equation (4.9c) over the complete range of velocity space and let the product of ρu( x, c, t ) f ( x, c, t ) vanish at the infinite integral limits; the transformation gives the momentum equation for each individual charged component: ∂ue + ( ue ⋅ ∇ )ue ] = −∇pe − ne e( E + ue × B ) + Re (4.11b) ∂t



ne me [



n+ m+ [

∂u+ + ( u+ ⋅ ∇ )u+ ] = −∇p+ − n+ e( E + u+ × B ) + R+ (4.11c) ∂t



n− m− [

∂u− + ( u− ⋅ ∇ )u− ] = −∇p− − n− e( E + u− × B ) + R− (4.11d) ∂t

Equations (4.11b) through (4.11d) describe the momentum balance for the plasma components by the contributions of the partial pressure gradient, Lorentz force, and momentum transfer with other components of the ionized gas. The momentum transfer terms Re, R+, and R–​ are derived from the interactions with other species of

12

4.4  Multi-temperature and Multi-fluid Models

121

the plasma from Equation (4.9d). These source terms will be vanished in electrically perfect conducting plasma (Shumlak and Loverich 2003). Similarly, take the second moment of the Boltzmann equation:

σ( x, t ) = ∫ [ci ( x, t ) − c j ( x, t )][ci ( x, t ) − c j ( x, t )] f ( x, c, t )d 3c (4.12a) The integral transformation yields the energy equation in thermodynamic variables in which temperatures are related to partial pressures of individual charged species by the equation of state:



ne ∂Te [ + ( ue ⋅ ∇ )Te ] = pe ∇ ⋅ ue (4.12b) γ − 1 ∂t



n+ ∂T+ [ + ( u+ ⋅ ∇ )T+ ] = p+ ∇ ⋅ u+ (4.12c) γ − 1 ∂t



n− ∂T− [ + ( u− ⋅ ∇ )T− ] = p− ∇ ⋅ u− (4.12d) γ − 1 ∂t Equations (4.12b) through (4.12d) describe the rate of change in temperature of plasma components that produces by the work done by the plasma in an adiabatic system. By defining the electric current density for each individual charged species J i = eni ui , the momentum equation Equations (4.12b) through (4.12d) can be rewritten in terms of the current densities of each individual component in divergence form: J J ∂J + ep I  e 2 n+ E  e  + ∇⋅ + + + +  = (J+ × B) +  m+  ∂t m+  m+  en+



J J ∂J − ep I  e 2 n− E  e  + ∇⋅ − − + −  = (J− × B) − (4.13a)  m−  ∂t m−  m−  en− J J ∂J e ep I  e 2 ne E  e  + ∇⋅ e e + e  = +   (Je × B) ∂t me  me  me   ene In Equation (4.13a), the symbol I represents the identity matrix. Combining Equation (4.12b) with Equation (4.13a) and assuming the plasma follows the law for equation of the state, the resultant equation appears as the conservation of electric current densities in divergence form. Taking a step further, by coupling the energy equation and the divergence equation of electric current density, Equation (4.13a), the Joule heating appears explicitly on the right-​hand side of the conservation of internal energy law for each component of the plasma:



∂ne J − ∇ ⋅ [( ee + pe ) e ] = J e ⋅ E ∂t ene ∂n+ J + ∇ ⋅ [( e+ + p+ ) + ] = J + ⋅ E (4.13b) en+ ∂t ∂n− J − ∇ ⋅ [( e− + p− ) − ] = J − ⋅ E en− ∂t

123

122

Plasma Dynamics Formulation

The three fluid model equations have often been simplified to consist of only the electrons, positively and negatively charged ions. Nevertheless, the equations system establishes the relation among the species number densities, the electric current densities, and the internal energy of each charged species. In this formulation, the species production/​depletion, electrostatic force, Lorentz acceleration, and Joule heating appear as source terms that are results of the electron–​ion collision process. These source terms of the motion for plasma significantly affect the evolution of the electromagnetic field. Thus the accurate solution of the multi-​fluid plasma model also requires the Maxwell equations to accommodate the modifications to the electromagnetic field after a solution advanced in time. However only Faraday’s induction law and Ampere’s electric circuit law are required in this formulation:



∂B +∇× E = 0 ∂t (4.14a) ∂E B εo − ∇ × ( ) − e( n+ u+ − ne ue − n− u− ) = 0 ∂t µo Gauss’s divergence laws for electric field strength and magnetic flux density are applied as constraints to eliminate nonphysical results:



∇ ⋅ B = 0 and ∇ ⋅ E = e( n+ − ne − n− ) ε o (4.14b) A limitation of the multi-​fluid model unfortunately excludes the direct treatment of Hall effects and diamagnetic terms. However, if necessary, these terms govern the ion current and the finite Larmor radius of the plasma components can be included. The Hall current has been added to the model equation that leads into the Whistler branch (Jackson 1975). In computational simulations of the Whistler branch, the dispersion relation shows wave velocity increases with frequency, thereby shrinking the envelope of computational stability. Physically the model represents the individual motions of charged particles, but preserves the globally neutral property of the plasma. In short, the multi-​fluid plasma model is more general than the classic MHD formulation. Specifically, this model treats the charged particles separately according to their polarities and can still preserve the intrinsic quasi-​neutral property of plasma. However, this feature actually is responsible for excluding the Hall and diamagnetic effects that are associated with the ion current and the different Larmor radii of the plasma constituents. Multi-​fluid plasma formulation has been widely applied to simulate the discharge by electronic impact ionization for flow control. The discharge generated by either direct or alternate current discharge, the total energy of plasma, is usually limited and up to a few electron volts and carried mostly by electrons alone. Therefore, only Equations (4.10b) through (4.10d) are needed for calculating the charged number densities and the source terms. The rates of production and depletion of charged species can be provided by either the chemical kinetics models or the inelastic collision quantum chemical models. A typical solution of the direct current discharge by solving the multi-​fluid equations, Equations (4.10b) and (4.10c), is depicted in Figure  4.3. The classical

123

4.4  Multi-temperature and Multi-fluid Models

123

Figure 4.3  Simulated direct current discharge by multi-​fluid formulation, ϕ  = 5.0 kV, p = 5.0 Torr, electrode gap distance H = 2.0 cm.

Poisson equation of plasma dynamics is used by satisfying simultaneously Gauss’s law for the electric field and Ampere’s circuit law. The production and depletion source terms for the conservation of electron and positively charged ion number density are modeled by the Townsend similar law for ionization (Surzhikov and Shang 2004). The normalized electron and ion number densities by a value of 1010/​cm3 are presented side by side for a parallel-​electrodes configuration. The direct current discharge is generated by an electric field intensity ϕ = 3.0 kV and at an ambient pressure of 5.0 Torr. The cathode is located on the lower flat plate opposite to the anode on the upper plate with a gap distance of 2.0 cm. In these presentations, the characteristic positive column of a normal discharge is displayed and especially the cathode layer is clearly revealed to have a dimension of 0.125 cm in comparison with the classic results of 0.138 cm (Von Engel and Steenbeck 1932). Equally important, the maximum values of the electric current density of the discharge are bracketed between the values of 1.29 

ε 1 . (5.2f) >> σ ωp

Therefore, the time scale for accurate MHD analysis is seen to be limited to microwave frequencies with a value around tens of gigahertz. Specially, the spectrum of microwaves is bonded at the upper limit by visible and infrared waves. Beyond this frequency range, the error of applying MHD equations to study plasma dynamics may become significant. From Ampere’s circuit law, Equation (5.2a), by neglecting the displacement current, the electric current density is exclusively induced by the curl of magnetic field intensity: J = ∇ × H . By means of a vector identity ∇ ⋅ ( ∇ × H ) ≡ 0, the assumption is equivalent to set a vanishing gradient of electric current density:

∇ ⋅ J = 0 (5.3a) As the natural consequence of the time-​independent Ampere’s law, J = ∇ × H , the charged conservation equation reveals another unique feature of MHD formulation:



∂ρe ∂t + ∇ ⋅ J = 0 (5.3b) Namely, the charged number density is therefore independent of time. According to Faraday’s induction law, the time derivative of divergence of magnetic flux density

15

150

Magnetohydrodynamics Equations

also vanishes by virtue of a vector identity: ∇ ⋅ ∂B ∂t = ∇ ⋅ ( ∇ × E ) ≡ 0. If the differentiation with time and space is interchangeable, we have, then: ∂( ∇ ⋅ B ) ∂t = 0 (5.3e)



The requirement of ∇ ⋅ B = 0 is Gauss’s law for the magnetic flux density and usually is imposed as an initial value, but in practice it is applied as a consistent constraint. In summary, when applying the MHD equation to investigate a plasma dynamic phenomenon, the medium needs to be a continuum and the characteristic time scale shall be greater than the microwave frequency. Equally important, the charge number density of the studied plasma shall also be time invariant. In addition, as a consequence of the globally neutral electric field of plasma, there is no free space charge. The Lorentz acceleration is therefore far greater than the electrostatic force | J × B |  | ρe E | . This condition generally holds except when charge separation occurs in plasma sheath and especially near electrodes. For study of the solar atmosphere, the ionosphere, or the earth magnetic sphere, the presence of plasma sheath is insignificant.

5.2

Generalized Ohm’s Law A simplified relationship between electric current and electric field intensity by the generalized Ohm’s law has been adopted in classic MHD. After omitting the displacement electric current, the remaining electric current arises from charges species’ random motion, which is characterized by each individual species’ velocity ui and mass-​averages velocity u components. The movement of charged particles generates the conductive and convective components, J t = ∑ ni ezi ui + u∑ ni ezi = J + uρe , i

i

and the notation zi in the electric current designates the number of elementary electric charges carried by different ionized species. In the global electrically neutral plasma, the convective current component is thus vanished; the conductive current becomes the sole component. In developing the MHD formulation, the generalized Ohm’s law offers a distinct advantage to a compact formulation, bypassing the complex and detailed description of electrical conductivity as a second-​rank tensor. The derivation and approximations of the MHD formulation are usually based on the two-​fluid, quasi-​neutral plasma that consists of electrons and singly charged ions. A standard approximation in MHD is the generalized Ohm’s law for electrical conductivity. The kinetic foundation of this law for charged particles must begin from the conservation law of mass and momentum of charged particles, namely the charged particle continuity and momentum transfer Equations (2.18b) and (2.18d):



∂ni mi + ∇ ⋅ ( ni mi ui ) = si ∂t (5.4a) ∂ni mi ui + ∇ ⋅ ( ni mi ui u j ) = ∇pi + ni e( E + ui × B)) + nmi g − M ij ∂t

15

5.2  Generalized Ohm’s Law

151

where the source term of the species continuity equation is denoted as si. The momentum of charged particles is driven by the partial pressure gradient, Lorentz force, possible gravitation, and average momentum exchange between colliding particles i and j, which is defined as:  mi m j  M ij = ni νij   ( ui − u j ) (5.4b)  mi + m j 



In this expression, Equation (5.4b), the average momentum transfer collision frequency can be given by the method of moments for the probability distribution function over the peculiar velocity space as:

νij = ni σij ∫∫∫ | u p − u | fi ( u p ) f j ( u )( d 3 u p )( d 3 u ) (5.4c) where the notation u p is the peculiar velocity of the Boltzmann distribution, and the average collision cross-​section can be given as:

σij = 2 3 ∫ r 2 e − xQij(1) ( 2 kT mij r )dr. (5.4d)



The derivation and approximations of the generalized Ohm’s law for electric conductivity are based on the collision kinetic equations for electrons and single charged ions. For this purpose, the mass density, averaged mass velocity, and current density are defined as follows:

ρ = mi ni + me ne ≈ n( mi + ne ) m u + me ue 1 u = ( ni mi ui + ne me ue ) ≈ i i (5.4e) ρ mi + me J = e( ni ui + ne ue ) ≈ ne e( ui − ue )



In Equation (5.4e), the symbols mi and me denote the unit mass of ions and electrons, respectively, and n is the number density of charged species. By examining the two-​ species plasma dynamic equation, the equation of charged particles’ motion in a steady applied electric and magnetic fields can be further simplified by the following observations. First, the mass-​averaged velocities of organized motion are negligible in comparison with the drift velocities of charged particles, ue > ui >> u , thus the convective terms for both electrons’ and ions’ motion can be neglected. Second, the viscosity of the charged fluid is also neglected for simplicity and the error is limited as long as the Larmor radius is much smaller than the other characteristic length scales of the charged particles’ motion; Equations (5.4a) of the charged particles’ motion now can be given as:

∂ui = en( E + ui × B ) − ∇pi + nmi g + Pie ∂t (5.5a) ∂u me n e = −en( E + ue × B ) − ∇pe + me ng + Pei ∂t

nmi

The two equations describe the motions of ions and electrons driven by the electromagnetic forces en( E + ui × B ) and en( E + ue × B ), respectively. In the foregoing

153

152

Magnetohydrodynamics Equations

equation, pe and pi represent the partial pressures of the electrons and ions, respectively, and the total momentum transfer between electrons and ions is designated as Pie and Pei. The sum of these equations is:

n ∂( mi ui + me ue ) ∂t = en( ui − ue ) × B − ∇p + n( mi + me ) g (5.5b) From Newton’s third law,Pei = − Pie , the net momentum transfers between charged particles negate each other. The equation of motion for a two-​species medium in an electromagnetic field is then:

ρ ∂u ∂t = J × B − ∇p + ρ g. (5.5c)



Multiply the equation of motion for ions by me and the equation of motion for electrons by mi, and subtract the latter from the former to obtain: mi me n



∂( ui − ue ) = en( mi + ne ) E + en( me ui + mi ue ) × B − me ∇pi + mi ∇pe ∂t − ( mi + me )Pei

(5.6a)

Invoke the definitions of number density, mass-​averaged velocity, and current density, and let total momentum transfer between ion and electron as Pei = n2 e 2 ( ui − ue ) / σ; the foregoing equation becomes:

mi me n ∂  J  J   = eρE − ( m i + m e )ne   − me ∇pi + me ∇pe + en( me ui + mi ue ) × B (5.6b) σ e ∂t n

The last term can be rearranged as:

me ui + mi ue = mi ui + me ue + mi ( ue − ui ) + me ( ui − ue ) = ρu n − ( mi − me )J en (5.6c) Dividing Equation (5.6c) by ρe yields:



E +u× B−J σ =

1  mi me n ∂  J     + ( mi − me ) J × B + me ∇pi − mi ∇pe  (5.6d)  ρe  e ∂t n 

Further simplifications are possible by considering that the temporal variation of the organized motion is vanishing or reaches a steady state, and the lower mass of electrons in comparison to ions is negligible: mi >> me . The resultant approximation is known as the generalized Ohm’s law, which describes the electric properties in an electrically conducting medium.

σ( E + u × B ) = J +

1 ( J × B − ∇pe ) (5.7) en

In the classic formulation of the MHD-​governing equation, the following expression for a high number density of charged particles is frequently adopted in MHD formulation. Then the last term of Equation (5.7) can be dropped.

J = σ( E + u × B ) (5.8) The physical interpretation of this equation is that the current density is driven by an effective electric field composed of the applied electric field intensity, E, and an induced field by the drift velocity of charged particles u × B .

153

5.3  Ideal MHD Equations

153

The generalized Ohm’s law has been modified to include the Hall effect, which in an electric current component flows in the direction mutually perpendicular to both the electric and magnetic induction fields (Mitchner and Kruger 1973). J = σ[ E + u × B − β( J × B )]. (5.9)



where σ is the scalar electric conductivity, and the Hall effect and Hall parameter have been defined previously as J × B and the Hall parameter has also been defined as the ratio between the cyclotron frequency and the electron-​heavy particle collision frequency:  β = eB me ν. For this reason, the Hall parameter sometimes has been interpreted as the approximate number of revolutions that an averaged particle may complete between collisions. Recall that the Larmor frequency of electrons and ions is given as ω e = eB m e and ω i = eB mi , respectively. In plasma, the Larmor frequencies of electrons and ions have a significant difference in which the electrons are nearly collisionless but the ions are not. Under this circumstance, the ion current may contribute the dominant amount to the total current by the E × B drift motion. This phenomenon is called ion slip; however, it can occur only in a partially ionized gas.

5.3

Ideal MHD Equations The classic ideal MHD formulation is an interdisciplinary equation system that combines the simplified Maxwell equations and the Euler equations. The approximated Maxwell equations are the result of the basic MHD premise in that the displacement current is considered negligible in comparison to the conductive counterpart, thus the current density depends exclusively on the curl of the magnetic field strength J = ∇ × H . By adopting the generalized Ohm’s law in a moving frame of reference, the current density becomes J = σ( E + u × B ), which establishes a simple relationship between the current density, electric field strength, and magnetic field intensity. Note that the Gauss law for the magnetic field, ∇ ⋅ B = 0, which eliminates the existence of a nonphysical magnetic dipole in the electromagnetic field, has been invoked repeatedly in derivations of the MHD equations, but not explicitly included in the equation system. A reinforcement of this zero divergence of magnetic field flux density constraint has been consistently imposed by computational simulations. The rationale of replacing the Navier-​Stokes equations with the Euler equations is the fact that the transport properties such as the molecular viscosity, thermal conductivity, and diffusion of an inhomogeneous gas are insignificant in comparison with the inertia of the fluid motion in a domain far from solid surfaces. This assumption is valid for the study of astrophysics and geophysics in the absence of a solid boundary. From the viewpoint of fluid dynamics, the approximation is achieved by setting the characteristic Reynolds number of the flow field to approach infinity, and this asymptotic limitation can be easily removed. The interdisciplinary governing equations have been given as (Brio and Wu 1988; Shang 2002):

15

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Magnetohydrodynamics Equations



∂ρ + ∇ ⋅ (ρu ) = 0 (5.10a) ∂t



∂B + ∇ × H = 0 (5.10b) ∂t



∂ρu + ∇ ⋅ (ρuu + pI ) = J × B (5.10c) ∂t



∂ρe + ∇ ⋅ (ρeu + u ⋅ pI ) = ( E + u × B ) ⋅ J (5.10d) ∂t where the symbol I denotes the identity matrix. The system of equations is not closed because there is an unbalanced number of unknowns and equations. Closure is achieved by the generalized Ohm’s law, and by the constitutive relationship:



B = µ H (5.11a) The magnetic permeability µ is treated as a slow variation parameter, so it can be moved outside of the spatial differential operators. It has been mentioned previously that Ohm’s law can also explicitly include the Hall current:



J = σ[ E + u × B − β( J × B )] (5.11b) and the term ( u × B ) is required for describing the electric field strength on a moving frame (Mitchner and Kruger 1973). It is an induced electric current contributed by the moving charged particles in a magnetic field. It may not be surprising that the MHD equations are a special class of hyperbolic partial differential equations because they are integrated results of two distinctive hyperbolic governing systems of the Maxwell and Euler equations. To analyze this equation system, the associated initial value and boundary condition are briefly summarized. For gas dynamics variables, the non-​permeable condition is required for the velocity components on the fluid–​solid interface, so the fluid velocity must be parallel to the interface:  nˆ ⋅ u = 0. The normal component of the magnetic flux density is continuous across the boundary, nˆ ⋅ ∇B = 0, and the discontinuity of the magnetic field intensity parallel to the interface must be balanced by the surface current, nˆ × ∇H = J s . On the other hand, the electric field intensity parallel to the interface is an invariant and the difference of the component normal to the surface must be balanced by the surface charged, Equation (3.10e). The MHD equations, Equations (5.10a) through (5.10d), are traditionally rearranged to achieve a more compact form to contain the magnetic flux density B alone; thus to highlight the unique features of electromagnetic force exerting on gas dynamics. The first step is to eliminate the electric current density from the Lorentz force and Joule heating through the generalized Ohm’s law and with the interlocked relationships between the electric and magnetic fields. After substituting the reduced Ampere’s electric law ∇ × H = J into the generalized Ohm’s law, it appears as:



E = ∇ × B σµ − u × B (5.12a)

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5.3  Ideal MHD Equations

155

Faraday’s induction law becomes:

∂B ∂t + ( ∇ × ∇ × B ) σµ − ( ∇ × u × B ) = 0 (5.12b) By invoking a vector identity and applying Gauss’s divergence law for magnetic flux density, ∇ ⋅ B = 0, we have ∇ × ∇ × B ≡ ∇ ⋅ ( ∇ ⋅ B ) − ∇ ⋅ ( ∇B ) = −∇ ⋅ ( ∇B ). The second term of Faraday’s law can be expanded by another vector identity to show ∇ × u × B ≡ ( B ⋅ ∇ )u − ( ∇ ⋅ u ) B − ( u ⋅ ∇ ) B+( ∇ ⋅ B )u. The last term of the identity drops out by virtue of Gauss’s law for magnetic flux density, and the remaining terms are regrouped to get ∇ × u × B = ∇ ⋅ ( Bu − uB ). Faraday’s induction law acquires the following expression:



∂B ∂t + ∇ ⋅ ( uB − Bu ) = ∇ ⋅ ( ∇B ) σµ (5.12c) The term on the right-​hand side of the foregoing equation is referred to as the Pierre-​Simon Laplace operator and is customarily neglected by virtue of the very high value of electric conductivity in most MHD environments. The Lorentz acceleration in the momentum conservation equation, Equation (5.10c), can be expressed as:



J × B = ∇ × ( B µ ) × B = [( ∇ × B ) × B ] µ (5.13a) again from two vector identities:  ( ∇ × B ) × B ≡ ( B ⋅ ∇ ) B − ∇( B ⋅ B ) 2 and ( B ⋅ ∇ ) B ≡ ∇ ⋅ ( BB ) − B( ∇ ⋅ B ). The last term of the second vector identity is dropped by Gauss’s law, and then the Lorentz acceleration by magnetic flux density becomes:



J × B = [ ∇ ⋅ BB − ∇( B ⋅ B ) 2 ] µ (5.13b) The Lorentz acceleration actually reveals the unique feature of the magnetic field flux density. The term BB is the well-​known Maxwell tensor or the magnetic tensor of rank two, and the term ( B ⋅ B ) is its normal stress component. It is also called the magnetic pressure, which is the counterpart of the hydrodynamic pressure of the molecular shear stress tensor. Substitution into the momentum Equation (5.10c) yields:



∂ρu + ∇ ⋅ [ρuu − BB µ + ( p + B ⋅ B 2µ )I ] = 0 (5.13c) ∂t The momentum equation of the MHD formulation clearly shows the magnetic tensor BB is the counterpart of the dyadic tensor of fluid dynamics, uu. The magnetic pressure is also parallel to the normal shear stress component or the hydrodynamic pressure of the fluid, Equation (5.10d). These internal forces within the control volume are the driving mechanisms of plasma motion. The rate of energy change of charged particles is the sum of the inner product of the electromagnetic and particle velocities; ∑ zi e( E + ui × B ) ⋅ ui . The last term is i

identically equal to zero, because the magnetic field cannot change the energy of a charged particle. Therefore, Joule heating is the only contributor to the change of electromagnetic energy in the energy conservation equation; it can be rearranged

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Magnetohydrodynamics Equations

by substituting the electric current with the time-​independent Ampere’s circuit law, J = ∇ × H , and the electric field intensity with the generalized Ohm’s law, E = ∇ × H σ − u × B, to give: ∇ × H  E⋅J =  − u × B  ⋅ ( ∇ × H ) (5.14a)  σ 



After applying three vector identities, ( u × B ) ⋅ ( ∇ × H ) ≡ ∇ ⋅ ( H × u × B ) H ⋅ ( ∇ × u × B ), the first term on the right-​hand side of the first identity can be further expanded to get ∇ ⋅ ( H × u × B ) ≡ ∇ ⋅ [ u( H ⋅ B ) − B( H ⋅ u )]. The second term by the third vector identity gives ∇ × u × B ≡ ( B ⋅ ∇ )u − B( ∇ ⋅ u ) − ( u ⋅ ∇ ) B + u( ∇ ⋅ B ). Gauss’s law for magnetic flux density will eliminate the last term of the third vector identity. From the prior development, the inner product of ( u × B ) ⋅ ( ∇ × H ) = ( u × B ) ⋅ [ ∇ × ( B µ )] can be further expanded and simplified to show: 

( u × B ) ⋅ ( ∇ × H ) = ∇ ⋅{[ u( B µ ) ⋅ B ] − [ B( B µ ) ⋅ u]} + ( B µ ) ⋅ [ ∇ ⋅ ( Bu ) − ∇ ⋅ ( uB )] (5.14b) Finally, the Joule heating is expressed as:



E⋅J =

  B ⋅ B ∇ × ( B µ )2 B ⋅ u   1  ∂( B ⋅ B )  B   ( ∇ ⋅ ∇B )  − ∇ ⋅ u   − B µ  −  2µ  ∂t +  µ  ⋅  µσ  σ  µ     (5.14c)

The leading and the last terms of Equation (5.14c) are customarily dropped in view of the rather great magnitude of the electric conductivity. The Joule heating can also be better understood from the Poynting theorem (Sutton and Sherman 1965); in the MHD formulation, the Joule heating can be given as E ⋅ J = H ⋅ ( ∇ × E ) − ∇ ⋅ ( E × H ). From this equation, the rate of change for plasma energy is the sum of the rate of variation from the electromagnetic field energy plus the influx of the Poynting vector E × H . Thus the energy exchanges of an electromagnetic field are derived from current flow through an external source, and coupling through capacitive and inductive mechanisms. Substitute all the derived expressions that eliminate the electric field intensity, electric current density, and magnetic intensity into the interdisciplinary governing equations, Equations (5.10a) through (5.10d). And rearrange these equations to produce the widely known ideal MHD equations system. One also needs to note that the rate of change for the specific internal energy of the MHD formulation now contains explicitly the magnetic pressure term.

∂ρ + ∇ ⋅ (ρu ) = 0 (5.15a) ∂t



∂B + ∇ ⋅ ( uB − Bu ) = 0 (5.15b) ∂t



∂ρu + ∇ ⋅ [ρuu − BB µ + ( p + B 2 2µ )I ] = 0 (5.15c) ∂t

157

5.3  Ideal MHD Equations



157

∂(ρe + B 2 2µ ) + ∇ ⋅ [(ρe + p + B 2 2µ ) u − ( B ⋅ u ) B µ ] = 0 (5.15d) ∂t A few comments concerning the MHD equations are needed here:  first, Gauss’s divergence law for magnetic flux density has been consistently invoked in the derivation, but not included explicitly. Second, the magnitude of electric conductivity is considered sufficiently large so the terms involved with the reciprocal of conductivity that appear in deriving Faraday’s induction law and the Joule heating are negligible (Zachary and Colella 1992; Powell et al. 1999). It must be also recognized that the ideal MHD equations are not a strictly hyperbolic type of nonlinear partial differential system and solutions can even be singular. This peculiar feature means that the analytic structure of the weak solution around some points is generally unknown (Brio and Wu 1988). Like that of the Maxwell equations, the coefficient matrix of these equations has duplicated eigenvalues, thus not all the eigenvectors are linearly independent. More important, at some locations of the field, the governing equations can be characterized neither as linearly degenerate nor as genuinely nonlinear. For this reason, the ideal MHD equations are classified as a nonconvex partial differential system. Therefore, solutions of ideal MHD equations may represent only the peculiarity of the differential system rather than the physics, similar to that of the Euler equations. The electromagnetic field through the Lorentz acceleration introduces the magnetic pressure B 2 2µ and the magnetic stress matrix BB to the conservation momentum equation. This physical phenomenon is clearly singled out by the ideal MHD equations. The magnetic stress matrix is the degenerate Maxwell stress tensor without the contribution from the electric field, and the magnetic pressure is but the normal stress components B 2 2µ of the magnetic stress tensor known as the magnetic pressure. The added physical phenomenon is the result from the basic MHD formulation by including the Lorentz force in the conservation momentum equation. The magnetic stress matrix is a second-​rank tensor similar to the dyadic of the velocity vector uu. In the Cartesian coordinates, the tensors in matrix forms are:



 Bx Bx  BB =  By Bx  Bz Bx 

Bx By By By Bz By

Bx Bz   By Bz  Bz Bz 

and

ux ux  uu =  uy ux  uz ux 

ux uy uy uy uz u y

ux uz   uy uz  (5.16) uz uz 

Thus the magnetic tensor exerts a tension parallel to the magnetic field and behaves as a compression transverse to the magnetic field. The additional components of the magnetic pressure or the normal magnetic stress also contribute to the definition of internal energy. The specific enthalpy now consists of four components: the thermal internal energy, kinetic energy, static hydrodynamic pressure, and magnetic pressure of plasma in the presence of a magnetic field.

h = e + u ⋅ u 2 + P ρ + B 2 2µ (5.17) The existence of the magnetic pressure in plasma has been indisputably demonstrated by Ziemer’s experiment depicted in Figure  5.1 (Ziemer 1959). In his

159

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Magnetohydrodynamics Equations

Figure 5.1  Magnetic field effect to bow shock standoff distance, Ziemer, R. American Rocket

Soc. Vol. 29, 1959.

experiment, a magnetized hemispherical nose cylinder is immersed in an electromagnetic shock tube and the fluid medium is a partially ionized gas moving at a supersonic speed. The electric conductivity of the steady ionized air is estimated to range from 28 to 55 mho/​cm. The magnetic field is generated by a coaxial magnetic coil embedded in the hemispherical nose and aligns with the oncoming partially ionized gas. The magnetic field is therefore parallel to the oncoming plasma in the stagnation region. Through this arrangement, the maximum magnetic field strength is 4.0 Tesla at the stagnation point. The activated magnetic field pushes the bow shock wave upstream in contrast to the deactivated condition, but the quantification of the stand-​off distance is not ascertained due to the optical distortion of the emitting glow. When the current supply to the magnetic coil is ceased, the bow shock retracts toward the blunt body. The stand-​off distance is accurately predicted by aerodynamics to be proportional to the density jump across the shock and the radius of the hemispherical nose. When the magnetic field is actuated, a magnetic pressure is generated and combines with the static hydrostatic pressure. The combined pressure decreases the gas density in the shock layer, leading to an outwardly displaced stand-​off distance of the bow shock, and is recorded by the photograph. In the stagnation region, the combined pressure is essentially a constant, since the magnetic pressure attains the maximum value at the stagnation point over the magnetic pole and the hydrodynamic pressure will decrease proportionally to the increasing magnetic flux density as the flow approaches the pole. The increased stand-​off distance in the plasma is, then, the result of a non-​monotonic and a lower post-​shock gas density in the shock layer. As a consequence, a large stream tube cross-​section parallel to the bow shock is required to ingest a large amount of gas mass into the shock layer

159

5.3  Ideal MHD Equations

159

Figure 5.2  Numerical results of MHD equations simulated in the observation by Ziemer.

to satisfy the continuity condition at the bow shock: ρ∞ u∞ = ρu. All known numerical simulations also duplicate this phenomenon, but consistently under predict the shock wave stand-​off distance. It has been pointed out that a part of the observed discrepancy in the bow shock stand-​off distance from the experiment is exaggerated by the illumination of the discharge glow. Two numerical computations have duplicated Ziemer’s experimental condition, and the results are presented in composite form in Figure 5.2. On the right-​hand side of the graph, the magnetic field intensity is set to nil value. The numerical result in the absence of an externally applied magnetic field shows the density jump across the shock wave according to the Rankine-​Hugoniot relationship and a continuous compression toward the stagnation point in the shock layer. The bow shock flow structure represents the pure aerodynamic phenomenon, and the numerical result reproduces the exact bow shock stand-​off distance predicted by the classic theory of aerodynamics. The distance is directly proportional to the product of the density jump across the shock and blunt-​body nose radius. On the left-​hand side of the graph in Figure  5.2, a concentrated high-​density gradient region is displayed just ahead of the stagnation point. The high-​density gradient domain reflects precisely the varying magnetic field intensity from the magnetic pole at the stagnation point. As the externally applied magnetic field is actuated, an additional magnetic pressure component is appended to the hydrodynamic pressure to reduce the density jump across the bow shock. In contrast to the deactivated magnetic condition, the density decreases along the stagnation

16

160

Magnetohydrodynamics Equations

streamline as the flow decelerates to a stop by an increasing magnetic pressure. As a consequence, the density distribution along the stagnation streamline follows a non-​monotonic path, and more important, it possesses a lower value in the shock layer. The stand-​off distance of a bow shock is inversely proportional to the density jump across the shock by satisfying the continuity condition. The magnetic pressure decreases the density jump across the bow shock, thus an expanding shock layer thickness or stand-​off distance of the bow shock is required to accommodate the same amount of mass flow. This phenomenon is a canonical behavior of the supersonic MHD flow over a blunt body and fully verified by experimental observations and computational simulations. All numerical simulations using the MHD equations have reproduced the experimental observation. Unfortunately, the numerical result only duplicates the essential physics of the outward-​moving shock envelope by the experimental observation, but is unable to accurately quantify the data reported by Ziemer. This glaring discrepancy between numerical simulations and experimental data has also been reported by all numerical simulations using full MHD formulation (Poggie and Gaitonde 2002). The deficit is consistently attributable to the overly simplified Ohm’s law in describing the partially ionized gas. A unique characteristic of the ideal MHD equation is revealed by the rearranged Faraday induction equation. Specifically, it is clearly displayed by the divergence of the interwoven magnetic and kinetic fields in which the diagonal elements are vanished: ∇ ⋅ ( uB − Bu ) = [ 0 + ∂( Bx uy − By ux ) ∂y + ∂( Bx uz − Bz ux ) ∂z ] i

+ [ ∂( By uz − Bz uy ) ∂z + 0 + ∂( By ux − Bx uy ) ∂x ] j (5.18) + [ ∂( B u − B u ) ∂x + ∂( B u − B u ) ∂y + 0 ]k z x

x z

z

y

y z

The intrinsic property of the magnetic field is that it acts perpendicularly to the electric field, which creates some unique behavior and challenges for analytic formulation. This behavior can be easily appreciated by examining the divergence operator in the derived Faraday induction equation, Equation (5.18), namely, the changes of the magnetic flux density component along the respective coordinates are conspicuously absent. It means that the rate of change of the scalar magnetic flux density along the evaluated spatial coordinate is time invariant. As a consequence, the corresponding elements in the coefficient matrix will appear as a null value to complicate eigenvalue and eigenvector analyses.

5.4

Eigenvalues and Electromagnetic Waves The governing equations of an ideal MHD constitute a non-​strictly hyperbolic system that is nevertheless an initial-​value and boundary condition problem; the solutions to the governing equations can be piecewise continuous. Therefore, the most effective solving procedure is still the approximate Riemann solver by splitting

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5.4  Eigenvalues and Electromagnetic Waves

161

the flux vectors according to the sign of its eigenvalue (Powell et al. 1999). This basic approach is a characteristic method that builds on the eigenvector and eigenvalue structures of the equation system. A substantial amount of effort has been devoted to developing a complete set of eigenvectors for the three-​dimensional ideal MHD equations into three one-​dimensional space-​time domains. The objective is to keep the eigenvectors well defined even in domains where the equations are degenerated or no longer strictly hyperbolic (Brio and Wu 1988). In applications, the ideal MHD equations are frequently applied in supersonic flow regimes, which always encounter shock waves. The higher-​order extension of the Godunov (1959) method for the solutions of gas dynamics including shock jumps emerges as an attractive solving procedure for the ideal MHD problem (Zachary and Colella 1992). The numerical algorithm requires detailed information of eigenvalues and eigenvectors of the equation system. According to the number of the independent variables of the ideal MHD equations, a complete set of eight eigenvalues and eigenvectors is presented. Since the diagonalization of a matrix is limited to one time-​space domain at a time, the equation system is first expressed in flux vector form and then solved in the one-​dimensional time-​space domain to search the eigenvalues. The ideal MHD equations constitute the perfect flux vector in divergent form: ∂U + ∇ ⋅ F = 0 (5.19a) ∂t



The ideal MHD equation in flux vector form is simply the time rate of change balanced by the divergence of the flux vectors; Equation (5.19a). The dependent variables U (ρ, ρu, B, ρe + B 2 2µ ) and flux vectors F have been given by Equations (5.15a) through (5.15d) to summarize as:



ρ     B  U = ρu    ρ( e + B 2 2µ )

ρu     uB Bu −   F= (5.19b) ρuu + ( p + B 2 2µ )I − BB µ    ρeu + ( p + B 2 2µ )I ⋅ u − ( B ⋅ u ) B µ   

The governing equation in flux vector form is best cast in Cartesian coordinates, because scale factors of all coordinates for differential operators have values of unity, which simplifies the formulation. On the fundamental Cartesian coordinates, the flux vectors can be described by three one-​dimensional time-​space domain. The peculiar magnetic field structure highlighted by Equation (5.19b) is easily displayed by the one-​dimensional projection of the flux vector of ideal MHD equations. It becomes obvious that the one-​dimensional flux vectors along the three coordinates reveal that the ideal MHD equations must be solved as a multidimensional problem. The temporal advancement of the magnetic flux density is not included along the corresponding spatial coordinate. In other words, there is no physically realistic MHD problem that can be properly implemented in a single temporal-​ spatial dimension.

163

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Magnetohydrodynamics Equations

ρux     0     ux By − uy Bx   ux Bz − uz Bx  (5.20) Fx =   ρux2 + p + Bx2 2µ   ρux uy − Bx By µ     ρux uz − B x Bz µ   (ρe + p + B ⋅ B 2µ )ux − ( Bx ux + By uy + Bz uz )Bx µ  To determine the eigenvalue of the foregoing equations system, the flux vector in each Cartesian coordinate is required to be expressed in coefficient matrices and solved in a one-​dimensional time-​space domain. The process is essentially to derive the coefficient matrix via the chain rule of differentiation of the flux vectors with respect to the dependent variables, which is simply the Jacobian of the flux vector with respect to the dependent vector U:



∂Fy ∂U ∂U ∂F ∂U ∂F ∂U ∂U ∂U ∂U +( x ) +( ) +( z ) = [ Ax ] + [ Ay ] + [ Az ] =0 ∂U ∂y ∂t ∂U ∂x ∂U ∂z ∂x ∂y ∂z (5.21) The coefficient matrices Ax, Ay, and Az or the Jacobians in each spatial coordinate ( ∂Fx ∂U ), ( ∂Fy ∂U ), and ( ∂Fz ∂U ) are obtained by differentiating with a quasi-​ linearized approximation. Since all the coefficient matrices are similar, only the coefficient matrix in the x coordinate ( ∂Fx ∂U ) is presented in the following:



0 u ρ 0 u 0  0 0 u  0 ∂Fx  0 0 =  0 0 0 ∂U  0 B − B  y x 0 B 0 z  0  0 γ p

0 0 0 u 0 0 − Bx 0

0 0 0 + By ρ + Bz ρ 1 ρ − Bx ρ 0 0  0 − Bx ρ 0  (5.22) 0 0 0  −v u 0 0 −w 0 u 0  −( γ − 1)u ⋅ B 0 0 u  0 − Bx ρ − By ρ − Bz ρ 0

By counting the number of elements of the coefficient matrix, there shall be a complete set of eight eigenvalues and eigenvectors. All the eigenvalues are real for the hyperbolic partial differential equations system, and if the coefficient matrix [A]‌  = [∂Fi ∂xi ] or Jacobian matrix is nonsingular, then it can be diagonalized: [A]=[S][λ ][S]–1. The eigenvalues of the equation system are obtained by solving the coefficient matrix by the determinant.

det{[ Ai ] − λ I } = 0 (5.23) The type of waves and the wave speeds can be uniquely determined from the eigenvalues of the governing equations. However, elements of the fifth row of the

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5.4  Eigenvalues and Electromagnetic Waves

163

coefficient matrix, Equation (5.22), are zero; the matrix is singular. In fact, the coefficient matrices of ideal MHD equations on other coordinates all exhibit the identical singular behavior, which has been pointed out in Equation (5.18). The approach is, then, to modify the governing equations so as to make the matrix nonsingular. The criterion is that the one-​dimensional Riemann solver remains unchanged. For a one-​dimensional formulation of the flux vector that is nonphysical, the coefficient matrix is singular. The degenerated Jacobian reduces to a 7 × 7 matrix and up to five out of the seven eigenvalues are duplex. A set of eight linearly independent eigenvectors for the ideal MHD equations has been found by modifying the singular coefficient matrix, Equation (5.22). The rational approach is imposing explicitly Gauss’s law for magnetic flux density: ∇ ⋅ B = 0. Jeffrey and Taniuti (1964) as well as Brio and Wu (1988) have found the eight waves solution. The eigenvalues are:

λ1 = u (5.24a)



λ2 = u +

Bn2 (5.24b) ρµ



λ3 = u −

Bn2 (5.24c) ρµ









2     ∂p   ∂p   Bn2   Bn2 Bn2    ∂p  λ 4 = u +   + +   +  − 4      2 (5.24d)  ∂ρ  s  ρµ    ∂ρ  s ρµ   ∂ρ  s ρµ  2     ∂p   ∂p   Bn2   Bn2 Bn2    ∂p  λ 5 = u −   + +   +  − 4      2 (5.24e)  ∂ρ  s  ρµ    ∂ρ  s ρµ   ∂ρ  s ρµ  2     ∂p   ∂p   Bn2   Bn2 Bn2    ∂p  λ 6 = u +   + −   +  − 4      2 (5.24f)  ∂ρ  s  ρµ    ∂ρ  s ρµ   ∂ρ  s ρµ  2     ∂p   ∂p   Bn2   Bn2 Bn2    ∂p  λ 7 = u −   + −   +  − 4      2 (5.24g)  ∂ρ  s  ρµ    ∂ρ  s ρµ   ∂ρ  s ρµ 

λ 8 = u (5.24h) In Equations (5.24b) through (5.24g), the symbol Bn is denoted as the magnetic flux density that is perpendicular to the direction of wave propagation. The term ( ∂p ∂ρ)s is actually the definition for the speed of sound by the differentiation of pressure with respect to density under isentropic condition. The added eighth eigenvalue λ 8 is a duplicity of λ1. They are interpreted as the entropy and the contact surface waves, respectively. The eigenvalues λ 2 and λ 3 are identified with the Alfven wave, which is a pure transverse wave. The fast electromagnetic waves are associated with eigenvalues λ 4 and λ 5. The slow electromagnetic waves are tied to the

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Magnetohydrodynamics Equations

1.6 1.4 1.2 1.0 U/C

164

Fast plasma wave Fast plasma wave Alfven wave Acoustic wave

0.8 0.6 0.4 0.2 0.0 0.0

45.0

90.0

135.0

180.0

Phase Angle Figure 5.3  Different MHD wave speeds ( Bn2 ρµ ) c 2 = 0.5.

eigenvalues λ 6 and λ 7. The fast and slow electromagnetic waves are complex waves constituted with both longitudinal and transverse wave components. All electromagnetic waves are created by a perturbation of the electromagnetic field by the acceleration of charged particles. These eigenvalues are actually the phase velocities of these distinctive electromagnetic waves. In the absence of a magnetic field, all the transverse wave components vanish. The electric field intensity E and the magnetic flux density B are both normal to the wave front and the electromagnetic energy flux is governed by the Poynting vector, E × H . Thus the electromagnetic energy always propagates in the direction of the wave. In Figure 5.3, the four electromagnetic wave speeds versus the phase angle of the magnetic flux density are displayed from 0 to 180 degrees, measured between the direction of wave propagation and the magnetic field (Shang 2002). The numerical results are obtained by assuming a homogeneous plasma field and all wave speeds are normalized by the speed of sound c = ( ∂p ∂ρ)s . The results are presented for the parameter ( Bn2 ρµ ) c 2 = 0.5. Based on this condition, the fast electromagnetic wave can propagate at 1.225 times the speed of sound in a transverse magnetic field, and this wave speed is greater than or equal to the speed of sound over the entire range of the phase angles. The slow electromagnetic wave, on the other hand, propagates at a rate slower than the speed of sound over the complete phase angle range. The highest ratio of the slow electromagnetic wave to the speed of sound is 0.7071 when the wave motion is parallel to the magnetic field. The slow wave ceases as the transverse magnetic field vanishes. The fast and slow electromagnetic wave

165

5.4  Eigenvalues and Electromagnetic Waves

165

speeds represent the upper and lower bounds for the propagating speeds of the Alfven and acoustic waves. The electromagnetic waves are compound waves for which the compression and rarefaction components can coexist as the direct consequence of the non-​convexity of the ideal MHD equations. The drastically different behavior from the hydrodynamic shock structure has been illustrated and identified by numerical simulations of a two-​ dimensional shock tube. To demonstrate the distinct wave propagating speed between media, a degenerate electromagnetic wave is produced by reversing the polarity of an externally applied transverse magnetic field at the midpoint of a shock tube. The computational results of the density distribution are presented when the diaphragm is ruptured between the two sections of different pressures (Canupp 2000). The computational accurate is verified by using two entirely different numerical algorithms by the flux splitting and compact difference schemes. The complex electromagnetic wave system is then presented in direct contrast to the pure longitudinal acoustic wave. The fast rarefaction waves propagate at speeds greater than the acoustic speed toward both ends of the shock tube. The similar left-​running fast rarefaction wave moving into the expanding section of the shock tube reveals a more complex wave system by a varying transverse magnetic field polarity that is unique in an MHD medium. At the location where the transverse magnetic field reverses its polarity, an intermediate or compound shock emerges at the instance of null value of the magnetic flux density. The compound wave actually shows that a slow rarefaction wave can be attached to a slow shock temporally in the plasma. This unique phenomenon occurs when the transverse magnetic field vanished during a polarity switches and the distinct electromagnetic wave degenerated. The numerical results that are produced by two different algorithms display the nearly identical behavior in Figure 5.4. However, the existence and physical significance of the compound wave are still uncertain because there is no known experimental observation that supports the theoretical result, but all independent computational simulations of ideal MHD equations have consistently exhibited this distinct behavior when a transverse magnetic field presents in the plasma field. However, these uncertainties of numerical simulation to fidelity for physics are completely removed in simulating the solar wind and the interaction of the expanding cometary atmosphere with a magnetized solar wind (Gombosi et  al. 1996; Powell et  al. 1999). The physical validation for the numerical simulations using the ideal MHD equations is supported by comparing well with all aspects of Giotto observations (Altwegg et al. 1993). In fact, the predicted ion mass density, the plasma speed, and the cometary ion temperature along the Giotto inbound pass versus the cometocentric distance reach good agreement with observations. In short, the classic MHD formulation was developed for the physical phenomena far from the dominant interface boundary conditions and satisfied all fundamental assumptions for which the equations were derived. In the solar atmosphere and interplanetary space, the electrical conductivity is in the order of 105 Ohm/​m, and the magnetic Reynolds number exceeds the value of 108. The ideal MHD formulation is therefore a physics-​based modeling for all astrophysics and territorial magnetic field problems.

167

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Magnetohydrodynamics Equations

Figure 5.4  Density distribution in an ideal MHD shock tube.

5.5

Full MHD Equations The full MHD equations are a set of nonlinear partial differential equations systems consisting of the reduced Maxwell equations in the time domain with the classic MHD assumptions and the compressible Navier-​Stokes equations. The difference from ideal MHD equations is that the fluid medium now possesses viscosity and heat conductivity for a more realistic modeling of the fluid medium, and for expanding the capability for engineering applications. From a pure fluid dynamic viewpoint, the assumption of infinity Reynolds number is removed. The fluid dynamics governing equations become a second-​order, nonlinear incomplete parabolic system (Shang 1985). The major changes for the differential system are the boundary conditions for the fluid medium; the non-​permeable condition for the velocity components on the solid surface is replaced by the nonslip condition and an additional condition for surface temperature is also required. A full description of the electromagnetic–​fluid dynamics interactions have been put forth, and a rich variety of the formulations has been derived as a consequence. The reduced Maxwell equations are consistently adopted like that of the ideal MHD formulation. The approximation includes Faraday’s induction law and Ampere’s circuit law by neglecting the displacement current, as well as the simple relationship between electric field intensity and current that is provided by the generalized Ohm’s law. The constitutive relation between magnetic flux density and magnetic field intensity is also introduced, and finally, by applying Gauss’s law in equation derivations, it is not explicitly imposed. In short, the difference between

167

5.5  Full MHD Equations

167

the ideal MHD and full MHD equations lies only in the description of transport properties for the flow medium, when the transport properties in molecular viscosity, conductivity, and diffusion are fully incorporated. The governing equations acquire the following form:

∂ρ + ∇ ⋅ (ρu ) = 0 (5.25a) ∂t



∂B + ∇ ⋅ ( uB − Bu ) = ∇ ⋅ ( ∇B ) µσ (5.25b) ∂t



∂ρu + ∇ ⋅ [ρuu − τ − BB µ + ( p + B 2 2µ )I ] = 0 (5.25c) ∂t



∂(ρe + B 2 2µ ) + ∇ ⋅ [(ρe + p + B 2 2µ ) u − κ∇T − u ⋅ τ − ( B ⋅ u ) B µ ] (5.25d) ∂t 2 = ( ∇ × B µ ) σ + ( B µ ) ⋅ [( ∇ ⋅ ∇B ) µσ ] Because the full MHD equations are frequently applied to supersonic flow regimes, the flow field will always encounter strong shock jumps. In a piecewise continuous discrete space, accurate computational simulations are obtainable by treating the discontinuity as an approximate Riemann problem through the splitting of flux vector, which is only valid for a hyperbolic differential system. For the full MHD equations the flux vector must be further split to derive this numerical advantage. The split flux vectors thus are separated into inviscid and viscous components and still write in flux vector form in each coordinate. The grouping of all inviscid terms is identical to that of the ideal MHD equations, which as a hyperbolic system has a clearly defined zone of dependence and the data transmission is followed by a pair of right-​and left-​running characteristics. The solving schemes are already established for the inviscid terms and are usually by a windward approximation to achieve desirable computational stability. The terms involving the shear stress and heating conduction are characterized by a diffusive nature, which does not have a preferred orientation for dispersion, and mimicked naturally by a second-​order spatial central numerical algorithm.



∂U ∂( Fx + Fx,v ) ∂( Fy + Fy,v ) ∂( Fz + Fz ,v ) + + + = S (5.26a) ∂y ∂t ∂x ∂z where the flux vectors Fx, Fy, and Fz contain only the inviscid terms of the compressible Navier-​Stokes or the Euler equations as the ideal MHD equations. The remaining terms of viscous stress, dissipation function, Fourier’s law of conductive heat transfer, and the Joule heating of the full MHD equations are expressed as Fx,v, Fy,v, and Fz,v. The eight dependent scalar variables of the split flux vector of Equation (5.26a) are:



T

U = (ρ, Bx, By, Bz ,ρux , ρuy , ρuz , ρe + p − B 2 µ ) (5.26b) and the split flux vectors in each spatial coordinate are given as the following:

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Magnetohydrodynamics Equations



ρux     0     ux By − uy Bx   ux Bz − uz Bx  (5.26c) Fx =   ρux2 + p + Bx2 2µ   ρux uy − Bx By µ     ρux uz − B x Bz µ   (ρe + p + B ⋅ B 2µ )ux − ( Bx ux + By uy + Bz uz )Bx µ 



0     0     [( ∂B y ∂x ) − ( ∂Bx ∂y )] µσ   [( ∂B z ∂x ) − ( ∂Bx ∂z )] µσ    (5.26d) τ xx =   τ xy     τ xz   ux τ xx + uy τ xy + uz τ xz + κ ∂T ∂x   {B [( ∂B ∂x ) − ( ∂B ∂y )] + B [( ∂B ∂x ) − ( ∂B ∂z )]} µ 2 σ  y x z z x  y 

Fx,v



ρuy     u B − u B y x x y     0   uy Bz − uz By   (5.26e) Fy =   ρux uy − Bx By µ   ρuy2 + p + B 2 2µ     ρux uz − Bx Bz µ   (ρe + p + B ⋅ B 2µ )uv − ( Bx ux + By uy + Bz uz )B y µ 



0     [( ∂B x ∂y ) − ( ∂By ∂x )] µσ     0   [( ∂B z ∂y ) − ( ∂By ∂z )] µσ    (5.26f) τ xy =   τ yy     τ yz   ux τ xy + uy τ yy + uz τ yz + κ ∂T ∂y   {B [( ∂B ∂y ) − ( ∂B ∂x )] + B [( ∂B ∂y ) − ( ∂B ∂z )]} µ 2 σ  x y z z y  x 

Fy,v

169

5.5  Full MHD Equations

169



ρuz     uz Bx − ux Bz     uz By − uy Bz   0  (5.26g) Fz =   ρux uz − Bx Bz µ   ρuy uz − By Bz µ     2 2 ρuz + p + Bz 2µ   (ρe + p + B ⋅ B 2µ )ux − ( Bx ux + By uy + Bz uz )Bz µ 



0     [( ∂B x ∂z ) − ( ∂Bz ∂x )] µσ     [( ∂B y ∂z ) − ( ∂Bz ∂y )] µσ   0    (5.26h) τ xz =   τ yz     τ zz   ux τ xz + uy τ yz + uz τ zz + κ ∂T ∂z   {B [( ∂B ∂z ) − ( ∂B ∂x )] + B [( ∂B ∂z ) − ( ∂B ∂y )]} µ 2 σ  x z y y z  x 

Fz ,v

The source terms on the right-​hand side of the governing equations, Equation (5.26a), are the results of the Pierre-​Simone Lapalcian operator of the magnetic flux density from Ampere’s electric circuit law and the Joule heating of the energy conservation equations:



0     2 2 2 2 ∂ Bx ∂x + ∂ By ∂y∂x + ∂ Bz ∂z∂x     ∂ 2 Bx ∂x∂y + ∂ 2 By ∂y2 + ∂ 2 Bz ∂z∂y   2 2 2 2 x ∂ z + ∂ B ∂ y ∂ z + ∂ B ∂ z B ∂ ∂ x y z     0   0 S=  (5.26i)   0   ( ∂Bz ∂y − ∂By ∂z )2 + ( ∂Bx ∂z − ∂Bz ∂x )2 + ( ∂By ∂x − ∂Bx ∂y )2    + Bx [ ∂ 2 Bx ∂x 2 + ∂ 2 By ∂y∂x + ∂ 2 Bz ∂z∂x ]     + By [ ∂ 2 Bx ∂x∂y + ∂ 2 By ∂y2 + ∂ 2 Bz ∂z∂y ]   2 2 2 2 + Bz [ ∂ Bx ∂x∂z + ∂ By ∂y∂z + ∂ Bz ∂z ]   The associated initial values and boundary conditions for the fluid dynamics variables are identical to the Navier-​ Stokes equations but with additional conditions for the electromagnetic variables. The nonslip condition is imposed for all velocity components on the fluid and solid interface u = 0. The density on the

17

170

Magnetohydrodynamics Equations

solid surface is derived from the intrinsic zero-​pressure gradient condition of the inner boundary layer nˆ ⋅ ∇p = 0. The surface temperature can be specified either by a given solid surface value or by the adiabatic condition T = Tw or nˆ ⋅ ∇T = 0. The boundary conditions for the electric variables are identical to that of the ideal MHD equations. In most engineering applications using MHD equations in a partially ionized gas, such as in remote laser energy deposition, interplanetary vehicle return to earth atmosphere, and plasma actuators for flow control, the assumption of negligible electric conductivity or low magnetic Reynolds number is fully justifiable. The assumption is usually imposed to the full MHD equations by dropping off nearly all the source terms. The widely adopted governing equations acquire the following form (Shang 2002):



∂ρ + ∇ ⋅ (ρu ) = 0 ∂t ∂B + ∇ ⋅ ( uB − Bu ) = 0 ∂t (5.27) ∂ρu + ∇ ⋅ [ρuu − τ − BB µ + ( p + B 2 2µ )I ] = 0 ∂t ∂(ρe + B 2 2µ ) + ∇ ⋅ [(ρe + p + B 2 2µ ) u − κ∇T − u ⋅ τ − ( B ⋅ u ) B µ ] = 0 ∂t The assumption is based on the fact that the inducted magnetic field is negligible in comparison with the externally applied magnetic field when the externally applied magnetic field is relatively weak or absent, which is frequently encountered in engineering applications. Under this condition, the magnetic Reynolds number is often less than the value of unity, and the simplification to computational stability and efficiency is also noticeable, especially as it eliminates a stiff condition for evaluating a fractional term with numerator and denominator that approach vanishing values simultaneously as in Equation (5.25d): ∇ ⋅ ( ∇B ) µσ. Indirectly this numerical behavior may reflect the fact that the electric current in a magnetic field always leads to discharge instability. The numerical methods adopted for solving the full MHD equations have evolved side by side with computational fluid dynamics. The system of equations is mostly solved in conservational form with an upwind scheme for convective terms based on the theory of characteristics to a hyperbolic partial differential equations system. The flux split approach of the Roe scheme to the approximate Riemann problem is frequently adopted because it stands out from its simplicity and clarity for the underlying physical modeling (Roe 1981). The solution is advanced in time by processing data across and balancing the normal flux vectors on each control surface to evaluate the split vectors. The more recent developments include the compact difference method and the primitive function flux reconstruction for finite volume method to achieve high numerical resolution (Gaitonde 2001). The higher-​order difference procedure has also been included in both the compact difference fourth-​and sixth-​order spatially implicit schemes. The well-​known time

17

5.6  Similarity Parameters of MHD

171

instability issue of the compact difference algorithm is effectively controlled by a low-​frequency filter up to tenth-​order accuracy. As the same time, the temporal integration is carried out by the classic fourth-​order Runge-​Kutta formula, thus the consistent temporal-​spatial high-​order solutions of the full MHD equations are achievable. The full MHD equations have been applied to explore and validate the mitigation of stagnation point heat transfer by imposing a dipole magnate to a hemispherical blunt body, at Mach number of 5.0 and Reynolds number of 80,000. The numerical study is conducted by invoking the low magnetic Reynolds number approximation (Poggie and Gaitonde 2002). The numerical results indeed reveal that the electromagnetic field reduces the heat flux transfer onto the body in the vicinity of the stagnation region, and the reduced heat transfer rate increases with increasing magnetic flux density. However, a more realistic varying electric conductivity may reduce the effectiveness of the degree of heat transfer mitigation; the quantification of the reduced heat transfer rate is uncertainty. In other words, the generalized Ohm’s law oversimplifies the modeling of plasma medium for this specific practical engineering application.

5.6

Similarity Parameters of MHD The magnetic field plays a very important role in a wide scope of MHD phenomena and also exhibits a drastically different characteristic in a varying degree of intensity to charge particles’ motions. Thus it will be very helpful to understanding the fundamental nature of magnetic flux density for incisive insights for problem solving. The best approach is by deriving the transport equation of the magnetic induction to explain the basic mechanisms of the transport equations. The analytic approach is achieved by examining Faraday’s induction law and its relation with the electric current, by taking the outer product of the generalized Ohm’s law to get:



∇ × J = σ[ ∇ × E + ∇ × ( u × B )] (5.28a) Since in MHD formulation, the electric current is induced exclusively by a varying magnetic field in space J = ∇ × B, and from Faraday’s induction equation, ∂B ∂t + ∇ × E = 0, the transport equation for magnetic flux density acquires the following form:



∂B ∂t + ∇ × ( u × B ) = [ ∇ × ( ∇ × B )] σµ (5.28b) By means of a vector identity ∇ × ∇ × B ≡ ∇ ⋅ ( ∇ ⋅ B ) − ∇( ∇ ⋅ B ) and invoking Gauss’s law for a magnetic field, ∇ ⋅ B = 0, Equation (5.28b) becomes:



∂B ∂t + ∇ × ( u × B ) = ∇2 B σµ (5.28c) The term on the right-​hand side of Faraday’s induction law is identified as the magnetic diffusion, and the inverse product of electric conductivity and magnetic permeability, 1 µσ , is referred to as the magnetic viscosity. The relative importance

173

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Magnetohydrodynamics Equations

of convection and diffusion mechanisms for transporting magnetic flux density is measured by the ratio of magnitudes between these two terms in the transport equation through a dimension analysis. The analysis is based on the Buckingham Pi theory of similarity by grouping physical phenomena of similar properties, so the general conclusion may be drawn for a class of problems of an identical similarity parameter by a single result. These two terms of Equation (5.28c) have the physical dimensions of uB L and B / µσL2 , respectively, and the ratio of these physical dimensions is µσuL. Therefore, the resultant dimensionless parameter is called the magnetic Reynolds number Rm = µσuL ; it can be shown that when the magnetic Reynolds number is very large, a small electric current will generate a large induced magnetic field. On the other limit, a moderate electric current will produce an induced field only as a small perturbation to the applied field (Sutton and Sherman 1965). For the situation where Rm is much less than unity, Rm < 1, most engineering applications utilizing plasma are implemented, such as electric propulsion, flow control by plasma actuators, remote energy deposition, and space vehicle reentry to earth’s atmosphere (Shang 2016). Under the low magnetic Reynolds number condition, the magnetic transport equation degenerates to:

∂B ∂t = (1 µσ ) ∇2 B (5.28d) The rate of change for the magnetic field is controlled only by the diffusive mechanism, and the characteristic decay time of an induced magnetic field can also be given by a simple dimensional analysis as τ d = µσL2. For a very strong spatial magnetic field, the decay period can be substantial, but for a low electric conductive medium like weakly ionized gas, the decay time is relatively short. On the other extremes, when the magnetic Reynolds number is much greater than unity, Rm >> 1, the strong magnetic field dominates the charged particles’ motion. From experimental observations, the highly electric conducting particles move freely along the magnetic line but are unable to cross the magnetic field. Literature often describes this phenomenon as though the charged particles are frozen in the magnetic field line by following the line of magnetic force, twisting and turning as the magnetic field dictates. This assessment has been demonstrated by considering the field as magnetic tubes in defining its orientation and magnitude in space. By means of the Stokes theorem relating the line integral around a control surface, the flux normal to the control surface will insist that a vanished velocity component is perpendicular to the magnetic flux density; it has been shown that (Bittencourt 1986):



d B( r, t )⋅ d s = 0 (5.28e) dt ∫ Equation (5.28e) is the mathematic expression of the assertion by Alfven that the magnetic field lines are frozen in the electric conducting fluid. It also has been pointed out by Bittencourt that this statement holds only for a medium that has infinity electric conductivity.

173

5.6  Similarity Parameters of MHD

173

Note that the term µσ or magnetic diffusivity is the counterpart of kinematic molecular viscosity. The value of induced magnetic field intensity by electric current is proportional to the product of the electric current, the magnetic permittivity of the medium, and the characteristic macroscopic length: Bind  LµJ . Again, the electrical current is proportional to the macroscopic charged particle velocity and the existent magnetic field intensity. Therefore the magnetic Reynolds number really is a measure of the ratio of induced and total magnetic field flux density. Another similitude in the MHD equations is the magnetic interaction parameter or the Steward number, which is a measure of the relative magnitude of the magnetic versus the electric field intensity. By an accurate definition, the Steward number is the ratio between the Lorentz force and the inertia of the moving medium. The similarity parameter of plasma in motion is given as:

St = LσB 2 ρu (5.28f) In general, the Steward number is small for most aerospace engineering applications using plasma as a fluid medium, because the electromagnetic force is often dominated by the gas dynamic inertia and operates in a relatively weak applied magnetic field. Especially in electrically neutral plasma the charged particles move in pairs even at high drift velocity, but will not result in a high electric current. The similarity parameter, especially the magnetic Reynolds number, can share important insights for solving the full MHD equations in engineering applications. The process is reducing the governing equations into a dimensionless form, normalizing the dependent variables with respect to their free-​stream value, and the magnetic pressure ( B∞2 µ ) by twice the value of the free-​stream dynamic pressure ρ∞ u∞2 . The non-​dimensional full MHD Equations (5.25a) through (5.25d) after the normalization of all dependent variables become: ∂ρ ∂t + ∇ ⋅ (ρu ) = 0 ∂B ∂t + ∇ ⋅ ( uB − Bu ) = − (1 R m )∇ × [(1 σ )∇ × ( B µ )]



∂ρu ∂t + ∇ ⋅ [ρuu − BB µ + pI ] = ∇ ⋅ τ R (5.29) 2 ∂ρe ∂t + ∇ ⋅{(ρe + p − BB µ )u − ( u ⋅ τ R ) − (κ∇T ) [( γ − 1)Pr M R ] + ( R Rm )[( B µσ )∇( B µ ) − ∇( B µ )( B µσ )]} = 0 In Equation (5.29), all dependent variables are dimensionless, the pressure term is the sum of the hydrostatic and magnetic components, p = p + ( B 2 2µ ), and the total internal energy is now e = [ p ( γ − 1)ρ] + ( B 2 2ρµ ) + u 2 2 . The symbols for M and Pr designate the free-​stream Mach number and the Prandtl number (Pr = c p µ g k ) of the moving medium. In this environment, the magnetic Reynolds number Rm and the Reynolds number R are very much greater than unity; the full MHD equations reduce to the ideal MHD approximation. The ideal MHD equation is perfectly suited for analyzing astrophysical and geophysical phenomena, because for the former the moving medium is a rarefied and fully ionized gas and for the later the magnetic poles are fixed with respect to

175

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Magnetohydrodynamics Equations

Table 5.1  Typical values and similar parameters of astrophysics. Phenomena

σ (mho/​m) B (Tesla) St

Ionosphere

10

Solar atmosphere

10

Solar corona

Rm

10–5

103

103

10

10

108

106

10–4

1016 1015

Interplanetary space

10

10

1014 1015

Interstellar space

103

10–10

1020 1021

3

5

–2

–9

8

the center of the earth. Both phenomena of interest occur farther away from the medium interface, and the characteristic length scale is enormous; the molecular dissipation effect is negligible. Especially for all the astrophysical investigations, the magnetic field intensity, although it is extremely low to have a value in the range from 10-​10 to 10-​2 Tesla, still is overwhelming over that of the inertia of charged particles’ motion in a rarefied gas regime. The typical astrophysics problems have a velocity range from 103 up to 106 m/​s, but their characteristic length scales in general are astronomical. Therefore the Steward number and the magnetic Reynolds number are on the opposite limit of the similarity parameters in contrast to the engineering applications (Mitchner and Kruger 1973). The typical value of the electric conductivity and the magnetic field in astrophysics is presented in Table 5.1. The two similarity parameters of electromagnetic field, the magnetic Reynolds number and Stewart number, are also included.

5.7

Modified Rankine-​Hugoniot Shock Conditions The shock discontinuity in perfect gas is well understood from the kinetic theory of gas and is summarized by the classic Rankine-​Hugoniot relationship in gas dynamics. The added mechanisms, such as Joule heating, electrostatic force, and Lorentz acceleration of an electrically conducting medium in an actuated electromagnetic field, certainly will alter the discontinuous shock behavior. Specially, plasma properties change rapidly across a shock, which may involve localized charge separation to produce a double layer with a large electric potential difference. This behavior is the direct consequence of the Debye shielding length, which has the smallest length scale in plasma. The electric field is constrained within the layer, and the double layer structures are often found in current carrying plasma (Torvein 1976). When the MHD shock velocities are approaching the speed of light, under this condition, a special relativistic treatment of this phenomenon becomes necessary. The simplest possible approach to study the shock wave is to assume the electric conductivity approaching infinity, which is equivalent to assuming the magnetic Reynolds number is infinite (De Hoffmann and Teller 1950). By this assumption, the fluid particles are free to move parallel to the magnetic field, but the motion perpendicular to the magnetic field is firmly attached to the lines’ magnetic force. Thus,

175

5.7  Modified Rankine-Hugoniot Shock Conditions

175

the transverse magnetic field to the shock front will produce the most drastic charge in the shock structure. The discontinuity of fluid behavior thus must be described by at least a two-​dimensional shock wave. In order to derive the equivalent shock jump condition, the energy equation needs to be rewritten in terms of the specific enthalpy h rather than the specific internal energy e:

∂ρh ∂p − + ∇ ⋅ (ρhu + q − u ⋅ τ ) = ( E + u × B ) ⋅ J (5.30a) ∂t ∂t In the MHD formulation, the globally neutral condition implies that there are no free space charges. A  fundamental law of Gauss’s divergent for electric field becomes:  ∇ ⋅ E = 0. By the generalized Ohm’s law, E y = J y σ − uw Bx + ux Bz and E z = J z σ − ux By + uy Bx . The cross-​product J × B can be manipulated to be a perfect divergence operator for the magnetic flux density. Finally, by a perfect electric conductivity assumption, the Joule heating also achieves a perfect divergent form in steady state:



[ E + ( u × B )] ⋅ J ≈ −∇ ⋅ [ u( B ⋅ B µ ) − B( B ⋅ u µ )] (5.30b) The conservation laws in steady state now appear as:



∇ ⋅ ρu = 0 ∇ ⋅ ( uB − Bu ) = 0 ∇ ⋅ [ρuu − BB µ − τ + ( p + B 2 2µ )I ] = 0 ∇ ⋅ [ρhu + κ∇T − u ⋅ τ − ( B ⋅ u ) B µ ] = 0

(5.30c)

Consider the simplest shock wave structure in steady state across the x-​coordinate and recall the electromagnetic field will not permit a realistic one-​dimensional phenomenon. Therefore, the shock jump condition must be formulated by the tangential and normal components across and parallel to a shock wave. The simplest shock relation that can be studied is essentially an oblique wave in the three-​dimensional electromagnetic field. The physical realistic macroscopic conservative equation can then be integrated across the shock. From the Maxwell equations, the normal components of the magnetic flux density B and the tangential component of the electric field intensity E must be continuous across any interface. From Equation (5.30c), all possible invariant shock jump conditions across the wave front in steady state for plasma become:



[ Bx ] = 0 [Ey ] = 0 [Ez ] = 0 [ u x By − uz Bx ] = 0 [ u x Bz − uz Bx ] = 0 (5.31) [ρux ] = 0 [ρux2 + p + ( By2 + Bz2 ) 2µ ] = 0 [ρux uy − Bx By µ ] = 0 [ρux uz − Bx Bz µ ] = 0 [ρux h + ( By2 + Bz2 ) 2µ − Bx ( uy By + uz Bz ) µ ] = 0

17

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Magnetohydrodynamics Equations

In this equation, all the dependent variables in square brackets are invariant properties perpendicular to the oblique shock. The shock jump relationship under these assumptions can be integrated across the shock Equation (5.31) because the rate of change by viscous dissipation and heat transfer has no influence on the jump condition. This observation can be understood on the basis of an equilibrium thermodynamic state. Since entropy is a function of the local thermodynamic state, the path of fluid particle motion is irrelevant, but only the condition at the end states matters. At the integral limits, upstream and downstream to the shockwave are:

(ρux )1 = (ρux )2 (5.32a)



( Bx )1 = ( Bx )2 (5.32b)



( ux By − uy B x )1 = ( ux By − uy B x )2 (5.32c)



( ux Bz − uz B y )1 = ( ux Bz − uz B y )2 (5.32d)



{ρux2 + p + ( By2 + Bz2 ) 2µ}1 = {ρux2 + p + ( By2 + Bz2 ) 2µ}2 (5.32e)



(ρux uy − Bx By µ )1 = (ρux uy − Bx By µ )2 (5.32f)



(ρux uz − Bx Bz µ )1 = (ρux uz − Bx Bz µ )2 (5.32g)



{ρux h + ux ( By2 + Bz2 ) 2µ − Bx ( uy By + uz Bz ) µ}1 (5.32h) = {ρux h + ux ( By2 + Bz2 ) 2µ − Bx ( uy By + uz Bz ) µ}2 In Equations (5.32a) through (5.32h), the subscripts 1 and 2 denote the quantities evaluated prior to and after the shock wave. Theses equations actually represent the jump condition across a magnetized shock wave. The second shock jump condition ( Bx )1 = ( Bx )2 is included to ensure that the normal magnetic flux density is an invariant across the interface boundary by the Maxwell equation. Similarly, a current sheet must also exist parallel to the shock to sustain the discontinuities of the transverse magnetic flux density to the shock. From the jump condition, Equations (5.32c), (5.32d), (5.32f), and (5.32g) reveal that the effects of an externally applied transverse magnetic field alter the classic shock jump condition across an oblique shock. In particular, the velocity components parallel to the shock front may also be modified, which is drastically different from the gas dynamic theory. For plasma with a perfect electric conductivity and a negligible Hall current, the three-​dimensional jump conditions are identical to those derived by Sutton and Sherman (1965). From the aforementioned jump conditions and the equation of state, the analog of the Rankine-​Hugoniot equation of gas dynamics with the jump conditions of the enthalpy, the gas density, and the magnetic field to MHD has been given by Sutton and Sherman as:



ρu{[ h ] + (ρu )2 [1 ρ2 ] 2 + ( uy2 + uz2 ) 2} + [ By2 + Bz2 )u µ − ( Bx µ )( By uy + Bz uz )] = 0 (5.33a) When this jump condition is rewritten in terms of internal energy of the system, the MHD equivalent of the Rankine-​Hugoniot equation becomes:

17

5.7  Modified Rankine-Hugoniot Shock Conditions

177

e2 − e1 + ( p2 + p1 ){(1 ρ2 ) − (1 ρ1 )} 2 + {( By,2 − By,1 ) + ( Bz ,2 − Bz ,1 ) }(1 ρ2 − 1 ρ1 ) 4µ = 0 (5.33b) 2

2

The modified shock jump condition in an electromagnetic field is seen clearly only associated with the applied transverse magnetic field. When the externally applied magnetic field is ceased, Equation (5.33b) will revert to the classic Rankine-​ Hugoniot jump condition.

e2 − e1 + ( p2 + p1 ){(1 ρ2 ) − (1 ρ1 )} 2 = 0 (5.33c) These equations summarize the changes to the plasma properties across a shock in the presence of an electromagnetic field. The key and sole contributor to the changing shock jump condition is the transverse magnetic field flux density to the shock wave. Since the discontinuous magnetic field strength must be balanced by the surface current in the shock layer, according to Equation (3.10e), the role of charged separation in the double layer across the shock is unfortunately unknown. The jump conditions nevertheless will revert to the classic Rankine-​Hugoniot relationship of gas dynamics when the applied magnetic field vanishes. Numerical solutions to the ideal MHD equation have demonstrated that the degenerate shocks occur either at the instances when the polarity of the magnetic field is switched or the applied magnetic field vanishes (Brio and Wu 1988; Gaitonde 2001. In the bounded region by the right-​running rarefaction precursor and a negative fast expansion wave, the varying magnetic field creates a negative large parallel velocity component to the planar shock. Thus, the transverse magnetic field induces a vortical structure in the post-​shock region. The vorticity generation mechanism by electromagnetic force to a planar shock can alter the entropy production across the shock. The altered velocity distribution markedly differs from the corresponding gas dynamic shock for which vorticity behind a planar shock wave shall be a null value. If the phenomenon of a shock wave under a varying magnetic field could be verified by experiments, the experiments could be an innovation for flow field manipulation. In theory, the added dilution generation mechanism by an externally applied magnetic field has shown the possibility to provide a new entropy production mechanism to a straight plasma shock and more complex vortical structure in the post-​ shock region. All the analytic results have produced the identical and supporting results of the ideal MHD formulation that overlooked the detailed plasma composition. However, it must be borne in mind that some simplified approximations have been imposed during the derivation process, thus experimental validation becomes paramount. From the classic theoretic study, the entropy increase across a shock is known to be produced by conductive heat transfer, viscous dissipation, and Joule heating, but the work done by the electromagnetic force in the double layer is far from clear (Shang 2016). Physics validation for the challenging phenomenon is not easily achievable because the complex interlocking electromagnetic–​fluid dynamics interactions are not fully understood and require nonintrusive and extremely accurate measurement technique in multiple dimensions.

179

178

Magnetohydrodynamics Equations

5.8

Classic Solutions of MHD Equations The MHD equations are difficult to solve analytically, but only analytical solutions can provide insight into electromagnetic force’s impact on fluid motions. The magnetic field interaction with fluid dynamics intrinsically can alter the flow behavior in an electrically conducting medium. The charged particles’ motion also modifies the electromagnetic field structure by induced intricate eddy electrical current loops in the medium that are totally unexpected. The classic Hartmann flow of an incompressible medium in a simple parallel channel and Ziemer’s experiment on shock stand-​off distance are exemplified for their invaluable contributions. Additional computational simulations have identified that the lack of detailed plasma composition can severely limit the fidelity to physics for practical engineering applications. However, the analytic finding by Hartmann and experimental observations by Ziemer have opened avenues for other to follow. The Hartmann solution is a generalization of the incompressible steady plane Poiseuille flow within a rectangular channel with an imposed uniform and constant transverse magnetic flux density Bz (Hartmann 1937; Hartmann and Lazarus 1937). Under this simplified condition, the relationship among the electromagnetic variables is sufficiently provided by the generalized Ohm’s law. From the eigenvalue structure of the MHD equations, a physically meaningful solution must be investigated in multiple spatial dimensions by any numerical simulation, thus the formulation is started from a three-​dimensional rectangular channel and systematically reduces the complexity by satisfying the compatible physical requirements. The electrode surfaces are thus prescribed on the y-​coordinate (height) and the insulated walls of a two-​dimensional channel, a transverse magnetic field is imposed along the z-​coordinate (width). The governing equations of Hartmann flow have been derived from the incompressible Navier-​Stokes equation with Ohm’s law for electric conductivity.



∇⋅u = 0 u ⋅ ∇u + ∇p − µ g ∇2 u = J × B (5.34a) J = σ( E + u × B ) For an incompressible fluid medium, the divergent condition of the continuity equation is satisfied identically and the conservation of energy equation becomes unnecessary. After applying the assumption of constant electric field strength across the channel, the divergence of the electric field vanishes: ∇ ⋅ E = 0. In steady state, the motion of the conducting medium transverse to the magnetic field will induce an electric current component J y perpendicular to both the flow and the width of the parallel channel. Equation (5.34a) reduces to:



∂p ∂x = J y Bz + µ g ∂ 2 u ∂y2 ∂p ∂z = − J y Bx (5.34b) J y = σ( E y − Bz u )

179

5.8  Classic Solutions of MHD Equations

179

In the absence of an externally applied magnetic field, the solution to the degenerated equations is the classic solution of the Poiseuille flow. The velocity profile across a channel with a width of 2w is parabolic under a constant streamwise pressure gradient: dp dx.

u = −(1 2µ )( dp dx )(w 2 − y2 ) (5.34c) From the conservation momentum equation, Equation (5.34b), the pressure along the channel at most can be a linear function of x, and must have a shear stress component in z to balance the Lorentz force J × B . Therefore, the velocity across the channel becomes a function of the z-​coordinate and the linear pressure gradient term along the channel becomes the driving force of the flow field. In addition, the induced x-​component magnetic flux density, Bx, becomes negligible by the symmetry condition, which leads to a zero-​pressure gradient across the channel. The momentum equation thereby reduces to a second-​order ordinary differential equation for the velocity distribution across the channel.



d 2 u dy2 − ( σBz2 µ )u = − [ σBz E y − ( dp dx )] µ (5.34d) The analytic forms of the non-​dimensional velocity and electric current profiles across the half channel width w have been found by Hartmann as:



u u = H [cosh( Hy w ) − cosh( Hy w )] [ H cosh( H ) − sinh( H )] (5.35a) Substitute the solution in Ohm’s law, Equation (5.34b), to get the analytic solution for the y-​component electric current:



J y ( y w ) = (ρuw µ )( dp dx ) H 2 + H [ H − tanh( H )] (5.35b) where the symbol H is the well-​known Hartmann number, H = Bl ( σ µ )1 2; it is the ratio of the Lorentz force to the viscous force. In fact, the Hartman number is related to the two classic similarity parameters of MHD; they are the magnetic Reynolds number (Rm = σµuL) and the Steward number (S = σB 2 L ρu). The relationship between three similarity parameters is given as (Mitchner and Kruger 1973):



H = ( Rm S )1 2 = Bz l ( σ µ )1 2 (5.35c) For the Hartmann flow, the characteristic length of this problem is the half channel width w: the original Hartmann number is defined as:



H = (w 2 ) Bz ( σ µ )1 2 (5.35d) Once the analytic solution of the velocity profile is known, the average velocity l across the channel is obtainable as u = (1 2w )∫ udy , and the detailed and coupled −l electromagnetic properties within the channel can be easily determined. Computational simulation of the Hartmann problem is rather straightforward by solving Equation (5.34d). The following graphs are generated by numerical simulations for the height and the width of the channel by a one hundred-​point mesh system over the channel. The numerical results are also fully verified by comparing with the accompanied closed form solutions by Hartmann. Through this

18

180

Magnetohydrodynamics Equations

Figure 5.5  Velocity profiles of Hartmann flow, 0.0 < H  ne , n+ , n− . Therefore the viscous force of the charged particles becomes negligible. At the same time, the characteristic time of the inter-​particle collision frequency are scaled by the speed of light. As a consequence, the inertia of the particles’ organized motion of Equation (6.16a) is negligible:



mi ni ∂ui ∂t + mi ni ( ui ⋅ ∇ )ui  0 (6.16b) In essence, the inertia of the charged particles’ motion is negligible in comparison to the electromagnetic force. From the definition of partial pressure of each charged species, pi = ni kTi , the three charged species of the approximated Equation (6.16a) are reduced to:



kTi ∇ni + eni ( E + ui × B ) + mi ni νin ( ui − un ) + mi ni νi + ( ui − u+ ) + mi ni νi − ( ui − u− ) = 0 (6.17) where the collision frequencies between species are denoted as νin , νi +, and νi −. Again, the huge disparity between the characteristic velocities of neutrals and charged particles by speed of sound versus speed of light makes the velocities’ differences among charged particles negligible. In addition, the number density of the charged particles is much lower than that of neutral particles, so that the

212

Ionization Processes in Gas

collision frequency between charged particles and neutral is orders of magnitude greater than the collision frequencies between charged particles: νen >> νe + , νe −, then Equation (6.17) can be given for electrons and positively and negatively charged ions as:

ne ue = − d e ∇ne − µ e ne ( E + ue × B ) (6.18a)



n+ u+ = −d + ∇n+ + µ + n+ ( E − u+ × B ) (6.18b) and



n− u− = −d − ∇ne − ( µ − ne E + u− × B ) (6.18c) From the definition and the Einstein relationship, it is recognized that the electron mobility µ e and the thermal diffusion d e of the electron:



µ e = e me νen and d e = kTe me νen (6.19a) Similarly the ion mobility and diffusion coefficients of the positively and negatively charged ions are recognized as:



µ ± = e m± ν± n and d ± = kT± m± ν± n (6.19b) For all practical processes, the masses of the positively and negatively charged particles are identical, thus the positively and negatively charged ion mobility and diffusion are equal. The foregoing equations actually define the diffusion velocities of an inhomogeneous, electrically conducting gas mixture. The leading terms in Equations (6.18a), (6.18b), and (6.18c) describe the ordinary diffusion due to local concentration of electrons, positively charged ions, and negatively charged ions. The second terms are the force diffusion by drift motion for an electrically charged species in an applied electric field. The last terms are also the force diffusion by the Lorentz acceleration. Once the species velocities are determined, the species conservation equation becomes a simplified approximation to the continuity equation, Equation (6.12a), for charge-​carried species of plasma. The diffusion-​drift model is equally applicable to all weakly ionized gas from thermal to electron impact ionizations, including glow, corona, and dielectric barrier discharges. This formulation has been adopted by most recent numerical simulations for electron impact ionizations (Shang et al. 2014). One of the weaknesses of this approximation is that the detailed distinctions between metastable species are no longer recoverable by this approximation. Instead, the weakly ionized gas is treated as a medium that consists of three charged and one neutral species. The species conservation equations for the three distinct charged particles are:



∂ne dw − ∇ ⋅ [ d e ∇ne + ne µ e ( E + ue × B )] = e (6.20a) ∂t dt



∂n+ dw+ − ∇ ⋅ [ d + ∇n+ − n+ µ + ( E + u+ × B )] = (6.20b) ∂t dt

213

6.5  Inelastic Collision Ionization Models



213

∂n− dw− − ∇ ⋅ [ d − ∇n− + n− µ − ( E + u− × B ) = (6.20c) ∂t dt where the terms on the right-​hand side of the equations, dwe dt , dw+ dt, and dw− dt, designate production and depletion rates of the considered ionized species. These rates have either been determined by some chemical kinetic models or are simplified approximations based on known ionization processes. In most formulations, the electron, positively charged ion, and negatively charged ion number density fluxes are designated by symbols Γ e , Γ + , and Γ −, which have been presented by Equation (2.22b). An externally applied magnetic field exerts a profound influence on the drift and diffusion velocities by the Lorentz acceleration. The altered drift-​diffusion number density flux vectors by a transverse magnetic field have been derived in Chapter 2.5, and are partially verified by experimental observations (Kimmel et al. 2006). The important connection between the coefficients of diffusion, di and the mobility, μi of drift velocity is given by the Einstein relationship: the relative magnitude is a function of the characteristic electron energy to the elementary electric charge |E|/​p (Surzhikov and Shang 2004):



µe =

e kT ≈ µ e ( E p ); d e = e µ e (6.21a) me νen e



µ± =

e kT± ≈ µ ± ( E p ); d ± = µ ± (6.21b) m± ν± n e

The drift velocity is given as a function of the reduced electrical field E /​p with a physical dimension of V/​cm-​Torr in the CGS system, which is the work done by transforming the field energy into the energy of molecular vibration. In the range of |E|/​p ratios up to 50, the electron drift velocity in ionized air has a typical value of around 107 cm/​s. From Equations (6.21a) and (2.21b), the drift velocity of ions is much slower than that of electrons; the drift velocity of ions is estimated to be 105 cm/​s due to the large differences between the masses of electrons and ions. Therefore, when considerable numbers of free space charges are formed by charged separation, the polarized electric field between charges restrains the motion of electrons from that of ions to appear as the ambipolar diffusion. The unique diffusion phenomenon of plasma is also known as a function of the ratio of electron energy and ambient thermodynamic states. The diffusion coefficients of charged particles have an explicit dependence on the characteristic temperatures of each species. Large differences in diffusion coefficients between electrons and ions can be anticipated in nonequilibrium plasma generated by electron impact, because in most partial ionized gases ions usually remain in the thermodynamic state of their surrounding environment. An enormous amount of energy is needed to generate localized volumetric plasma, because an ionization potential is 34 eV for electron beam, 65.7 eV for DCD, and 81 eV per ion–​electron pair for discharge at the radio frequency (Raizer

214

Ionization Processes in Gas

1991). In the electron impact processes for ionization, the positive and negative charged ions retain their ambient conditions. For this reason, the partially ionized gas is often identified as the low-​temperature plasma with a charge number density generally limited to a value of 1013/​cm3. As weakly partially ionized plasma by electron impact, the electromagnetic force usually exerts only a small perturbation to the mainstream flow, and the thermodynamic behavior is also significantly different from the plasma generated by thermal excitation. Therefore the plasma actuator for flow control is the most effective at flow bifurcations such as the onset of dynamic stall, laminar-​turbulent transition, and vortical separation (Shang et  al. 2014). However, the electromagnetic effect can also be amplified by an externally applied magnetic field or by inviscid–​viscous interaction at the leading edge of hypersonic control surfaces (Shang and Surzhikov 2005). Ionization by inelastic collision kinetics determines the net rates of generation and depletion of individual ionized species by the chemical-​physics modeling. This approach groups the charge species into global categories of electrons, positively charged ions, and negatively charged ions. The justification for this approach is to recognize that the charged species only appear in the tracing amount in molar or mass fractions and are dominated by a few radical components. In fact, the ionization by electron collisions occurs at the outer limit of the Maxwell-​Boltzmann distribution, where the gas energy is lower than the ionization potential (Raizer 1991). The dominant species of DBD are the metastable molecules N2 ( A3 Σ u+ )and O2(b1Σ d+ ). The main ionized processes for a nonequilibrium volumetric discharge consist of four types of reactions:  electron/​molecular, dissociative recombination, ion-​ion recombination, and electron attachment. At any instance during discharge, the ionized species concentration is the net balance between the ionization, detachment, attachment, and recombination processes. The inelastic collision ionization model by quantum chemical kinetics is therefore focused on these mechanisms. The classic formulation for electronic impact ionization is based on the similar law by Townsend, which is an empirical formula. The basic process is a complex chain of events; it involves charge accumulation on a cathode, penning penetration-​induced secondary emission, and electron cascading. In the formulation, the ionization coefficient α, which measures the number of ionizations by electron impact per unit distance, is a function of the reduced electrical field E p. This quotient is also a measure of the energy gained by a charged particle between collisions from the principle of similarity. It is remarkable that the coefficient α = A exp( − Bp / E ) of Townsend’s similarity law holds extremely well in comparison with a large group of experimental data, both in the ionization frequency and in the degree of ionization. For discharge in air of the E p range from 100 to 800 (V/​Torr cm) can be modeled by A = 15 and B = 365. At a higher value of E p, an accuracy improvement may be needed, but is not essential (Surzhikov and Shang 2004). The coefficients of ionization for air can be summarized as:

215

6.5  Inelastic Collision Ionization Models



215

  α 365   p  = 15 exp  − ( E / p )  , 1/cm × Torr (6.22)   The depletion of electron number density is accomplished by the recombination and attachment processes. The dissociative recombination is the fastest mechanism of the bulk recombination of a weakly ionized gas, and is a simple binary chemical reaction. The decay rate with time of plasma is often given as:



 dne   dn+   = −βn+ ne ;   = −βn+ ne (6.23a)  dt dis dt  dis and the typical value of coefficient β has been assigned a value of 2×10–​7 cm3/​s, and the characteristic decay time scale is less than 10–​3 second. Another main mechanism of charge neutralization is the ion-​ion recombination process. In a low-​pressure environment, the recombination takes place through binary collision and the reaction is similar to the charge transfer. The reaction rate constants have been estimated around the values of 10–10 cm3/​s for two-​body collisions. At a moderate pressure, the process proceeds through triple collisions, and the reaction rates are much slower, at a value of about 10–​25 cm6/​s (Raizer 1991). Again the rate of ion-​ion recombination can be approximated by collision kinetics:



 dn+   dn−   = −βi n+ n− ;   = −βi n+ n− (6.23b)  dt r dt  r For example, the recombination rate constants between O2− and O +4, as well as NO + and NO2− , are on the orders of magnitude of 10–​25 and 10–​26 cm6/​s. The maximum value of the ion-​ion recombination at the one atmosphere has a value of 10–​6 cm3/​s. The electron attachment and detachment are the key formation and depletion processes of negatively charged electrons in partially ionized air. The chemical reaction rates are uncertain and have a wide range of values from 10–​8 cm3/​s to 10–​30 cm6/​s, and especially for the triple-​collision reactions, and a strong electronic temperature dependence is noted (Gibalov and Pietsch 2004). For the strongly bound molecules, a sufficiently high energy is required for the dissociative attachment of an electron. The electron attachment is a main mechanism for removing the electron from the negatively charged ions. The lost electron number density can be given as:



 dne   = − νa ne (6.23c)  dt  a The attachment frequency of electrons in dry air at one atmosphere condition is around 108/​s. The electron attachment mechanism has been modeled to be proportional to the Townsend’s ionization process for a nonphysical, but created a computational stability concern. In this regard, some investigators adopt a similar basic collision formulation for the attachment and recombination mechanisms of the ionization process with empirically determined coefficients (Boeuf and Pitchford 2005). Others also model the negative ion ionization by splitting a portion of the electron generation as the electron attachment process (Unfer and Boeuf 2009). In

216

Ionization Processes in Gas

short, the physics-​based formulation for charge attachments by inelastic collision kinetics has been widely adopted for most DBD simulations (Shang 2016).

 dne   dn−   = kd nn ne ;   = − kd nn n− (6.23d)  dt  de dt  de The coefficients of the detachment of electrons, κ d , in partially ionized gas at room temperature have the value of 10-​10 cm3/​s. The value for the metastable molecules N2(A3 Σ u+ ) and O2(b1Σ d+ ) in air is unknown. But by an indirect estimate, the discharges are characterized by a value of 10–​14 cm3/​s. It is also interesting to note that weakly ionized plasma is sustainable in negatively charged gas at a lower value of E p than that of a short pulse discharge. From this brief discussion, the plasma generation and depletion processes through the complex chemical-​physics involved quantum mechanics are modeled by the inelastic collision for the multi-​fluid model (Surzhikov and Shang 2004; Shang and Huang 2014):



∂ne − ∇ ⋅ ( ne µ e E + d e ∇ne ) = α( E ) | Γ e | −βn+ ne − ν a ne + κ d nn ne ∂t ∂n+ + ∇ ⋅ ( n+ µ + E − d + ∇n+ ) = α( E ) | Γ e | −βn+ ne − βi n+ n− (6.24) ∂t ∂n− − ∇ ⋅ ( n− µ − E + d − ∇n− ) = ν a ne − κ d nn n− − βi n+ n− ∂t In fact, the sum of these equations satisfies fully the charge conservation law; the rate of change in charge number density must be balanced by the gradient of the electric current density, which is derived from the Maxwell equations, Equation (3.5b). Again in the foregoing approximation, Equation (6.24), the number density of neutral particles of air, nn , is treated as a constant to have a value of 2.69×1019/​ cm3. The attachment frequency νa is formally defined as νa = ( α a / p )ue ,drift ⋅ p , a commonly adopted value for the parameter ( α a / p ) is 0.005 / cm × Torr, and ue ,drift is the drift velocity of an electron. From the drift-​diffusion theory, the velocity can be consistently evaluated as ue ,drift = µ e E . The coefficient of ion-​ ion recombination βi is estimated from the chemical kinetics of ionized and molecular nitrogen and oxygen to have a value of βi = 1.6 × 10 −7 , cm 3 /s. The detachment coefficient is estimated to have a value range of 10 −14 < κ d < 8.6 × 10 −10 , cm 3 /s. This coefficient is important for the negatively charged ion number density computation because it’s the principal mechanism that governs the depletion of this species. Fortunately, its value can be verified by the discharged electrical current from experimental data. This model of electron collision ionization is actually replaced by the energy conservation equations of vibrational and electronic internal excitations, Equations (6.13a) and (6.13b), as well as, the law of mass conservation, Equation (6.12a), and Equation (6.9c) via chemical kinetics modeling. In all, the formulation for ionization by collision kinetics has been widely adopted for most direct current discharge (DCD) and dielectric barrier discharge (DBD) simulations (Shang and Huang 2014).

217

6.5  Inelastic Collision Ionization Models

217

Figure 6.7  Discharge domains of dielectric barrier discharge at opposite electric polarities, ϕ = ±3.0 kV, ν = 10 kHz, Gap = 0.0 mm .

In Figure  6.7, the time-​dependent computational charge number densities for dielectric barrier discharge are depicted at the instants when the peak values of the EMF of ±3.0 kV are applied. The externally applied AC electric field potential represents the lower range of the DBD operation, because the breakdown voltage of dry air over the overlapped electrodes is around 2.2 kV. The reduced electrical field potential within the DBD field is made evident by the charge number density accumulation over the dielectrics or the exposed electrode. At this instant in time, when the exposed electrode carries a positive voltage, the secondary electrons emission from the dielectrics, which acts as the cathode, creates an overwhelming positively charged particles accumulation over the barrier surface to reduce the electrical potential of the DBD field. At the same instant, all the electron propagate toward the anode and concentrate over the edge of the anode. When the electrical polarity in the AC cycle is reversed on the exposed electrode that becomes the cathode, all positively charged ions are now expelled from the dielectrics and concentrate near the lower edge of the exposed electrode. At the same time, the electrons are accelerated from the exposed electrode toward the dielectrics embedded electrode. The accumulated electrons over the dielectrics diminish the electrical intensity within the DBD field. The propagations of the positively charged ions and electrons within an AC cycle are clearly illustrated by the computational simulations. The final and the most important verification of the chemical-​ physics ionizing model is presented by a direct comparison with an experimentally measured

218

Ionization Processes in Gas

Figure 6.8  Comparison of DBD conductive current with experiment, EMF = 4 kV, ν = 5 kHz.

conductive current during the discharge (Shang et al. 2014). The ionization model is built on Townsend’s discharge mechanism together with the bulk and ion-​ion recombination, as well as the electron attachment and detachment. If the ionization model by the chemical-​physics kinetics cannot describe a reasonable discharge composition, the calculated conductive current of a DBD will be clearly erroneous. The experimental data are collected at an AC cycle of 5 kHz and an electrical potential of 4.0 kV. In Figure 6.8, the calculated conductive current is designated by a thick black line that lies within the scattering traces of the experimental data to reflect the correct magnitude and duration of the discharge. Equally important, the effects of different electrical permittivity and secondary emission coefficients on metallic electrode and dielectric surface are also accurately captured. In theory, the foregoing ionizing models can be further improved by expanding the experimental data base or by an ab initio computational approach to the quantum chemical-​physics. In this regard, the state-​of-​the-​art status in low-​temperature ionization modeling is similar to that of chemical-​physics models using the approximated chemical reaction rates, but it has a decided advantage in substantially reducing the computational resource required for simulations. On the other hand, the challenges using the ab initio approach encounter formidable difficulties both in concept and in required computational effort. The potential energy surface is the central issue of quantum computational chemistry and is conducted on an impossible-​to-​visualize n-​dimensional hypersurface. The potential energy surface

219

6.6  Database of Chemical Kinetics

219

is defined by the stationary points where ∂E ∂q = 0 for all geometric parameters q. Therefore the main tasks of computational chemistry become the determination factor for the structure and energy of molecules and the transition states. Although the Born-​Oppenheimer approximation has simplified, the application of the Schrodinger equation to molecules is still extremely difficult, thus it remains knowledge that is unquantifiable by the electronic energy and nuclear repulsion potential. These topics are truly beyond the scope of our discussion.

6.6

Database of Chemical Kinetics The energy transfer among the internal degrees of freedom of high-​temperature air consists of two major mechanisms, quantum jumps within the excited internal modes and the finite-​rate chemical kinetic processes. The chemical kinetic process has been further broken down into seven main mechanisms for energy transfer by Park (1993, 2001). They are ionization, electron impact dissociation, elastic collision between electrons and heavy particles, interaction between electrons and the vibrational mode, energy exchange by the formation of species, associative ionization, and, finally, radiation losses. These processes are complex and our understanding of the physical chemical interaction in the molecular and atomic scales is extremely limited. Nevertheless, there are continuous developments of chemical kinetics data starting from the early 1960s until the 2000s (Kang and Dunn 1973; Park 1993; Olynick et  al. 1999; Park, Jaffe, and Partridge 2001). According to the kinetic theory of dilute gas, the translational and rotational excitations can reach an equilibrate state after a few numbers of collision between different chemical species at the standard condition, thus these internal excitations are considered to be in thermal equilibrium condition. However, the higher degrees of internal freedom require far more numbers of collision between species to accumulate a sufficient amount of energy to attain the equilibrium condition, which leads to a time-​delay for reaching the final thermal equilibrium state. These relaxation phenomena must be explicitly modeled by the theories of Landau and Teller (1936), Treanor and Marrone (1962), Millikan and White (1963), and Chernyi and colleagues (2004). The chemical kinetic models of most aerospace engineering analyses are adopted based on the unique flow features immediately downstream to a strong shock. In this environment, the level of vibrational energy is far lower than that of the translational energy by shock compression and the energy exchange among the internal modes is dominated by the adjacent quantum level (ladder climbing). In addition, the vibrational quantum state is limited to the first few above the ground state. Therefore, the anharmonic correction is not necessary, but the relaxation phenomenon persists. From the principle of detailed balance, the classic result of Landau and Teller (1963) becomes the backbone for the present kinetic modeling. The following kinetic models have guided the development of the present approach.

Ionization Processes in Gas

220

Table 6.2  Finite chemical reaction rate coefficients of twenty-​six-​species nonequilibrium air model by Kang and Dunn. No.

Af , cm3/​ n f (mole×s)

E f , K

Ar , cm3(6)/​ nb (mole×s)

E b, K

1

N2 + N2 ⇔ N

+ N

+ N2

3.80E+19 –1.0

0.1132E+06

2.50E+18 –1.0 0.0

2

N2 + O2 ⇔ N

+ N

+ O2

1.90E+19 –1.0

0.1132E+06

1.00E+18 –1.0 0.0

3

N2 + NO ⇔ N

+ N

+ NO 1.90E+19 –1.0

0.1132E+06

1.00E+18 –1.0 0.0

4

N2 + N

⇔ N

+ N

+ N

1.30E+20 –1.0

0.1132E+06

7.00E+18 –1.0 0.0

5

N2 + O

⇔ N

+ N

+ O

1.90E+19 –1.0

0.1132E+06

1.00E+18 –1.0 0.0

6

O2 + N2 ⇔ O

+ O

+ N2

2.30E+18 –1.0

0.5940E+05

1.90E+16 –0.5 0.0

7

O2 + O2 ⇔ O

+ O

+ O2

2.30E+18 –1.0

0.5940E+05

1.90E+16 –0.5 0.0

8

O2 + NO ⇔ O

+ O

+ NO 3.00E+18 –1.0

0.5940E+05

2.50E+15 –0.5 0.0

9

O2 + N

⇔ O

+ O

+ N

0.5940E+05

2.50E+15 –0.5 0.0

10

O2 + O

⇔ O

+ O

+ O

8.50E+18 –1.0

0.5940E+05

7.10E+16 –0.5 0.0

11

NO + N2 ⇔ N

+ O

+ N2

2.40E+17 –0.5

0.7550E+05

3.20E+18 –1.0 0.0

12

NO + O2 ⇔ N

+ O

+ O2

2.40E+17 –0.5

0.7550E+05

3.20E+18 –1.0 0.0

13

NO + NO ⇔ N

+ O

+ NO 2.40E+17 –0.5

0.7550E+05

3.20E+18 –1.0 0.0

14

NO + N

⇔ N

+ O

+ N

0.7550E+05

3.20E+18 –1.0 0.0

⇔ N

+ O

+ O

15

NO + O

16

O2 + N2 ⇔ NO + NO

17

N2 + O

18

NO + O

19

N

20

O

21

3.00E+18 –1.0

2.40E+17 –0.5 2.40E+17 –0.5

0.7550E+05

3.20E+18 –1.0 0.0

2.00E+14

0.0

0.6160E+05

1.00E+13

0.0 0.40E+05

⇔ NO + N

6.80E+13

0.0

0.3775E+05

1.50E+13

0.0 0.0

⇔ O2

+ N

4.30E+07

1.5

0.1910E+05

1.80E+08

1.5 0.33E+04

+ O

−

⇔ NO + E

1.30E+08

1.0

0.3190E+05

2.00E+19 –1.0 0.0

+ O

⇔ O2+ + E−

6.00E+08

0.5

0.8080E+05

5.00E+19 –1.0 0.0

N

+ N

⇔ N2+ + E−

8.50E+09

1.0

0.6770E+05

5.00E+18 –0.5 0.0

22

N

+ E-​ ⇔ N

+ E

−

+ E

2.70E+13

0.5

0.1686E+06

5.30E+20 –1.0 0.0

23

O

+ E-​ ⇔ O+

+ E−

+ E−

1.64E+13

0.5

0.1578E+06

2.80E+20 –1.0 0.0

24

NO + E-​ ⇔ NO + E

−

2.50E+13

0.5

0.1073E+06

2.00E+24 –1.5 0.0

25

O2 + N2 ⇔ O2+ + N2 + E−

5.00E+04

1.5

0.14000E+06 2.00E+15 –1.5 0.0

+

−

+

+

−

+ E

The earliest chemical kinetics model for high-​temperature air by Kang and Dunn (1973) was composed of twelve species and sixty-​four elementary chemical reactions and was later simplified to eleven species and twenty-​six reactions. The chemical kinetic model by Kang and Dunn is presented in Table  6.2. For these calculations, all internal degrees of excitation are assumed to retain thermal equilibrium condition. The reactions involving only the neutral species are known from the accumulated research results. The rates of coefficients for the recombination of charged species NO + , O + , N + , O2+ , and N 2+ are measured from the application conditions. Unfortunately, the important reaction rate coefficients for the charge transfer and ion–​molecule reactions are uncertain, because they have a significant influence on the nonequilibrium process. In general, the calculated electron number

21

6.6  Database of Chemical Kinetics

221

Table 6.3  Chemical reaction rate coefficients of eighteen-​species nonequilibrium air model by Olynick, Chen, and Tauber. Af , (cm3/​mole)/​s

nf

Ef, K

nr

Er , K

1

N 2 +M ↔ N+N+M

0.70E+22

–1.6

0.113E+06

−0.6

0.

2

O2 +M ↔ O+O+M NO+M ↔ N+O+M

0.20E+22

−1.5

0.598E+06

−0.5

0.

0.50E+16

0.0

0.755E+05

1.0

0.

0.37E+15

0.0

0.690E+05

1.0

0.

5

C 2 +M ↔ C+C+M CO+M ↔ C+O+M

0.23E+21

–1.0

0.129E+06

0.0

0.

6

CN+M ↔ C+N+M

0.25E+15

0.0

0.710E+05

1.0

0.

7

CO2 +M ↔ CO+O+M

0.69E+22

–1.5

0.633E+05

-​0.5

0.

8

3 4

C3 +M ↔ C 2 +C+M

0.63E+17

–0.5

0.101E+06

0.5

0.160E+05

9

H2 +M ↔ H+H+M

0.22E+15

0.0

0.483E+05

1.0

0.

10

NO+O ↔ O2 +N

0.84E+13

0.0

0.195E+05

0.0

0.348E+04

11

N 2 +O ↔ NO+N

0.64E+18

–1.0

0.384E+05

−1.0

0.467E+03

12

CO+O ↔ O2 +C

0.39E+14

–0.18

0.692E+05

−0.18

13

CO2 +O ↔ O2 +CO

0.21E+14

0.0

0.278E+05

0.0

0.240E+05

14

CO+C ↔ C 2 +O CO+N ↔ CN+O

0.20E+18

–1.0

0.580E+05

–1.0

0.933E+03

0.10E+15

0.0

0.386E+05

0.0

0.

–0.11

0.232E+05

–0.11

0.

17

N 2 +C ↔ CN+N CN+O ↔ NO+C

0.11E+15 0.16E+14

0.1

0.146E+05

0.1

0.377E+03

18

CN+C ↔ C 2 +N

0.50E+14

0.0

0.130E+05

0.0

0.

19

HCN+H ↔ CN+H2 C+e- ↔ C+ + e- + e-

0.80E+12

0.017 0.835E+05

0.017 0.930E+05

20

0.635E+16

0.0

0.138E+06

1.0

0.403E+04

21

N+e- ↔ N + e- + e-

0.508E+17

0.0

0.121E+06

1.0

0.

22

O+e- ↔ O+ + e- + e-

0.635E+16

0.0

0.106E+06

1.0

0.

15 16

+

0.

densities using Kang and colleagues’ model for the RAM-​C-​II probe slightly over-​ predicted the flight data at the low altitude and under-​predicted the flight data at the lower altitude on the reentry trajectory. The most widely adopted Olynick-​Chen-​Tauber chemical kinetics model is a continued improvement from Park’s original effort and extends its application to including ablation phenomena (Olynick et al. 1999). The reaction coefficients are frequently applied to the simulations involving chemical reactions with empirical different temperatures for the nonequilibrium internal degrees of freedom (Park et al. 2001). Carbon monoxide, CO, will always be formed in the shock layer from the carbonaceous heat shield around any space vehicles entering into the atmospherics of Mars, Venus, or Earth. Carbon monoxide is known to produce strong radiation in the vacuum ultraviolet spectrum by its Fourth Positive system. Therefore, for computational simulations of interplanetary flights, the chemical composition shall

222

Ionization Processes in Gas

consist of C, N, C3, N2, O2, CN, CO, NO, CO2. Park’s chemical kinetic model has also been adopted to apply the Mars Sample Return Orbiter (MSRO) program. The escape velocity for planet Mars is 5.0 km/​s. When the minimum-​energy trajectory is adapted from Earth to Mars, then the Mars reentry is estimated to be around 5.8 km/​s. Park’s chemical kinetic model has been applied to the Mars reentry; a systematic modification by eighteen, thirty, seventy-​nine, and eighty-​ three elementary chemical reactions was implemented. A series of computational tests was conducted successfully and validated against available shock tube data by the European Space Agency–​sponsored program (Surzhikov 2006). In Table 6.2, the Dunn and Kung model containing chemical kinetic constants for the forward and backward reactions are included as the benchmark information for the chemical kinetics process:  E f (b)  k f ( b ) = Af ( b )T η f ( b ) exp   KT  (6.25)



Table  6.3 presents the chemical kinetic Olynick-​Chen-​Tauber model for atmosphere air, including the carbonaceous elements of the front heating shield.

References Boeuf, J.P., Pitchford, L.C., Electrohydrodynamic force and aerodynamic flow acceleration in surface dielectric barrier discharge, J. Appl. Phys., Vol. 97, 2005, pp. 103307-​1-​10. Bogdanov, E.A., Kudryavtsev, A.A., Kuranov, A.L., Kozlov, L.E., and Tkchenko, T.L., 2D Simulation of DBD plasma actuator in air, AIAA 2008-​1377, 2008. Chen, Y.K. and Milos, F.S., Ablation and thermal response program for spacecraft heatshield analysis, J. Spacecr. Rockets, Vol. 36, 1999, pp. 475–​483. Chernyi, G.G., Losev, S.A., Macheret, S.O., and Potapkin, B.V. Physical and chemical and plasmas. Vol. 2.  Progress in astronautics and aeronautics (Paul Zarchan, Ed.) Vol. 197, 2004. Clarke, J.F., and McChesney, M., The dynamics of real gases, Butterworths, Washington, DC, 1964. Dunn, M.G. and Kang, S.W., Theoretical and experimental studies of reentry plasma, NSA CR 2232, April 1973. Eletsky, A.V. and Smirnov, B.M., Elementary nonradiative processes, basic phenomena physics I, edited by Galeev, A.A. and Sudan, R.N., North-​Holland Publishing Co., Amsterdam, 1983, pp. 49–​70. Elisson, B. and Kogelschatz, U., Nonequilibrium volume plasma chemical processing, IEEE Trans. Plasma Sci., Vol. 19, 1991, pp. 1063–​1077. Gibalov, V.I. and Pietsch, G.J., Dynamics of electric barrier discharges in coplanar arrangement, J. Phys. D Appl. Phys., Vol. 37, 2004, pp. 2082–​2092. Herzberg, G., Atomic spectra and atomic structure (2nd edn.), Dover Publisher, New York, 1944. Herzberg G. The spectra and structure of simple free radicals, Cornell University Press, Ithaca, NY, and London, 1971. Howatson, A.M., An introduction to Gas discharge (2nd edn.), Pergamon Press, Oxford, 1975. Jones, L.J. and Cross, A.E., Electrostatic probe measurements of plasma parameters for two reentry flight experiments at 25,000 feet per second, NASA TN D 66-​17, 1972.

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Josyula, E., Bailey, W.F., and Suchyta, C.J., III, Dissociate modeling in hypersonic flows using state-​to-​state kinetics, J. Thermophys. Heat Transfer, Vol. 25, No. 1, 2011, pp. 34–​47. Kang, S.-​ W., Jones W.L., and Dunn, M.G. Theoretical and measured electron-​ density distributions at high altitude, AIAA J., Vol. 11, 1973, pp. 141–​149. Kimmel, R.L., Hayes, J.L., Menart, J.A., and Shang, J., Effect of magnetic fields on surface plasma discharges at Mach 5, J. Spacecr. Rockets, Vol. 42, No. 6, 2006, pp. 1340–​1346. Landau L. and Teller, E., Zurtheorie der shallispersion, Physik Z. Sowjetunion, B., Vol. 10, 1936, p. 14. Lighthill, M.J., Dynamics of a dissociating gas –​Part I Equilibrium flow, J. Fluid Mech., Vol. 2, 1958, pp. 1–​32. Meyerand, R.G., and Haught, A.F., Gas breakdown at optical frequencies. Phys. Rev. Lett., Vol. 11, No. 9, 1963, pp. 401–​403. Millikan, R.C., and White, D.R., Systematics of vibrational relaxation, J. Chem. Phys., Vol. 39, No. 12, 1963, pp. 3209–​3213. Olynick, D.R., Chen, Y.K., and Tauber, M.E., Aerodynamics of the Stardust Sample Return Capsule, J. Spacecr. Rockets, Vol. 36, No. 3, 1999, pp. 442–​462. Pancheshnyi, S.V., Starikovkaia, S.M., and Starikovskii, A.Yu., Role of photoionization processes in propagation of cathode-​directed streamer, J. Phys. D Phys., Vol. 34, 2001, pp. 105–​115. Park, C., Review of chemical kinetics problems of future NASA missions, I Earth entries, J. Thermophys. Heat Transfer, Vol. 7 No. 3, 1993. Park, C., Jaffe, R., and Partridge, H., Chemical-​kinetic parameters of hyperbolic Earth entry, J. Thermophys. Heat Transfer, Vol. 15, No. 1, 2001, pp. 76–​90. Rafatov, I., Bogdanov, E.A., and Kudryavtsev, A.A., On the accuracy and reliability of different fluid models of the direct current glow discharge, Phys. Plasmas, Vol. 19, 2012, pp. 033502-​1-​12. Raizer, Yu.P. Breakdown and heating of gas under the influence of laser beam, Soviet Physics Uspekhi, Vol. 8, No. 5, 1966, pp. 650–​673. Raizer, Yu. P., Gas discharge physics, Springer-​Verlag, Berlin, 1991. Raizer, Yu.P. and Surzhikov, S.T., Diffusion of charges along current and effective numerical method of eliminating of numerical dissipation at calculations of glow discharge, High Temper., Vol. 28, No. 3, 1990, pp. 324–​328. Saha, M.N., Ionization in the solar chromosphere, Phil. Mag., Vol. 40, No. 238, p. 472, 1920. Shang, J.S., Computational electromagnetic-aerodynamics, IEEE Press Series on RF and Microwave Technology, John Wiley & Sons, Hoboken, NJ, 2016. Shang, J.S., Adrienko, D.A., Huang, P.G., and Surzhikov, S.T., A computational approach for hypersonic nonequilibrium radiation utilizing space partition and Gaus quadrature, J. Comp. Phys., Vol. 266, 2014, pp. 1–​21. Shang, J.S. and Huang, P.G., Surface plasma actuators modeling for flow control, Prog. Aerosp. Sci., Vol. 67, 2014, pp. 29–​50. Shang J.S. and Surzhikov, S.T., Magnetoaerodynamic actuator for hypersonic flow control, AIAA J., Vol. 43, No. 8, August 2005, pp. 1633–​1643. Shang, J.S. and Surzhikov, S.T., Nonequilibrium radiative hypersonic flow simulation, J. Prog. Aerosp. Sci., Vol. 53, 2012, pp. 46–​65. Singh, K.P. and Roy, S., Modeling plasma actuators with air chemistry for effective flow control, J. Appl. Phys. Vol. 101, 2007, pp. 123308 1-​8. Solov’ev, V., Konchakov, A.M., Krivtsov, V.M., Aleksandrov, N.L., Numerical simulation of a surface barrier discharge in air, Low-​Temperature Plasma, Vol. 34, No. 7, 2008, pp. 594–​608.

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Surzhikov, S.T., Convective and radiation heating of MSRO, predicted by different kinetic models, Radiation of high temperature gases in atmospheric reentry, Rome, Italy, Sept. 2006. component plasma model for two-​ dimensional Surzhikov, S.T. and Shang, J.S., Two-​ glow discharge in magnetic field, J. Comp. Phys., Vol. 199, No. 2, Sept. 2004, pp. 437–​464. Surzhikov, S.T. and Shang, J.S., Fire-​II flight test data simulations with different physical-​ chemical kinetics data and radiation models, Front. Aerosp. Eng., Vol. 4, No. 2, 2015, pp. 70–​92. Surzhikov, S.T., Sharikov, I. Capitelli, M., and Colonna, G. Kinetic models of nonequilibrium radiation of strong air shock wave, AIAA Preprint 2006-​0586, 2006. Treanor, C.E. and Marrone, P.V., Effect of dissociation on the rate of vibrational relaxation, Phys. Fluids, Vol. 5, No. 9, 1962, pp. 1022–​1026. Unfer, T. and Boeuf, J.P., Modeling of a nanosecond surface discharge actuator, J. Phys. D Appl. Phys. Vol. 42, 2009, pp. 194017-​1-​12.

25

7

Plasma and Magnetic Field Generation

Introduction Plasma and magnetic field generation for engineering application are addressed in this chapter, and the focus is put mostly on laboratory and explorative conditions for research and development activities. Under this premise, the ionization by electron impact is the most important plasma generation process and is also the most studied procedure. The illuminating and scholarly effort by Raizer (1991) has established a solid foundation in classifying the type of discharges by a unique relationship between electric current and voltage in different ambient conditions. The discharge structure and its characteristics have been detailed and highlighted in specific applications from the near vacuum to atmosphere conditions. The discharge electric current is usually operating in the mille-​ampere per square centimeter range by voltage from few hundred to tens of kilovolts except before the discharge transition to arc. The power supply to the gas medium is often limited to less than a few hundred watts and the generated electromagnetic force is around one Newton per cubic meter; therefore, the electromagnetic effect must be amplified either by an ensuing fluid dynamic interaction or by an external applied magnetic field to be effective for engineering applications. A  special emphasis of our discussion is focused on the most recent applications of discharge by an altering electric current in the form of dielectric barrier discharge. This newly acquired information will be infused into the basic knowledge for flow control and plasma micro jet, as well as a possible extension applied to plasma-​assisted combustion and combustion stability enhancements. Among traditional techniques for plasma generation, shock tube is a principal tool for studying nonequilibrium high-​enthalpy chemical kinetics by experimental observation at extremely high pressure and the enthalpy condition. The hypersonic flow condition is reproduced by rapid expansion and the ensuing reflected compression shock waves in an enclosed environment. The thermodynamic state of gas medium reaches the necessary condition for ionization as well as for initiation of nuclear chain reaction. The shock tube actually provides a renewed incentive for plasma dynamic research; the Maxwell stress has been known for years through theoretic work, but the effect and existence of its normal stress component is observed experimentally by the work of Ziemer (1959). In fact, the first convincing demonstration of a notable electromagnetic phenomenon was performed by him. He used

27

226

Plasma and Magnetic Field Generation

a shock tube to generate a partially ionized gas flow past a magnetized hemisphere-​ nose model to display for the first time the magnetic pressure that displaces the enveloping bow shock wave upstream. It is a consequence of the sum of magnetic and hydrodynamic pressure in the shock layer reducing the local gas density. In order to satisfy the continuity condition across the shock, the stand-​off distance of the bow shock must increase to accommodate the mass flow through the lower shock layer density by the increasing magnetic field strength. Today the shock tube facility can achieve a stagnation pressure and temperature of 7,000 mPa and 11,000 K with a stagnation enthalpy at 26 MJ/​kg (Lu, Liu, and Wilson 2005). However, the extremely high thermodynamic condition for gas mixture can be sustained only up to 10 microseconds, but the plasma generation processes are even shorter in the nanoseconds time frame. The available testing duration shall provide a sufficient time for data collection and evaluations. The magnetohydrodynamic channel, also known as the MHD electric generator, is an electromagnetic device developed for electricity generation. It is designed for converting thermal and kinetic energy directly into electricity. The ionization is enhanced by using seeding materials; the electric energy extraction from a heated gas mixture is through a transverse magnetic field by the Hall or the Faraday current. The quality of the generated plasma is not outstanding because the corrosive seeding material is included, usually with potassium or alkali, at an operational temperature less than 3,000 K. The electric conductivity of the generated plasma rarely exceeds 10 Ohm/​m and an electric current density of few Ampere/​cm2. Under these conditions, the magnetic Reynolds number is in the order less than ten under most operational conditions. In this type of electric generator, the most effective energy transformation device is the so-​called disc generator, and the magnetic field is generated by a pair of Helmholtz coils across a disc channel to become a combined Hall and Faraday generator. In all, the maximum electric power generation capacity is known over 100 megawatts with peak efficiency about 30.2 percent. The plasmatron produces the low-​temperature and density plasma at the steady states, including the arc and radio frequency (RF) induction in the spectrum between 1 and 3 MHz or at a wavelength from 1 mm to 1 m. The applied microwave system is thus in the range from 300 MHz to 3,000 MHz. The plasma-​generated power varies from a few hundred watts to thousands of kilowatts; the most common application belongs to the plasma torch. The arc torch uses the metal to process as the anode; the ionization processes are similar to both the arc and RF plasmatron, but the ionization is completed by the inductively coupled plasma torch. The plasma generator must be cooled by a tangential injected gas in a helical motion, because the temperature along the axis can reach as high as 10,000 K. The microwave plasmatron is the most efficient gas-​ionizing process but is also the most difficult to control, although it is similar to the waveguide discharge scheme and the key issue is rested on the antenna design and subjected to the challenges of discharge instability. Remote energy deposition by microwave is built on a focused microwave beam; the energy conversion process is carried out by concentrating a single frequency or a broad-​band microwave into a small space domain. The energy conversion

27

7.1  Direct Current Discharge

227

is basically a reverse bremsstrahlung radiation. First, the microwave energy is released to electrons by multiphoton ionization and electrons absorb the microwave radiation through an avalanching process (Raizer 1966; Morgan 1974). The energy conversion process is unstable because at the plasma-​focused point there is a sequence of shedding vortices from the high-​energy release region that becomes a strong plasma-​aerodynamics interacting phenomenon. In practical applications, the remote energy is released in pulsating mode at a frequency of 10 kHz and the energy transmitted per pulse is limited to around a few hundred mille-​Joules (Adelgren et al. 2005). As we have mentioned previously, the strengthened electromagnetic effects for application require an externally applied magnetic field. The magnetic field plays a very important role in applying plasma for flow control for two reasons: because the Lorentz force always has a component, σ( E + u × B ) × B , which retards the velocity normal to the applied magnetic field (Resler and Sears 1958). The strong magnetic field is also constrained by the charged particles along the magnetic lines, thus a transverse magnetic field can change the flow direction of an oncoming stream without a solid surface deflection. This phenomenon is unique to the electromagnetic field, which gives a wide berth for innovations by manipulating the charged particle motion for supersonic flow control; the flow deflected by the electromagnetic force triggers expansion or compression waves. In laboratory experiments, the magnetic fields have been generated by permanent magnets by the compact neodymium rare earth or NdFeD magnets. On the pole surface, the magnetic flux density has been measured as high as 0.47 Tesla, but all the magnets have a Curie temperature limitation; beyond the maximum allowable temperature, the magnetic flux density diminishes rapidly. Traditionally, the magnetic induction coils are widely used; the known maximum magnetic flux density is as high as 7.0 Tesla across the magnetic poles, but its weight limitation and high electric current requirement, at present, is limited to ground-​based facilities. However, the super-​conducting material research will eliminate this restriction. The detailed and practical plasma and magnetic field generation processes are described in this chapter. For computational modeling and simulations, the governing equations and their associated initial and boundary condition are presented and delineated. The experimental setups are also presented for laboratory plasma generation, including implementations and data collection.

7.1

Direct Current Discharge The gas discharge is initiated by secondary emission after the electric breakdown between electrode gaps and is sustained by a small current. The main mechanism was discovered by Penning in 1928: the accumulated field of positively charged ions on the cathode creates a thin potential barrier by strong electric field intensity on the order of nuclear binding. Electrons from the cathode, either by a conductor or by dielectrics, immediately tunnel into the ion layer to neutralize it. The ion-​electron

29

228

Plasma and Magnetic Field Generation

Table 7.1  Ionization potential of selected gases E/​p (V/​cm-​Torr)

E/​p (kV/​cm-​atm)

Ar

3.6

2.7

He

13

10

H2

26

20

N2

46

35

O2

40

30

Air

42

32

emission has been characterized by a coefficient with an empirical formula and its magnitude is strongly dependent on the surface material but relatively insensitive to the externally applied electric field intensity. The maximum value for the secondary emission of 0.21 is found for tungsten, and for platinum the coefficient is as low as 0.005. The tunneling action leads to an electron secondary emission from the cathode; the process transforms a dielectric medium between electrodes into a conductor under a sufficiently strong electric field. The process is rapid and usually takes place at time scales from 10–​4 to 10–​9 second. The electric breakdown is a transformation process from a dielectric medium between electrodes into a conductor. The electron avalanche is the next essential step for the electric breakdown but makes the process very complicated because it acts like a nuclear chain reaction. An electron transfers energy to the molecule and ionizes it to produce more electrons. It then starts the chain of ionization and drifts systematically, receiving energy from an electric field. The breakdown has a distinct threshold behavior beyond a specific electric field strength known as the ionization potential; above this value the ionization will dramatically increase. Raizer (1991) has developed an empirical expression, based on the theory of Engel and Steenbeck (1932), for the ionization potential as a function of the ambient pressure and electrode gap distance between electrodes for the discharge.

ϕi = B ( pd ) [C + ln( pd )] (7.1) The product pd of ambient pressure p and the electrodes’ gap distance d is a manifestation of a similar law made evident through Paschen curves. The peak values of the ionization potential for some gases at atmosphere pressure have been experimentally measured  as  exhibited by Table 7.1. The optimal condition for ionization is defined as the energy needed to create one pair of ions and electrons known as the Stotetov constant. For air, the constant is 66 eV per pair of ions and electrons, and the reduced electric field potential has been interpreted as the energy transfer between collisions. Another outstanding phenomenon of the discharge is the pattern of light emission to appear as stratification into dark and bright luminous layers. The pattern is particularly pronounced at the low-​pressure environment below 0.1 Torr;

29

7.1  Direct Current Discharge

229

Figure 7.1  Cathode glows of abnormal discharge at p = 10 Torr, I = 300 mA & 600 mA.

the stratification consists of Aston dark space, cathode glow, cathode dark space, negative glow, and Faraday dark space over the cathode. Beyond these layers are the positive column, anode dark space, and anode glow next to the anode. When the ambient pressure is increased up to 10 Torr or more where most engineering applications occur, the cathode layer dominates and the cathode glow becomes very bright. The cathode glows of abnormal discharge between parallel electrodes under the ambient pressure of 10.0 Torr are displayed in Figure 7.1. The electric current flow between electrodes can reach a value of 600 mA by an applied electric potential up to 500 Volts at the lower ambient pressure condition. Ionization by electron impact is the most important mechanism for plasma generation for engineering applications. The electron density is limited to a maximum value not to exceed 1014 number of particles per cubic centimeter (n = 1014 cm 3 or n = 1020 m 3 ). Thus the degree of ionization is rather low in view of the dry air number density at the standard condition of 2.504 × 1019 cm3 . In general the plasma is not in thermodynamic equilibrium, that is, the excited electron usually achieves a temperature around a few electron volts, but the ion retains the temperature of the ambient condition. The rather weak electrostatic force by charge separation is acting normally on the electrode in the cathode layer, and the Joule heating is also concentrated mostly over the cathode within the cathode layer. However, the heat transfer mechanism is vastly different from the conductive electrode heating, because the energy releases directly to the gas medium adjacent to the solid surface. Therefore Joule heating becomes the principal electromagnetic effect for the direct current discharge (DCD). The rate of discharge based on kinetics theory is proportional to the product of ionization frequency and electron number density:

dne dt = νi ne (7.1a) As always, the temperature of the electron-​impact gas discharge is substantially lower than the ionization potential. For an example, at the ambient pressure of 50 Torr, temperature of 300 K, and electron number density of 1.7 × 1013 cm 3 the ionization collision frequency is around 510 s. The electron temperature can be as high as 11,600 K or more, but the temperature of ions remains the same as the ambient condition.

230

Plasma and Magnetic Field Generation

For DCD, the energy distribution, the mean electron energy, and the drift velocity of the electrons are all functions of the E/​P ratio, known as the reduced electric potential. Townsend’s similarity law for discharge is actually an empirical formula and widely used:

dne dt = Ap exp[ − B ( p E )] (7.1b) where the parameters A = 12, B = 342, A = 15, and B = 365 for nitrogen and air, respectively. These values are determined from experimental data in the E/​ P range from 100 to 800. Townsend’s similarity law has been adopted in a wide range of plasma computational simulations (Surzhikov and Shang 2004; Shang 2016). The electric field breakdown at high pressure or in a high-​density environment is identical to that occurring in the low-​pressure environment. However, it is not feasible to determine the electrode gap distance from the Paschen curve at high pressure, because the minimum breakdown electrode gap distance is too small to measure. For example, the breakdown field strength for atmospheric air is already in the range of 33 kV/​cm, and the breakdown voltage is more than 2 kV to give a very thin gap distance. Breakdown at high pressure in a nonuniform field is markedly dependent on the geometry of the electrodes and on their polarity for corona discharges (Raizer 1991; Moreau 2007). Corona discharge takes place in a highly nonuniform electric field, causing the much smaller characteristic electrode dimension r than the gap distance d between electrodes. For parallel-​wires electrode arrangements, the limit ratio is d/​r > 5.83, when the ratio is greater than that, the discharge produces a spark and transits from the corona discharge. When a high voltage is applied to the cathode, the discharge is a negative or anode-​directed corona; otherwise it is a positive or cathode-​directed corona. Although the corona is basically a Townsend charge, the sustaining mechanisms for the continuous ionization depend on the polarity of its electrodes. The maximum electrical field intensity can be estimated by:



E max = (V 2r ) ln( d r ) (7.1c) In principle, the ionization of the negative corona does not differ from the Townsend breakdown. However, the negative corona generated by a needle-​to-​plate electrode arrangement is associated with a periodic discharge pattern known as Trichel pulses, which have a repetition rate greater than in positive coronas. These pulses are connected with streamers starting from the pointed electrode toward the flat counterpart with a repetition frequency of about 1 kHz, and the pulsation is easily observed from an Oscillogram. When the voltage is increased further, the pulses vanish and a steady-​state corona will be sustained until the spark arc breaks down within the discharge gap. For multiple positive corona discharges, the electron reproduction is by the so-​ called secondary photo process, and displays a luminous filament emitting from the electrode as streamers or micro discharges. The streamer is a moderately weak ionized channel formed by the primary avalanche in a sufficiently strong electric

231

7.1  Direct Current Discharge

231

field between electrodes. Again in the case of a positive corona, the electrons are mostly produced by photo ionization and lead to an avalanche. The ionization zone has a dimension of a few tenths of a millimeter and displays luminous filaments from the anode. The streamer is conveyed by ion drift to have a charge number density of 109 to 1013/​cm3. When the current or voltage exceeds a certain threshold, there will be a corona-​to-​arc transition. The cathode emission of the arc is different from the normal discharge in which the cathode fall is significantly shallow to have a small value of less than 10 V, in contrast to several hundred volts of a normal direct current discharge. The basic mechanism is also a complex combination of thermionic and field emissions. An arc has a large electric current up to 105 Amperes and the current density has a range from 102 to 104 Amp/​cm2. The high electric current is sustained at very low electric potential around 30 Volts, because of the vaporized electrode material, which substantially increases the electric conductivity. The consequence of eroding and vaporizing the electrodes by arcing has severely limited it for engineering applications. The specific ionization rate has been calculated equally effectively by the chemical kinetic process (Section 6.3) (Elisson and Kogelschatz 1991; Solov’ev et  al. 2008) and the inelastic collision model (Section 6.5) (Surzhikov and Shang 2004). For the chemical kinetic formulation, the ionization falls into four groups:  the electron-​ molecule, the atom-​ molecule, the decomposition, and the synthesis process. Chemical kinetics models for low-​temperature plasma generation span a range of complexity (Shang 2016). The rate of plasma production and depletion is based on the Arrhenius formula through the finite-​rate chemical reaction. The most significant species of the chemical kinetic model for air are:  O2, O3, N2O, NO, N(4S), O(3P), N 2 ( A3 Σ u+ ) , O2 ( A1 ∆ ) , O*3, O2 ( B1Π ), N(2D), and O(1D). Even more elaborate chemical kinetic models have considered more than one hundred species, including the metastable species such as N 2 ( A3 Σ u+ ) and O2 ( B1Σ d+ ) . However, the physical fidelity of chemical kinetic models for electron impact ionization completely depends on the chemical kinetic models. The verification of the calculated results by comparing them with experimental observations is extremely limited. The plasma model by the inelastic collision model is much simpler by restricting the plasma to four components: neural particles, electrons, and positively and negatively charged ions, as discussed in earlier chapters. The major processes in consideration for inelastic collisions are the ionization, dissociation recombination, ion-​ion recombination, electron attachment, and electron de-​attachment. These physical processes have been modeled reasonably well by the kinetic models for direct current discharge (DCD) and dielectric barrier discharge (DBD) (Surzhikov and Shang 2004; Shang and Huang 2010). The first DCD computational modeling simulation was introduced by Surzhikov and Shang (2004) for electrons and ions in the presence of a transverse magnetic field along the z-​coordinate. The formulation is based on the combination of the drift-​diffusion theory and the inelastic collision ionization models:

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Plasma and Magnetic Field Generation

Figure 7.2  Direct current discharge between parallel electrodes, p = 3 & 10 Torr.



∂ne ∂Γ e ,x ∂Γ e , y = α( E , p ) | Γ e | −βn+ ne (7.2a) + + ∂y ∂t ∂x



∂ne ∂Γ + ,x ∂Γ + , y = α( E , p ) | Γ e | −βn+ ne (7.2b) + + ∂y ∂t ∂x



∂2 ϕ ∂2 ϕ = 4 π( ne − n+ ) (7.2c) + ∂x 2 ∂y 2 The required boundary conditions are the electron flux number density for the secondary emission on the cathode and the vanishing value on the anode. All flux number density must vanish asymptotically at the far field.



( Γ e )c = γ ( Γ + )c (7.2d) ( Γ e )a = 0 The computational simulations DCD between parallel electrodes using the inelastic model are displayed in Figure 7.2 (Shang et al. 2009). The simulated discharge structures show a strong dependence on the ambient pressures from 3.0 and 10.0 Torr. The gap distance between electrodes is 20.0  mm and the discharge is maintained by an electric field potential of 2.0 kV. The width of the cathode layer shrinks from 26.6 to 5.8 mm and the positive discharge columns are also reduced accordingly as the ambient pressure is elevated, which reflects physics correctly and yields the experimental trend described by Equation (7.1b). More important, the

23

7.1  Direct Current Discharge

233

Figure 7.3  Effects of direct current discharge voltage in transverse magnetic field.

predicted electric current across the electrode per unit area of the electrode is 4.88 mA, which compares well with the classic result by von Engel and Steenbeck (1932) of 4.85 mA (Surzhikov and Shang 2004). In view of the low current density generated by DCD, the effect of the Joule heating for flow control must be amplified either by a subsequent viscous-​inviscid aerodynamics interaction or by augmentation with an externally applied magnetic field (Shang 2005). The effects of an applied transverse magnetic field on DCD generation are presented by the experimental observations by Kimmel, Hays, Menart, and Shang (2006) and depicted in Figure 7.3. The data are collected from a supersonic plasma channel, M  =  5.25, and at a Reynolds number of 9.4 × 10 4 for the fixed discharge electric current densities of 10.0 and 25.0 mA. The current data are collected under uniform but different transverse magnetic field strengths from minus to positive polarities of one tesla; B = ±1.0 T . The DCD is generated between the side-​by-​side electrodes arrangement; both the electrodes have a dimension of 31.8×6.4 mm and are separated by a distance of 38.1 mm. At the lower ambient pressure of 2.02 Torr, the constant discharge electric current densities of 10.0 and 25.0 mA can be maintained in the externally applied transverse magnetic field of both polarities with a lower voltage than without an applied magnetic field. Thus, the data indicate a higher DCD current density can be obtained with a weak applied transverse magnetic field. However, the difference diminishes at the higher ambient pressure condition of 3.52 Torr and the discharge

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Plasma and Magnetic Field Generation

instability occurs, which reflects the known fact that the discharge becomes unstable with an electric current. For the magnetic field of the negative polarity, the suppressing Lorentz force − u × B even prohibits the ignition of DCD. Additional research deems necessary to ascertain these experimental observations. In summary, the DCD for aerospace applications for flow control is mainly based on Joule heating. In applications for flow control, the electromagnetic–​ aerodynamic interaction is derived from the thermal energy release to decrease the local density, which in turn increases the displacement thickness of the boundary layer. The outward displaced boundary layer deflects the external flow onto the oncoming supersonic or hypersonic streams to trigger compression waves. The electromagnetic–​aerodynamic interaction has been successfully demonstrated to be used as a virtual control surface. However, the effective DCD flow control can only take place at a lower ambient condition, thus its operational regime is restricted to the stratosphere and mesosphere to become only a possible special-​purpose application (Shang 2016).

7.2

Dielectric Barrier Discharge The dielectric barrier discharge (DBD) is just another electron impact ionization procedure, but it operates by an alternative electric current and has a unique self-​ limiting feature that prevents its transition to arc at atmospheric pressure. DBD has been known in ozone production as the silent discharge since the middle of the nineteenth century. Recently Corke, Enloe, and Wilkinson (2010) demonstrated a promising approach for flow control to delay aerodynamic bifurcation for dynamic stall at low Reynolds number condition. In applications, a DBD can operate under atmospheric pressure with a gap distance between electrodes up to a few centimeters, but the electrodes are always separated by a thin dielectric film up to few micron-​meters in thickness. At a large combined value range of ambient pressure and gap distance (pd), the DBD operates in a streamer mode. An alternating voltage is required to support the random transient streamer formations in the electrode gap and quenching by a localized charge build-​up on the dielectric surface. In essence, the DBD discharges are operated in the micro discharge mode where the streamer discharge’s lifetime is governed by the gas medium, electric field intensity, and capacitance of the dielectric barrier. An important feature of plasma generation by a DBD is that the low-​pressure restriction imposed by direct current discharges is removed. A schematic of DBD and a typical electric field contour are shown in Figure 7.4. Two electrodes are placed side by side; one of the electrodes is exposed and the other is encapsulated beneath a thin dielectric film of a few microns in thickness. For flow control, the exposed-​embedded electrodes are aligned in the direction to coincide with the induced so-​called electric wind (Enloe et  al. 2004; Corke et  al. 2010), which is just a wall-​jet stream formed by collisions between ions and neutral particles. The electrode arrangements are usually slightly overlapped, but often can

235

7.2  Dielectric Barrier Discharge

235

Figure 7.4  Schematic of DBD electrode arrangement and computed electric field contour.

be separated by a gap distance of a few millimeters. DBD is powered by an AC source in the microwave frequency spectrum (from 3.0 up to 12.0 kHz) and by an electric potential in the tens of kilovolts range. The adopted dielectric films usually are polystyrene or other isolators and have a value of electric permittivity greater than 7.0. The accompanied instantaneous electric field contours are also included in the graphic presentation just to show that the overall electric field is symmetrical with respect to the juncture of the electrodes, but the symmetric property does not exist over the small discharge domain where the charge separation occurred. In an AC cycle, when the externally applied electric field intensity exceeds the electric breakdown voltage threshold, the plasma will ignite. Ignition depends on the gas medium, the electrode gap distance, and the electric permittivity of the dielectric. In dry air, the breakdown voltage of a typical DBD has a value slightly above 2.0 kV. After the breakdown, a conductive current appears between electrodes, which is contributed by multiple micro discharges or streamers randomly in space and time. The electric field intensity across the electrode and the dielectric that is dropped from the electromotive force (EMF) of the external source by charge accumulation over electrodes usually retains nearly a constant value because of the self-​ limiting characteristic of a DBD, which prevents the discharge from transitioning to an arc. This unique feature has been mentioned earlier and is further delineated in the following discussion. The discharge ceases when the applied voltage falls below the breakdown value, and the voltage–​current relationship returns to the applied electric AC field. The phenomenon repeats itself as soon as the absolute magnitude of the externally applied voltage is greater than the breakdown value in both phases of positive and negative polarities. This voltage–​current relationship is routinely recorded by experimental observations and computational simulations with physics-​based plasma models (Shang and Huang 2010; Shang 2015). The cyclic voltage and current relationship is well known and thus will not be repeated here. At atmospheric pressure, the streamer, or the micro discharge, is a weakly ionized channel of the primary avalanche in a strong electric field. The space charges induce an additional electric field potential, which makes it different from the corona discharge. For DBD, one of the electrodes is encapsulated in dielectric film and the

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dielectric serves two purposes: it limits the amount of charged transport by a single micro discharge, and it distributes the micro discharges over the entire electrode surface. The displacement current can pass through the dielectric film by an alternative electric potential source, thus the circuit is closed and the required voltage is in the range from 3 to 20 kV in AC cycles of the microwave range. However, the most outstanding feature of DBD is the accumulated charges over the dielectrics that effectively reduce the electric field intensity that provides the self-​limiting behavior from transition to spark. The governing equations for DBD computational simulations again consist of two distinctive approaches via the chemical kinetic models (Elisson and Kogelschatz 1991; Solov’ev et al. 2008), discussed in Chapter 6.3. The other formulation is based on the combination of the classic drift-​diffusion theory and inelastic kinetic models (Surzhikov and Shang 2004; Boeuf and Pitchford 2005; Shang, Roveda, and Huang 2011). The detailed formulation has been given in Chapter 6.5. The self-​limiting characteristic preventing DBD transition to arc is illustrated by the periodic boundary conditions of the Maxwell equations over the plasma and electrode/​dielectric interface. In the positive polarity ϕ(t ) > 0 AC cycle, the exposed metallic electrode acts as an anode. According to the electric circuit law, the electric potential across the electrode gap is the difference between the externally applied EMF and the voltage drop over the discharge current achieved by satisfying the electric circuit equation, Chapter 3.4. The boundary conditions on the electrodes are:

ϕ(t ) = EMF sin( ωt ) − R ∫ e( Γ + − Γ − − Γ e )dv (7.3a)



  n ⋅ ( ∇ϕ ) p = n ⋅ ( ε d ε p )( ∇ϕ )d + ( e ε p )∫ ( n+ − ne − n− )d ∆ (7.3b) In Equation (7.3b), the electric current density and surface charge are denoted by J = e( Γ + − Γ − − Γ e ) and qe = e( n+ − n− − ne ), respectively. It is clearly indicated that the electric field intensity in the discharge domain is reduced by the surface charges’ accumulation over the dielectric, the second term of the right-​hand side of Equation (7.3b), and the difference in electric permittivity between plasma and dielectric (ε p and ε d ). During the negative polarity ϕ(t ) < 0 AC cycle, the role of the electrode and the dielectric surface reverses; now the exposed electrode becomes the cathode. The identical plasma and dielectric interface boundary condition applies for the voltage drop across the discharge: Equations (7.3a) and (7.3b). The charge accumulation on the dielectric is the same as Equation (7.3b), even if the encapsulated metallic electrode is grounded. A previously unnoticed feature of DBD is that the micro discharges have distinct characteristics within a complete AC cycle (Enloe et al. 2004; Enloe, Mcharg, and McLaughlin 2008). The micro discharges are initiated from electrodes similar to either the positive or the negative corona discharges. The avalanche transforms the discharge into a streamer before it reaches the receiving electrode. The negative streamers propagate against the direction of the electron drift velocity, thus they

237

7.2  Dielectric Barrier Discharge

237

Figure 7.5  Micro discharges or streamers of DBD, Enloe et al. 2004.

tend to be more diffusive than the positive streamer. On the other hand, positive streamers require traveling a longer distance to reach the cathode. This distinct behavior, displayed in Figure  7.5, has been captured by the high-​speed, image-​ intensified photographs in each phase of the DBD by Enloe and colleagues (2004). The nonequilibrium chemical composition associated with the DBD is well known and is thoroughly studied by treating 143 reactions among thirty reacting species (Elisson and Kogelschatz 1991). The complex chemical kinetics of a weakly ionized gas is fully recognized, but the most profound feature for aerodynamic applications is the transport property of the ionized gas. In spite of the long history of DBD applications, the critically important electromagnetic field parameter for electromagnetic–​aerodynamic interactions is still largely uncertain, especially the quantification of momentum transfer between charged and neutral particles. Nevertheless, the electromagnetic–​aerodynamic interaction through elastic collisions between ions and neutral particles produces a wall-​jet-​like air stream over the dielectric. The jet stream is often referred to as the electric wind of a DBD and is located within the plasma sheath. The maximum speed of the wall jet can be as high as a few meters per second. This energetic wall jet has been shown to be very effective for flow control. There is a very limited amount of basic knowledge on the interaction of the charged and neutral particles via collisional momentum exchanges in an electromagnetic field. But the source of the driving force is completely understood from charge separation over electrodes, which has played a dominant role for DBD applications to flow control. In dry air, the breakdown voltage of a typical DBD for flow control has a value slightly above 2.0 kV. After the breakdown, a conductive current appears between electrodes by the electrons’ motion. The electric field intensity across the electrode and the dielectric drops from the EMF of the external source, but retains nearly a constant value because of the self-​limiting characteristics of a DBD, which prevent the discharge from transitioning to an arc. The discharge ceases when the applied voltage falls below the breakdown value, and the voltage–​current relationship returns to the externally applied AC field. The phenomenon repeats itself as

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Plasma and Magnetic Field Generation

Figure 7.6  Electrical voltage and current data for DBD.

soon as the absolute magnitude of the externally applied voltage is greater than the breakdown value in both positive and negative polarities. Figure 7.6 depicts the detailed composition of the electrical-​current components during a complete AC cycle. The solid line denotes the constant displacement electrical current within the AC circuit operating over a spectrum upward from 5.0 kHz. The magnitude of the displacement electric current, ∂D ∂t is the rate of change for the electric displacement with respect to time. Thus the voltage–​current relationship traces a straight line with a constant slope. When the measured total electric current departs from the straight line, it is an indication that breakdown occurs. At the breakdown voltage, the magnitude of the conductive current is extremely small, as shown by the results of computational simulations. Therefore current is not an accurate criterion to determine the exact breakdown voltage, but presents irrefutable evidence of the DBD ignition. The integrated conductive electrical current through plasma is less than a few mille-​amperes per unit area and is comparable to the magnitude of a direct current discharge. The DBD operates by an AC voltage up to 20 kV and in the microwave spectrum, but most known experimental efforts are operated around 10 kV. At an externally applied electrical potential of 4.0 kV across overlapping electrodes, the maximum total power imparted to plasma is merely 0.60 watt. At the higher applied AC voltage of 10 kV with a 1.0  mm thick dielectric film of the electric permittivity of 6.7 (rubber), the conductive electric current has only a small magnitude up to

239

7.2  Dielectric Barrier Discharge

239

Figure 7.7  Instantaneous and time-​average force vectors over dielectrics of DBD.

6.0 mA. Therefore the magnitude of the electromagnetic effect generated by a single DBD is rather limited, and the most outstanding phenomenon of DBD is the so-​ called electric wind, which is produced by charge separation and is deeply imbedded within the plasma sheath. In addition, the periodic electrostatic forces within the positive and negative phases of an AC cycle are opposite to each other, even if the instantaneous force during the discharge has a value as high as 2.46 N/​cm3 (Beouf et al. 2007; Shang et al. 2011). When the instantaneous force is integrated over the complete AC cycle, the net force is reduced to about 2.4 × 10 −3 N cm3 . The selected instantaneous and periodic electrostatic force in vector plots and the time-​average force parallel to the dielectric surface due to charge separation are given in Figure 7.7. The computational simulations are obtained by using the multiple-​ fluid plasma formulation based on the drift-​diffusion theory and the inelastic collision ionization model (Surzhikov and Shang 2004; Huang, Shang, and Stanfield 2011). Note that all the electrostatic forces exist in the plasma sheath region that has the vertical dimension less than the exposed electrode thickness: d = 1.0 mm. Since the DBD is a periodic event, only the peak values of the changing polarities are selected when the externally applied voltages are at ±8.0 kV . When the exposed electrode acts as an anode in the positive polarity, ϕ = +8.0 kV , the positively charged ions are expulsed from the anode moving toward the dielectrics and colliding with the neutral species. A motion is created by the periodic force from the exposed electrode toward the dielectric. When the polarity is reversed, ϕ = −8.0 kV , the exposed electrode acts as a cathode; the ions’ movement is reversed and also generates neutral particles’ motion from dielectrics to the exposed electrode. So the periodic forces are opposing each other during an AC cycle. The time-​average force over the entire period and the complete discharge domain is included at a frequency of 5.0 kHz and is presented in the right-​most panels to show a small net counteracting electrostatic force from DBD. The magnitude of the time-​average force is merely around a thousandth of Newton per cubic centimeter. The calculated forces agree

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Plasma and Magnetic Field Generation

in the same order of magnitude by all computational simulations, but cannot be verified with experimental data. Nevertheless, the small time-​average value is the driving force for the so-​called electric wind. From these experimental observations and computational simulations, the magnitude of the electromagnetic effect generated by a single DBD is rather limited and the most outstanding phenomenon of DBD is the so-​called electric wind that is deeply imbedded within the plasma sheath. Therefore DBD can only be very effective for flow control at the fluid-​dynamic bifurcation because it affects the innermost structure of the shear layer. The electromagnetic–​fluid dynamic interaction can be amplified by multiple implementations of DBD, or further amplified by fluid dynamic inviscid–​viscous interaction or by an externally applied magnetic field. The amplified effects may be necessary for wide-​range practical applications.

7.3 Shock Tubes A unique feature of a shock tube is its design to store energy over a period of time and to release the accumulated energy suddenly; the feasibility of storing and releasing huge amounts of energy at an instance have no other viable alternative. Therefore a shock tube is an impulse experimental facility, in direct contrast to continuous operating ground-​based wind tunnels. The original function of a shock tube is investigation by abrupt changes of thermodynamic conditions at the shock front to study high-​speed transient aerodynamic phenomena in a jolting dynamic and thermal event. At the high-​enthalpies condition, it offers opportunities for analyzing dissociation and ionization. More recently, the shock tube has been used as a plasma generator and as a device to study radiative energy transfer, as well as a magnetohydrodynamic accelerator. In this conjunction, pressure and temperature rise have been augmented from magnetic compression by Lorentz acceleration (Zel’dovich and Raizer 2002). Through this arrangement the generated plasma in a shock tube is further compressed by the so-​called magnetic piston through a vertical chamber connected to the convenient shock tube. The compression shock wave is stronger than that in the absence of a magnetic field, but the gain also increases the complexity for evaluation. The flow field structure for a traditional shock tube is complex because the flow is three-​dimensional and the partition diaphragm bursting between the driven and the low-​pressure section is highly irregular and takes place in a finite time frame. Viscous effects of the gas mixture must be considered by the formation of a boundary layer over the shock tube surface; the relaxation of real gas phenomenon also trails the moving shock. If the reflection shock wave from the end wall is desired to gain even higher gas thermodynamic conditions, then the flow field structure is nearly impossible to analyze accurately. However, a simplified formulation by inviscid theory has been developed over the years for analyzing the shock tube performance, which still can provide a reasonable estimation of the key parameters of the shock tube.

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7.3  Shock Tubes

T4

T2 T

241

T1

T3 p4

P2=P3 p

P1

Contact surface

Rarefaction waves

Shock wave t

X Figure 7.8  Shock tube waves diagram.

The simplified fluid motion described by the inviscid theory in one dimensional time-​space is presented by Figure 7.8. The basic parameter of the shock tube is the diaphragm pressure ratio p4/​p1 separating the high-​and low-​pressure sections of a shock tube. When the diaphragm ruptures, a compression shock wave propagates to the low-​pressure section of the tube while a rarefaction wave expands toward the high-​pressure section. The interface between the two sections is the contact surface, which is the original partition diaphragm location of the two gas sections. From the inviscid theory, the contact surface permanently separates the high-​and low-​pressure domains. The temperatures and densities of the gas medium across the contact may be different, but the pressures and the velocities on each side of the contact surface must be identical. The compression shock moves at the shock speed at the wave front, us at a hypersonic Mach number and follows a highly compressed gas (p2, T2) with a gas-​specific heat ratio γ 2 . Immediately following the shock and in this domain, region 2, gas is ionized. During this period, the thermodynamic state of the gas is dependent upon the shock speed and the length of the low-​pressure section of the shock tube, and it usually lasts only a few microseconds. When the shock wave reaches the end wall, the reflected wave increases the pressure and temperature of the plasma further, but the viscous–​inviscid interaction involves a complex vortical structure and a transition to turbulence. At this stage in time, the flow field within the compression section of the shock tube is beyond the reach of the simple inviscid analysis. On

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the expansion section of the shock tube, the rarefaction wave motion constitutes a fan of straight lines of constant propagation speed ue. These lines are equally interesting, but they do not directly associate with plasma generation; therefore, they are not included in our discussion. From these physics phenomena, the basic shock tube equations are obtained from the Rankine-​Hugoniot shock condition (Liepmann and Roshko 1958) to determine the pressure and temperature rises and the wave propagation speeds:

p 2 p1 = ( p4 p1 ) {1 − [( γ 4 − 1)(c1 c4 )( p2 p1 − 1)] [ 2 γ 1 2 γ 1 + ( γ 1 + 1)( p2 p1 − 1) ]}2 γ 4

( γ 4 −1)



(7.5a)



T2 T1 = [1 + ( γ 1 − 1) ( γ 1 + 1)( p2 p1 ) [1 + ( γ 1 − 1) ( γ 1 + 1)( p1 p2 )] (7.5b)



us = c1 ( p2 p1 − 1) 2 γ 1 [( γ 1 + 1)( p2 p1 ) + ( γ 1 − 1)] (7.5c)



ue = 2c4 ( γ 4 − 1)[1 − ( p2 p4 )( γ 4 −1) 2 γ 4 ] (7.5d) The thermodynamics states for plasma generation are fully described by Equations (7.5a) and (7.5b). The specific heat ratios γ 1 and γ 4 of the driven gas and test medium can be found from the real gas tables at the initial conditions. The pressure and temperature in the compression section can be estimated reasonably accurately with the known diaphragm pressure ratio and the composition of gas media. Finally, the shock tube operational time for data collection purposes is obtainable from the dimensions of the shock tube, the shock propagating speed us, and the contact surface propagation speed ue. These wave propagation speeds are provided by Equations (7.5c) and (7.7d). More recent shock tube developments have been improved by the ideas of free piston, detonation technology, and magnetic compression to achieve even higher spontaneous enthalpy for the ionized gas. The energy content of the compressed gas in the shock tube is measured by the stagnation enthalpy of the gas medium; at a temperature of 8,000 K, it has attained an enthalpy value as high as 15.6 mJ/​ kg. In fact, the typical energy supplied to the shock tube is around 20 to 25 J for an average duration of 60 microseconds, but the electric conductivity of the generated plasma is rather low because of the high-​pressure condition. For examples, the maximum electric conductivity of Ziemer’s electromagnetic shock tube is recorded at 55 mho/​cm. In most experimental conditions, the electric conductivity of air increases a thousand fold or more by using seeding potassium or cesium compound (Lu et al. 2005). At the convenient shock tube operational conditions, the electric conductivity with a 1 percent Cesium carbonate seeding by weight is 423 mho/​m at one atmospheric condition, but drops to a value of 95.3 mho/​m at the pressure level of 100 atmospheres. The more recent innovative recent development for the shock tube can be identified by the hypervelocity expansion tube (Dufrene, Sharma, and Austin 2007). The gas is compressed by consecutive sections partitioned by two diaphragms; after the rupture of the second diaphragm, the gas is further accelerated and can reach a speed approaching the Mach number of 30. The gas in the hypervelocity shock

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7.4  MHD Electric Generators

243

tube attains the hypervelocity speed up to 6 km/​s and at the altitude of 100 km with an enthalpy of 12.5 mJ/​kg (Holden et al. 2007). In the test section of the hypervelocity shock tube, the thermodynamic condition duplicates the flight envelope of aerospace vehicle, including the space shuttle and the maneuvering reentry vehicles for a data collection period around 4 milliseconds. The detailed information for the ionized gas generated by the shock tube such as the degree of ionization or the charged number density depends strongly upon the gas media and the transient nonequilibrium thermodynamic states of the shock tube. The performance of a shock tube depends on its system. Another more recent progression in shock tube technology is achieved by combining shock detonation engine techniques for ionization with the shock tube operations (Litchford, Thompson, and Lineberry 2000; Lu et al. 2005). It may be clear; the primary purpose of a shock tube is not for plasma generation, but rather as a facility that can produce the environments replicating the ionized gas at extremely high speed and at a high altitude of atmosphere.

7.4

MHD Electric Generators Magnetohydrodynamic (MHD) channels or MHD electric generators were originally developed for electrical power generation. The principle of the electricity-​ generating process is identical to that of a conventional generator  –​relying on an electric conductor moving through a magnetic field. Instead, the electrically conducting working gas medium is elevated to a plasma state, and this is usually accomplished by thermal ionization via the combustion process with easily ionizable substances such as the salts of alkali metals to increase its electric conductivity. The ionized gas mixture is further accelerated by a converge-​and-​divergent nozzle to speed up the charged gas particles flowing through an externally applied transverse magnetic field so the electricity can be extracted. For analytic study and numerical simulations of MHD electric generators, the process is based on ionization by chemical kinetics as discussed in Chapter  6.4. The governing equations for analyzing the performance of a MHD generator are the coupled three-​dimensional compressible gas dynamics Equations (6.12a) through (6.12e), with Maxwell Equations (3.1d), (3.2d), (3.3d), and (3.5b). The chemical kinetics model depends on what fuel is used, a detailed description of the chemical composition is required for the complete system numerical simulation. However, in most practical applications, the products of combustion and the thermodynamic state are prescribed as the entrance condition for the MHD channel; the degree of ionization can be determined from these data. In order to make the investigation trackable, the generalized Ohm’s law is frequently adopted as a valid approximation. The operational principle is based upon the generalized Ampere’s law in that a conductor like plasma moving through a magnetic field will generate electric current. The electromagnetic force in the momentum conservation law for plasma

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Plasma and Magnetic Field Generation

motion is approximated by the Lorentz force. The pertaining basic electromagnetic equations are:

J = σ( E + u × B ) − b( J × B ) (7.6a)



F = J × B (7.6b) where b is the scalar Hall parameter, which has been introduced before, and it is defined as the ratio between the Larmor or gyro frequency and plasma collision frequency, b = ω b ν. It is essential to maintain a high electric current density so that the ratio of the Larmor frequency and the characteristic collision frequency of plasma is minimized. The electrical power delivered by an MHD generator per unit volume is P = J ⋅ E and the rate of energy that can be extracted from the plasma per unit volume is u ⋅ ( J × B ). The efficiency of a MHD generator is given as:



ηe = J ⋅ E [ u ⋅ ( J × B )] (7.6c) The electric generation efficiency of an MHD generator is essentially the ratio between the energy that can be delivered by the plasma and the possible total energy possessed in the flowing plasma. The velocity of the ionized gas motion is an important factor together with the electric current density of the generated plasma. The externally applied magnetic flux density B is equally important for possible high-​energy output of an MHD electric generator. The MHD electric generator for electricity generation is designed to convert thermal and kinetic energy into electricity. In order to increase the electric conductivity and the electric current of the flow medium, the ionization is enhanced by using seeding materials. The electric energy is extracted from a heated gas mixture through a transverse magnetic field by the Hall or the Faraday current. The common and more efficient arrangement of the Hall current than the Faraday current is placing arrays of electrodes in the plasma channel, but only loading the outermost electrodes. By a different electrode loading arrangement, the Faraday current induces a powerful magnetic field within the plasma and produces an arch-​ shaped current flow between the outer electrodes pair. However, the most effective energy transform device for extracting electricity is the so-​called disc generator. The plasma is guided to flow between two concentric circular discs and the magnetic field is generated by a pair of Helmholtz coils across a disc channel to become a combined Hall and Faraday generator. In all, the maximum electric power generation is known to be more than 100 megawatts with peak efficiency about 30.2 percent. However, the MHD generator can be an environmental hazard, because the generated plasma is contaminated by corrosive and toxic chemical elements and the exhaust gas temperature usually is around 1,000 k. Numerical simulations have been applied to assess the performance of MHD electric generators. At an operational temperature less than 3,000 K and using the seeding enhancement, the electric conductivity of the plasma can exceed 100 mho/​m and an electric current density of few ampere/​cm2. Hence, the magnetic

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7.4  MHD Electric Generators

245

Reynolds number Rm = σµuL of most applications remains within an order less than ten for most operational conditions. Under this condition, the solutions via numerical simulations are produced from either the full MHD or the low magnetic Reynolds approximations. The simulated computational results from the two different approaches have produced comparable accuracy. The heat sources of the plasma generation process have a wide range of possibilities from burning fossil fuels to the coolant from a nuclear reactor, but the most efficient process is known as the hypersonic vehicle electric power system (HVEPS) (Lineberry et al. 2006; Shang 2015). The system consists of a scramjet by combusting ethylene in oxygen and enriching the product with potassium carbonate or cesium seeding in the presence of a magnetic field. The mass flow rate of a typical HVEPS is around 0.6  kg/​s with a fuel rate less than 0.16  kg/​s. The ionized gas is operated at a pressure range from 0.20 to 4.0 atm and at a temperature around 2,400 K. The plasma is characterized to have a maximum enthalpy up to 6 × 106 J/​kg with an electric conductivity varying from 10 to 40 mho/​m. When the step-​function arrangement of a transverse magnetic field is applied with magnetic field intensity varying from 0.4 to 1.5 Tesla, the peak Hall parameter assumes a value of 1.5, and the Faraday current density yields a value of 4.0 Amp/​cm2. For the plasma flows at a speed of 2,000 m/​s, the energy density has reached a value of 2.35 × 105  J/​cm3 and the power output from the HVEPS attends a value of 80 kW. The schematic of a scramjet magnetohydrodynamic electric generator or HVEPS is presented in Figure  7.9. The fuel and high-​speed air are introduced into the scramjet inlet; after initial compression by the compression ramp and sidewalls, the fuel mixture passes through the isolator. Then the fuel is ignited in the combustion section with an injection of salts of alkali metals for enhancing the electric conductivity of the ionized gas. The plasma is accelerated by nozzle flows to a supersonic speed then through the segmented electrodes in a transverse magnetic field. The electricity is then extracted from the plasma and exhausts through the exit of the scramjet. Recently the MHD electric generator has been applied to electrically accelerate an ionized gas medium for propulsion (Gurijanov and Harsha 1996). The physical process is just reversing the function of an MHD electric generator when a voltage is applied between electrodes in that a transverse magnetic field is applied to accelerate the ionized gas. For this reason, the electromagnetic accelerator is often referred to as the Hall current accelerator. A transverse Hall current is created to interact with the cross-​flow magnetic component and yields the accelerating force. In this sense, the basic approach is a cross-​field accelerator or the Hall current accelerator. The electronic energy is converted directly into kinetic energy via the Lorentz force. The force is created by mutually perpendicular electric and magnetic fields, and the device will operate at a relatively high pressure. By this arrangement the induced magnetic field will be negligible but the Hall effect may not, thus the segmented electrodes are used to enhance the conversion efficiency. The electromagnetic accelerator operated on this arrangement is often referred to as the Hall

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Figure 7.9  Schematic of a scramjet MHD channel.

current accelerator. The Lorentz force along the axial direction is given as (Sutton and Sherman 1965): F = [ σuBz2 (1 + b2 )]( Kb2 − 1) (7.7a)



where Bz is the transverse component of the magnetic field perpendicular to the applied electric field intensity and K is known as the accelerator loading factor, K = E y uB . The optimized acceleration is found to have a value of the loading factor K: K opt = (1 + 1 + b2 ) b2 (7.7b)



The arrangement of this device is feasible in a coaxial or annular geometry, which is suitable for high-​speed propulsive systems. This idea is attractive because according to the classic Hartman (1937) solution, the transverse magnetic field force tends to accelerate the flow more uniformly than by the pressure gradient, and can avoid completely the thermal choking phenomenon. However, a detailed analysis using the generalized Ohm’s law in computational simulation is warranted for further applications.

7.5

Arc Plasmatron The arc plasmatron is another plasma-​generating device that can produce steady discharge by supplying cold gas, though it and the gas are ionized by arc, other than radio frequency discharge or microwaves radiation. The arc plasmatron is the most common plasma generator with power output in the range from hundreds of Watts to kilowatts and even to megawatts. The arc plasmatron is the most widely

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7.5  Arc Plasmatron

247

used industry tool for welding and cutting purposes, in operation the material to be processed often acts as the anode. The arc discharge distinguishes from the normal glow discharge, in that the cathode fall is much lower in the range of tens of Volts rather than hundreds of Volts. The difference is derived from the cathode emission mechanisms. The combination of complex thermionic, field electron, and thermionic field emissions is capable of supplying a greater electron current from the cathode, and the electron current nearly equals the total discharge current. The behavior of plasma is obviously different from typical equilibrium plasma in that the temperature of arc usually is the same as the electron temperature. The arc temperature has a range of 6,000 to 10,000 K, which is far above the melting temperature for most metallic materials. And the electron number density has an upper limit around 1014/​cm3. Most of all, the arc discharge is characterized by large currents from 1 to 105 Ampere. Thus, the cathode current density exceeds a value of 102 A/​cm2, in some special cases, it can reach a value more than 107 A/​cm2. For this reason, a protection precaution for cathode from erosion and evaporation must be implemented. Analytic study and numerical simulations of arc plasmatron are rather limited, because the swirling velocity component is presented for electrodes’ protection; the governing equations must couple the Maxwell equations with three-​dimensional compressible gas dynamics equations, as provided in Chapter 6.4. However in order to make the investigation practical, most analyses are carried out by assuming the generated arc exists in the absence of an externally applied magnetic field for which the Hall effect is negligible. Under this circumstance, the simplified Ohm’s law, Equation (1.31a), becomes a valid approximation. A series of computational results has been performed for studying the counter-​flow plasma injection, but the property of plasma is produced by a simplified plasma model supported by experimental observation (Shang 2002). The plasma torch used in counter-​flowing injection is presented in Figure 7.10. The maximum rated electric current output from this plasma torch is 35 Ampere, with an AC input single-​phase voltage of 208. However, in the displayed application, the unit is operating in the starting mode. Therefore, the power output is far below the rated value. The arc starting circuit has a high-​frequency generator that produces an AC voltage from 5,000 to 10,000 V at a frequency of approximately 2 MHz. The pilot arc within the torch head is initiated in the gap between the cathode and the positively charged tip. The pilot arc ionizes the compressed air passing through the torch head and exits through a small orifice in the torch tip with a swirling velocity component. The plasma is further expanded by a conical nozzle built in the model. The plasma jet exits the nozzle at a Mach number of 3.28 at a sonic condition at its throat and the nozzle flow is fully attached to the side walls of the nozzle. The computed simulated flow field of the plasma torch at a stagnation pressure of 243.3 kPa and stagnation temperatures of 4,400 K (7,260 R) is depicted in Figure  7.11. The computational simulation of the plasma torch is intended to duplicate the experimental observation by a simple two-​fluid plasma model and

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Figure 7.10  Composite presentation of a cut-​out and schlieren photograph is depicted for a plasma torch operated at supersonic flow.

the overall thermal condition of the plasma torch. Two simplifications are made by the numerical simulation; first, the swirling velocity of the jet for the torch is not implemented. Second, even the nonequilibrium vibration, dissociation, and electron are considered, but the plasma model is limited to nitrogen and only includes the radical reaction of C 2 π u+  X 2 π +g . Therefore the consequence of the simplification is uncertain; regardless, the numerical result exhibits that the temperature contours of the ionized counter-​flow jet from the plasma torch have a similar penetration distance in comparison with the experimental data. The thermal diffusion of the plasma torch is dominant over the room-​temperature air injection; the high-​ temperature regions are clustered on the nozzle surface and the forward free stagnation point in the jet stream. From the numerical simulation, the swirl flow velocity component is seen to be critically important to prevent erosion and evaporation of the electrodes.

7.6

Induction Plasma Generators Plasmatron by magnetic field induction is probably the simplest plasma generator; it does not require any additional heat source but a high-​current, electric-​inducting coil over a tubercular conductor. The theory of the inductively coupled plasma torch was developed in the 1960s (Raizer 1991); an AC current is applied to the surrounding coil over a metallic tube at the radio frequency spectrum (300 kHz to 3 Hz) by an electric power input around 100 kW. The generated plasma usually

249

7.6  Induction Plasma Generators

249

Figure 7.11  Temperature contours of a plasma torch.

has a temperature up to 10,000 K with an exit speed more than tens of meters per second, leading to an exit flow rate about 1,000 cubic centimeters per second at atmospheric condition. To protect the RF plasmatron from overheating, a tangential inflow arrangement is required, creating a centrifugal force to keep the hot plasma from the plasmatron surface. In Figure 7.12, a schematic of the RF plasmatron is given. According to Lieberman and Lichtenberg (2005), the power of inductively coupled plasma is transferred from the electric field to the charged particles within a skin-​depth layer by Ohmic dissipation and a collision-​less heating process in which the electrons receive energy from the oscillating inductive electric field. The former mechanism is the result of collisional momentum transfer between the oscillating electrons with the neutral particles. The latter mechanism is known as stochastic heating in which electrons are accelerated and subsequently thermalized in a capacitive radio frequency sheath. The time-​averaged stochastic power flux must be

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Figure 7.12  Schematic of radio frequency inductive plasmatron.

determined from the kinetic equation from the probability distribution function of the electrons. The structure of the flow field by induction plasma generator can still be analyzed by a steady axial-​symmetric formulation. In the following formulation, the only externally applied electric current is a periphery component that encircles the plasma generator, Jθ = σEθ , and the governing equations on the cylindrical polar coordinate system (r, θ, z ) are: ∂ρuz 1 ∂( rρur ) + = 0 (7.8a) ∂z r ∂r



ρuz

∂uz ∂u ∂p 1 ∂( r τ rz ) ∂τ zz + ρur z = − + + − Jθ H r (7.8b) ∂z ∂r ∂z r ∂r ∂z



ρur

∂ur ∂u ∂p 1 ∂( r τ rr ) ∂τ rz + ρur z = − + + + Jθ H z (7.8c) ∂r ∂r ∂r r ∂r ∂z



∂ρe 1 ∂( rρe ) ∂q 1 ∂rq  ∂ur ∂u   ∂u ∂u  + = + −  τ rr + τ zz z  − τ rz  r + r   ∂r ∂z r ∂r ∂z r ∂r  ∂z ∂z  ∂z  (7.8d)  1 ∂( r τ rr ) ∂τ rz   1 ∂( r τ rz ) ∂τ zz  2 − + σ − ur  + u + E z θ  r ∂r ∂z  ∂z   r ∂r



(1 r ) ∂( rEθ ) ∂r = − ∂( µH z ) ∂t (7.9e)



∂Eθ ∂z = ∂( µH r ) ∂t (7.9f)



∂H r ∂z − ∂H z ∂r = σEθ (7.9g) where the stress tensor components are τ rr ,τ zz and τ rz . In the following definitions of the stress components, the symbols µ g and λ denote the molecular and bulk viscosity of the gas:

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7.6  Induction Plasma Generators



τ rr = 2µ g

∂ur  2   1 ∂( rur ) ∂uz  +  λ − µg   +  ∂r 3   r ∂r ∂z 



τ zz = 2µ g

∂uz  2   1 ∂( rur ) ∂uz  +  λ − µg   + ∂z  3   r ∂r ∂z 



251

 ∂u ∂u  τ rz = µ g  r + z   ∂z ∂r  In the process of magnetically induced ionization, the electric field supplies energy directly to the electrons, which then transfer the energy to the neutral and heavier gas particles. The discharge is sustained by the radio frequency current and voltage applied to electrodes that encircle the plasma generator. The AC electric field creates a high-​voltage capacitive sheath between the electrodes and the plasma stream. The current flowing across the sheath leads to stochastic or collision-​less heating in the sheath and Ohmic heating in the bulk of the plasma. Since the energy transfer process mostly resides in the shear-​layer region, a suitable physics-​based ionization model is required for accurate computational simulations. The appropriate interface boundary conditions for the electromagnetic variables that have been developed in Chapter 3.4 must also be rigorously enforced. For the fluid dynamic variables, the non-​slip condition for velocity components u(ur, uz) = 0 is imposed, together with the specified wall temperature. The stochastic heating can only be evaluated by the kinetic theory of discharge by determining the energy diffusion coefficient via quasi-​ linear theory (Aliev, Kaganovich, and Schluter 1997). The spatial radio frequency length scale shall be much smaller than the electron mean free path for the low-​pressure discharge. By a separation of space scales in the Boltzmann equation, the approximated diffusion coefficient has been found (Lieberman and Lichtenberg 2005):



d = ( e 2 π 2 m2 uz2 | uz |) | u ⋅ E k |2 (7.9a) where the subscript k denotes the ratio between the radio frequency and the plasma velocity and is designated as k = ω uz . The time-​averaged stochastic power flux can be approximated as:



Ps = − m∫∫∫ dux duy duz uz d

∂f e (7.9b) ∂uz

Due to the huge difference in mass ratio of these two types of colliders, only a smaller fraction of the electron energy is capable of transferring by the elastic collision. In fact, the averaged differential scattering cross-​section is proportional to mass ratio between collision partners (2 m1 m2 ) and in the order of magnitude of 10-​4 (Lieberman and Lichtenberg 2005). Hence particles of the same mass can transfer energy very efficiently, but not between ions and electrons. Therefore, the initial ionization requires a sufficient relaxation time, but the induction period decreases rapidly as the degree of ionization increases. In general, the ionization is achieved mostly by transferring the energized electron translational energy to

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heavier gas particles; the resultant electron current is proportional to the temperature difference between electrons and neutrals, as well as the collisional cross-​section for momentum transfer and frequency. Thus, the high electron density and electron impacts are the main contributors for ionization. Based on the microscopic balancing principle, the degree of ionization mostly depends on the electron temperature and, to a lesser degree, on the temperature of neutral particles. This behavior is easily observed from experimental data in which the discrepancy between theory and data diminishes when the electron density is high in RF plasmatron. In this environment, the combined Faraday and Hall electric fields induced by charged particles’ motion may further enhance the required electric field.

7.7

Microwave Plasmatron The microwave plasmatron is operated on the same principle as the discharge from a waveguide, in which a gas is pushed through a dielectric tube or in an open space from an antenna. The generated plasma usually exists in a medium temperature in a range from 4,000 to 6,000 K at low pressure around 20 Torr; the plasma generation efficiency has been known to attain a value as high as 90 percent. The frequencies of microwave are in the X (16.5 GHz) and Ku (9.5 GHz) bands, which are commercial, readily available microwave generators. The electron density becomes an important parameter that controls the microwave radiation transfer to discharge. At low degrees of ionization, the space variation of the plasma structure is estimated by the emission intensity from the (0-​0) band, at 337.13  nm from the second positive system of molecular nitrogen (Exton et al. 2001). The breakdown to discharge is dependent on frequency at different ambient pressures; for the lower-​frequency microwave, the discharge occurs at a higher pressure. For example, the X-​band microwave breaks down at 17 torr by a pulse duration of 3.5 microseconds with a power input at 210 kW. The volumetric plasma is generated along the beam path between an antenna and a reflector that consists of a bright, thin planar plasma layer a few millimeters thick near the antenna aperture and is followed by trains of plasmoids. The spatial pattern corresponds to the standing wave radiation structure of the Fresnel or near-​field regime. These plasmoids have the irregular shapes of conic discs and are separated by a distance of about half a wavelength. The trains of plasmoids are presented over the illuminated funnel-​shape glow path connecting the surface plasma sheet from the exit of the antenna to an optical reflector. The sketch of the plasmoids and an insert of a photograph record of plasmoids in free space are depicted in Figure 7.13, and the degree of ionization by the microwave-​generated plasmoids is not uniform. The electron number density of the discharge domain has a two-​order magnitude of variation from 1011/​cm3 to 1013/​cm3. The plasma also is characterized by rotational and vibration temperatures as high as 1,200 k and 4,300 K, respectively. Another typical microwave discharge setup in the laboratory is displayed in Figure 7.14. The microwave discharge system consists of the power supply system

253

7.7  Microwave Plasmatron

253

Figure 7.13  Microwave-​generated plasma between antenna and reflector.

Figure 7.14  Typical microwave setup in laboratory.

and the electric circuit control loop. The power supply system includes the high-​ and low-​voltage power sources and modulator that are governed by a transformer and a regulator. After modulation by the controller, the power is transmitted to a magnetron check by a reflector then passes through the waveguide with the lens finally discharged into a vacuum chamber. The residual power after the microwave discharge is dissipated by an absorber. The power side of the microwave discharge system is displayed by the thicker line to reflect the two-​way feedback. The circuit control presented is rather simple for a closed loop, but a more elaborate arrangement also may include a registration system and a microwave probe. The plasma structure from a microwave breakdown in free space is not limited by the train of plasmoids, but can produce a complex structure in the form of multiple twisting, entwining streamers. Figure 7.15 displays the continuous elongation of the channel in space by Brovkin and Kolesnichenko (2004). The complex

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Plasma and Magnetic Field Generation

Figure 7.15  Streamer structure of microwave breakdown courtesy of Kolesnichenko through private communication (p = 230 torr, I = 104 w/​cm2, t = 30 μsec).

structure exists in the ambient pressure environment up to around 300 Torr through the action of pulsing microwave radiation. One general feature is the unified self-​ ignition, and the simulated microwave discharge always propagates toward the radiating microwave source. This phenomenon is revealed by the waveguide technique in which a transmitting, high-​power, pulsing microwave causes breakdown and leads to plasma creation and propagation in a waveguide. The physical mechanisms for discharge propagation in super-​breakdown and sub-​breakdown conditions are quite different. For the former, the process is very sensitive to the electromagnetic field and the initial electron distribution. On the other hand, the sub-​breakdown is strongly affected by the self-​organization processes. However, the most effective microwave application is derived from the resonant interaction. At low pressures, when the angular frequency of the microwave, ω , exceeds the electron elastic collision frequency, νc, the resonance with the plasma field will lead to a random discharge pattern. At high pressure, when ω  νe , the resonance is suppressed by dissipation and eventually ceases. In general, the microwave structure consists of curved and branched channels. A discharge channel in an electromagnetic field with an electric intensity significantly lower than the breakdown value is often referred to as a streamer, and the mechanism of propagation is also known as a streamer. This behavior is an indication of stochasticity in the formation of the discharge structure, which is unique to the system formed by the streamer mechanism. In this sense, the microwave breakdown discharge is often referred to as a vacuum torch. Thus, the microwave discharge is a mechanism that can be applied to enhance combustion stability in a low-​pressure environment by acting as a catalyst and imparting additional energy to overcome the activity energy barrier of chemical reactions.

25

7.8  Plasma by Radiation

7.8

255

Plasma by Radiation Plasma generation by radiation is also known as a plasma-​generation process by optical means. When using a continuous wave (CW) CO2 laser as the energy source for ionization, it is fundamentally a reverse free-​free transition of radiative energy of an electron or ion by emitting or absorbing a photon to acquire additional kinetic energy, known as bremsstrahlung radiation. The electron will slow down in the presence of ions but loses a part of its energy through radiation. It is interesting to note that the bremsstrahlung process has a continuous emission and absorption spectrum (Zel’dovich and Raizer 2002). When the electron is decelerating, the radiation becomes more intense as the peak intensity shifts toward higher frequencies. According to kinetic theory, the bremsstrahlung absorption coefficient in partially ionized gas is linearly proportional to the product of the electron and ion number density, thus the plasma density. The scattering cross-​section can be represented by the Thomson scattering cross-​section as σ ≈ 6.65 × 10 −25 cm 2 and the integrated emission coefficient for bremsstrahlung is:



∞

I = ∫ I ν d ν = (32 π 3)( 2 πkTe 3m )1 2 ( z 2 e 6 mc3 h )n+ ne (7.10a) 0

where the symbol z denotes the number of charges of the ion and Te characterizes the electron temperature. The integrated emission coefficient has the physical dimension of erg-​sec/​cm3. The important absorption coefficient of the bremsstrahlung radiation can be approximated as:

kν ≈ 4.1 × 10 −23 z 2 [ n+ ne Te3 2 ( hν kTe )3 ] (7.10b) In cgs systems, the absorption coefficient has a dimension of cm-​1. The plasma generation by radiation usually takes place due to a single frequency microwave or laser with sufficient intensity; ordinarily transparent gas mediums like air will break down to produce free electrons (Meyerand and Haught 1963). To achieve this result, a very high concentration of radiation energy fluxes is required by focusing the laser beam with a lens. In 1966, the unsteady physical phenomenon was first described by Raizer (1966) in detail; his finding opens a new avenue for plasma dynamics research and application. The laser breakdown process is very complex; it starts with the multiphoton ionization, which is carried out by a series of photon energy transfer cascades. The depleted energy of the initiating electrons for ionization is refurnished by radiative absorption of the ionized gas. For this reason, the plasma domain generated by a highly focused laser beam does not have a regular shape. The amount of the energy released by the laser beam at the focal point is huge and at a very short duration that leads to a blast wave. In application to remote energy deposition, the flow field responds to the suddenly released energy and the adverse pressure gradient leads to the rotational motion of fluid and generates a formation of a toroidal vortex along the incident wave path. The breakdown threshold for plasma generation by radiation is extremely sharp and is always characterized by the intensity of the electromagnetic field by

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Plasma and Magnetic Field Generation

microwave. For the electron-​avalanching process, the breakdown occurs at an electric field strength of values from 106 to 107 V/​cm. Although the focal region has a small dimension, the distribution and the shape of ionized gas are neither uniform nor regular near the focus. Particularly the avalanche is developed by electrons absorbing photons from collision with neutral particles and meanwhile accumulating sufficient energy for ionization. The process is like a chain reaction and the rate of ionization is accelerated by radiation, the number of electrons in the avalanche increases exponentially with time. The highly ionized gas may have an exceptional electron temperature as high as 600,000 K in atmospheric air. It has been conjectured that breakdown occurs at the focal spot of the narrowest part of the wave beam, but as soon as the ionized layer becomes optically opaque, it absorbs the radiative energy strongly and moves away from the beam-​focused location. The ionization process propagates along the beam path as an absorptive and heating wave. As the heated gas expands, a shock wave moves to all directions like a detonation. The detonation wave propagation can be assessed at the Chapman-​ Jouguet point by the local speed of sound, which can attain a speed up to 100 km/​s (Meyerand and Haught 1963; Adelgren et  al. 2005). However, the breakdown charge velocity is merely 6 km/​s, and the dimension of the radiation breakdown is generally within a few millimeters. The outstanding sequence of events is reproduced by Surzhikov (2005) by computational simulations. A carbon dioxide laser with a wavelength of 10.6 micrometers is sustained by a power of 5.0 kW and focused by an optical lens at a distance of 2.9 cm from the antenna. The pulsing power is transmitted to an air stream moving at a velocity range from 1.0 to 10.0 m/​s at duration of one to five microseconds. The plasma is ignited at the focus point and propagates along the beam path. The maximum temperature has reached a value of more than 10,000. K. The thermally excited air plasma is depicted in Figure 7.16, and a train of compression and expansion waves is detectable in this presentation. A trademark of toroidal vortical structure at the laser focus for plasma generation is produced by viscous–​inviscid aerodynamic interaction with a suddenly released point thermal source like a blast wave (Surzhikov 2005). The computational simulations have captured the laser-​ generated plasma phenomenon as displayed in Figure 7.17. All these physical phenomena associated with ionization by radiation are observed in the study of remote energy deposition by Adelgren and colleagues (2005). For their laser plasma-​generation experiments, a beam from a pulsed 1.0 cm diameter Nd:YAG 532-​nm laser was focused at one diameter (25.4 mm) upstream of a spherical model supported by a stem. Their experiment results establish not only the evidence of laser-​induced optical breakdown in air but also the interaction of remote energy deposition for flow modification. The instrumented model is submerged in a supersonic stream with an oncoming Mach number of 3.45; at the freestream temperature of 77.8 K, the unperturbed stream moves at a velocity of 610 m/​s. The laser beam was focused by a 250-​mm focal length lens, resulting in a focal diameter of approximately less than 0.1 mm. The energy deposition size,

257

7.8  Plasma by Radiation

257

Figure 7.16  Laser discharge at the laser focal point.

Figure 7.17  Periodic toroidal vortical structure downstream of laser beam focus.

however, is estimated at a volume of 3 ± 1 mm3. The incident laser is ignited with a temporal pulse width of about 10 ns and a repetition rate of 10 Hz. For the pulsing duration of 10 ns, the maximum delivered laser energy is 283 mJ per pulse. A sequence of schlieren photographs from the ignition of the laser pulse beyond 120 µs was recorded. For our discussion, only three images are included in

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Plasma and Magnetic Field Generation

Figure 7.18  Selected schlieren images for remote energy deposition via plasma generation by laser (Adelgren et al. 2005).

Figure 7.18, at the instants of 20, 70, and 120 µs to delineate the basic structures of ionization by radiation and the aerodynamic interactions. At the time elapse of 20 µs after the ignition, the blast wave generated by the suddenly converted thermal energy from the laser is clearly displayed. The three-​dimensional spherical shock wave front does not show discernable distortion by the oncoming stream, which indicates that the blast wave propagates at a much higher velocity than the freestream. From the consecutive schlieren record by Adelgren and colleagues (2005), the blast wave speed is estimated at the least to be 5.08 km/​s, but not to exceed 10 km/​s. However, the blast wave strength cannot be determined from these schlieren images. At the time frame of 70 µs, the complex blast and bow shock wave interaction ensues; the shock wave system is the result of interactions between blast wave, bow shock, and the blast wave reflected from the model surface. The resultant shock wave has a highly irregular shape in contrast to the undisturbed single bow shock. The most interesting features, however, are the multiple slipstreams originating from the interception of the spherical blast wave and the toroidal vortex along the incident wave path, and accentuated in the shock layer. It is the unique flow structure produced by the remote energy deposition application. At the time frame of 120 µs, the generated blast wave associated with the laser passes the spherical model and the complete flow field structure is returning to its unperturbed state until the arrival of the next pulse. In all, the essential physics of ionization by radiation is captured by the remarkable experimental effort by Adelgren and his team (2005).

7.9

Magnetic Field Generations The effect of an applied magnetic field on plasma dynamics simply cannot be overstated. The magnetic flux density increases the mechanisms for electromagnetic

259

7.9  Magnetic Field Generations

259

Figure 7.19  Effect of transverse magnetic field on surface direct current discharge.

effects by introducing the Lorentz acceleration, Larmor gyration, and Hall current, which in turn allows the possibilities of plasma confinement, altered plasma sheath thickness, and amplification of electromagnetic–​aerodynamics interactions. The most vivid illusion is revealed through the influence on the domain of discharges of a direct current plasma generation as depicted in Figure 7.19. The side-​by-​side electrode arrangement put the cathode on the left-​hand side of the anode, and the electrodes’ separation distance is five times the electrode width. A  constant discharge current of 100 mA at an ambient pressure of 10 Torr is preset for all three direct current discharges under different externally applied magnetic fields. The externally applied magnetic flux density varies from negative polarity of -​0.5T, 0.0T, and positive polarity +0.5 T for the data collection process. In the absence of the externally applied magnetic field, a steady glow is accentuated mostly over the cathode and can be maintained by the steady electric potential of 400 volts. When the negative magnetic flux density is applied, the Lorentz acceleration of the electromagnetic field restrains the charged particles from leaving the electrodes and the dielectric surface. The negative polarity is defined by the right-​hand rule; the magnetic field is pointed perpendicular outward from the view plane. Under this circumstance, J × B < 0, the charged particles are pushed toward electrodes by the Lorentz force and the discharge domain is suppressed by the transverse magnetic field; meanwhile, the discharge current of 100 mA now must be maintained by an oscillating voltage around 1090 Volts. When the polarity of the applied transverse magnetic field is reversed, J × B > 0, the Lorentz acceleration now expels the plasma from the electrodes, and the required electric field intensity is slightly lower than the applied negative transverse magnetic field, an unsteady electric intensity by a value of about 980 volts. Under the influence of the transverse magnetic field, the discharge domain spreads nearly uniformly over the electrodes and dielectric surface. At the same time, the discharged currents increase in comparison with the condition in the absence of an externally applied magnetic field. As the consequence of an additional and positive electromagnetic force to expand the discharged domain, the thickness of the plasma sheath is increased and the frequency of collisions between ions and neutral particles is enhanced for momentum transfer. Thus the transverse magnetic field increases the effectiveness of a plasma-​based actuator for flow control.

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Plasma and Magnetic Field Generation

A key element in generating an electromagnetic–​aerodynamics interaction in flight or in a ground testing facility is the strength of the applied magnetic field. Based on the Stuart number, (σB 2 ρu ), the low electron number density in most applications must be compensated for by the applied magnetic field strength B (Shang et  al. 2005; Shang et  al. 2008). Magnetic field generation has two widely adopted procedures: either by using permanent magnets or by steady state electrical solenoid. In fact, it is also the most effective control parameter because the interaction parameter is increased according to the square power of the magnetic field intensity. One of the most effective means to introduce an applied magnetic field immediately adjacent to a solid surface is by an array of neodymium rare-​earth (NdFeB) permanent magnets. The NdFeB magnet has a higher maximum energy product (BH)max than most other rare-​earth permanents. These magnets are commercially available in a wide variety of shapes and sizes. These magnets are usually produced in disk or square block shape with a thickness less than 3.2 mm and a diameter of 19.0 mm. The pole surface area is as large as 1,451 mm2, and at the pole surface the magnetic field strength is greater than 0.47 T. However, a weakness of the NdFeB permanent magnet in comparison with other magnets using an aluminum-​nickel-​cobalt alloy (Alnico) is a much lower Curie temperature. When the temperature of an application environment exceeds this limit, the magnetic field intensity diminishes very rapidly. Therefore, the maximum working temperature for NdFeB permanent magnets is 455 K. This shortcoming should not be overly detrimental for most practical applications and can be overcome by a suitable cooling arrangement. Another attractive feature of the permanent magnets in application is the slower magnetic field decay rate with respect to the distance from the pole surface, in addition to its lower weight, and simple installation by the solenoid. Several direct measurements of the magnetic field strength by a Gauss meter show the decay rate is lower than that for a field generated by a steady state solenoid of a compatible strength on the pole cap. The most common adopted magnetic generator is the helical electric induction coils or the solenoid winding around a ferrite core and operating by a steady electric current with water cooling. If the length of the solenoid is much greater than its radius, the magnetic flux density at the center of the solenoid is proportional to the permeability of the medium, the number of turns of the coil, and the current through the solenoid, and inversely proportional to the length of the solenoid; B = µnI l . It is also known that at the poles of the solenoid, the magnetic flux density is reduced to one half of the value at the center of the solenoid. The coils, when connected in a series and when their electric resistance is kept to a value lower than 0.5 Ohm, can achieve a maximum power rating of 110 kW (140 ampere, 80 volt). The solenoid then can attain a self-​inductance up to 80 m Henry over the pole cap diameters ranging from 25 to 250 mm. The theoretical decay rate of a solenoid-​ generated magnetic flux density is proportional to the inverse cubic power to the pole gap distance, but the field strength is a function of the pole gap and the pole cap diameter. In practical applications, the pole gap distance usually has a range up to 160 mm. For example, for a pole gap of 10 mm and a pole diameter of 25 mm,

261

7.9  Magnetic Field Generations

261

Figure 7.20  Characteristic current discharge for an impulse solenoid.

the uniform field strength can be as high as 3.5 Tesla on the magnetic pole, but at a distance of 0.33 mm from the pole, the strength is reduced by half. In practice, the solenoid is always operated at the minimum pole gap and maximum pole cap diameter. Under an optimistic condition, the transverse magnetic field by a coupled poles arrangement can have a nearly uniform magnitude of several Tesla. Since the generated magnetic flux density is proportional to the current flows through a solenoid coil, a desired magnetic field strength can be achieved by a high surging electric current. Thus an impulse solenoid can deliver a greater magnetic flux density on the pole cap by a fine-​tuned circuit of a capacitor, resistor, and inductor; a fully damped circuit can be designed for a required impulse solenoid. The pulse current is deliverable from a bank of capacitors with a peak current of more than 900 Amps; these capacitors can sustain a discharging current of this magnitude for the duration of a fraction of millisecond. The current discharge characteristic of an impulse solenoid is given by Figure 7.20; the electric current rises rapidly to reach the peak current exceeding 920.0 amps and drops to a value around 100.0 Amps at 26.0 µs later. The generated magnetic flux density sustains a value greater than 2.0 Tesla, taking less than 1.7 µs. The fully damped current discharge behavior from the experimental data to an impulse induction coil is compatible

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Plasma and Magnetic Field Generation

Figure 7.21  Magnetic field over a hemispherical-​nosed probe by an impulse solenoid, t = 68 µ  s.

with Ziemer’s (1959) experience and should be sufficient for most laboratory electromagnetic experiments. The impulse solenoid-​generated magnetic field by intensity contours over a hemispherical probe is presented in Figure 7.21. A recessed fourteen-​gauge cooper wire winding is wrapped around a ferrite core to be accommodated by the interior space of a probe whose outline is depicted by the thick grey line. The probe is submerged in a hypersonic flow field characterized by a Mach number of 5.8 and a Reynolds number of 4.52 × 10 4 / m, and the enveloping bow shock is depicted by the dashed line with a stand-​off distance of 7.62 mm. At the instant of t = 68.0 µs, the constant-​value magnetic flux density contours are recorded to span a range from 0.2 to 2.0 Tesla with the maximum value at the pole surface. It is noted that at the bow shock envelope along the stagnation streamline, the magnetic field intensity has already reduced to the half value at 1.0 Tesla from the magnetic pole surface.

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References

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The field strength within the shock layer diminishes rapidly away from the axial line of the ferrite core as the typical behavior of a solenoid-​generated magnetic field. The experimental observation reveals that the externally applied magnetic field can rarely be uniformly distributed. On the axis of symmetry, the theoretic magnetic field decay rate for a dipole is inversely proportional to the cubic power of the distance from the pole. According to Ziemer’s steady-​state experimental measurement with a Gauss meter, the magnetic field decay rate of a solenoid is found actually as: B = Bo ( rb r )3.61 (7.11)



This empirical formulation has also been confirmed by another independent measurement. The most powerful steady state solenoid for magnetic field generation without a question is by using the superconducting material for its winding coil because of the possibility of enormous electric current flow. The superconductivity is a quantum mechanical phenomenon in which the magnetic field line is ejected from an electric conductor at nearly absolute temperature known as the Meissner effect (Bardeen, Copper, and Schriffer 1957), and at this condition the electric resistant has completely vanished. Theoretically, an infinite amount of electric current can be realized by applying a low electric potential. The most widely adopted superconducting material is either niobium-​titanium (NBTI) or niobium-​tin (NBSN), and the critical temperature of the superconductor for NBTI and NBSN is 10 K and 18 K, respectively. The required operational condition must keep the wire temperature at the cryogenic temperature of 4.2 K in an insulated container, and to maintain the cryo-​s tate, the temperature needs to be at least around 40 K. The highest known magnetic flux density generated by a super magnet is around 45 Tesla, but it is common to be around 20 Tesla. These super magnets at present are widely used in medical imaging and in focusing magnetic fields for extremely high-​energy particle acceleration, but have not been very practical for most aerospace engineering applications. References Adelgren, R.G., Yan, H., Elliott, G.S., Knight, D.D., Beutner, T.J., and Zheltovodov, A.A., Control of Edney IV interaction by pulsed laser energy deposition, AIAA J., Vol. 43, 2005, pp. 256–​269. Aliev, Y.M., Kaganovich, I.D., and Schluter, H., Phys. Plasma, Vol. 4, 1997, p. 2413. Bardeen, J., Copper, L., and Schriffer, J.R., Theory of superconductivity, Phys. Rev., Vol. 8, 1957, p. 1178. Boeuf, J.P. and Pitchford, L.C., Electrohydrodynamic force and aerodynamic flow acceleration in surface dielectric barrier discharge, J. Appl. Phys., Vol. 97, 2005, pp. 103307-​1-​10. Brovkin, V.G. and Kolesnichenko, Yu.F., Structure and dynamics of simulated microwave gas discharge in wave beam, J. Moscow Phys. Soc., Vol. 5, 1995, pp. 23–​38. Corke, T.C., Enloe, C.L., and Wilkinson, S.P., Plasma actuators for flow control, Annu. Rev. Fluid Mech., Vol. 42, 2010, pp. 505–​525.

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Dufrene, A., Sharma, M., and Austin, J.M., Design and characterization of hypersonic expansion tube facility, J. Propul. Power Vol. 23, No. 3, 2007, pp. 1185–​1193. Elisson, B. and Kogelschatz, U., Nonequilibrium volume plasma chemical processing, IEEE Trans. Plasma Sci., Vol. 19, 1991, pp. 1063–​1077. Engel, A.V. and Steenbeck, M., Elektrische Gasentladungen. Ihre Physik und Technik. Springer-​Verlag, Berlin GmbH, 1932. Enloe, C.L., Mcharg, M.G., and McLaughlin, T.E., Time-​ corrected force production measurements of the dielectric barrier discharge plasma aerodynamic actuator J. Appl. Phys. Vol. 103, 2008, 073302. Enloe, C.L., McLaughlin T.J., Vandyken, R.D., Kachner, K.D., Jumper, E.J., Corke, T.C., Post, M., and Haddad, O., Mechanisms and responses of a single dielectric barrier plasma actuator, Plasma morphology, AIAA J., Vol. 42, 2004, pp. 595–​604. Exton, R.J., Balla, R.J., Herring, G.C., Popovic S., and Vuskovic, L. Volumetric near-​field microwave plasma generation, AIAA Preprint, 2003-​4181, 2003. Gurijanov, E.P. and Harsha, P.T., Ajax:  New directions in hypersonic technology, AIAA 1996-​4609, Nov. 1996. Hartman, J., Hg-​dynamics I: Theory of the laminar flow of an electric conductive liquid in a homogeneous magnetic field. Danske Videnskab, Mat-​fys. Medd., Vol. 15, No. 6, 1937. Holden, M.S., Wadhams, T.P., MacLean, M., Mundy, E., and Parker, R.A., Experimental studies in LENS I  and X to valuate real gas effects on hypervelocity vehicle performance, AIAA Preprint, 2007-​1204, 2007. Huang, P.G., Shang, J.S., and Stanfield, S.A., Periodic electrodynamic field of dielectric barrier discharge, AIAA J. Vol. 49, No. 1. Jan. 2011, pp. 119–​127. Kimmel, R.L., Hays, J.L., Menart, J.A., and Shang, J., Effect of magnetic fields on surface plasma discharges at Mach 5, J. Spacecr. Rocket, Vol. 42, No. 6, 2006, pp. 1340–​1346. Lieberman, M.A. and Lichtenberg, A.J., Principle of plasma discharges and materials processing, John Wiley & Sons, New York, 2005. Liepmann, H.W. and Roshko, A., Elements of gasdynamics, John Wiley & Sons, New York, 1958. Lineberry, J., Begg, L., Castro, J., and Litchford, R., Scramjet driven MHD power demonstration –​HVEPS Program, AIAA Preprint, AIAA 2006-​3080, June 2006. Litchford R.  J., Thompson B.  R. and Lineberry J.  T., Pulse detonation magnetohydrodynamic power, J. Propul. Power Vol. 16, 2000, pp. 251–​262. Lu, F.K., Liu, H-​C., and Wilson, D.R., Electric conductivity channel for a shock tube, Measur. Sci. Technol., Vol. 16, 2005, pp. 1730–​1740. Meyerand, R. and Haught, A., Gas breakdown at optical frequencies, Phys. Rev. Lett., Vol. 11, 1963, pp. 401–​403. Morgan, G.C., Laser induced breakdown of gases, Rep. Pro. Physics. Vol. 38, 1974, pp. 621–​655. Moreau, E., Airflow control by non-​thermal plasma actuator, J. Phys. D. Appl. Phys., Vol, 40, 2007, pp. 605–​635. Raizer, Yu., Breakdown and heating gases under the influence of a laser beam, Sov. Phys.-​ Usp, Vol. 8, 1966, pp. 650–​673.   Gas discharge physics, Springer-​Verlag, Berlin, 1991. Resler, E.L. and Sears, W.R., The prospects for electro-​aerodynamics, J. Aeronaut. Science, Vol. 25, 1958, pp. 235–​245, 258.

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Shang, J.S., Plasma injection for hypersonic blunt –​body drag reduction, AIAA J., Vol. 40, No. 6, June 2002, pp. 1178–​1186.   Computational electromagnetic-​aerodynamics, Wiley IEEE Press, Piscataway, NJ, 2016. Shang, J.S. and Huang, P.G., Modeling AC dielectric barrier discharge, J. Appl. Phys., Vol. 107, No. 11. 2010, pp 113302-​1-​7. Shang, J.S., Huang, P.G., Yan, H., and Surzhikov, S.T., Computational simulation of direct current discharge, J. Appl. Phys., Vol. 105, 2009, pp. 023303-​1-​14. Shang, J.S., Kimmel, R., Hayes, J., Tyler, C., and Menart, J. Hypersonic experimental facility for magnetoaerodynamic interactions, J. Spacecr. Rocket, Vol. 42, 2005 pp. 780–​789. Shang, J.S., Kimmel, R., Menart, J., and Surzhikov, S.T., Hypersonic flow control using surface plasma actuator, J. Propul. Power, Vol. 24, No. 5, 2008, pp. 923–​934. Shang, J.S., Roveda, F., and Huang, P.G., Electrodynamic force of dielectric barrier discharge, J. Appl. Phys., Vol. 109, No. 11, 2011, pp. 113301-​1-​8. Solov’ev, V., Konchakov, A.M., Krivtsov, V.M., and Aleksandrov, N.L., Numerical simulation of a surface barrier discharge in air, Low-​Temperature Plasma, Vol. 34, No. 7, 2008, pp. 594–​608. Surzhikov, S.T., Fundamental of radiative gas dynamics, Von Karman Lecture Series, Von Karman Institute for Fluid Dynamics, Rhode-​ST-​Genese Belgium, 2005. Surzhikov, S.T. and Shang, J.S., Two-​component plasma model for two-​dimensional glow discharge in magnetic field, J. Comp. Phys., Vol. 199, No. 2, Sept. 2004, pp. 437–​464. Sutton, G.W. and Sherman, A., Engineering magnetohydrodynamics, McGraw-​Hill, New York, 1965. Zel’dovich, Ya.B. and Raizer, Yu.P., Physics of shock wave and high-​temperature hydrodynamic phenomena. Dover Publications, Inc., Mineola, NY, 2002. Ziemer, R.W., Experimental investigation in magneto-​aerodynamics, J. Am. Rocket Soc., Vol. 29, 1959, pp. 642–​647.

8

Plasma Diagnostics

Introduction Plasma parameters such as the degree of ionization, composition, and energy content are essential for evaluating the effectiveness of applied electromagnetic effects to engineering applications. A  distinctive feature of plasma is the accentuation of instability and collision frequency that manifest both in the multiple components and in the energy distribution of charged particles. Furthermore, the distribution functions for microscopic particle dynamics for determining macroscopic properties of plasma are usually not describable by the Maxwellian distribution. Thus the data must be estimated from the ionized components of plasma and the partition functions of internal degrees of freedom of plasma. In general, the processes of plasma diagnostics are classified by whether invasive probes, spectroscopic display, and optical effect by free electrons are used. The spectroscopic data collection can be further categorized into the passive and active means, depending on whether the procedure is based on either the Doppler shift and atomic spectrum broadenings or the absorption and emission of spectroscopy. The third technique considered is developed from microwave probing. It is important to bear in mind that all aforementioned procedures have very restrictive conditions under which accurate data can be obtained. The low-​energy or low-​temperature and nearly collision-​free plasma occupies a sizable operational domain; the intrusive electrode probe is widely used, particularly the Langmuir probe (Mott-​Smith and Langmuir 1926). The requirement for using the probe is minimal, because the probe must function only as an antenna for electric voltage discharge and reception. The measured electric voltage and current relation will yield the electron number density, which can be further deduced to get the electron temperature. Serious complication also arises when using an intrusive probe in the presence of an externally applied magnetic field. A  combination of probes has been used to measure the electron temperature by the difference between the screened and unscreened probe potentials. In the measuring process, the balanced saturation currents of electron and ion are acquired by a ceramic-​shield probe, thereby screening off an adjustable current from the probe. If the magnetic field is unsteady, the net temporal rate of change can be measured by a coil, and the induced voltage is recorded to give the current flowing in the plasma. These newer techniques are innovative and require additional research. In any event, the

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Introduction

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theoretical foundations of probe measurements are built on the Poisson equation of plasma dynamics and the collision-​free Boltzmann equation. To measure the plasma potential in magnetized plasma, a novel idea has been discovered by using the intrusive electrode probes. As we discuss the Langmuir probing procedure later, when the electron saturation potential reaches the same magnitude of the ion saturation current, the floating potential of the measuring system is identical to that of the plasma potential. The requirement for this technique is that the size of the isolated probe electrode must have a smaller dimension than the gyro radius of the electrons. At the same time, the electron temperature is obtainable from the knowledge of the plasma potential and the floating potential of the Langmuir probe. However, the measuring technique needs a high temporal resolution power supply system. Plasma emits radiation over a wide frequency extending from microwaves over infrared and ultraviolet into the X-​ray domain. The diagnostic technique using spectroscopic distribution of plasma radiation is often referred to as the passive technique, because the process does not introduce any perturbations to the measured medium. The diagnostic data are collected only from the spectral line intensity, which is characterized by the power emitted by atomic transitions. These measurements from spectral line intensities can determine the plasma composition and the temperatures of each component. In order to make quantitative analysis of plasma composition, the absolute measurements of spectral line intensities are necessary. The visible and the common extensions to the ultraviolet and infrared spectra can cover the wavelength range from roughly 380 nm up to 1 µ m. From the experimental view point, this wavelength is the best choice in plasma spectroscopy; because the air is transparent, the quartz window can be used by the diagnostic system with the most available light source. When the wavelength is below 200 nm, the quartz is no longer transparent and the oxygen in the air absorbs light, leading to the requirement of an evacuated light path. When the wavelength is above 1 µ m, the thermal background will overwhelm the signal strength. Fortunately, the adverted spectral computational ability has permitted a direct comparison of data with computational results to greatly facilitate the diagnostic process. Applying the spectral line profile analysis, in particular, the Doppler broadening has been successfully used for moderate number density of charged particles, and the Stark broadening is suitable for high-​density plasma. The Doppler effect is developed on the basis that the shape of spectral lines is directly connected to the thermal motion of radiating particles. Any thermal perturbation will be reflected by the spectral line profile. When the particle velocity distribution is the Maxwellian, the Doppler frequency shifts, leading to a Gaussian line profile, and the line width is related to the temperature of the radiating particles. On the other hand, the Stark effect is the consequence of the interaction and collision of charged particles with an electric field. For the high-​charged number density plasma, 1015/​cm3 < ne < 1017/​cm3, the Stark broadening is greater than that of the Doppler broadening, which offers some advantage for data analysis. In the optical frequency range, the continuum radiation is mostly generated from bremsstrahlung radiation via electron

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Plasma Diagnostics

collisions. The averaged electron density is therefore obtainable by the radiation intensity along the line of observation. However, some prior information of the radiating frequency range is very helpful to improve the accuracy of diagnostics. Finally, the Zeeman effect can also be used to measure the magnetic field of plasma, when the magnetic field is perturbed by the plasma current. The microwave probing diagnostic technique is built on the attenuation characteristics of electromagnetic wave propagation in plasma. When a plane electromagnetic wave propagates through a medium of different permeability, varying degrees of amplitude attenuation and phase shift will take place. In fact, it is defined by a complex propagation coefficient and a function of the plasma frequency ω p , the collision frequency for momentum transfer ν, and the incidence microwave frequency ω . If the incidence microwave frequency is lower than the plasma frequency, ν < ω < ω p, the plasma will be reflected at the media interface and an electromagnetic wave becomes an evanescent standing wave behaving like a waveguide below the cut-​off frequency. This phenomenon is often observed as that of the communication blackout. When the probing microwave frequency is higher than the plasma frequency, the plasma behaves as a relatively low-​loss dielectric. At this condition, the reflective index is quite insensitive to collisional damping; the wave attenuation and dispersion of the incident microwave are rather small but measurable, which becomes the basis for the microwave diagnostic technique. For diagnostics using microwave probing with a magnetic field, the situation is considerably more complicated because the magnetic force is perpendicular to a moving charged particle; the charge motion is now governed by the Langevin equation. Accordingly, the complex propagation coefficient has an additional function of electron cyclotron frequency or gyro frequency, which describes the angular frequency of an electron gyrating in a magnetic field. The refractive and attenuation indices of these cyclotron waves now can be calculated from the dielectric constants. The plasma diagnostic techniques cover an extremely broad range of scientific disciplines that have a deep root in quantum mechanics and optical physics. Driven by the increasing technical needs, progress and innovation are happening daily; the diagnostic methods considered in our discussion are naturally incomplete. The illuminating works by Lochte-​Holtgreve (1968) and Heald and Wharton (1965), and the more recent efforts by Hutchinson (2005) and Fantz (2006) are highly recommended.

8.1

Electrode Arrangements of Langmuir Probe A Langmuir probe is operated on the principle that in a plasma sheath there is a unique relationship between plasma current flowing into a surface and the local electric potential drop. This condition prevails when the probe potential is greater than the plasma potential, then no electrons are reflected by the sheath and the electron current is fully saturated. From the measured electric current, the electron

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8.1  Electrode Arrangements of Langmuir Probe

269

temperature, electron number density, and plasma potential can be derived. The basic construction of a Langmuir probe is inserting a small electrode in a plasma domain that is sustained through an external electric circuit with an electrode of a larger surface area than the probe. The probe consists of an electric-​conducting surface as a sensor to measure the supply voltage and current flow. There are many possible shapes, and multiple materials have been used for the sensing surface; the sensor materials often use tungsten, tantalum, or platinum wires several thousandths of a centimeter in diameter. The platinum probe is frequently used for plasma being generated in atmospheric condition because it does not oxidize readily at high temperatures. The size of the probe preferably will not overly perturb the plasma and has a high melting point that can withstand the highest power load. Three critical length scales in making accurate Langmuir probe measurements are the radius of the probe, the mean free path of the charged particles (ions and electrons), and the Debye shielding length. The size of the pertaining length scales determines the applicable theory for the collisional or collision-​less plasma formulations for data reduction (Chung, Talbot, and Touryan 1975). Applying a Langmuir probe in the presence of a magnetic field for diagnostics becomes more complex, because the project area of the electrode must be considered in order to take the cross-​field currents into account. The simplest modification is adopting a projection area greater than the surface area of the electrode; the basic scaling rule is based on the ion Larmor radius. The most widely used Langmuir probes are single or double sensors. The basic idea behind the single probe is to sense the current flowing through the probe as a function of the applied voltage. The collected data yield the characteristic of electric current versus voltage. From this set of measurements, the electron temperature, the electron number density, and the floating potential can be obtained. By taking a probe measurement at two closely placed positions, the electric field strength of the plasma can be determined. A  fair amount of data scattering should be anticipated by the single probe measurements, because the probe creates a significant disturbance to the plasma. On the other hand, the double probe measures the current flowing between the two probes as a function of the applied voltage between them; the perturbation to the electric field is less than the single probe. Similarly, the electric current and voltage relationship will reveal the electron temperature, the electron number density, and the electric field strength in the plasma. A schematic of the electric circuit arrangement of a single Langmuir is presented in Figure 8.1. The single probe immersed in the plasma is connected to a power supply system and a resistor, and across the resistor is a voltmeter. The voltmeter and the resistor are used to measure the net probe current. The other end of the electrical circuit is connected to the anode, which serves as a reference voltage, and the circuit is closed by the plasma domain. The power supply system usually consists of a bipolar power supply covering the positive-​negative polarities over hundreds of volts. In addition, a unidirectional power supply is often adopted to alter the reference level for the bipolar power supply. The unipolar power supply is

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Plasma Diagnostics

Figure 8.1  Schematic of single Langmuir probe.

Figure 8.2  Schematic of double Langmuir probe.

not needed if a bipolar power supply with a greater voltage range is available. The sensing area of the probe is easily adjusted by a movable ceramic sleeve. A schematic of the typical electrical circuit of a double Langmuir probe is shown in Figure 8.2. Two sensing probes are immersed in a given position to detect the plasma current. Similar to the single probe, connected between the two sensing surfaces are a power supply and a resistor. The voltmeter across the resistor is used to measure the net current flowing between the two sensing surfaces. Unlike the single probe, no reference voltage is required and thus the electric circuit does not require connection to the anode. Both sensing surfaces together are always operating in the floating mode. This means that they draw no net current from the measured plasma, no matter what the applied voltage between them. This is an advantage over the single probe because it causes less disturbance to the plasma. The single probe removes electric current from the plasma, whereas the double probe replaces as much current as it removes. The separation distance between

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8.1  Electrode Arrangements of Langmuir Probe

271

Figure 8.3  Typical electrical current and voltage data by a double Langmuir probe, Ep = 8.82 V/​cm, Ip = 0.21 Amps, Vp = 298 V.

probes is not overly critical, but the gap shall be a calibration subject of experiment for the data-​collecting process. From the Langmuir probe, a typical electrical current versus the voltage characteristic by a single or double probe is very similar. Particularly the outstanding features of the collected data are nearly identical, thus only the results from a double probe are depicted in Figure 8.3. At the location denoted as ISC, the negative current is the ion saturation current to one of the sense surfaces. When voltages applied to the probe are less than that value, all the current measured is due to the positively charged ions. As the applied voltage is increased from this point, more energetic electrons start to reach the probe surface. Therefore the net amount of current in the probe circuit is approaching zero, and the measured voltage is known as the floating potential. For the Langmuir single probe, this point is easily identifiable, as appeared in this data set. As the voltage is further increased from this value, the electron current increases rapidly and all ion and electron current is completely collected by the probe. This point on the current-​voltage characteristic is referred to as the completely collected current of electrons and ions (CCI). Applying higher voltage to the probe causes ions to be repelled, no ions can reach the probe, and the electron saturation current is measured to the other sensing surface; this point is identified as ESC in Figure  8.3. The counterpart of ESC is the ion saturation current (ISC) on the electric current and voltage characteristic. Beyond the electron saturation point (ESC), the plasma potential is determined and the electron number density can be evaluated from the electron saturation current. For electron

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temperature determination, the data reside on the current-​voltage characteristic between the CCI and zero current location. The single probe detects a much wider range of electron energies than the double probe, because the double probe only detects the more energetic electrons. This data range between the voltage polarities’ switching of the double probe tends to be rather small, and it is one of the disadvantages of the Langmuir double probe compared to the single probe. However, it is more than compensated for this shortcoming by a much reduced measured data scattering that reflects a lower level of perturbations to the plasma. A triple Langmuir probe has been developed; its configuration consists of two electrodes biased with a fixed voltage, and the third electrode is floating. The selected bias voltage is important for determining the electron temperature because the negative electrode draws the ion saturation current, which is measured directly together with the floating potential. Since the biased configuration is floating, the positive probe can draw at the most an electron current only equal to the same magnitude of the ion saturation current by the negative probe. The Langmuir triple probe has the advantage of simple biasing electronics, data analysis, and excellent time resolution, and it is relatively insensitive to potential fluctuations. Like the Langmuir double probe, these multi-​electrode arrangements are sensitive to the gradients in plasma parameters. Arrangements with tetra or penta probes have been used; however, the advantage of their application over the double probe is not completely ascertained.

8.2

Data Reduction for Langmuir Probes The fundamental formulation for Langmuir probes’ data reduction is built on the three following assumptions. First and foremost, the electron energy distribution in the plasma is Maxwellian. Second, the mean free path of the electrons is greater than the ion sheath around the tip and radius of the probe. This requirement limits the size of the probe to make the electrodes a micro instrument. Finally, the probe sheath sizes are much smaller than the probe separation distance; therefore, the electric current flowing into probes can be composed of the high-​energy electron saturation current and the ion saturation current. For practical applications, the formulations for Langmuir probe data reduction have been further refined to interpret accurately the electric current-​voltage characteristic according to relative length scales between the mean free path of electrons, the Debye length, and the radius of the cylindrical probe. The probe operations have been categorized into two major domains as the continuum and non-​continuum regions (Chung et al. 1975). In the continuum region, the measuring condition is further classified as the collisional thin sheath, thick sheath, and collision-​less thin sheath regimes. Similarly in the non-​continuum region, the operational conditions have been grouped into the conventional thin sheath and the orbital limit thick sheath, as well as the collisional thick sheath regimes. Corresponding to these operational regions, the collision-​ less, transition, and collisional models have

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8.2  Data Reduction for Langmuir Probes

273

Figure 8.4  Curve fitting of electrical current-​voltage characteristic with the collision-​less

model.

accomplished data reduction. The detailed derivation processes are elaborative and the formulas are rather complex; more information can be found in the works of Thornton and Chung and colleagues(Thornton 1971; Chung et al. 1975). The data reduction process using the Langmuir probes for plasma in the collision-​ less domain is rather straightforward, because the interrelationship between the electron-​saturated current and the voltage can be represented by a Boltzmann distribution.

I e = I esc exp( eϕ p kTe ) ϕ p < 0 (8.1) In Equation (8.1), the symbol ϕ p represents the potential between the probe sensing surfaces or the voltage drop across the electrodes, and e denotes the elementary electron charge. The electron saturated current is designated as I esc and can be calculated as:



I esc = ene A( kTe 2 π me )1 2 (8.2) where the ne is the electron number density of the undisturbed plasma, A is the total effective sensing area of the probe, and me is the mass of the electron. The curve-​fitting result of the collision-​less model for the electric current-​voltage data is depicted in Figure 8.4. The Langmuir probe data are taken in plasma generated in quiescent air at a low pressure of 6 Torr and the surrounding temperature of 330 K. The overall current delivered to the plasma varies from 0.15 to 0.31 Amps. The gap of the electrodes is 4.5 cm and the anode is a flattened hemisphere while the cathode is a large, flat plate. The Langmuir probe consists of two identical 0.5 mm diameter platinum wires each with an explored length of 3.2 mm. The separation

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distance of the two sensing surfaces is 3.2 mm. At the testing condition with the plasma number density of 1012/​cm3, the mean free path of ions is estimated to be 3 µ m, the mean free path of electrons is 17 µ m, and the Debye length is 7 µ m. The probe radius is 250 µ m. Therefore the Langmuir probe is operated in the continuum thin sheath regime; although the overall curve-​fitting result seems reasonable, the collisions still could affect the data reduction process. The electron temperature can be easily obtained from taking the derivative with respect to ϕ p from Equation (8.1) to get:

Te = e ( k d ln I e d ϕ p ) ϕ p < 0 (8.3) For data reduction to determine the electron density, the slope of the electric current-​voltage characteristic is required. The best practice for the Langmuir single probe is to curve fit the electron current versus voltage between the locations of the net zero electrical current and the CCI by an exponential function. The specific function chosen is that of a Boltzmann distribution in Equation (8.1). Note that this functional form is accurate only when the electrons have a Maxwellian distribution of velocity.



I e = c1 exp(c2 ϕ ) (8.4) In Equation (8.4), the electron current I e is obtained from the measured current by subtracting the ion saturation ion current from the measured electrical current: I e = I − I isc . From Equation (8.3), the electron temperature is easily calculated from the curve-​ fitting coefficient c2 as:



Te = e kc2 (8.5) The ion current velocity satisfies the Bohm sheath criterion if the sensing surface is biased by a large negative voltage. The Bohm inequality is usually marginally fulfilled and the ion velocity at the sheath edge is simply the speed of sound:



ui = [ k ( zTe + γ Ti ) mi ]1 2 (8.6a) In most circumstances, the ion temperature of plasma is negligible in comparison with that of the electronic temperature by electronic impact ionization. Thus, the ion velocity can be given as:



ui = ( kzTe mi )1 2 (8.6b) where the symbol z denotes the averaged charge number per ions. The value of z creates uncertainty in the analysis of the Langmuir probe data and is difficult to resolve. Approximations are frequently relied on. The ion saturation current density on the sensing surface is mostly due to ions, defined as I i = ni eui , and has been widely approximated as:



I i = 0.4ni eA( 2 zTe mi )1 2 (8.7) Once the electron temperature is known, the ion number density can be determined.

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8.2  Data Reduction for Langmuir Probes



275

ni = I i [ 0.4eA( 2 kTe mi )1 2 ] (8.8) In Equation (8.8), the plasma sheath area A is approximated by the total area of the sensing surface, and the mass of the ions is the averaged value of all the components of the plasma. It is dependent on the plasma composition, thus additional information regarding the specific composition and mass fraction of the plasma is needed. Based on the fundamental global neutrality property of plasma, the electron number density shall be identical to that of the ion: ne ≡ ni . For the single Langmuir probe, the major portion of the electrical current voltage characteristic below the plasma potential is dominated by the electron current. In contrast to the double probe, the slope evaluation is more complicated because the ion current is important for the entire characteristic. Thus, the electron temperature determination is more complicated because the ion electrical current is important for the entire characteristic. The slope evaluation for determining the electron temperature for a double probe has been suggested by Hershkowitz (1989) as:



Te = ( e k ){[ I e ,1I e ,2 ( I i ,1 + I i ,2 )][1 ( dI d ϕ )]}v = 0 (8.9) In Equation (8.9) all quantities in the bracket are evaluated at the location where the measured voltage ϕ is vanished. The electron and ion currents flowing into the sensing surfaces 1 and 2 are designated as I e ,1, I e ,2, I i ,1 and I1,2, respectively. For data reduction of the Langmuir double probe measurements, the best practice is conducting a least square curve fit for the electron temperature with a third-​ order polynomial of the applied voltages. This is the lowest polynomial that can be used to accommodate an inflection point on the segment of current-​voltage characteristic over the net zero electrical current (ϕ = ϕ o = 0) and the CCI locations. The slope of the current-​voltage characteristic is evaluated at the vanished current position.



I = c1 + c2 ϕ + c3 ϕ 2 + c4 ϕ3 (8.10) ( dI d ϕ )vo = c2 + 2c3 ϕ + 3c4 ϕ 2 The typical results of a double Langmuir probe are presented in Figures 8.5 and 8.6. A direct current discharge is generated between two plate electrodes with equal length and widths of 45.7  mm and 6  mm. The electrodes are relieved 5 degrees from parallel placement, the minimal distance between the electrodes is 50.8 mm, and the plasma is generated in an ambient condition with an air temperature of 300 K and a pressure of 7 Torr. A  Langmuir double probe is used for the data collection, consisting of two 0.2  mm diameter platinum wires 1.6  mm long; the separation distance of probes is 0.5 mm. Overall the deduced data exhibit a noticeable scattering band. Three factors affect the measurement accuracy and more significantly the data reduction process; the leading error may arise from a non-​ Maxwellian electron distribution because the discharge takes place in the transient between the collision-​less and collisional domains. This deficiency is a fundamental concern in gas kinetic theory and presently unresolved. The other concerns are attributable to the unsteadiness of plasma discharge and the contamination of the probe surface.

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Plasma Diagnostics

Figure 8.5  Ion number density in a 50 mA direct current discharge.

The ion number density distributions between electrodes for a direct current discharge are depicted in Figure  8.5 at 50 mA by an applied electrical potential of 1.2 kV. The distributions along the centerline of the electrodes at three survey positions are located on the distances of 0.55, 0.82, and 1.11 electrode lengths from the leading edge of the electrode. The data are displayed in normalized distances by the one-​half gap distance, d/​2 between electrodes and over the cathode surface. As has been anticipated, the ion number density should increase toward the cathode in the positive column to reach a peak level greater than 6.4 × 1011 cm 3 . The discharge domain also extends beyond the range over the electrodes. In Figure  8.6, the corresponding electron temperature distributions are given. Electron temperatures are generally uniform in the gap between electrodes except close to the cathode layer. The electron temperatures lie between about 4,000 K to 9,000 K, and spike to a value up to 25,000 K in the cathode layer. The electron temperature drops sharply away from the positive column and away from the electrodes. The measured value and the general behavior agrees well with the classic observations for a direct current discharge; the maximum electronic temperature in the cathode layer is around one eV (Petrusev, Surzhikov, and Shang 2006).

8.3

Emission Spectroscopy Spectroscopy of plasma is one of the long-​established diagnostic tools in astrophysics and plasma dynamics research (Griem 1964; Lochte-​Holtgreven 1968). Radiating atoms, molecules, and charged particles can reveal real-​time information

27

8.3  Emission Spectroscopy

277

Figure 8.6  Electron temperature in a 50 mA direct current discharge.

for plasma processes and ionized gas parameters for practical applications. In particular, the visible emission spectra are readily obtained by simple and reliable experimental means, but its interpretation of the observations can be fairly complex in thermal nonequilibrium conditions at low temperatures and in low-​pressure environments. Nevertheless, the intensity of emission is related to the particle density in the excited quantum states, while the radiative absorption/​emission is also correlated with the ground state of its element. In other words, the atomic structure of atoms and molecules is closely tied to the energy level of the ionized gas and directly related to the emission and absorption spectra. The rotational and vibrational temperature of ionized air can be measured using atomic or molecular emission spectroscopy. The dissociated and ionized high-​ temperature gas consists of atoms, electrons, ions, and neutral molecules. The continuous absorption and emission spectra of ionized gas appear as the result of bound-​free (photoelectric absorption emission) and free-​free (bremsstrahlung emission and absorption) transitions. For bound-​free and free-​free transitions, the absorption and the emission transition in a high-​enthalpy condition lead to continuous spectra. A bound-​bound (discrete) transition of atoms produces line spectra and the same transition of molecules yields a formation of band spectra. The spectra band actually is a group of a much greater number of spectral lines that are closely spaced with different intensities and frequencies. Under certain conditions, the individual lines are so closely clustered that they can be partially overlapped to appear as quasi-​continuous. The light emission spectrum is the fundamental optical property of gas and follows the allowable energy states of atomic and molecular systems (Herzberg 1950; Zel’dovich and Raizer 2002).

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The frequency or the wave number of an emission is uniquely related to the energy state of atoms or molecules. Therefore, once a spectra band of a chemical species is detected, the energy distribution of this molecule is easily determined. The temperatures of internal degree excitations can then be deduced by known distribution functions. For air plasma, the most adopted absorption and emission transitions for determining the vibrational and rotational temperatures of partially ionized gases are: the Schumann-​Runge band O2 ( B 3 Σ u− ) ↔ O2 ( X 3 Σ −g ) of oxygen; the first positive band N 2 ( B 3 Π g ) → N 2 ( A3 Σ u+ ) of nitrogen; and the second positive band N 2 (C 3 Π u ) → N 2 ( B 3 Π g ) of nitrogen, as well as the β band NO( B 2 Π ) ↔ NO( X 2 Π ) of nitro-​oxide. These choices are preferred because the theoretical results of these transitional bands are well known and the computational procedures for these spectra have been developed and fully validated. The rotational temperature of gas species is obtained by comparing the measured spectrum of the nitrogen second positive 0-​2, 0-​3, or 1-​3 transitions to their corresponding theoretical spectrums. In practice, the rotational temperature is determined by the best match of the calculated theoretical results and the measured spectrum. Assuming an optically thin collection cone, the intensity of a rotational line can be written as (Herzberg 1950):

I νν′" = ( N ν′ Aνν′" Qr )Sνν′" exp( − eν′ kT )( hc λ νν′" ) (8.11) In Equation (8.11), N ν′ and Aνν′" are the molecular population in the upper rotational level and the so-​called Einstein coefficient, respectively; the symbol Sνν′" denotes the rotational line strength, and the notation λ νν′" is the wavelength of the transition. The approximation of the measured spectrum is obtained from Equation (8.11) by summing up all the rotational lines and combining them into the instrument line shape function p( λ, λ νν′" ). The different broadening mechanisms of the measured spectrum produce a half-​width that often is negligible in comparison with the half-​width of the instrumental line function. The instrumental line shape function is obtained by measuring and curve fitting the spectra by a mercury discharge tube at a selected value. For the nitrogen second positive transitions, a reference wave number is often picked to be 435.833 nm. The measured spectrum can be improved by refining the instrumental line function. Therefore the half-​widths of the different broadening mechanisms in the measured data can be approximated by a Dirac delta function. Thus the measured spectrum is given by:



ν"

I νν′" = ( N ν′ Aνν′" hc Qr ) Σ[Sνν′" exp( − eν′ kT )[ p( λ, λ νν′" ) λ νν′" ] (8.12) ν′

where the instrument time function is designated as p( λ, λ νν′" ), the rotational line strength is characterized by the symbol as Sνν′" of the transition wavelength λ νν′". However, it must be borne in mind that Equation (8.12) is truly valid for the measured plasma that has a uniform rotational temperature in a data collection control volume. For nonuniform plasma, the error can only be reduced by minimizing the size of the data collection volume.

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8.3  Emission Spectroscopy

279

Figure 8.7  Setup of typical spectrometry measurement.

The common spectrometer has a typical setup and is depicted in Figure 8.7. The light emission from a mercury lamp passing through an ionized gas is collected by a quartz lens and focused on the entrance slit of a spectrometer. The focus length of the spectrometer together with the size of the grating defines the aperture of the system and determines the spectral resolution. The width of the entrance slit is important for the light throughput in the intensity of light, which impacts the spectral resolution. The resolution of 0.01  mm is commonly recommended. Usually twenty rotational lines are simulated at each of fifteen vibrational transitions for suitable data resolution. The output of the spectrometer is then fed to a photomultiplier tube (PMT), which is mounted behind the exit slit. The overall sensitivity of the spectrometer is strongly dominated by the type of detector. The photomultiplier scans a specific wavelength range with different cathode coatings and sensor types. Spatial resolution can be achieved by the adaption of a two-​dimensional detector. In general, the photomultiplier has a faster response time than the charge-​couple device (CCD), thus it is frequently adopted for spectrometry. The resulting signal is conditioned by an amplifier/​discriminator and transmits to a Log/​Linear rate meter. This rate counter converts the individual electrical impulses into an analog voltage, which subsequently is digitized and recorded on a computer. The typical setup displayed in Figure 8.7 can follow the temporal behavior of an emission line that has a relatively poor spectral resolution of ∆λ ≈ 1 − 2 nm, but with a good resolution in time. The spectral resolution is characterized by the choice of grating or the spectral line per millimeter; the optical components and the measured data are greatly improved with a focal length from 0.5 to 1.0 meter and with a grating of 1,200 lines per mm.

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Plasma Diagnostics

The most used method for determining the temperatures for molecular internal degrees of freedom is analyzing rotational lines of a vibration band of molecular nitrogen, which is a simple diatomic molecule. The rationale is that the energy level between rotational quanta in one vibrational state is usually around a tenth of an electron volt, thus the rotational population density can be reasonably characterized by its rotational temperature. In fact, the rotational temperature is obtained either from a Boltzmann plot or by a comparison of the measured vibration band with computational simulations. In the latter case, the gas temperature is determined by fitting the measurements of a vibrational band of molecular nitrogen to the computational simulation. The accuracy and reliability of internal degree temperature measurement are derived from the rotational line shape fitting of molecular species of plasma. Three different vibration transitions of nitrogen in air plasma, N 2 (C 3 Π u ) → N 2 ( B 3 Π g ), have been recorded to ensure measurement accuracy and repeatability. The rotational temperatures are obtained from contour analysis of the spectra, because the individual rotational quantum states are not resolvable as a result of the complexity of the spectra. The validating experiments are conducted in an evacuated glass cell. A  glow discharge is generated at a pressure of 3 Torr and a plasma current of 9 mA at approximately 1.3 kV. Theoretical rotational lines in a nitrogen second positive vibrational band are generated as a function of temperatures. These lines are then convolved with a Gaussian instrument function to produce a modeled band head spectrum. A least-​square-​fit to the measured spectrum results in a temperature measurement. A typical fit of measured data and the computed results is presented in Figure 8.8. The plasma emission is measured with a charge coupled device detector array with a 400 ms time constant. The temperature measurement accuracy can be ensured by comparing the plasma-​induced emissions of several vibrational manifolds. In the curve-​ fitting process to obtain temperatures of internal degrees of freedom, the collected spectrum is convolved with the slit function of the spectrometer, and the mean-​square-​root error between the predicted and measured spectrum is computed. If the curve-​fitting procedure is conducted in 10 K increments, then it effectively creates a 10 K resolution in the measured temperature. The curve-​ fitting results taken in a nitrogen discharge tube are compared as functions of the discharge current for N2 plasma at a pressure of 10 Torr. The measurements are collected for three different vibrational bands (0-​2), (0-​3), and (1-​3) by the averaged radiative emission. The results of Figure 8.9 also exhibit the temperature estimates from the on-​axis and the end-​view emission measurements. The fitted gas rotational temperature revealed a data scattering of ±40 K up to the gas temperature of 800 K. The discrepancy between the two sets of data is incurred by the radiation trapping of the low J lines of the low rotational quantum number. This conclusion is also supported by the electrical current axial electric field measurement. The displayed results also show that a careful analysis of several vibrational bands, along with a data fit for only high J lines, is needed. The gas rotational temperature measurement accuracy can be further ensured by comparing the plasma-​induced

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8.3  Emission Spectroscopy

281

Figure 8.8  Comparison of measured and simulated emission N2 (0-​3) transition.

emission of additional vibrational manifolds. However, note that the presented results are collected in a tightly controlled nitrogen discharge tube, which may not be achievable in practical environments. The vibrational temperatures of each chemical species are determined using the well-​known Boltzmann-​plot method (Howatson 1976). For air plasma, the most adopted approach is again to measure the spectrum of the second positive group of nitrogen transition N 2 (C 3 Π u ) → N 2 ( B 3 Π g ). The intensity of a vibrational transition is given by:

I ν′ , ν " = DN ν′ ν4ν′ , ν "Sν′ , ν " (8.13) All symbols are identical to that of Equation (8.11). Assume a Boltzmann distribution relating the upper level of population and temperature to get:



N ν′ = [ N Qν (T )] exp[ − G ( ν′ )hc kT ] (8.14) Substituting Equation (8.14) into Equation (8.13) provides a straight-​line relationship between the normalized intensity of the vibration transition and the vibrational temperature; we have:



ln[ I ν′ , ν " ν4ν′ , ν " Sν′ , ν " ] = −[ hcG ( ν′ ) kT ] + const (8.15)

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Figure 8.9  Rotational temperatures data measured in N2 discharge tube, p = 10 Torr.

The normalized intensities of vibrational transitions are plotted as a function of the normalized transition energy. A straight line is curve-​fitted through the measured data and the rotational and vibrational temperatures are extracted from the slope of the fitted line. In data processing, the transitions with intensities less than 5 percent of full scale are rejected. In the high-​frequency domain, the rotational lines are not fully resolved by the spectrometer, thus rotational temperatures are estimated by comparing a computational simulated spectrum to the measured spectrum. As an illustration, the spectral measurements taken in a nitrogen discharge tube are presented in Figure 8.10, to data collected from a discharge between two electrodes in a small, hypersonic MHD channel. The discharge conditions are vastly different from each other; the MHD channel operates at a static temperature of 43 K, a pressure of 9 Torr, and an air flow velocity of 670 m/​s. The lower temperature of the MHD data is qualitatively discernible from the spectra. Rotational bands of the MHD data in comparison with the discharge tube spectra are pushed to longer wavelengths, indicating a higher population in the lower rotational states and a depopulation of the higher rotational quantum states. Also, the spectral peaks with the MHD data are shifted to longer wavelengths, probably due to the increased intensity from P-​branch rotational transitions at wavelengths near the band head. Analysis of the spectral confirms these qualitative observations and yields a rotational temperature of approximately 60 K for the MHD measurement, compared

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8.3  Emission Spectroscopy

283

Figure 8.10  Compare data from the charge tube and the MHD channel, N2 second positive transition.

to approximately 490 K for the discharge tube data. Vibrational temperatures for the MHD data are also lower, approximately 3,100 K compared to 6,700 K for the discharge tube. The result of a practical application of emission spectroscopy for vibration temperature measurement is shown in Figure  8.11. The vibration temperatures are collected in a hypersonic low-​density and low-​temperature MHD channel measured for a range of discharge currents. The data are measured on the centerline of two parallel discharging electrodes and at a distance of 32.0 mm from the leading edge of the electrodes. The centerline of the electrodes is located in the positive discharge column over the cathode (y = 12.7 mm) but far above the cathode layer. Unsteadiness as measured by the root-​mean-​value for the vibrational temperature is quite high for discharges above 100 mA. Vibrational temperatures range from about 4,000 K at 50 mA, and increase up to about 200 mA. Above this discharge current, the vibrational temperatures appear to be saturated at around 5,000 K to 7,000 K and the data scattering band width also increases. The increased data scattering is probably due to the nonuniformity of the direct current discharge. In these conditions, the discharge exhibits a more complicated variation than that observed at lower discharge currents. The direct current discharge becomes an abnormal discharge when the power unit cannot supply the minimum discharge current; the spark will repetitively ignite in the discharge gap. The spark is made evident by an occasional local bright spot that

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Figure 8.11  Vibrational temperatures measured by emission spectroscopy.

has characteristics more akin to an arc than a glow discharge. In an arc, a greater amount of local heating takes place. At the highest discharge current of 400 mA with a discharge voltage of 300 V, the power expended is 120 W. The data also show a greater amount of uncertainty has been confirmed by extensive surveys in the gap distance between the electrodes. Finally, note that the precise spectral line measurements for line broadening are important plasma diagnostic tools through which a myriad of parameters can be obtained, but the methodology demands a high spectral resolution of a spectroscopic system. Therefore, it is not usually utilized in practical engineering applications. The spectral line-​broadening phenomena include Doppler broadening, pressure broadening, electrical field-​induced Stark broadening, and changing the line shape by the magnetic field, known as the Zeeman effect. The Stark and Zeeman broadenings are critically important to study high-​electron density plasma and in strong magnetic fields; therefore, they are considered out of the scope of our discussions. More detailed information on line-​broadening mechanisms can be found from the works by Griem (1977). However, it should be borne in mind that the diagnostic technique of spectroscopy is extremely effective for developing unique insights on plasma properties, because it is the only method that provides the line-​of-​sight averaged plasma parameters.

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8.4  Microwave Attenuation in Plasma

8.4

285

Microwave Attenuation in Plasma A microwave is simply an electromagnetic wave generated by electric excitation and synchronized with perpendicular electric and magnetic fields. The waves are represented as vector fields, so they may interact through constructive or destructive interference by the property of superposition. Electromagnetic waves also possess the properties of refraction, diffraction, and reflection as light; when they cross from one medium to another, they will scatter from an interface of media (Krause 1953). Furthermore, the driven mechanisms of electromagnetic waves are dependent on the electric and magnetic properties of the medium, which have the controlling influence on how the waves disperse and dissipate, especially when the medium contains charged particles that can alter the electromagnetic field. Therefore in a semi-​conductor, the wave can be completely reflected from a medium interface or the wave amplitude of the incident wave can be attenuated in a very short distance within the electrically conducting medium. This phenomenon is known as communication blackout in microwave communication. This salient feature of microwave attenuation in plasma has been used for plasma diagnostics. The absorption is measured by the microwave power loss after the microwave is transmitted from an antenna and passed through plasma in application. The microwave power is recorded as an averaged voltage of the wave signal over the beam cross-​section and during several periods of the probing wave. The total power transmitting through a control volume is defined by the surface integral of the Poynting vector. From the Poynting theorem, the depth of the plasma and the wavelength of the interrogating microwave are crucial parameters for plasma diagnostics. Equally important, the recorded data are the accumulative information along the beam path, and by assuming the plasma has a uniform property; the edge diffraction from the antenna and wave reflections from the media interfaces is negligible. In spite of all the aforementioned uncertainties, the dominant phenomena for plasma diagnostics are that the microwave amplitude decays exponentially and exhibits a linear phase shift in an electrically conducting medium. A linearly polarized transverse electromagnetic wave in a dielectric or electrically nonconducting medium is the principal or the zero-​order electromagnetic wave. The generalized Ampere law gives a temporal and spatial relationship between the magnetic and electric field strengths. In a dielectric medium, these relationships can be manipulated to separate the electric and magnetic components of the temporal and spatial derivatives for the plane electromagnetic wave equations. These results are also referred to as D’Alembert equations; they have been presented in Chapter 2.3:



∂2 E y ∂t 2

=

∂2 H z 1 ∂2 E y 1 ∂2 H z , = (8.16) ∂t 2 µε ∂x 2 µε ∂x 2

The general solutions to these equations, Equation (8.16), are simply the harmonic functions in time and space:

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Plasma Diagnostics

E y = E0 [sin β( x + vt ) + sin β( x − vt )] (8.17a) H z = H 0 [sin β( x + vt ) + sin β( x − vt )] The general solutions are written in exponential form in terms of amplitude and wave frequency to appear as:



E y = E0 exp( α x ± ωt ) (8.17b) H z = H 0 exp( α x ± ωt ) One of the arguments of exponential function can be rewritten as βvt = ωt, because the phase velocity is a function of wave frequency and wavelength: v = λ f . Then the circular frequency can be defined as: βv = ( 2 π λ ) f λ = 2 π f = ω . The phase velocity of the electromagnetic wave is:



v = ω β = λf =1

µε (8.18a)

and the group velocity of the electromagnetic wave can be given as a function of the phase velocity and its rate of change with the wavelength:

ug = d ω d β = β ( dv d β ) = v − λ dv d λ . (8.18b) From the general solutions of the plane electromagnetic wave, the ratio of E y H z = µ ε is the so-​called intrinsic impedance of the lossless medium. The energy transfer by a traveling electromagnetic plane wave can be evaluated by the electric and magnetic energy density De = εE 2 2 and Dm = µH 2 2; both have the physical unit of Joules per cubic meter. The energy increase within a control volume must be balanced by an inflow of energy. Therefore, the energy per unit area passing per unit time at any location is required to satisfy a divergent condition:



∇ ⋅ S = − ∂ ∂t( De + Dm ) (8.19a) Recall from our earlier discussion on plane electromagnetic wave equations that:



S = E × H (8.19b) In fact, Equation (8.19b) is the definition of a Poynting vector, which gives the rate of energy flow per unit area in a wave and perpendicular to the electric and magnetic fields. The power transmitted by the TE1,0 wave in a waveguide can be computed easily by the Poynting vector formulation, Equation (8.19b). For the dominant wave, the calculated component of the Poynting vector along the waveguide, Pz, is depicted in Figure 7.12. as an example. Media of three different normalized electric conductivities, σ ωε of 0.0, 0.0625, and 0.25 were simulated. For the dielectric medium, the power transmitted by the TE1,0 wave stays constant, as it should, over the entire length of the waveguide. However, the electromagnetic energy is mostly dissipated in 8.5 wavelengths in a medium with the lower electric conductivity of 0.06. At the highest normalized electric conductivity condition simulated, σ ωε = 0.25, the electromagnetic energy flow essentially ceases after the wave has propagated a short distance of 2.5 wavelengths (Figure 8.12).

287

8.4  Microwave Attenuation in Plasma

287

Figure 8.12  Attenuation of Poynting vector in a waveguide.

An electromagnetic wave attenuates rapidly in a conducting medium, and the wave penetrates only to a very shallow depth in a perfect electric conducting medium. The electromagnetic plane wave described by the generalized Ampere law in an electric conducting medium and by applying Ohm’s law yields:

∇× H = J +

∂D ∂D = σE + (8.20) ∂t ∂t

By taking the derivative of Equation (8.20) with respect to time and assuming the electric permittivity and magnetic permeability are slow-​varying functions of time and space, the equation of electromagnetic wave propagation becomes:

εµ

∂2 E ∂J −µ = ∇2 E − ∇( ∇ ⋅ E ) (8.21a) ∂t 2 ∂t

The general solutions of the electric strength and electric current density are expressible as harmonic functions in time and space:

E = Eo ei ( ωt ± kr ) (8.21b) J = J o ei ( ωt ± kr ) Since the wave is harmonic in time, we have then the electric complement of the plane wave equation in an electrically conducting medium as:

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Plasma Diagnostics

∂2 E y ∂x 2

− (i µσω − µεω 2 )E y = 0 (8.22a)

By letting γ 2 = i µσω − µεω 2, which is a complex wave number, the equation of plane wave motion in a conducting medium becomes:

∂2 E y ∂x 2

− γ 2 E y = 0 (8.22b)

Again, assume that the temporal varying of the electric field can be represented by a simple harmonic motion E y = Eo ei γ x. For a conductor, then:

γ = (1 + i ) ωµε 2 = α + iβ (8.23) The real part α is associated with wave attenuation and the imaginary component β describes the phase relation. The attenuation factor is given as e − ωµσ 2 x , and the depth of wave penetration in a conducting medium is defined by the magnitude of Ey decreasing to the original value of 1/​e = 0.368 times. The penetration depth of the electromagnetic wave into a lossey medium is (Krause 1953):



δ=

1

πµσ f

(8.24)

It is clear that an electromagnetic wave will be dissipated when propagating in an electrically conducting medium such as plasma, and at a high frequency, the wave will be damped out more quickly than at a low-​frequency counterpart. Using microwave for plasma diagnostics, the electromagnetic wave is required to propagate across media with a different electric permittivity, with distinct wave speeds in different regions. For our present purposes, the phenomena at the media interface are eliminated by studying the plane wave motion in a collection of time sequences. The different electric permittivity between the initial condition of an electric conducting medium and the computational domain leads to different wave speeds normalized by µε from 0.75 to 1.0. The effect of the varying intrinsic impedance µ ε of a lossless medium can be achieved by changing the permeability or the permittivity of the medium; it changes the attenuation factor of wave propagation. The net result is analogous to changing the electric conductivity of the medium. The computational results display the electromagnetic wave dissipation and dispersion from the input data in a parametric form of σ ωε. The solution of the wave in a dielectric medium is set as the initial value and includes a reference for the purpose of comparison. Temporal evolutions of numerical results are depicted over a span of twenty periods of the wave motion. The computational results are depicted in Figure 8.13 to show the transient is permitted to evolve in a uniform medium of a different permittivity. The wave motion shows a drastic attenuation and phase shift. At the end of twenty periods, the microwave is completely dissipated across the computational domain.

289

8.5  Microwave Dispersion in Plasma

289

Figure 8.13  Computational simulations of microwave attenuation.

8.5

Microwave Dispersion in Plasma One of the most dramatic manifestations of plasma is the dispersion of electromagnetic wave propagation. The amplitude of an incident wave always attenuates in a weakly ionized gas, because moving charged particles must interact with the local electromagnetic field and the energy of the wave will be dissipated. A  complete reflection of the microwave at the interface of the media also can occur when the frequency of the incident wave is lower than the plasma frequency; it becomes the same as the cut-​off frequency. Another unique feature of wave motion in plasma is that it is not solely dependent on the collision mechanism as in an acoustic wave. Multiple modes of the transverse wave also exist such as the Alfven wave associated with ions and the Whistler wave related to electrons. Both transverse waves are generated by an initial disturbance perpendicular to the magnetic field. For this reason, these transverse waves are often regarded as the vibration of a line of magnetic force. All the direction-​dependent disturbance propagations lead to a very complex wave system and introduce fascinating physics in resonance and energy damping. The amplitude and phase angles of an electromagnetic wave propagating in an electrically conducting medium are describable by a complex wave number as

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shown by Equation (8.21). According to Heald and Wharton (1965), the interaction of discrete ions and neutral molecules in plasma can be represented by a viscous damping term. The consequence is that the electron equation of motion has an additional parameter νc, the collision frequency for momentum transfer. A careful examination of this approximation shows this damping term is correct only when the collision frequency is independent of the electron velocity. By this approach the complex wave number becomes:

ω 2p   ω 2p ( νc ω )   γ = α + iβ = 1 − 2 −i  2 (8.25) 2  2   ( ω + νc )   ω + νc  where ω and ω p designate the frequency of the incident microwave and the plasma frequency, respectively. A  nearly identical expression is derived by Mitchner and Kruger (1973) by adopting the generalized Ohm’s law to describe the electric current density and the electric field intensity, from which the temporal behavior of the electric current density is approximated as: ∂J = νc ( σ E − J ) (8.26) ∂t



Recall that the electric conductivity is defined as σ = ne e 2 me νc . By means of the complex wave number, the refractive index and attenuation index can be derived (Heald and Wharton 1965). Electromagnetic wave propagation in plasma of a uniform permeability has different behaviors in three different frequency domains. When the plasma frequency is much greater than the collision frequency, ω p  νc , and the collision frequency is greater than the incident microwave, νc > ω , the plasma behaves according to the elementary theory of a conductor, and the plasma penetration depth is approximately:

δ=

1 c  2 νc  =   α ωp  ω 

12

 ω  1 − 2 ν  (8.27) c

In the intermediate frequency domain, ω p > ω > νc, the complex wave number becomes purely imaginary and the general solution E = Eo exp − ( ωt ± kr ) can no longer describe a propagating wave. It is known as an evanescent wave, and on average it does not transport energy and will be reflected from the media interface. Under this condition, the microwave will not propagate in the plasma, also known as the communication blackout phenomenon. In the high-​ frequency domain, the incident wave has a higher frequency than the plasma frequency, ω > ω p, plasma behaves as a low electric-​conducting medium. It is also the operational condition for plasma absorption diagnostics. The electron number density has a square-​ root-​ dependent relationship to the 2 12 plasma frequency, ω p = ( ne e me ε ) . The electron number density of the probed plasma can be experimentally determined from a detected incident and transmitted voltage through the plasma. The fundamental optical properties of plasma in the frequency range are well known. The refractive index, attenuation

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8.5  Microwave Dispersion in Plasma

291

index, and penetration depth of a microwave provided by Heald and Wharton (1965) are: 12



2 12 2  ω 2p  1  ω 2p   ω 2p νc    1  µ =  1 − 2 + 1 − 2 + 2    2 2 2  2  ω + νc  2  ω + νc   ω + νc ω   

(8.28a)

12



2 12 2  ω 2p  1  ω 2p   ω 2p νc     1 (8.28b) χ =  − 1 − 2 + 1 − 2 + 2    2 2 2  2  ω + νc  2  ω + νc   ω + νc ω   

δ=

c ωp

 2ω 2   ω 2p   ν ω  1 − ω 2  (8.28c)  c p

Equations (8.28a), (8.28b), and (8.28c) are the basic formulas for understanding and evaluating the microwave-​probing diagnostics. The attenuation of the microwave is very small for the investigated condition and the ratio of the plasma frequency to the incident microwave plays a dominant role. It is interesting to note that the refractive index is relatively insensitive to collisional damping. All the indices also change smoothly within boundaries of the three frequency domains, with the exception of the resonance of the incident microwave and the plasma frequency. The abrupt cut-​ off behavior is very distinct. The situation of plasma immersed in a static magnetic field is considerably more complicated because the Lorentz acceleration, u × q , associates with moving charges. The simplest case considered is usually addressing the wave propagation along the magnetic field in the positive z-​coordinate, then by splitting the normal components of the electrons’ motion into the x-​y symmetry plane of a circularly polarized field. The gyrating motion of electrons is described by equivalent components of electric intensity perpendicular to the direction of an externally applied magnetic field. By this conventional approach, two conductivities for the circular polarized field are obtained:

σ x = ( ne 2 me )( νc + j ( ω + ω b )−1 (8.29) σ y = ( ne 2 me )( νc + j ( ω − ω b )−1 In Equation (8.29), the electron cyclotron frequency or gyro frequency is introduced as ω b = eB me . This frequency is the angular frequency of an electron as it gyrates around the magnetic field. The equivalent complex wave number is readily obtained, and in the absence of a magnetic field is:



ω 2p νc ω 2p ( ω ± ω b )     γ = 1 − − i (8.30)   2 2 2 2  ω[( ω ± ω b ) + νc   ω[( ω ± ω b ) + νc  Similarly, the refractive and attenuation indices of the electron cyclotron wave can be calculated. The formulations of a microwave propagating in the transverse magnetic field are akin to that in the absence of a magnetic field within different frequency domains. The behaviors for plasma propagation in a steady transverse magnetic field now must include the cyclotron frequency of an incident microwave in the x-​y plane for evaluations. The results of optical indices of a microwave

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propagating in a transverse magnetic field, and that in an externally applied magnetic field of general arbitrary direction, can be found in the works of Heald and Wharton (1965).

8.6

Microwave Probing Simulations For microwave probing, the incidence wave must have a higher frequency than that of the plasma, or the incident wave wavelength should be much smaller than the plasma dimension. Therefore, there is a practical upper limit to using microwave probing as a diagnostic tool for high-​density plasma (ne < 1015 / cm 3 ). Beyond this range, an optical probing is more suitable to operate at a wavelength under one micron or a frequency exceeding 300 THz. Using microwave probing, the incident microwave at a millimeter wavelength operational range is transmitted by means of waveguides and emitted through plasma using antennas. A pair of pyramidal horns is preferred because they can collimate the wave and reduce diffraction on the beam path through the plasma region. The operating principle is based on the energy dissipated by a microwave when passing through plasma. The electron number density of plasma in the microwave path can be measured by the difference between the voltage or power of the incident microwave and the voltage or power transmitted through the plasma. The microwave probing diagnostics by experimental and computational means are presented in the following. A typical microwave attenuation measurement arrangement around 18 GHz is depicted in Figure  8.14; it consists of a microwave generator usually with built-​ in transmitted and reflected energy detectors. The transmitted voltage signal is measured by a high dynamic range diode detector, while a diode detector coupled with the output line measures the reflected voltage signal. Through a coupler, a frequency counter is connected to the microwave generator to determine the operational frequency. The output of the microwave generator is transmitted to a waveguide and antenna by a semi-​rigid coaxial cable. The microwave antennas frequently adopted consist of a pair of pyramidal horns operated in the X band (8.20 to 12.4 GHz) or the K µ band (12.4 to 18 GHz) with suitable aperture dimensions. The antennas are placed and aligned by an optical guide. The output from the microwave detector is exclusively dependent on the voltages inputted to the antenna and received from the antennas. The data from the receiving antenna are measured by a transmitted voltage or power detector. Prior to sending this signal string to the oscilloscope, the data are usually conditioned by a combination of filters and amplifiers. In order to minimize the noise-​to-​signal ratio, a digital oscilloscope is often required to collect voltage readings over an extensive period of time and to calculate an average sampling data. The widely used data-​processing procedures are a single-​frequency or a dual-​ frequency method (Frederick, Blevins, and Coleman 1995). For the single-​ frequency method, the electron number density is determined by the voltage or

293

8.6  Microwave Probing Simulations

293

Figure 8.14  Schematic of microwave probing system.

power difference between the incident and transmitted signal and knowledge of the plasma depth.

ne = ( me cε e 2 )( νc d )( ϕinc − ϕtrn ) Vinc (8.30) = ( me cε e 2 )( νc d )( Pinc − Ptrn ) Pinc In Equation (8.30), the subscript inc and trn designate the voltage or power incident to and transmitted through the plasma. The notation d denotes the depth of the plasma domain. The more accurate dual-​frequency method requires a transmission coefficient of the wave propagating through a medium, including the attenuation index and the power of incident transmitted. The electron number density is obtained by solving the index of attenuation, Equation (8.28b). However, the microwave probing system is designed to provide voltage data, while the data-​reduction process requires values in power units. The interrelationship can be generated by a microwave generator in terms of power in milliwatts (dBm), and the calibrated function is best prepared in a logarithmic data fit. In the dual-​frequency data-​reduction process, the experimental data can be reduced to two unknowns: collision frequency and the electron number density. The indeterminacy of the unknowns to the number of equations is resolved by collecting data from two different incident microwave frequencies. The dual-​frequency method from theoretical consideration is more accurate for plasma in collision-​dominated realms. The overall accuracy of the microwave attenuation measurements vary greatly from less than 10  percent to 100  percent, depending on the discharge condition. However, it may be worthwhile to point out the rather large data accuracy uncertainty in an appropriate

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perspective, because the results are generally within the range of 109 to 1013 per cubic centimeters. A factor of two differences in measurements may be acceptable for most circumstances. The computational simulation capability for plasma microwave diagnostics is a multidisciplinary endeavor in which the interactions between electromagnetics, gas dynamics, and chemical kinetics are involved. Under practical circumstances, ionization and recombination are far from thermodynamic equilibrium and the required knowledge in these disciplines has been gained mostly at the phenomenological level. To alleviate these presently irresolvable complexities, the nonequilibrium ionization process is not included in the numerical simulation. Furthermore, the nonlinear gas dynamics interactions are also excluded, so the computational simulated microwave attenuation is focused only on the plasma diagnostics in quiescent air. The governing equations for the computational simulation are built on the solution to the three-​dimensional Maxwell equations in the time domain. The closure of the partial differential equation system is achieved by the constitutive relationship for describing the electric current density and the electric field intensity. Therefore, the plasma is described by global behavior in the form of its electric conductivity. The rate of change for the electric charge density is derived from the generalized Ohm’s law, Equation (8.26).

∂B ∂t + ∇ × E = 0 (8.31a)



∂( ε E ) ∂t − ∇ × ( µ B ) = − J (8.31b)



∇ ⋅ B = 0 (8.31c)



∇ ⋅ ( ε E ) = ρe (8.31d)



∂J ∂t − νc ( σ E + J ) = 0 (8.31e) The two Gauss laws, Equations (8.31c) and (8.31d), are automatically satisfied for the computational simulations, because the initial value of the solution is a homogeneous function and the global charged particle number density vanishes in the plasma. Thus the two divergence equations can be eliminated from the solving procedure (Shang 2006). The governing equations constitute a hyperbolic partial differential system that must be analyzed in an unbounded computational domain. An effective means to eliminate the wave reflection from a finite computational domain is using the unique characteristics of the hyperbolic differential system, because a group of variables is invariant along the characteristics of the wave motion. For computational microwave diagnostic simulations, the radiation field from a pair of antennas cannot be described by a rectangular coordinate system, but can be easily described by body-​ oriented transformed coordinates. The transformed coordinates map the surfaces of the pyramidal horn and its aperture on the constant value coordinate lines to facilitate the boundary condition implementations.

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8.6  Microwave Probing Simulations

295

The complete equations system cast into the flux vector form generalized curvilinear coordinates by a coordinate transformation, ξ( x, y, z ), η ( x, y, z ),  and ζ( x, y, z ):

∂U ∂t + ∂Fξ ∂ξ + ∂Fη ∂η + ∂Fζ ∂ζ = − J (8.32) where the dependent variable U is the function of U(B, E, J), and the flux vectors are functions of the metrics of the transformed coordinates and the flux vector in Cartesian coordinates Fξ = Fξ ( ξ x Fx , ξ y Fy , ξ z Fz ) , Fη = Fη ( ηx Fx , ηy Fy , ηz Fz ), and Fζ = Fζ (ζ x Fx , ζ y Fy , ζ z Fz ). The metrics of the coordinate transformation from Cartesian (x,y,z) to a generalized curvilinear, body-​oriented coordinate ( ξ, η, ζ ) are denoted as ξ x, ηy, ζ z, etc. The flux vectors of the time-​dependent Maxwell equation on the Cartesian frame are column vectors along the Cartesian coordinates:



Fx = [ 0, − E z , E y , 0, Bz µ, − By µ ]T Fy = [ E z , 0, − E x , − Bz µ , 0, Bx µ ]T (8.32a) Fz = [ − E z , E x , 0, By µ, − Bx µ , 0 ]T For the hyperbolic partial differential equations, a domain of dependence is bounded by characteristics and reflects the domain of dependence of wave physics. The direction of wave propagation is governed by the eigenvalue and eigenvector structure of the system of equations. The split flux vectors F+ and F-​ are uniquely associated with the sign of the eigenvalues of the flux Jacobian matrix, ∂F ∂U . Physically, the splitting procedure duplicates the direction of signature propagation in the domain of dependence by the signs of the eigenvalues: λ +  and λ − .



Fη = Sη (U )( λ +η + λ −η )S −1 (U ) = Fη+ (U L ) + Fη− (U R ) Fζ = Sζ (U )( λ ζ+ + λ ζ− )S −1 (U ) = Fζ+ (U L ) + Fζ− (U R ) (8.32b) Fξ = Sξ (U )( λ ξ+ + λ ξ− )S −1 (U ) = Fξ+ (U L ) + Fξ− (U R ) The notations S and S–1 are the similarity matrices and its right-​hand inverse, which have been derived from the diagonization process of the flux vectors F. By the split flux vector formulation into the right-​running and left-​running components, this far-​field boundary condition is imposed by setting the incoming or the reflected flux vector component to a null value, Equation (8.32b). In essence, at the boundaries of the computational domain, all reflected waves are suppressed using the signs of the local eigenvalue as the discriminator (Shang and Fithen 1996).



F ζ− = 0 (8.32c) On the media interfaces of the pyramidal horn and plasma and air, the pertaining boundary conditions in the absence of an externally applied magnetic field are:



nˆ × ( E1 − E2 ) = 0 nˆ ⋅ [( ε E )1 − ( ε E )2 ] = ρs (8.32d) The computational plasma diagnostic simulations are produced by the monotone upstream center scheme for conservative laws (MUSCL), through the flux vector reconstruction across the control surface. The spatial resolution yields third-​order

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Figure 8.15  Validating microwave attenuation in isotopic plasma, by a TE1,0 wave ω = 4GHz, 0.0 < σ ωε < 0.25 .

accuracy, and a fourth-​order Rung-​Kutta method is adopted for time integration (Shang 2006). The first verification step for the computational simulation is conducted for a microwave propagating through a waveguide. For a transverse plane electromagnetic wave in the TE1,0 mode and traveling in the z-​direction, the wave motion is adopted as the reference for comparison. The numerical results are generated on a (25 × 25 × 197) grid system and displayed in Figure 8.15. The behavior of a microwave traveling in plasma depends strongly on the electric conductivity σ and electric permittivity ε of the medium, as well as the transmitting frequency ω of the electromagnetic wave. In fact, the parameter σ ωε is the criterion for classifying the medium as dielectric, quasi-​conductor, and conductor. The x-​component of the magnetic field intensity Hx over a range of electric conductivities at fixed values of microwave frequency and plasma electric permittivity is given together with theoretical results in the range of 0.0 < σ ωε < 0.25. The simulated microwave in the transverse electrical T1,0 mode propagates through the plasma at a frequency of 4.0 GHz with a wavelength of 7.495 cm. The computational electromagnetic wave propagating through air is unimpeded without any attenuation and the difference with theory is negligible. The microwave traveling in the plasma is devoid of a media interface and reveals a monotone attenuation proportional to the magnitude of the electric conductivity σ ωε = 0.125 . By doubling the value of electric conductivity,

297

8.6  Microwave Probing Simulations

297

Figure 8.16  Instantaneous electric field intensity contours in air between antennas ω = 12.5 GHz, σ ωe = 0.0 .

σ ωε = 0.25 , the wave is completely dissipated in 4.75 wavelengths. The numerical results of wave attenuation are in excellent agreement with the theory; the maximum discrepancy is less than 1 percent. Taking advantage of the symmetric configuration of the pyramidal antennas with respect to the microwave propagation direction, only a half-​plane of the entire radiating field is simulated on the mesh system of (75 × 5 × 142) for the entire computational domain. The transverse wave is also assumed to be uniformly across the y-​coordinate, thus only five planes in the y-​coordinate should be sufficient to capture the essential physics. The normalized physical dimensions of the computational simulation by the minimum height of the aperture at 0.938 cm are (15.74 × 4.71 × 26.89). All computational simulations are conducted at a microwave frequency of 12.5 GHz (wavelength of 2.938 cm), thus the total length of the antenna is 3.307 wavelengths. The microwave is required to travel 10.71 wavelengths to cover the entire distance from the entrance of the transmitting pyramidal horn to the exit of the receiving pyramidal horn. The microwave propagation is strictly a time-​dependent phenomenon that may have a periodic asymptote. The instantaneous radiating field of a microwave propagating in air between the transmitting and receiving antennas is presented in Figure 8.16. The density contours of the electric field intensity ( E x2 + E y2 + E z2 )1 2 are depicted at forty equal increments from the entrance of the antenna. The TE1,0 wave transmitted from the waveguide exhibits the development of a spherical wave front

29

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Plasma Diagnostics

within the pyramidal horn and the wavelength decreases as the wave approaches the antenna exit. As the wave completely leaves the antenna into the free space, the edge diffraction is clearly indicated. The radiating field reveals a very complex wave interference pattern, including the diffractions at the edges of the transmitting and receiving antennas. The numerical result makes the phase shift of microwave propagations in the transmitting antenna and those in the free space clearly observable. The exiting microwave therefore is not in phase with the emitting microwave and has a different wavelength in free space than that within the waveguide, which is in perfect accord with the antenna theory. Equally interesting is that the multiple reflecting waves originating from the receiving horn lead to a strong wave interference with the incoming wave. However, the edge diffraction from the antenna is not detectable from the computed result of the y-​component electric field strength Ey. It is important to know that even for the microwave propagating through free space without any attenuation, the time-​ averaged ratio of power received by the receiving horn from the transmitting horn is around 0.1905. In other words, only one-​fifth of the microwave energy of the TE1,0 wave is received for the plasma diagnostic arrangement and the rest has been propagated and reflected into free space. This observation has correlated reasonably well with experimental measurements. The length or the thickness of the microwave’s path through a plasma domain is a critically important input for microwave probing simulation, because it is the length that the wave amplitude attenuated. Uncertainty of this information in experimental measurement often becomes a dominant error for an oscillating plasma discharge. Computational simulation can easily demonstrate this observation. At the microwave frequency of 12.5 GHz, a total of four different plasma sheets ranging from the thickness of 0.46 to 1.82 wavelengths are conducted by the magnitude of electric field intensity. The contours are depicted by all instantaneous values that are recorded at t = 28.25 s by forty equal increments from the entrance of the transmitting horn. The increased microwave attenuation is easily appreciated by the diminished number of the contours. The interacting field of a TE1,0 microwave at the plasma sheet of thickness of 1.823 wavelengths across antennas stands out. In Figure 8.17 by assigning an electric conductivity at 69.5 mho/​m, the simulated parameter σ ωε acquires a value of 0.25. The leading and trailing edges of the plasma sheet are located at z λ  = 4.338 and z λ  = 6.133, respectively. Substantial microwave attenuation is clearly visible when propagating through the plasma sheet and the reflected wave from the walls of the receiving pyramidal horn is barely identifiable. The observation is further substantiated by the fact that the suppressed absolute value of the Poynting vector along the wave path essentially vanishes at 1.66 wavelengths from the transmitting antenna aperture. The plasma in a small hypersonic microwave channel is characterized by a very low electric conductivity beneath 20 mho/​m, and the thickness of the wave path also has a highly oscillating discharge domain; the numerical accuracy therefore requires an extremely demanding high resolution. A preliminary experimental and

29

8.6  Microwave Probing Simulations

299

Figure 8.17  Instantaneous electric field intensity contours in plasma, ω = 12.5 GHz, λ = 2.398 cm, σ = 69.5 Mho m.

computational side-​by-​side investigation effort has been conducted for the plasma diagnostic project by using pyramidal horn antennas across the hypersonic MHD channel (Kurpik et  al. 2003; Shang 2006). In addition, the computational simulation must assume the plasma is uniform in the MHD channel and let the averaged electrical conductivity be the lowest estimated value of one mho/​m across the discharge domain. Nevertheless, the time-​averaged microwave energy of 4.2 microwatts received by the antenna is comparable to the measured data. Figure 8.18 depicts the computational microwave power distributions in the receiving horn antenna and the transmitted microwave power in free space is included as a reference comparison. In relatively low electrical conductivity conditions, the electromagnetic fields exhibit very little difference from the microwave propagation in free space. It becomes evident by direct contrast to the microwave power transmission in free space that the power is continuously reflected from the radiating field and the energy is persistently dissipated in the plasma. Again, the reflected wave from the media interface is negligible, and the effects of diffraction at the apertures of the antenna are accounted for, but are not separable from the integrated result. Due to the large data scattering error and the lack of computational resolution, only a general agreement was observed from the independent efforts due to the limited amount of experimental data and insufficient numerical accuracy. However, the feasibility of applying microwave probing for plasma diagnostics has been convincingly demonstrated. The numerical resolution of the more recently developed

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Plasma Diagnostics

Figure 8.18  Microwave power distribution in the receiving antenna.

compact-​ difference algorithm and the high-​ resolution orthogonal polynomial refinement technique offer a realizable potential (Shang et al. 2014).

8.7

Retarding Potential Analyzer The retarding potential probe is a diagnostic tool used to measure the ion energy distribution of plasma in high-​temperature conditions. Frequently, the Langmuir probe and the retarding potential analyzer (RPA) are applied simultaneously to verify the plasma properties in the plume of an ion engine. The diagnostic tool is extensively adapted mostly to determine the energy distribution of the thruster plume (Capacci et al. 1999). The probe is designed to operate in electron temperatures in the range from 0.5 to 5.0 ev, and a maximum electron number density around 1013 per cubic centimeter. The specially developed diagnostic tool is mostly for measuring the plasma parameters of the ion thruster for optimizing the performance of ion engines. The basic idea is built on the Debye shielding length by a series of grids, which are simply the electrically conducting screens and an ion collector. The wire-​ screen grid must be close-​knit to have gaps shorter than the Debye length in order to discriminate electrons from ions in the plasma. The major function of the grid is first to repel all the electrons of the plasma then to retard the ions and retain only the ions with an energy-​to-​charge ratio greater than the applied voltage, which can

301

8.7  Retarding Potential Analyzer

301

Figure 8.19  Schematic of retarding potential probes.

pass the retarding grid to reach the collector. Multiple levels of the applied voltage on different grids are then applied, so the collected data yield the derivative of the voltage-​current characteristic, which is proportional to the ion energy distribution. The relationship between the voltage-​current and the ion energy and ion charge number density and the effective collector area can be given as (Hutchinson 2005):

f ( Ei qi ) = ( mi qi2 e 2 ni A)( − dI d ϕ ) (8.33a) In the data collection procedure, only the applied voltage on the grid varies, thus the ion energy distribution is uniquely related to the energy distribution of ions in the plasma:



f ( Ei qi ) = f ( ϕ ) (8.33b) The RPA applies a series of grids to selectively filter ions in the thrust plume of an ion engine and samples the ion energy distribution, as its schematics is depicted in Figure  8.19. The basic construction of the retarding potential probe consists of four layers of grids and an ion collector beneath them. The outermost grid is exposed to the plume to minimize the perturbation by the exhausting stream and the probe. The first grid is held at the floating potential of the plasma to be measured; its purpose is to isolate the perturbation in the incoming plasma. Meanwhile it provides the additional attenuation of the plasma, which leads to an increasing Debye length of the internal probe by reducing the charge number density. An equally important function is also preventing the interaction of electric

30

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Plasma Diagnostics

potentials from different grids before ions reach the collector. The second grid holds at a high negative electric potential, which repels all electrons from the plasma and admits ions passing through. The grid spacing is therefore determined by the Debye shielding length, and the operational principle is based on the fact that the ions have a much lower mobility than the electrons of the plasma. The common practice for grid spacing is installing all grids with a spacing around four Debye lengths. After the filtering by the second grid, the number of electrons of the analyzed plasma shall be null. The third grid of the retarding potential probe has an applied swept voltage from zero increasing to create a high positive bias. Initially at the low applied voltage, all ions are permitted to pass through; once the applied electric potential is sufficiently high, the lower-​energy ions will be blocked out, resulting in a lower number of ions flowing through the grid. Therefore only the ions with energy-​to-​charge ratios greater than the grid voltage can pass through the retarding grid. The fourth grid acts as an additional electron repulsion grid to remove the secondary electron emissions from the bombardment of the ions on the collector. This grid is important to reduce the data scattering and improves measurement accuracy. All ions passing through the four different grids will reach the collector to create an electric current exclusively of ions. The derivative of the electric current-​voltage characteristic is proportional to the ion energy distribution. Thus the energy distribution function f ( Ei qi ) and the electric current are uniquely related to the discriminator voltage, the number density, and the area of the effective collector aperture. Usually, the retarding potential probe and the classic Langmuir probe are operated together to verify the performance characteristics of ion thrusters. In order to minimize the perturbation of the retarding potential analyzer to the measured plasma field, the probe aperture diameter has a limited dimension of a few millimeters. The size of the probe aperture is governed by the Debye shielding length of the plasma field, and in turn the sizes of the grids are related closely to the aperture opening. Thus the analyzer is an intrusive microprobe; at the same time, it must accommodate a cooling system and an isolator between the grids, as well as the data-​collecting cable, which poses significant challenges in design and fabrication. In addition, the most important gap distance between the second and the third grids must be carefully calibrated for the electron-​repelling voltage. Nevertheless, the retarding potential analyzer has been designed and operated mostly in laboratory conditions and recognized as an effective instrument for plasma ion energy per unit charge distribution measurement. References Capacci, M., Matticari, G., Noci, G., and Severi, A., An electric propulsion diagnostic package for the characterization of the plasma thruster/​spacecraft interactions on the Stentor satellite, AIAA Preprint 1999-​2377, 1999. Chung, P.M., Talbot, L., and Touryan, K.J., Electric probes in stationary and flowing plasma: Theory and application, Springer-​Verlag, New York, 1975. Fantz, U., Basics of plasma spectroscopy, Plasma Sources Sci. Technol., Vol. 15, 2006, pp. S137–​S147.

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Frederick, R.A., Blevins, J.A., and Coleman, H.W., Investigation of microwave attenuation measurements in a laboratory-​scale rocket motor plume, J. Spacecr., Vol. 32, No. 5, 1995, p. 923. Griem, H.R., Plasma spectroscopy, McGraw-​Hill, New York, 1964.   Principles of plasma spectroscopy, Cambridge University Press, Cambridge, 1997. Heald, M.A. and Wharton, C.B., Plasma diagnostics with microwaves, John Wiley & Sons, New York, 1965. Hershkowitz, N. How Langmuir probes work, in Plasma diagnostics, Vol. 1, edited by O. Auciello and D.L. Flamm, Academic Press, New York, 1989, pp. 113–​184. Herzberg, G., Spectra and molecular structure I, Spectra of diatomic molecules (2nd edn.), Van Nostrand Reinhold, New York, 1950. Howatson, A.M., An introduction to gas discharges, Pergamon Press, New York, 1976, p. 183. Hutchinson, I.H., Principle of plasma diagnostics (2nd edn.), MIT Press, Cambridge, MA, 2005. Krause, J.D., Electromagnetics (1st edn.), McGraw-​Hill, New York, 1953. Kurpik, A., Menart, J., Shang, J., Kimmel, R., and Hayes, J., Technique for making microwave absorption measurements in a thin plasma discharge, AIAA Preprint 2003-​3748, 2003. Holtgreven, W. (ed.), Plasma diagnostics, North Holland Publishing Company, Lochte-​ Amsterdam, 1968. Mitchner, M. and Kruger, C.H., Partially ionized gases, John Wiley & Sons, New York, 1973. Mott-​Smith, H.M. and Langmuir I., The theory of collectors in gaseous discharges, Phys. Rev., Vol. 28, No. 4, 1926, pp. 727–​763. Petrusev, A.S., Surzhikov, S.T., and Shang, J.S., A two-​dimensional model for glow discharge in view of vibrational excitation of molecular nitrogen, High Temp., Vol. 44, No. 4, 2006, pp. 814–​822. Shang, J.S., Simulating microwave radiation of pyramidal horn antenna for plasma diagnostics, Comm. Comp. Phys., Vol. 1, No. 4, 2006, pp. 677–​700. Shang, J.S., Andrienko, D.A., Huang, P.G., and Surzhikov, S.T., A computational approach for hypersonic nonequilibrium radiation utilizing space partition algorithm and Gauss quadrature, J. Comp. Phys., Vol. 266, 2014, pp. 1–​21. Shang, J.S. and Fithen, R.M., A comparative study of characteristic-​based algorithms for the Maxwell equations, J. Comp. Phys., Vol. 125, 1996, pp. 378–​394. Thornton, J.A., Comparison of theory and experiment for ion collection by spherical cylindrical probes in a collisional plasma, AIAA J., Vol. 9, No. 2, 1971, pp. 342–​344. Zel’dovich, Ya.B. and Raizer, Yu.P., Physics of shock waves and high-​temperature hydrodynamic phenomena, Dover Publications, Mineola, NY, 2002.

9

Radiative Energy Transfer

Introduction Radiative energy transfer always associates with a quantum transition that occurs in high-temperature  environments to  a medium in a highly excited energy state; therefore, it is an essential and unique feature of plasma. Unfortunately, it is often overlooked as an intrinsic physical phenomenon of ionized gas. Traditionally radiation has been described as a particle phenomenon for energy transmission by photons or light quanta, but in the framework of an electromagnetic field, radiation is described by wave dynamics not only by its energy content but also by its optical behavior. The radiative field in time and space must be described by the distribution of its intensity with respect to the frequency and direction of the energy transfer. The radiation spectrum covers a wide range from radio wave, infrared, visible, ultraviolet, x-​ray, and gamma-​ray frequencies from 103 to 1022 Hz; by wavelengths, it varies from 103 to 10–​12 m.  The portion of spectra associated with plasma for thermal energy transfer, however, is mostly concentrated in the infrared, visible, and ultraviolet spectra. For visible radiation, the characteristic medium temperatures are in the order from 7,000 K to 13,000 K, as shown in Figure 9.1. The radiative energy transfer occurs when a molecule, atom, or ion is in a highly excited energy state at the instant of a quantum jump. The energy depleted or gained is transmitted by photons or light quanta. An instant radiation field is always described by the distribution of the intensity for radiation by its frequency, location, and the direction of the energy transfer; the intensity is a unique function of the photon particle distribution f ( ν, s, Ω, t ). The propagation of radiating energy in a medium, therefore, is best described on the theoretic base of an electromagnetic wave. All quantum transitions for energy transfer have been divided into three groups according to the continuity criterion of its energy spectrum by the initial and final states of the transition as bound-​bound, bound-​free, and free-​free (Zel’dovich and Raizer 2002). The bound-​bound transitions take place in atoms and molecules from one discrete state to another. These transitions begin from line spectra to emit or absorb energy according to the atomic and molecular structure that follows the permissible quantum jumps. For molecules, the transitions are followed by changes of quantum states of its internal degrees of freedom. In case there are no exchanges in the electronic state, the transition involves very low energy and occurs in the

305

Introduction

305

Figure 9.1  Spectrum of thermal radiation.

infrared spectrum. The energy transfer is limited to the unit of hν or the product of the Planck constant and radiating frequency (h = 6.626 × 10 −34 m 2 kg/s ), and the frequency is limited to an extremely narrow range. The amount of the transferred energy is actually the difference between the two energy levels of an atom and is referred to as selective absorption. The cross-​sections of absorption are associated with short photon mean free paths, at the standard atmospheric condition, the mean free path of a photon is around 10–10 cm. The bound-​free transition results from the amount of energy transfer that exceeds the bounding energy of an electron to an ion, atom, or molecule; the electron becomes free. If the input energy is greater than the bounding energy, the excessive energy is transformed into the kinetic energy of the free electrons. A free electron can receive more energy than the binding potential, thus it can have continuous absorption and emission spectra. In the case of the reverse transition, the bonded electron will release the energy by photons. The free-​free transition is also known as the bremsstrahlung, because the motion of free electrons is slowing down in the field of the ion and loses a part of its continuous spectrum. This energy transition is the key element of ionization via radiation, as has been discussed in Chapter 7.8. From the previous discussion, note that absorption and emission are important mechanisms of radiation energy transfer together with the scattering process. The coefficient of bound-​bound and bound-​free absorptions is proportional to the number of absorbing atoms per unit volume of gas, which is the unique property of an absorbing atom. The controlling parameter is the absorption cross-​section defined as σ a = kνa N , in which kνa is the absorption coefficient and is the reciprocate of the absorption mean free path, and the number of atoms per unit volume is designated by N. In fact, the absorption coefficient is defined by the spectral radiation intensity, I ν with a frequency ν:dI ν = −κ ν I ν ds . The collision cross-​section of an

307

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Radiative Energy Transfer

atom or any other particle for the scattering phenomenon can be defined identically as that for absorption. In the case of free-​free energy transition, the absorption must be accomplished by collision with ions. Therefore, the coefficient of bremsstrahlung absorption is proportional to the number of ions and the number of free electrons in a unit volume. The coefficient of free-​free transition is approximately an order of magnitude smaller than the bound-​free counterpart. However, this behavior is different in a fully ionize plasma (Zel’dovich and Raizer 2002). The scattering cross-​section of electrons is also known as the Thomson scattering cross section, which is relatively small with a value σ s = 6.65 × 10 −25 cm2. In partially ionized plasma, the scattering mean free path of a photon resides in a continuous spectrum that is always greater than the corresponding absorption mean free path. Therefore the scattering becomes significant only in an extremely rarefied and fully ionized gas. In most engineering applications, except for conditions of astrophysics, the scattering can always be neglected in comparison with absorption. At the same time, the scattering process in radiation can play a significant role in multiphase media. In view of the far-​ranging physical phenomena, the radiating energy transfer involves extremely wide intertwining scientific disciplines from quantum physics, quantum chemistry, optical physics, and thermodynamics to aerodynamics. In our discussion, we must limit the topics to the reasonable scope of the formulations of the radiative energy transfer and the methods of analysis for theoretical and computational simulations for aerospace engineering applications. Therefore, we focus on the variations of multi-​flux formulations, methods of ray tracing, and the Monte Carlo method of photon dynamics. The detailed approximation techniques such as the narrow band, wide band, and view-​factor approaches are not included.

9.1

Fundamental of Thermal Radiation A radiation field is always described by the distribution of the intensity of radiation by its frequency ν, location s, and instance in time t, as well as the direction Ω of the energy transfer; the intensity is a unique function of the photon particle distribution f ( ν, s, Ω, t ). When the radiative energy emits from one medium into another, the energy flux frequently has a different strength in different direction due to attenuation, diffraction, reflection, and refraction. The energy flux may have infinitely many directions and traditionally is described by the vector in terms of spherical coordinates as shown in Figure  9.2. The total surface area of a unit radius is enclosed by the solid angle above the control surface and the polar angle θ, which is measured from the surface outward normal, and the azimuthal angle ϕ from the axis of the vector projects onto the surface. The domains of the two angular measures for a hemisphere are 0 ≤ θ ≤ π 2 and 0 ≤ ϕ ≤ 2 π , thus the solid angle Ω is the project area of the vector. The direction of the position vector is associated with an infinitesimal solid angle defining the infinitesimal area on the hemispherical surface, d Ω = sin θd θd ϕ . The spectral radiant intensity of a

307

9.1  Fundamental of Thermal Radiation

307

Figure 9.2  Definition of solid angle.

spectral interval d ν and within the domain of solid angle dΩ along the unit vector Ω is I ν ( s, Ω, t )d νd Ω = hν f ( ν, s, Ω, t )d νd Ω. The spectral radiant energy density, U ν ( s, t ), is obtained by integrating the spectral radiative intensity over the entire range of the solid angle: U ν ( s, t ) = hν ∫ f ( ν, s, Ω, t )d Ω = ∫ I ν ( s, Ω, t )d Ω, (0 < Ω < 4 π). By dividing the domain 4π

4π

into the right and left hemispheres, the spectral radiant energy flux passing the control plane is obtained by integrating the spectral intensity with direction cosine cos ϑ with respect to the solid angle Ω and the plane outward normal. The integral result is a vector quantity known as the spectral radiant energy density or the radiating energy flux, Uv(s,t). A radiation field is always described by the distributions of absorption, emission, and scattering intensities of radiation by its frequency, its position, and the direction of the energy transfer. These mechanisms are keys for attenuation of a radiative emission from the absorption of the medium and the colliding interaction with participants. According to the kinetic theory, all the radiating intensities are uniquely related to the photon distribution function f ( ν, s, Ω, t ). The radiant intensity in a spectral interval d ν and a range of solid angle dΩ along the direction of unit vector Ω is I ν ( s, Ω, t )d νd Ω = hν f ( ν, s, Ω, t )d νd Ω, and the interrelationship is just the product of the Planck constant and the collision frequency hν. The overall contribution to radiation by emission, absorption, and scattering is described by the integro-​differential Boltzmann equation for the radiation intensity. The classic theories of radiative energy transfer are derived mostly from the equilibrium radiative condition. This condition is characterized by the radiation

309

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Radiative Energy Transfer

Figure 9.3  Planck distribution function over wavelength.

emitted by the medium per unit time and volume at a given frequency in a given solid angle Ω that is identical to that absorbed by the medium. The radiating field is then isotropic; the field is a universal function of the frequency and temperature. From the quantum statistics, the spectral energy density has been derived by Planck (Modest 1993; Siegel and Howell 2002), and it is the amount of energy per unit volume radiated at equilibrium at a frequency ν.

U b, ν = (8πhν3 c3 ) ( e hν kT − 1) (9.1a) The spectral intensity in the isotropic radiation condition is given by the Planck distribution function or simply Planck’s function, and the Planck formula in terms of wavelength is depicted in Figure 9.3:



I b, ν = ( 2 hν3 c 2 ) ( e hν kT − 1) (9.1b) In the low-​ frequency region, hν > kT , the spectral energy density, Equation (9.1a), yields Wien’s displacement law:

309

9.2  Integro-differential Radiation Transfer Equation



309

U ν = (8πhν3 c3 ) e hν kT (9.1d) When the equilibrium radiant density is integrated over all frequencies, it becomes the well-​known expression for equilibrium radiation density:



∞

U = ∫ U ν d ν = 4σT 4 c (9.1e) 0

where the Stefan-​Boltzmann constant is σ = 2 π5 k 4 15h3 c 2. In the cgs system, it has the value of 5.67 × 10 −5 erg/(cm 2 × sec × K5 ). The amount of radiant energy given by the amount per unit time and area over all frequencies is therefore σT 4 . For the following discussions of radiative energy transfer, the concept of the local thermodynamic equilibrium (LTE) condition is very important. The LTE condition is defined as when the radiation at each point of a medium with a uniform temperature is close to equilibrium; then the radiation intensity can be described by the Planck function at the temperature of the point in consideration. In fact, the Planck spectral distribution of emissive power is derived from quantum mechanics; it describes the equilibrium between the spontaneous emission and absorption. It is of great importance, because the Planck distribution provides quantitative values for spectral radiation from a black body. The Planck function distribution over the wavelength from 10–​2 to 10 microns in the temperature range of 2,000 K to 20,000 K is depicted in Figure 9.3. The magnitude of the Planck function reaches its maximum around the wavelength of one tenth of a micron at the temperature of 20,000 K, and shifts to more than one micron at the lowest-​temperature condition. In other words, the equilibrium point of spectral absorption and emission of a medium moves to a shorter wavelength radiative spectrum at an increasing temperature condition. Another outstanding characteristic of the propagating radiating ray in a participant medium is its opaqueness, which is caused by the amplitude attenuation of electromagnetic waves by the collision and non-​collision processes. The latter is incurred by scattering photons by diffraction or by the penetration of photons into particles and by changing direction from refraction. As a consequence, the medium of radiation propagation has been classified as the optical thick and optical thin. A brief discussion of these characteristics is provided in the following discussions.

9.2

Integro-​differential Radiation Transfer Equation There are five basic mechanisms in the radiation heat transfer process: the spectral absorption, emission, scattering, reflection from media interface, and reabsorption. The absorption rate per unit volume in a point in space at coordinate s for radiation propagated in elementary solid angle dΩ can be given as kν ( s )I ν ( s, Ω, t )d Ω. The spectral emission is more complex; traditionally it has been categorized as spontaneous, induced, or medium emissivity, because it depends on the state and properties of the emitting medium. The total emission rate per unit volume at

31

310

Radiative Energy Transfer

LTE conditions I νem ( s, t ) is obtained by invoking Kirchhoff’s law, which expresses the general principle of detailed balancing applied to the emission and absorption of light. Therefore, when the absorption coefficient is known, then the emission coefficient of a medium can be calculated as follows:  I νem ( s, t ) = κ ν ( s )I b, ν [T ( s )], where I b, ν [T ( s )] is the spectral intensity of black body (the Planck’s function). Note that in this case, the spectral intensity does not depend on the solid angle Ω. The spectral radiation scattering coefficient σ ν ( s ) gives the scattering rate per unit volume for radiation propagated in an elemental solid angle dΩ, therefore, it follows σ ν ( s )I ν ( s, Ω, t )d Ω. Attenuation of radiative energy in a medium can be caused by absorption, reflection, and also by scattering; the opacity of a medium is measured by the attenuation of the transmitted radiation energy. By definition the spectral scattering indicatrix p( s, Ω ′, Ω, ν′, ν) is a function that characterizes the probability of scattering to direction from Ω to direction Ω ′ and to frequency ν ’ from a frequency ν. The rate of energy transfer between two states will have a value of (1 4 π ) σ ν ( s )I ν ( s, Ω ′, t ) p( s, Ω ′, Ω, ν′, ν)d Ω ′d ν ’. The reflection of radiating energy transfer from the media interface must also be considered specifically for each individual problem; it is strictly an initial value and boundary condition problem. In summary, the radiating mechanism is generated by the kinetics of the photon or light quanta, which is directly related to the radiative intensity by its probability distribution function with a constant value multiplier hν. Therefore is treatable by the differential-​integral Boltzmann equation for photon distribution function in phase space. From the kinetic theory for the probability distribution function of light quanta, which is related to the radiative intensity at a given frequency:



1 ∂I ν ( s, Ω, t ) ∂I ν ( s, Ω, t ) + + [ kν ( s ) + σ ν ( s )] I ν ( s, Ω, t ) c ∂t ∂s (9.2a) σ (s) ∞ = I νem ( s,tt ) + ν ∫ ν ν ν p ( s , Ω , Ω , , ) I ( s , Ω , t ) d Ω d ′ ′ ′ ′ ′ ν′ 4 π ν = 0 Ω =∫4 π In Equation (9.2a), the two leading terms on the left-​hand side of the equal sign are the total derivative of the radiative intensity, and the convective velocity is the speed of light along the direction of the solid angle. The third term represents the spectral intensity associated with the absorption and scattering with the coefficients of kν ( s ) and σ ν ( s ), respectively. The first term on the right-​hand side of the equation represents the amount of spectral intensity of the medium, and the last term describes the contribution to the rate of change for radiative intensity by scattering. The symbol p( s, Ω ′, Ω, ν′, ν) denotes the phase function of radiative scattering or the scattering indicatrix described previously. It intuitively obvious that the time-​dependent term of the radiative rate equation is scaled by the speed of light; therefore, the partial derivative with respect to time, 1 c ∂I v ( s, Ω, t ) ∂t , is negligible in comparison with the spatial derivative. Thus all the dependence of radiative intensities on time becomes negligible; the time-​ independent radiative rate equation acquires the following form:

31

9.2  Integro-differential Radiation Transfer Equation



311

∂I ν ( s, Ω ) + [ kν ( s, t ) + σ ν ( s )] I ν ( s, Ω ) ∂r (9.2b) σ (s) ∞ = I νem ( s ) + ν ∫ p s Ω Ω ( , , , ν , ν ) I ( s , Ω ) d Ω d ν ′ ′ ′ ′ ′ ν ′ 4 π ν = 0 Ω==∫4 π Note that the time dependence of all functions in Equation (9.2b) now reflects the dependence on variation of thermodynamic and optical properties of media in space only. Equation (9.2b) is the fundamental governing equation for the rate of change by radiative transfer, and it is derived from the Boltzmann equation of the photon probability distribution function. The important and widely adopted simplification by the theory of radiative heat exchange rate equation is to neglect the dispersion of radiation over the frequency of the electromagnetic wave propagation. This is the so-​called approach of coherent dispersion. Under this condition, photons of radiation incident on the elemental physical volume are diffused on the same frequency of electromagnetic radiation. As has been mentioned before, the radiative energy transfer occurs at the quantum transition of atoms and molecules and at electrodynamic interactions of electrons with atoms, molecules, and their ions. The intensity of absorption and emission is controlled by the medium excited state and the binding energy of species. Quantum transitions accompanying the energy absorption and emission are subdivided into three types: the bound-​bound transition has a discrete spectrum in atoms that produces line spectra, and in molecules, it appears as band spectra that contain a large number of rotational lines. The bound-​free transition leads to photoelectric absorption. The free-​free transition is referred to as the bremsstrahlung emission and absorption. Therefore, the spectral absorption coefficient distribution will reflect distinctive characteristics at different thermodynamic states of a radiating medium. The typical air spectral absorption coefficient distributions are displayed in Figure 9.4 at two different temperatures of 5,000 K and 20,000 K, but at the same pressure of one atmosphere. It is clearly shown that the magnitudes and distributions of absorption spectral coefficient as a function of wave number are entirely different from each other. The difference is definitely not merely a shifting of the distribution shape in the spectrum but also drastic change in intensities to reflect the different excited internal energy states of the air mixture. In practical applications, the discrete and huge numbers of atomic lines and molecular bands spectra are usually approximated as groups by an averaged frequency. Specifically, the multi-​group, wide-​band, and narrow-​ band spectra, as well as the line-​by-​line integration over a spectrum model, are commonly used for a specific radiative phenomenon (Surzhikov 2002; Shang and Surzhikov 2012). Theoretic derivation of the spectral absorption coefficient can only be accomplished by approximations to the quantum mechanics and statistic quantum mechanics through a perturbation approach. As an example, the total coefficient of the bound-​free adsorption of a monotonic molecule (hydrogen-​like), together with

31

312

Radiative Energy Transfer

Figure 9.4  Typical spectral absorption coefficient of air at one atmosphere pressure and

different temperatures.

the free-​free transitions in the field of residual ions, has been derived by Surzhikov, Capitelli, and Colonna (2002) as: 

κ Σν = 0.197 ⋅10 −6

∞ 1 I Z 2ϕ n n e i e ∑ 3 3 32 ω Te n = n∗ n

n2 kTe

+ 0.137 ⋅10 −22

Z2 1 n n g ( ne ,Te ) , 3 12 i e cm ω Te (9.3a)

where ni and ne are the number densities of ions and electrons and the symbols ϕ and n are the potential of ionization and the general quantum number. The notation ω designates the wave number, Te is the temperature of electrons, Z indicates the number of charges attaching to ions, and g is Gaunt’s function. The approximated and simplified result offers significant insights on the dependence of number densities of electrons and ions, as well as radiative frequency and electron temperature to the spectral absorption coefficient. There is a distinctive difference in absorption coefficient kν and scattering coefficient σ ν , but both have physical units of reciprocal length dimension. These coefficients are strongly dependent on the composition and thermodynamic state of the transmitting medium, and are referred to as linear coefficients, but they are really the volumetric coefficients. When the combined coefficient has a constant value β ν = k ν + σ ν , then it represents the reciprocal of the radiation mean penetration distance of the medium. The dimensionless parameter, the optical thickness of a medium, is obtained by integrating the combined absorption and scattering coefficient along a beam path to become a measure of the penetration distance of radiation:

s

τ ν = ∫ β ν ( s )ds (9.3b) 0

If τ ν ( s ) >> 1, the beam path in the medium is considered optically thick, and the Rosseland diffusion approximation is applicable to radiation. Whereas the dimensionless parameter is very small in comparison with unity, τ ν ( s ) 0,

Yn− m ( µ, ψ ) = P ( µ ) cos mψ , m ≤ 0, (9.7b) − 1 ≤ µ < 1, 0 < ϕ < 2 π ( m) n

The infinite series approximation at the upper limit of n → ∞, the spherical harmonics approximation, is an exact solution to the rate equation of radiation transfer. The function ϕ nm ( r ) is a location-​dependent coefficient and will be determined by the solution satisfied at the boundary conditions. The term Ynm ( µ, ψ ) is the normalized, angular-​independent spherical harmonics. The angular dependence is defined by the complex wave function eimϕ, which provided the trigonometry functions cos mϕ and sin mϕ. The symbol δ 0,m in Equation (9.7a) is designated as the Kronecker delta. The directional components of the radiation intensity and interrelated angular displacements are depicted in Figure 9.5 (Surzhikov 2002, 2008). In practical applications, the odd-​order number of the Legendre polynomial series is preferred over the even-​order number, because the even-​order polynomials are difficult to resolve for boundary conditions implementation. Therefore the truncated finite N terms are mostly odd numbers for engineering radiative transfer evaluation, such as the P1 and P3 approximations, which have been found adequate for most applications. The higher-​order polynomial is more accurate for the approximation, but the added complexity frequently makes it impractical. The P1 spherical harmonic method represents the first-​order approximation to the solution of the radiative rate equation, which is the most efficient for radiation energy transfer associated with an emitter into a simple layer. For this reason, the P1-​approximation is commonly used in the radiation energy transfer for the forebody heat shield analysis associated with interplanetary reentry applications. To formulate the approximation for an arbitrary geometry, we need to consider only the radiation heat transfer equation for a non-​scattering medium in local thermodynamic equilibrium:

319

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Radiative Energy Transfer

z

µ θ

Ω ξ

Θr

ϕ

r

η y

Φ x

Figure 9.5  Spherical coordinates.

Ω ⋅ ∇I ν ( r , Ω ) + κ ν ( r ) I ν ( r , Ω ) = κ ν ( r ) I b, ν ( r ) (9.8a)



The approximate series solution to Equation (9.8a) retains only the first two terms of the infinite series for Equation (9.7a) to appear as the following representation for the radiative intensity: I ν ( r , Ω ) = ϕ1, ν ( r ) + ϕ 2, ν ( r ) ⋅ Ω  4 π (9.8b)



where ϕ1, ν ( r ) , ϕ 2, ν ( r ) are the unknown functions to be determined by the boundary condition. The physical significance of these functions, ϕ1, ν ( r ) and ϕ 2, ν ( r ) is clearly defined by integrating the radiation intensity over all solid angles, and the first moment of the radiative intensity with respect to the solid angle is:

∫ I ν ( r , Ω) dΩ = ϕ ν ( r ) = cU ν ( r ) , 1,



4π

(9.8c)

∫ Ω I ν ( r ,Ω) dΩ = ϕ ν ( r ) = Wν ( r ). (9.8d)



2,

4π

For the P1 approximation, the radiation intensity can be presented in the form of (Surzhikov 2002): I ν ( r , Ω ) = cU ν ( r ) + 3Wν ( r ) ⋅Ω  4 π . ∈ (9.8e)



The first equation for the required analysis is obtained by integrating the radiation heat transfer Equation (9.8a) over the solid angle to get: ∇ ⋅ Wν ( r ) + κ ν ( r ) cU ν ( r ) = κ ν ( r ) cU b, ν ( r ) , (9.9a)

where cU b, ν ( r ) =

∫ I ν ( r ) dΩ. b,

4π

319

9.4  Spherical Harmonic (PN) Method

319

The second equation can be obtained after integration of Equation (9.9a) with respect to Ω: Wν ( r ) = −



c ∇U ν ( r ) . 3κ ν ( r ) (9.9b)

Equation (9.9b) shows that the radiation flux vector is proportional to the gradient of the volumetric density of radiative energy, which reveals the outstanding characteristic of the diffusion process. Therefore, the coefficient 1 3κ ν ( r ) is often called the radiation diffusion coefficient. Substitute the radiation flux Equation (9.9b) into Equation (9.9a) to achieve the following form:  1  −∇ ⋅  ∇U ν ( r ) + κ ν ( r )U ν ( r ) = κ ν ( r )U b, ν ( r ) (9.9c)  3κ ν ( r ) 



Typical boundary conditions of an arbitrary geometry configuration for the P1 approximation usually consist of two possibilities. The first physically meaningful condition is that, if there is no external radiation at boundary surfaces, at ( Ω ⋅ nˆ ) < 0, I ν ( r,Ω ) = 0. Where nˆ is the unit normal to surface s. In this case, the surface outward radiation flux becomes: c Wνn ( r ) = ( n ⋅ Wν ( r ))s = − U ν ( r ) (9.10a) 2

or

2 −1 cκ ν ( r ) ∇U ν ( r ) = −U ν ( r ) . (9.10b) 3 s



The second permissible boundary condition is for a cylindrical geometry with axial symmetry or a radial symmetry for spherical geometry, rˆ ⋅ Wν ( r ) = 0, where rˆ is the unit vector in radial direction. The boundary condition can be rewritten as: ∂U ν ( r ) ∂r = 0. (9.10c)



From the simple analytic formulation of the P1 spherical harmonic approximation, it is of great practical interest to examine the plasma medium for the distinctive features associated with extremely thin or thick optical depths. For an optically thin medium, τ ν = ∫ [ kν ( s ) + σ( s )]ds < 1; the exact equation, s

Equation (9.9c), gives:

∇ ⋅ Wν ( r ) = κ ν ( r ) c U b, ν ( r ) − U ν ( r ) . (9.11a) At the condition of extremely thin optical depth τ ν  1, and without external radiating source U ν ( r ) = 0, Equation (9.11a) becomes:



∇ ⋅ Wν ( r ) = cκ ν ( r )U b, ν ( r ) . (9.11b)

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Radiative Energy Transfer

Upon integrating Equation (9.11b) with respect to radiation frequency, we obtain the relationship of the integrated divergence with the Planck integral coefficient: ∞

∫ ∇ ⋅ Wν ( r ) dν = ∇ ⋅ W ( r ) = 4πκ ( r ) σT ( r ) , 4

P



(9.11c)

0

where ∞

κ P (r ) =

c ∫ κ ν ( r )U b, ν ( r ) dν 0 ∞

∫ ∫ I ν ( r ) dΩ dν

∞

b,

=

b,



∫ κ ν ( r ) I ν ( r ) dν 0

∞

∫ I ν ( r ) dν b,

0 4π

0

.

For the condition of an optically thick medium, τ ν = ∫ [ kν ( s ) + σ( s )]ds > 1, the specs

tral energy density is mostly contributed by the black body radiation. U ν ( r ) ≈ U b, ν ( r ) . (9.11d)



Note that under this circumstance, the result of ∇ ⋅ Wν ( r ) by Equation (9.9a) may be in error because the values of U b, ν ( r ) and U ν ( r ) are very close. The result of ∇ ⋅ Wν ( r ) will be reflected by a negligible value. For an accurate result, one must use the following relation by Equation (9.9b): Wν ( r ) = −



c ∇U b, ν ( r ) . 3κ ν ( r ) (9.11e)

Upon integrating Equation (9.11e) with respect to radiation frequency, we obtain the association of integral flux for radiation with the Roseland mean coefficient (Surzhikov 2008). ∞

c 16 ∇U b, ν ( r ) dν = − σT 3 ( r ) κ R−1 ( r )∇T ( r ) , 3 κ 3 r ν( ) 0 (9.11f)

W (r ) = −∫

where

∞



κ R−1 ( r ) =

∫ 1 κ ν ( r ) dU ν (T ) dT  dν b,

0

∞

∫ dU ν (T ) dT  dν

. (9.11g)

b,

0

For practical applications, note that the P1 approximation of the spherical harmonic method has poor accuracy at a very thin optical thickness limit. To overcome this shortcoming, one needs to proceed with the practical application as follows: the P1 method is used only in spectral regions where the optical depth is thick; otherwise the emission approximation for radiation heat transfer with no absorption should be used.

321

9.5  Method of Discrete Ordinates

321

The next higher-​order spherical harmonic method is the P3 approximation; this method has been frequently applied to analyze radiation heat transfer over axial symmetric configurations. The series solution to the radiation rate equation can be systematically derived by substituting Equation (9.7a) into Equation (9.2b) with the aid of the recursive formula of the Legendre polynomials. The resultant series become a boundary-​value problem relating to the boundary conditions for the radiative intensity. Specific details of the series solution are rather lengthy and can be easily found in the works of Siegel and Howell (2002) and Surzhikov (2002), so they will not be repeated here.

9.5

Method of Discrete Ordinates The discrete ordinates method is also related to the multi-​flux methods that we have discussed. This method is like the half-​moment and spherical harmonic methods, which may be carried out in any arbitrary order of accuracy as needed. The basic idea is transforming the coordinate of the radiation rate equation for a gray medium from the Cartesian frame of reference onto the coordinates of the propagating radiative wave path by the chain rule of differentiation. The orientation of the radiation intensity is further discretized into many directions in the transformed coordinates (Fiveland 1988; Menart 2000) and into a set of simultaneous partial differential equations. A solution to the radiation rate equation is then found by solving the rate equation by a set of discrete directions spanning the total solid angle range of 4 π . d I ν  ∂x  d I ν  ∂y  d I ν  ∂z  d I ν (9.12a) +  +  =  ds  ∂s  dx  ∂s  dy  ∂s  dz



The metrics of coordinate transformation, ∂x ∂s, ∂y ∂s, and ∂z ∂s are direction cosines (α, β, γ ) that link the transformed coordinates to the Cartesian frame. The transformed rate equation of radiation including the scattering mechanism acquires the following form:

α

∂I ν ∂I ∂I σ 4π + β ν + γ ν = kν I b, ν − ( kν + σ ν ) I ν + ν ∫ I ν ( s, Ω ) p( s, Ω, Ω′ )d Ω ∂x ∂y ∂z 4π 0

(9.12b)

Equation (9.12b) represents the general transfer relations for the radiative intensity through which the angular average is projected over each transformed ordinate direction. The integral results over the incident angular directions are approximated by the sum of phase functions, p( s,Ω,Ω′ ). The outgoing and incoming angular directions are designated by the superscript m and m’, meanwhile the integral of the scattering can be approximated by the summation over the directional cosines with a weight function, wm. The equation of transfer in the m direction becomes:

α m I Ωm + β m I Ωm + γ m I Ωm = kν I b − ( kν + σ ν ) I Ωm + ( σ ν 4 π )∑ wm ′ I Ωm ′ p mm ′ (9.12c) m′

32

322

Radiative Energy Transfer

Along the propagation path in the direction m, the initial value at the source of emission must be specified. The only one condition of nˆ ⋅ Ω m > 0 is required; thus the equations system Equation (9.12c) is constructed by m number of discretized first-​order differential equations. When the scattering is presented (σ ν ≠ 0), then the equations are coupled and an iterative procedure becomes necessary. In the absence of scattering and with an initial emitting condition, the solution of the discrete radiation intensity will be the exact solution. The boundary condition written in terms of incident intensities is the sum of the intensity leaving and reflecting from the surface along each ordinate direction m or m’: I 0m = ε I b, ν +



1− ε ∑ α m′ wm′ I νm′ (9.12d) π m′

where α m′ is the direction cosine between the m’ direction and the outward normal nˆ of the emitting surface, and ε is the emissivity of the surface. The discrete ordinates methods have been developed for three-​dimensional rectangular and cylindrical polar coordinates for radiative transport in a participating medium (Fiveland 1988; Surzhikov 2002). Specially, Ramankutty and Crosbie (1998) present a modified discrete ordinates solution of three-​dimensional radiative transfer in enclosures with localized boundary-​loading conditions. Another discrete ordinate method has been used to find the spectral characteristics for thermal radiation for a two-​dimensional, axisymmetric, free-​burning argon arc (Menart 2000). A similar method has also been applied to study radiative heating of the internal surface for the air and hydrogen laser-​supported plasma generator (Surzhikov 2002). An illustrative development of the discrete ordinates method for evaluating radiative intensity is provided by Surzhikov (2002). In computational domains on Cartesian coordinates, the discrete ordinates method is divided into finite numbers of non-​overlapping tetrahedral cells. On discrete ordinates, the radiation heat transfer equation without the scattering mechanism can be given as:

αm

∂I m ∂I m ∂I νm + β m ν + γ m ν = κ ν ( I b, ν − I νm ) (9.13a) ∂y ∂z ∂x

where I νm is the spectral radiation intensity, which is a function of position and direction Ω m , and I b, ν is the spectral black body radiation at a given temperature of the medium. As before, the symbol κ ν denotes the spectral absorption coefficient of the medium, and α m , β m , γ m are the directional cosines of the direction Ω m ; finally, ν is the radiative frequency. The condition of the vanishing radiative intensity on the boundary Γ within a computational domain is as simple as:

I νm = 0, r ∈ Γ (9.13b) If the surface of the emitting surface Γ ′ is assumed to be gray and the emission and reflections are diffuse, then the radiative boundary condition on the surface can be given as:

32

9.5  Method of Discrete Ordinates

I νm = ε I b Γ +



323

1− ε ∑ ω m′ nΓ Ω m′ I νm′ r ∈ Γ ′ π n Γ Ω m′ < 0 (9.13c)

Usually spectral values at different frequencies are replaced by group values in practical applications within a control volume by Equation (9.13a). The numerical result can be obtained by integrating the transfer equation over the tetrahedral volume as follows: 4

∑ ( nˆ Ω ) S I i



m

i =1

where Iim =

1 Si

∫I

m

m i i

= κ pVp ( I b, p − I pm )

(9.13d)

dσ is the side area of an averaged group intensity; the subscript

Si

i identifies the number of tetrahedral faces and the area of the control surface is designated as Si . By integrating the discretized intensity over the control volume, 1 the average group intensity of the cell is I pm = I m dυ and Vp is the volume of the Vp V∫p tetrahedral cell. In Equation (9.13d), nˆi denotes the outward normal unit vector to the side i of the tetrahedral cell, and κ p is the group absorption coefficient in the cell. The group black body radiation at the temperature in the cell p is characterized by I b, p . Solving Equation (9.13d) for intensity at the center of a cell I pm may be evaluated as the sum of all components of the intensities: I pm = I b, p −

1 κ pVp

4

∑ (n Ω ) S I i

i =1

m

m i i

(9.14a)

Therefore, to solve Equation (9.14a), all auxiliary relations among average intensities on the control surfaces of a tetrahedral cell are required. Three possibilities for radiation arise for a tetrahedral cell and are depicted in Figure 9.6, depending on the manner in which the cell is approached by radiation. These possibilities were derived in detail by Sakami, Charette, and Le Dez (1998). If only one control surface receives radiation from the three other faces (Figure 9.6a), then intensity on receiving face 4 is given as:

S  S S I 4m =  14 I1m + 24 I 2m + 34 I 3m  χ p + I b, p (1 − χ p ) (9.14b)  S4  S4 S4 For the second possibility, when two faces receive radiation from other two faces (Figure 9.6b), then intensities on the receiving faces become:



S  S I 2m =  12 I1m + 32 I3m  χ p + I b, p (1 − χ p ) (9.14c)  S2  S2



S  S I 4m =  14 I1m + 34 I3m  χ p + I b, p (1 − χ p ) (9.14d)  S4  S4

325

324

Radiative Energy Transfer

Figure 9.6  Three possibilities of radiative transport in a tetrahedral cell.

The third possibility describes the situation when three faces receive radiation from the fourth face (Figure 9.6c). Intensities on the receiving faces are given as:

I1m = I 2m = I 3m = I 4m χ p + I b, p (1 − χ p ) (9.14e) Equations (9.14b) through (9.14e) include the function χ p for prescribing the optical thickness of the medium, which can be expressed as:



χp =

2 τp

 1 − e−τp  1 − τ  (9.14f)   p

where the notation τ p is the maximum optical thickness in the cell p along the direction of Ω m. The accuracy of the discrete ordinates solution depends on the choice of the quadrature scheme. Although the choice in principle is arbitrary, a completely symmetric quadrature is preferred in order to preserve the geometric invariance of the solution. The quadrature schemes used in the work of Sakami and colleagues (1998) are based on the “moment-​matching” technique, whereby the ordinates are chosen so as to integrate as many moments of intensity distribution as possible. Although any type of quadrature can also be applied, the preferred algorithm was created to identify the type of Ωm orientation in the cell. This algorithm permits appropriately characteristic Equations (9.14a)  –​(9.14d) for finding the unknown average face intensity. If the solver is applied to the cells in the correct order, all terms on the right side of Equation (9.14a) will be known, so the cell-​center intensity can be found by direct substitution. This approach allows the solution to be obtained by moving from cell to cell in the optimal order. For the solution generated on an irregular grid, the order is not obvious. To find a solution on irregular grids, the equation can be solved by repeatedly sweeping across the grid without regard to the optimal sequence.

325

9.6  Governing Equations of Gas Dynamics Radiation

325

One of the most thorough assessments of the discrete ordinate methods is provided by Balsara (2001). He has shown it is possible to discretize the radiative rate equation using the discrete ordinate method of the total variation diminishing (TVD) scheme, which splits the radiative intensity via a genuinely multidimensional discretization. In this regard, the Newton-​Krylov numerical algorithm can provide a natural way of treating the advection and scattering terms in the rate equation without resort to operator splitting. This feature permits the maximal coupling between the two different energy transfer mechanisms. It has been demonstrated that this coupling enhances convergence even in the strong scattering limit where previous methods have been known to perform poorly. The genuinely multidimensional discretization takes account of the direction of characteristic propagation, which represents the physics with greater fidelity. By an accurate analysis, the numerical results can attain close to second-​order accuracy in spite of the fact that the chain rule of differentiation is only a first-​order differential operator.

9.6

Governing Equations of Gas Dynamics Radiation The advent of high-​performance computational  technology enhances a different approach to radiation transfer calculations by directly solving the combined radiation rate equation with conservation laws of gas dynamics. Gas dynamic radiation is an interdisciplinary science involving aerodynamics, nonequilibrium chemical kinetics, quantum physics, and radiation. The coupling of radiation with high-​ temperature gas dynamics has been studied since the early 1960s (Vincenti and Traugott 1971; Zel’dovich and Raizer 2002). The radiative energy transfer originating from the quantum transition from molecules and atoms of a gas medium is beyond the realm of gas kinetic theory. Thus the distinct physical phenomena must be described by chemical composition, atomic and molecular structure of each species, and nonequilibrium thermodynamic state of the medium by quantum transitions. The interesting and significant fact is that the wavelength of thermal radiation is compatible with typical dimensions of atoms and molecules and the corresponding time scale is very short. The transition of quantum state takes place on the scale of speed of light in contrast to convective and diffusive motions of gas that proceed at the speed of sound. In view of the drastically different mechanisms and time scales, the close or loose coupling between gas dynamics and radiation is no longer a critically important issue. Thermal radiative energy exchange with gas motion is controlled by energy balance with surrounding interplanetary exploration for earth reentry and in hypersonic flows. In these environments, the radiating energy transfer contributes up to a quarter of the total heat flux in the spectrum range, mostly from the far infrared to ultraviolet bands. In practical aerospace engineering applications of gas dynamics including radiation, the ionization processes are accomplished by thermally enhanced inelastic collisions between heavy molecules and electron elastic collisions. In both processes, a substantial amount of energy is exciting the

327

326

Radiative Energy Transfer

molecules and atoms by elevating them from a more stable ground state. It is also realized that under most conditions, the ionized gas usually occupies only the first few quantum states of electrons and ions with a relatively low degree of ionization. The contribution from the kinetic energy of the ionized components to the interaction is negligible in comparison with the counterpart neutral components; the kinetic energy of ionized species varies from the relative order of magnitude of 10–5 to 10–6 in the temperature range from 10,000 K to 100,000 K. Similarly, the partial pressure generated by the colliding ionized components in comparison with the neutral counterpart is also very low, around an order of relative magnitude of 10–5 at 10,000 k, and increases to less than 10–1 at the extremely high-​temperature condition. In all, the energy coupling between radiation and gas dynamics appears only as a heat source or sink to the radiating flow field, and the significant amount of radiative energy is transmitted across the intermediate boundary. On the Newtonian frame, the essential physics of gas dynamic radiation interaction can be effectively approximated by a simplified governing equation based on the mass, momentum, and energy conservation laws outlined in Chapter  6.4. The system of interdisciplinary equations is essentially the unsteady nonequilibrium compressible Navier-​Stokes equations that couple with radiative energy and electromagnetic effects. The complete governing equation system includes the simplified rate equation for radiation and a source or sink term to the conservation energy equation:



∂ρi dw + ∇ ⋅ [ρi ( u + u i )] = i (9.15a) ∂t dt ∂ρu + ∇ ⋅ (ρuu + p I − τ ) = ρe E + ( J × B ) ∂t (9.15b)



∂ρe + ∇ ⋅ [ρeu − κ∇T + ∑ ρi ui hi + qrad + u ⋅ pI + u ⋅ τ] + Qvt − Qet = E ⋅ J (9.15c) ∂t



∂I ν ( s, Ω, t ) + kν ( s, t ) I ν ( s, Ω, t ) = I νem ( s, t ) (9.15d) ∂s The coupling of gas dynamics and radiation is through the global energy conservation law, Equation (9.15c). In this equation, the radiative energy transfer ∇ ⋅ qrad appears as either a sink or a source for the energy balance. The radiation energy transfer alters the thermodynamic state of gas along its beam path, which is actuated by the quantum transition. When the transition takes place, the immediately adjacent quantum state is referred to as the climbing-​the-​ladder transition, which is a common occurrence in a high-​enthalpy flow field. When the quantum jump occurs by multiple quanta, then it is known as the big-​bang transition. Under the latter condition, the amount of energy emitted or absorbed can be substantial, which may happen at shock jump locations where the pressure and temperature of the gas can change abruptly within a few mean free lengths of the gas medium. The radiation equation, Equation (9.15d), is simplified from Equation (9.2b) by omitting the scattering mechanism. The spectral-​induced emission is usually

327

9.6  Governing Equations of Gas Dynamics Radiation

327

evaluated by the Kirchhoff law through the local thermodynamic equilibrium (LTE). The radiative energy flux actually closes the global energy conservation equations by performing the integration of the radiative intensity over all solid angle and frequency: ∞



qrad = ∫0 d ν∫4 π I ν ( s,Ω )Ωd Ω (9.15e) From the kinetic theory of gas, the translational and rotational modes of molecules need a few collisions to achieve equilibrium, thus it is always considered in thermal equilibrium condition with the translation mode. The energy transfer of all internal degrees of freedom beyond the translation mode is described by the following two equation systems. The quantum energy transfers are approximated by empirical models to the physics, discussed in Chapter 6.4. The vibrational energy conservation equations for polyatomic molecular species and the electronic conservation equation are given by Equations (9.15f) and (9.15g):



∂ρi eiV dwi + ∇ ⋅ [ρi ( u + ui )eiV + qiV )] = eiV + QV ,Σ . (9.15f) ∂t dt



∂ρi ee dw + ∇ ⋅ [ρi ( u + ui )ee ] = ee i + Qe ,Σ (9.15g) ∂t dt In a multicomponent gas mixture system, the mass, momentum, and energy fluxes conservation are driven by the identical transport mechanism. In fact, there is a unique and critical coupling between mass and energy exchange by diffusion in a high-​temperature gas mixture. A landmark of the kinetic theory of diluted gas mixture is its ability to describe the transport property for any combination of gas mixture. In order to be consistent with the kinetic model of the internal structure of gas, the transport properties of the gas mixture for diffusion, viscosity, and thermal conductivity are evaluated from the Boltzmann equation by the Chapman-​Enskog expansion (Chapman and Cowling 1964). An extensive amount of effort has been devoted to study and to develop the transport property for high-​temperature gas and to simplify it for practical use. The progress is remarkable and impressive, but the most recent approaches in basic research are adopting directly the formulations by gas kinetic theory (Capitelli et al. 1996). The kinetic theory includes the diffusion, viscosity, and thermal conductivity for gas mixture by the collision cross-​sections and collision integrals. A formulation for a heterogeneous gas mixture is also widely used and known as the Wilkes mixing rule (Bird, Stewart, and Lightfoot 1960), Equation (4.18c). Although substantial progress has been made in the kinetic theory of gas for calculating transport properties, a specific comparison of the computational results still yields a significant discrepancy up to 20 percent in viscosity and thermal conductivity of high-​temperature air by using different sets of inter-​particle potential in the temperature range from 300 K to 30,000 K (Levin and Wright 2004). In the presence of charged species, the force diffusion added by the electromagnetic effects has been effectively described by the screened Coulomb potential or by the classic

329

328

Radiative Energy Transfer

draft-​diffusion theory. In certain application conditions, the ambipolar diffusion can be implemented as necessary. For studying radiative energy transfer, the optical property is uniquely determined by the composition and energy state of the medium, which is needed on a line-​of-​the-​sight coordinate for a photon beam path, which is different from the nonequilibrium thermos-​chemical evaluation. Therefore, radiation simulation presents two fundamental challenges:  the generation of emissive, absorptive, and scattering properties of an optical activity medium, and the extraction of these intrinsic properties as input along the radiating path for the radiation rate equation. In principle, the optical data-​generation process can only be accomplished by the ab initio application of quantum mechanics or some physical-​based approximation from the molecular and atomic spectral structures. However, the most commonly accepted databases are provided under the local thermodynamic equilibrium (LTE) approximation (Whiting et  al. 1966; Surzhikov and Shang 2012). The remaining challenge resides in computational algorithms. Since the solving procedures for the unsteady, compressible Navier-​Stoke equations are well known and documented by the computational fluid dynamics community from the later 1960s, only the historical development and the most recent innovations for solving the radiative rate equations Equations (9.2b) or (9.4d) are detailed in our discussions. Among the most efficient numerical algorithms for solving the radiative rate equations are a broad range of approaches from the classic half-​moment, spherical harmonic, discrete coordinates, and multiple-​spectral group, to the more recently developed Monte Carlo or ray-​tracing methods (Galvez 2005; Siegel and Howell 2002; Surzhikov and Shang 2012). Among them, the gist of multiple-​spectral group techniques for the rapidly varying optical property of a medium is by separating the spectrum into many subspectra, which is a sensible engineering technique. The idea is dividing the entire spectral domain into multiple spectral subgroups according to its frequency and summing over all the frequency-​independent solutions as a whole. The numerical accuracy can be improved by an increasing number of spectral subgroups. However, this approach cannot accurately simulate a medium of rapidly varying opacity. On the other hand, the Monte Carlo or the ray-​tracing method is very appealing because the intrinsic computational procedure closely mimics the physical process of radiation (Surzhikov and Shang 2012).

9.7

Ray-​Tracing Procedure The ray-​tracing technique is based on optical physics, and the procedures of geometric optics can be applied to the radiative exchange process. The radiative energy flux is actually evaluated on each single radiating ray, which is usually different from the coordinate on which the optical properties are determined. After a sufficient number of rays are traced, the complete radiating energy is summed over the entire

329

9.7  Ray-Tracing Procedure

329

radiating field of interest. In the situation where the beam path encounters multiple reflective surfaces, the reflecting ray is dictated by the property of the reflector. If the reflection is a specular surface, the incident ray is reflected symmetrically about the surface normal by an angle identical to the incidence. For thermal radiation study, the reflecting surfaces have a finite absorptivity, thus they will attenuate the radiative energy, but they will also emit energy. The relevant physics are complex, but the ray-​tracing method is widely adopted in aerospace engineering because the method is naturally suitable for parallel computational simulation. In fact, the ray-​ tracing technique is embarrassingly easy to map on a high-​performance concurrent or massively parallel computing architecture and the computational efficiency is scalable to tens of thousands of multi-​computers (Shang 2016). The radiation process occurs at the speed of light rather than sonic speed for aerodynamic phenomena. Therefore, the most convenient approach to address the additional energy exchange mechanism is integrating the radiative heat transfer as a heat source or sink in the global energy conservation equation. Again in view of the drastically different mechanisms and time scales from diffusion and convection to radiation, the instantaneous temporal behavior of thermal radiation is usually not sought after. Rather the coupling is conducted at the time scales of aerodynamics for nonequilibrium radiation simulation. The radiation transport equation (RTE) satisfying the law of detailed balance can be expressed as an integral-​differential equation in space. The following equation is derived from Equation (9.2b) with some approximations:

d I ν ( s, Ω ) σ = κ ν I b, ν − (κ ν + σ ν ) I ν ( s, Ω ) + ν ∫ σ ν ( s ) I ν ( s, Ω, Ω′ )d Ω′ ds 4π 4 π

(9.16)

In this equation, the symbol σ ν is the scattering cross-​section, and the last term on the right-​hand side of the equation represents the radiative scattering originating from other domains. The dependent variable I ν ( s, Ω ) is the radiation spectral intensity at the location s and moves along the direction of a unitary vector Ω at a frequency ν. The symbol κν denotes the absorption coefficient of radiation (cm–1), and dΩi is the infinitesimal element of the solid angle. A  large group of numerical schemes for solving the multidimensional RTE has been devised over the years and can be broadly categorized into the deterministic and stochastic approaches (Modest 1993; Siegel and Howell 2002; Surzhikov 2002). The methods of computational physics by the deterministic approach are highly developed and are built on the evolution of computational fluid dynamics (CFD). The critical heat loading of most interplanetary reentries takes place at the upper atmosphere and under a rarefied gas environment (Surzhikov and Shang 2012). The local thermodynamic equilibrium (LTE) condition and a non-​scattering medium are the accepted engineering practice for all radiative heat transfer simulations for interplanetary entry. These engineering practices when applied in analyzing reentry heat transfers have been verified reasonably well by flight data (Feldick 2011; Liu et al. 2010; Shang and Surzhikov 2012). The appropriate radiation transport

31

330

Radiative Energy Transfer

equation simplifies to the mostly widely adopted formulation in applications (Siegel and Howell 2002): d I ν (r , s) = κ ν I b , ν − kν I ν ( r , s ) ds (9.17a)



The general solution to the foregoing first-​order partial differential equation is: s

I ν ( s ) = I ν ( s0 )e − τ ν ( s0 ) + ∫ κ ν ( s ′ )I b, ν ( s ′ )e − τ ν ( s ′ ) ds ′ (9.17b)



s0

As we have discussed previously, the integral of the absorption coefficient along the distance of the beam path is an approximate optical penetration depth, s

τ ν ( s ′ ) = ∫ κ ν ( s ")ds ", which is a dimensionless parameter for measuring the opacity s′

of a medium excluding scattering mechanism at the location s'. For solving Equation (9.17a), the absorption coefficient of radiation, κ ν, along the position vector is needed and must be derived locally from an emitting/​absorbing medium. These spectral optical properties are uniquely related to the thermodynamic state and chemical composition of the radiating medium, and must be derived from the nonequilibrium flow field. In the process of interpreting the optical property of the medium, the effectiveness of the nearest neighbor search plays a pivotal role in computational efficiency. In general, the optical data-​generation process can be accomplished by the ab initio application of quantum mechanics or some physical-​ based approximation from the molecular and atomic spectral structures. However, the most commonly accepted databases under the local thermodynamic equilibrium (LTE) approximation have been provided by NASA (Whiting et  al. 1966), NIST (Ralchenko et al. 2006), and the Russian Academy of Sciences (Surzhikov 2002). These macroscopic spectral optical databases are adopted for most radiative simulations and the present study. For nonequilibrium radiative simulations, the aerodynamic conservation equations of a flow field must be solved iteratively in conjunction with the radiative energy transfer equation to achieve a global energy balance. The required optical properties associated with the quantum transition between energy states also alter the internal energy distribution of the medium. The coupling of the thermodynamic state with the optical properties of any gaseous medium is unique for the fundamental bound-​bound, bound-​free, and free-​free quantum transition processes (Zel’dovich and Raizer 2002). The numerical methods for solving the radiation transfer equation, Equation (9.15), as we have discussed, constitute a broad range of approaches from the classic spherical harmonic, half-​moment, multiple-​spectral group, to the more recently developed Monte Carlo or ray-​tracing methods (Galvez 2005; Kong, Ambrose, and Spanier 2008). Among them, the basic idea of a multiple-​spectral group technique for the optical property is a sensible engineering technique dividing the entire spectral domain into multiple spectral subgroups according to its frequency and

31

9.7  Ray-Tracing Procedure

331

Figure 9.7  Radiation flux distributions by a 148-​spectral group within a plasma laser

generator at five axial locations, 0.46, 0.93, 1.79, 4.1, and 10.8 cm.

summing over all the frequency-​independent solutions as a whole. The numerical accuracy can be improved by increasing the number of spectral subgroups. An illustrative example using the multiple spectral groups is provided by Surzhikov (2002). The computational simulated result for the interior of a CW CO2 laser generator is presented in Figure 9.7. The generator operated at one atmosphere provides a power of 200 kW by a flowing medium at the speed of 30 m/​s; the generator has a length dimension of 11 cm and a radius of 1.3 cm. The numerical results are developed by a 148-​group radiation flux model via the ray-​tracing method. A total of five solutions are displayed at the axial locations of 0.46, 0.93, 1.79, 4.1, and 10.8 cm, and have provided a sufficient resolution for simulation. However, the multiple spectral group approach is cannot accurately simulate a medium of rapidly varying opacity. The alternative approach is adopting the Monte Carlo or the ray-​tracing method; these approaches are very appealing because the intrinsic computational procedure closely mimics the physical process of radiation. The radiation flux vector used by ray tracing, like all other solutions solving the radiative rate equation, is a measure of energy exchange per unit time per unit area, and can be calculated as: ∞ 4π



w( s ) = ∫

∫ I ν ( s, Ω)d Ωd ν

0 0

(9.18a)

3

332

Radiative Energy Transfer

The infinitesimal increment of solid angle dΩ of this equation is defined by the azimuthal and angle displacements of direction cosines for the flux vector from the surface outward normal, dφ and dθ, respectively. The definition for solid angle integration associated with a single ray of radiation is straightforward. The ray-​tracing technique replicates the electromagnetic wave propagation by issuing a group of rays from a designated point on the radiating surface over the entire range of the solid angles. The radiation flux density is obtained by integrating the radiative spectral intensity over the solid angles and over all radiating spectra. Typically a few hundred surface points are processed in a cross-​flow plane of an earth reentry simulation, and a total of several hundred points along each tracing ray is calculated. Finally, at the least one hundred spectral subgroups of the spectral data are integrated over the entire spectrum; as a consequence, a hundred million calculations must be performed within an internal iteration. The coupling process demands a substantial amount of computational resources in addition to a large number of optical data shipments through the nearest neighbor search process for the ray-​tracing calculations. In a conventional ray-​tracing method, a group of tracing rays is designated by its solid angle Ωm,n and has been widely adopted in applications (Shang and Surzhikov 2012; Surzhikov and Shang 2012). The spectral flux density is often obtained by a simple first-​order approximation as: 2π



π

2

wν ( s ) = ∫ d ν ∫ d ϕ ∫ I ν ( s, Ω ) cos θ sin θd θ 0 m

0

n

= ∫ d ν∑ ( ϕ m +1 − ϕ m )∑ I ν ( Ω m,n )[sin θ n +1 cos θ n +1 − sin θ n cos θ n ](θ n +1 − θ n ) 1

1

(9.18b) The integration over the solid angle ΔΩ in a uniformly angular displacement space can also be obtained by the product sum with a weight function. In fact, the three-​ point Simpson’s rule generates a more accurate result than the aforementioned first-​ order approximation, and can be described as: wν ( s ) = ∫ d νΔΩ∑ wn I ν,n ( sn ) ∞



n

Δθ [ I ν,n ( sn ) cos θ n sin θ n + 4 I ν,n +1 ( sn +1 ) cos θ n +1 sin θ n +1 (9.18c) 3 m n ∞ Δθ5 ( 4 ) + I ν,n + 2 ( sn + 2 ) cos θ n + 2 sin θ n + 2 + I ν,n ( s )] 90

= ∫ d ν∑ Δϕ ∑

There are also known formulations for an even higher-​order Simpson’s rule; however, one needs to break down the finite number of the discrete data by the groups of three-​point or more segments and perform an algebraic sum. This integrating rule also has restrictions for an even number of intervals or an odd number of base points over a constant incremental value of ΔΩ. Although the higher-​order integration is desirable, the choice of the integrating method should

3

9.7  Ray-Tracing Procedure

333

be compatible with the accuracy of the interpolated spectral radiation intensity Iv ( Ωm,n ). As the radiating beam propagates through an inhomogeneous optical path, the spectral intensity needs to take into account the opaque effect along the beam path of the wave motion. In general, this can be accomplished by including the integration formulation as a modifier (Surzhikov 2002): r

s

rb

0

I ν ( Ω m,n ) = ∫ {I b, ν ( s )κ ν ( s ) exp[ − ∫ κ ν ( s ′ )ds ′ ]}ds (9.19a)



In Equation (9.19a), the integration limits rb and r are the beginning and ending locations along the ray, and the I b, ν is the source term of the emission from the body surface, which is the classic black body radiation intensity. Finally, the radiative flux density by the product sum with a weight function acquires the form:

m

n

r

s

1

1

r0

0

wν ( s ) = ∫ d ν∑ Δϕ m ∑ Δθ n wn cos θ n sin θ n {∫ I b, ν ( s )κ ν ( s ) exp[ − ∫ κ ν ( s ′ )ds ′ ]} (9.19b) ∞

In some practices, the local radiation spectral intensity I b, ν ( s ) and the absorption coefficient κ ν ( s ) have been approximated by the averaged values between the start and the end points of a tracing ray. Surzhikov (2002) has also recommended additional simplification by performing the integral as: k



I ν ( Ω m,n ) = ∑ I b, ν,k + 1 ( τ k +1 − τ k ) (9.20a) 2

k =1

In this equation, the radiation spectral intensity I b, ν,k + 1 is obtained by an average 2

temperature along the tracing ray between the positions of sb and s. This simple approximation is essentially the trapezoidal rule and is fully compatible with the mean-​value theorem. The symbol τ k in Equation (9.20a) denotes the optical depth sk

at these locations: τ k = exp[ − ∫ κ ν ( s ′ )ds ′ ]. s0

In short, the divergence of the radiation heat transfer rate in the global conservation energy equation Equation (9.4c) appears as either a heat sink or source; it now can be given as:



∞

4π

0

0

∇ ⋅ qr = ∫ κ ν [ 4 π I b, ν −

∫ I ν ( s,Ω)d Ω]d ν

(9.20b)

For applications of the ray-​tracing method, the thermodynamic state and gas composition are generated first by the solutions of nonequilibrium aerodynamic conservation laws on a body-​oriented coordinate, and then the optical property is interpreted along the tracing rays. The selected rays and the body-​oriented coordinate, either on a structured grid over a blunt forebody of a reentry space vehicle or on an unstructured grid within a scramjet combustor, are depicted in Figure 9.8.

35

334

Radiative Energy Transfer

Figure 9.8  Radiative rays on coordinates for flow field computations.

A typical coupling coordinates system along the radiating ray from a surface point is characterized by a set of solid angles over the frame of the nonequilibrium flow field. The radiative rays are emitted from a solid surface either uninterrupted toward outer space or bouncing from other reflectors, then exit the control volume. The coupling process between data on different coordinates is the nearest neighbor search (NNS), commonly referred to as the proximity search. It is an optimization problem for locating the closest discretized points among the two coordinates in metric space. This area of efforts can open new research and bring a new level of computational performance to multidisciplinary modeling and simulation for nonequilibrium radiation. A major portion of efforts in using the ray-​tracing method is focused on the optical data interpolation from the nonequilibrium flow field and projects onto the line-​of-​the-​sight tracing rays. The numerical resolution and computational efficiency improvement has been demonstrated by the space partition algorithm of the nearest neighbor search between the two coordinate systems for optical property determination. The basic idea is akin to both the principal axis trees (PAT) by NcNames (2001) and the binary trees (K-​D trees) by Freidman, Bently, and Finkel (1977) by reducing the data search domain. The more recent proximity search algorithm is put forward by Shang and colleagues (2014); the uniqueness of the space partition algorithm (SPA) is built on the vector projection of a tracing ray with

35

9.7  Ray-Tracing Procedure

335

Figure 9.9  Radiative heat transfer distributions on the stagnation region of the RAMC-​II

probe at different reentry stages.

respect to its surface outward normal. The optimizing technique is expected to reduce an order of magnitude of computational resources required for the optical data interpolation. An efficient ray-​tracing (Monte Carlo) method can be realized by jointly applying the line-​by-​line spectral data integration through the Gauss-​ Lobatto quadrature via the local resolution refinement. An innovative approach for nonequilibrium radiation simulation utilizing the ray-​tracing technique has been accomplished. A space partition algorithm for the nearest neighbor search process combining with a better placement of spectral points along the tracing rays by the Gauss-​Lobatto polynomial has been successfully implemented and coupled with the governing equations for nonequilibrium hypersonic flows. These two unique features of the approach have achieved a significant efficient computational gain over the existing methodology for the interdisciplinary computational simulation. On the relatively small databases of 15,416 and 60,636 discrete points over the reentering RAMC-​II probe, factors of 20.82 and 44.38 times computational efficiency gain are realized for nonequilibrium radiating hypersonic flow simulations. The magnitudes of the stagnation point heat transfer by the radiative mechanisms at five different trajectory stages by the ray-​tracing method are depicted in Figure 9.9. The computational results at different locations from the stagnation point are

37

336

Radiative Energy Transfer

indicated by the inset and over the wave number in the range of 10 4 ≤ w ≤ 105 cm −1 . The surface heat transfer distributions along the probe surface are generated at the altitudes of 61, 71, and 81 km on the fine mesh system of (326×186) and with a constant surface temperature of 1,000 K.  The radiative energy exchanges are calculated by a total of 3,721 tracing rays. It is clearly shown that the radiation heat transfer is essentially ceased from a distance of 0.2 m along the probe surface originating from the stagnation point. At the higher altitudes, 71 and 81 km, the radiating heat transfer rates are 36.38 and 29.13 W/​cm2. At the altitude of 61 km, the rate increases to a value of 38.61 W/​cm2 to become the peak radiative heating condition. The conductive-​convective heat transfers at different altitudes on the stagnation point of the probe are 74.99, 106.53, and 181.59 W/​cm2 at the altitudes of 81, 71, and 61 km, respectively. At the maximum heating load, which occurs at the altitude of 61 km, the radiative heat transfer contributes to 21.26 percent of the conductive-​ convective heating rate. The computational efficiency improvement is mostly gained from a space partition algorithm for the nearest neighbor search process. The better placement of the data along the tracing ray on the roots of Gauss-​Lobatto polynomial is also reducing the computational resource by a factor of five (Shang et al. 2014). The accuracy of numerical results has been systematically verified with flight data and previous numerical results. Additional improvement is still possible to achieve by a spectral-​like integral accuracy over the solid angle of spectral emission, and the idea of utilizing the weight function for high-​resolution integration is very promising.

9.8

Monte Carlo Method The Monte Carlo method is based on a statistical sample technique and can be applied to simulate the dynamics of photons or light quanta. For most thermal radiation applications, the collisions between particles and photons occur over many mean-​free paths of photons. Therefore, solving the radiation rate Equations (9.2b) or (9.16) is really tracing the history of a statistically meaningful random sample of photons for the past encounter or its source of emission. Computational simulations by the Monte Carlo method have great potential for developing the universal computational procedure for prediction of spectral emissivity and absorption of radiation by different aerospace vehicles. The Monte Carlo method has been applied to solve the radiation rate equation, as well as used to simulate directly the photon trajectories. In solving specially the radiation rate equation, the acceleration of the convergence by the Monte Carlo algorithm has been enhanced through coupling the forward and adjoining radiation transport equations (Kong et al. 2008):



Ω ⋅ ∇I ( s, Ω ) + k ν I ( s, Ω ) = σ ν ( s )∫ I ( s, Ω′ )p( s, Ω, Ω′ )d Ω′ + I νem ( s, Ω ) (9.21a) Ω

37

9.8  Monte Carlo Method

337

The solution of the radiation intensity can be expressed in terms of a volume Green’s function, which leads to an adjoining integro-​differential equation:

−Ω ⋅ ∇I' ( s, Ω ) + k ν I' ( s, Ω ) = σ ν ( s )∫ I' ( s, Ω′ )p( s, Ω, Ω′ )d Ω′ + I'νem ( s, Ω ) (9.21b) Ω

The boundary conditions for I ′( s, Ω ) will be dual to that for I ( s, Ω ), because they must satisfy a reflecting boundary condition as the case for light transport where Fresnel and Snell’s law applied (Hecht 2002); we have, then:

∫ I ( s, Ω ) I ν

em

Ω

′( s, Ω )dsd Ω = ∫ I' ( s, Ω ) I νem ( s, Ω )dsd Ω (9.21c) Ω

The approach of applying the Monte Carlo method to solve the radiation transport equation directly is mathematically rigorous and can automatically adapt itself to the specific needs of individual radiation problems. Another more fundamental approach using the Monte Carlo simulation for radiative energy transfer is tracking photon groups over an averaged spectral band by scientists in the 1990s (Lee, Hollands, and Rathby 1996; Surzhikov and Howell 1998). The methods can be applied to radiating phenomena such as light-​ scattering plumes of rockets, and emissivity of reentry space vehicles in light-​ scattering and non-​scattering atmospheres, trailing in the wake of meteors. The direct computational simulations analyze the energy transfer over thousands of spectral bands in which the optical thickness always varies by orders of magnitude. Performing simulations associated with complex configurations and including scattering phenomenon is beyond the capability of conventional computing systems. However, high-​performance parallel computers with distributed memory and concurrent calculations allow solving radically the main problem of the Monte Carlo imitating algorithms and reduce the dispersion of numerical simulation results. However, this method always incurs statistical error after tracing N number photon bundles of the sampling results S(N). The exact answer is theoretically possible by sampling infinitely many energy bundles S(∞), but it will not be practical. The error estimate is achievable by the mean square root deviation for the number of bundles and the number of subsample of radiative intensity: ε 2 = ∑ [S ( Ni ) − S ( N )]2 N ( N − 1). Through a series of comparative study and some high-​order spatial discretizing schemes (Gokmen et al. 2005; Wu, Modest, and Haworth 2007) for studying combustion phenomena, the computational method has ventured into previously unattainable research simulations. The Monte Carlo method for solving the radiative rate equation is in good agreement with either the tracing method or the discrete ordinate method, but the computational resource requirement is greater. The advantage of the Monte Carlo technique is unique, from which the statistical and discretization errors can be isolated and quantified. Therefore, from a purely computational consideration, it may limit the number of photon bundles that can be used in practice. In order to reduce the required computing resources, the accuracy of spatial discretizing can be restricted to a lower-​order scheme.

39

338

Radiative Energy Transfer

The basic Monte Carlo imitative algorithm for radiation energy of heat transfer in an arbitrary inhomogeneous volume of light-​scattering media has been considered by Surzhikov and Howell (1998). As an example, to calculate the radiative energy in the infrared spectrum (from 500 to 10,000  cm–1), the resolution of the half-​width of a rotational line and the width of a vibrational band of air are important for predictive accuracy. The half-​width of the rotational band spans a range from 0.01< γ < 1 cm–1, and the width of the vibration band Δω varies from 200 cm–1 to 500  cm–1. Therefore, for description of the molecule structure in the infrared spectrum, the wave number resolution must be finer than the size of the rotational spectral line, but the calculating incremental wave number shall not contain too many rotational lines to degrade the numerical efficiency. From this estimate, it is easily observed that the accurate simulation depends only on the number of statistical samples chosen, and the required number samples are huge. In short, a line-​ by-​line approach integrating the radiative heat transfer equation on a spectrum of rotational lines, as well as the line-​by-​line application to trajectories for a limited number of photons, must be conducted. In most spectral regions, the mediums are characterized by a wide variation of thin and thick optical thickness. Several numerical algorithms such as the hybrid statistical method, the method of smoothing coefficients, and the two-​group method have been developed with the thin optical approximation to reduce the number of photon bundles required (Surzhikov and Howell 1998). The evaluation criterion is to determine the reflectivity, emissivity, and transmissivity of an isotropic medium in a scattering and selective adsorbing plane layer. For lack of a rigorous database and a unified baseline for physical fidelity, the line-​by-​line spectral data computational becomes a defector standard bearer. In general, when the medium has a thin optical thickness, τ  0.1, all developed algorithms yield reasonable affinity with respect to results from line-​by-​line calculation. The discrepancy grows with the absorption and scattering with increasing optical thickness, as is expected. These basic research efforts open new avenues for innovations, but for now our discussions focus on the computational efficiency of different numerical algorithms. In the basic approach, the given spectral range is divided into N number of spectral subregions Δω g in terms of wave number. The following averaged absorption coefficient is introduced in the limits of each subregion as:

κg =

1 Δω g

Nl  p  g κ ω κ ω κ il ( ω ) dω (9.21) + + ( ) ( )  ∑ ∫ i  Δω g

where κ il ( ω ) is the spectral volumetric absorption coefficient of i-​th line in terms of wave number; N l is the number of lines located in the subregion Δω g . The spectral volumetric absorption of media either of solid or gas or liquid are designated as κ g ( ω ) and κ p ( ω ), respectively. If one assumes that all lines have a Lorentzian contour, then it can be given as:

κ i (ω ) =

Si γ L ,i 2 π ( ω − ω 0,i ) + γ 2L ,i   

(9.22)

39

References

339

where Si , γ Li are the intensity and half-​width of i-​th Lorentzian line, and ω 0,i is the wave number at the center of the i-​th line. Monte-​Carlo simulation for photon trajectories are performed for each spectral group Δωg as that for any “grey” medium (Surzhikov 2002a). It should be emphasized that the absorption of nonuniform medium coefficients κ g , κ p , κ ic , κ l , κ i ( ω ) is a function of wave number. The intensity Si and the half-​width Lorentzian line γ are functions of spatial location. Numerical dispersion of direct statistical simulation results is noticed, which has accumulated in the solution of similar problems in “grey” medium. Nevertheless, a satisfactorily accurate simulation is obtainable, as a guide, for simulation up to about N f = 10 4 − 5 × 10 4 trajectories. For the case of relatively low albedo, ω = σ ( σ + k ) < 0.9, with moderate optical thickness (τ < 1), this number can be reduced by an order of magnitude. On the contrary, for multi-​scattering mediums of greater optical thickness, this number shall be further increased. When there is a need to find averaged radiating characteristics over the spectral region Δω g , it is possible to use the following approach:  first find a solution that simulates trajectories only of N f ~ 10 4 photon groups. The energy of each new photon group is then estimated statistically and the spectral absorption coefficient can be evaluated at any spectral point within the spectral region Δω g . The algorithm can be modified as follows: the range Δω g is broken into many elementary spectral subregions. The number of these subregions must be sufficient for the detailed description of the line structure in the spectrum. For example, at Δω g = 20 cm−1 and an average size of half-​width γ = 0.1 cm−1 it is sufficient to consider around 1,000 spectral subregions Δω ′g . One can apply standard procedures of line-​by-​line spectral data integration to calculate averaged radiating characteristics, but must be within the bound of each spectral subregion Δω ′g . In general, the computational simulation can achieve reasonable accuracy by simulating trajectories not to exceed 104 photon groups (as in the regular line-​by-​line method), and usually only up to ten to fifteen photon groups should be sufficient. It must be stressed that it is impossible to determine the spectral characteristics inside Δω g in this manner, but to achieve only the averaged characteristics in the spectral region Δω g with an acceptable accuracy. In summary, the Monte Carlo method for solving the radiative rate equation can realistically represent the genetic nature of radiation energy transfer via the photon dynamics. The random sampling technique has opened a new avenue to study and understand radiation on a microscopic scale that is not achievable by any existing method known to us. References Balsara, D., Fast and accurate discrete ordinates method for multidimensional radiative transfer. Part I, basic methods, J. Quant. Spectr. Radiat. Transfer, Vol. 69, 2001, pp. 671–​701. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. Transport phenomena, John Wiley & Sons, New York, 1960.

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Capitelli, M., Celiberto, R., Gorse, C., and Giordano, D., Transport properties of high temperature air components: A review, Plasma Chem. Plasma Process., Vol. 16, 1996, pp. 267S–​302S. Chapman, S. and Cowling, T.G., The mathematical theory of non-​uniform gases (2nd edn.), Cambridge University Press, Cambridge, 1964. Eddington, A.S., The internal constitution of the stars, Dover Publications, New York, 1959. Feldick, A.M., Modest, M.F., and Levin, D.A., Closely coupling flowfield  –​radiation interactions during hypersonic re-​entry, J. Thermophys. Heat Transfer, Vol. 25, No. 4, 2011, pp. 481–​492. Filipskii, M.V. and Surzhikov, S.T., Numerical simulation of radiation heat transfer in laser plasma generators, AIAA Preprint 2004-​0988, 2004, Reno, NV. Filipskii, M.V. and Surzhikov, S.T. Discrete ordinates method for prediction of radiative heating of space vehicle, AIAA Preprint 2005-​4948, 2005, Reno, NV. Fiveland, W.A., Three-​dimensional radiative heat transfer solutions by the discrete-​ordinates method, J. Thermophys. Heat Transfer, Vol. 2, 1988, pp. 309–​316. Freidman, J.H., Bently, J.L., and Finkel, R.A., An algorithm for finding best matches in logarithmic expected time, ACM trans. Math. Softw., Vol. 3, No. 3, 1977, pp. 209–​226. Galvez, M., Ray-​tracing model for radiation transport in three-​dimensional LTE system, Appl. Phys., Vol. 38, 2005, pp. 3011–​3015. Gokmen, D., Faruk, A., Nevin, S., and Isil, A. Comparison between performance of Monte Carlo method and method of lines solution of discrete ordinates method, J. Quant. Spectr. Radiat. Transfer, Vol. 93, 2005, pp. 115–​124. Hecht, E., Optics (4th edn.), Addison Wesley, New York, 2002. Kong, R., Ambrose, M., and Spanier, J., Efficient, automated Monte Carlo methods for radiation transport, J. Comp. Phys., Vol. 227, 2008, pp. 9463–​9476. Lee, P.Y.C., Hollands, K.G.T., and Rathby, G.D., Reordering the absorption coefficient within the wide band for predicting gaseous radiative exchange, J. Heat Transfer, Vol. 118, No. 2, 1996, pp. 394–​400. Levin, E. and Wright, M.J., Collision integrals for ion-​neutral interaction of nitrogen and oxygen, J. Thermophys. Heat Transfer, Vol. 18, 2004, pp. 143–​147. Liu, Y., Prabhu, D., Trumble, K.A., Saunders, D., and Jenniskens, P., Radiation modeling for the reentry of the Stardust sample return capsule, J. Spacecr. Rockets, Vol. 47, No. 5., 2010, pp. 741–​752. Menart, J., Radiative transport in a two-​dimensional axisymmetric thermal plasma using the S–​N discrete ordinates method on a line-​by-​line basis, JQSRT, Vol. 67, 2000, pp. 273–​291. Mihalas, D. and Mihalas, B., Foundations of radiation hydrodynamics, Oxford University Press, New York, 1984. Milne, F.A., Thermodynamics of the stars, Handbuch der Astrophysics, Vol. 3, Springer-​ Verlag, Berlin, 1936, pp. 65–​635. Modest, M.F., Radiative heat transfer, McGraw-​Hill, New York, 1993. NcNames, J., A fast nearest-​neighbor algorithm based on a principal axis search tree, IEEE Trans. on Pattern Analysis & Machine Intelligence, Vol. 23, No. 9, Sept. 2001, pp. 964–​976. Ralchenko, Yu, et al., NIST atomic spectra data base, Version 3.10, July 2006. ordinates solution of radiaRamankutty, M.A. and Crosbie, A.L., Modified discrete-​ tive transfer in three-​ dimensional rectangular enclosures, JQSRT, Vol. 60, 1998, pp. 103–​134.

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Ratzel, A. and Howell, J.R., Two-​dimensional energy transfer in radiatively participating media with conduction by the PN approximation, Proc. 1982 International Heat Transfer Conf., Vol. 2, 1982, pp. 535–​540. Ripoll, J.F. and Wray, A.A., A half-​moment model for radiative transfer in a 3D gray medium and its reduction to moment model for hot, opaque source. JQSRT, Vol. 93, 2005, pp. 473–​519. Sakami, S., Charette, A., and Le Dez, V., Analysis of radiative heat transfer in enclosures of complex geometry using the discrete ordinates method, Proceedings of the Second International Symposium on Radiation Transfer, Kusadasi, Turkey, July, 1997, M. Pinar Mengüç Editor, Begell House, 1998, pp. 253–​270. Shang, J.S. Computational electromagnetic-​aerodynamics, IEEE Press Series on RF and Microwave Technology, John Wiley & Sons, Hoboken, NJ, 2016. Shang, J.S., Andrienko, D.A., Huang, P.G., and Surzhikov, S.T., A computational approach for hypersonic nonequilibrium radiation utilizing space partition algorithm and Gauss quadrature, J. Comp. Phys., Vol. 266, 2014, pp. 1–​21. Shang, J.S. and Surzhikov, S.T., Nonequilibrium radiative hypersonic flow simulation, J. Prog. Aerosp. Sci., Vol. 53, 2012, pp. 46–​65. Siegel, R. and Howell, J.R., Thermal radiation heat transfer (4th edn.), Taylor & Francis, New York, 2002. Surzhikov, S.T., Direct simulation Monte-​Carlo algorithms for the rocket exhaust plumes emissivity prediction, AIAA Preprint 2002-​0795, Reno, NV, 2002a.   Radiation modeling and spectral data, Von Karman Lecture Series, 2002-​7, Von Karman Institute for Fluid Dynamics, Rhode-​ST-​Genese Belgium, 2002b. Surzhikov, S.T. Radiative modeling in shock-​tubes and entry flows, RTO-​AVT-​VKI Lecture Series, Von Karman Institute for Fluid Dynamics, Rhode-​ST-​Genese Belgium, 2008. Surzhikov, S.T., Computational physics of electric discharges in gas flows, De Gruyter, Berlin, 2013. Surzhikov, S.T., Capitelli, M., and Colonna, G., Spectral optical properties of nonequilibrium hydrogen plasma for radiation heat transfer, AIAA Preprint, 2002–​3222, Maui, HI, 2002. Carlo simulation of radiation in scattering Surzhikov, S.T. and Howell, J.R., Monte-​ volumes with line structure, J.  Thermophys. Heat Transfer, Vol. 12., No. 2., 1998, pp. 278–​281. Surzhikov, S.T. and Shang, J.S., Coupled radiation-​gasdynamic model for Stardust earth entry simulation, J. Spacecr. Rockets, Vol. 49, No. 5, 2012, pp. 875–​888. Vincenti, W.G. and Traugott, S.C. The coupling of radiative transfer and gas motion, Ann. Rev. Fluid Mech., Vol. 3, 1971, pp. 89–​116. Whiting, E.E., Park, C., Liu, Y., Arnold, J.O., and Paterson, J.A., NeqAIR96, Nonequilibrium and equilibrium radiative transport and spectral program: User manual, NASA Report, 1389, Dec. 1966. Wu, Y., Modest, M.F., and Haworth, D.C., A high order Monte Carlo method for radiative transfer in direct numerical simulation, J. Comp. Phys., Vol. 223, 2007, pp. 898–​922. Zel’dovich, Ya.B. and Raizer, Yu.P., Physics of shock waves and high-​temperature hydrodynamic phenomena, Dover Publications, Mineola, NY, 2002.

10 Applications

Introduction As early as the nineteenth century, the active chemical-​reacting characteristic of plasma had been recognized. In particular, the dielectric barrier discharges were known in Europe as the silent discharge, which generates metastable chemical components such as ozone and had been applied to purify water and pasteurize dairy products (Elisson and Kogelschatz 1991). However, the first application of plasma to aerospace engineering is attributed to Goddard; he actually conducted the first experiment of an ion engine in the time frame of 1916 to 1917 (Jahn 1968). The ion thruster technology became one of the most sustained engineering developments using plasma for interplanetary exploration and satellite station keeping. Different types of ion engines became the single most practical plasma dynamics application to aerospace engineering. The research and development effort to improve the propulsive efficiency of ion thrusters is continuing even today. Another well-​known plasma application is for electrical power generation, which was started in 1938 by industry; unfortunately, the technology development was interrupted by the Second World War. In a magnetohydrodynamics (MHD) power generator, the electricity is extracted from flowing plasma across an applied magnetic field. The research efforts for a more efficient MHD generator are active worldwide, but its large-​scale application is hindered by the relatively high life cycle investment. A wide range of practical applications utilizing plasma dynamics in aerospace engineering was initiated in the late part of the 1950s. Resler and Sears (1958) advocated the potential application of electromagnetics to enhance aerodynamic performance. Their observations were strongly supported by the pioneering research by Bush (1958), who derived the theoretic couplings between the aerodynamic and electromagnetic forces and heat transfer in hypersonic flights. Ziemer (1959), on the other hand, had demonstrated for the first time that the magnetic pressure in the shock layer over a blunt body can displace the bow shock wave outward for supersonic flow by increasing the stand-​off distance of the enveloping shock. In the same time frame, Meyer also found that the Lorentz force always has a component, σ( u × B ) × B , that will decelerate the flow. Across the nonuniform velocity distribution in a shear layer, the decelerating force increases in magnitude with distance

34

Introduction

343

from the magnetic pole as the local velocity approaches the external flow. In general, the net result is a reduction of the velocity and temperature gradients within the shear layer. The fattening of the velocity and temperature gradients diminishes both heat transfer and skin friction on the contacting surface (Meyer 1958). In the following five decades, space exploration propelled the development of computational simulation techniques to provide thermal protection for space vehicles reentering the atmosphere (Park 2001). Very complicated and sophisticated ionization numerical modeling and simulation methodologies have been developed for the nonequilibrium weakly partially ionized air based on finite-​rate chemical kinetics; some are expanded to include the composite ablative surface materials (Chen and Milos 1999). The radiative heat transfer technique is also applied to understand the effectiveness of alleviating the tremendous heating load that a reentry vehicle must endure. These intensive research and development efforts enrich the knowledge of plasma dynamics and are exclusively applied to aerospace engineering. In the intermediate years, the unique features of plasma that can concentrate a huge amount of energy on a designated location and the chemically reactive discharge have been quickly recognized. High-​temperature plasma is applied to a wide range of engineering applications from the initiation of a nuclear chain reaction to cutting and welding tools. Since the degree of ionization and the component of ionized gas are easily controlled, plasma is also used to modify the surface properties of materials. Manufacturing industries have adopted plasma-​processing technology to itch out delicate and complex integrated circuit boards on a large scale (Lieberman and Lichtenberg 2005). Again, the emission of plasma covers a spectrum from inferred, visible light, to extra violet; it is applicable for display panels and visual devices. Beyond aerospace engineering, the engineering applications using plasma have benefited automobile manufacturing. Magneto-​aerodynamic research was rejuvenated in the mid-​1990s by the innovative AJAX hypersonic concept vehicle (Gurijanov and Harsha 1996). A  very promising innovation is the idea of an MHD-​bypass scramjet engine (Fraishtadt, Kuranov, and Sheikin 1998), which, if realized in principle, can increase the propulsive efficiency by transferring a part of the energy from ionized air in the inlet to the exhaust nozzle, thereby using the energy more effectively. First, the electricity from the compressed ionized air within the inlet is extracted by an MHD electrical generator, then the generated electricity is applied to accelerate the exhausting jet like an ion engine. The innovation seems to fit perfectly to address the technical needs of the National Aero-​Space Plane (NASP) program. The goal of the NASP program was to develop and demonstrate the feasibility of a single-​stage-​to-​orbit vehicle that can take off from conventional airfields, accelerates to hypersonic speeds, enters orbit by a single-​stage launching operation, delivers useful payloads to space, and returns to earth with only air-​breathing propulsive capabilities. Stimulated by the renewed interest in plasma dynamics research, many exciting innovations have been put forth in areas of remote energy deposition for drag

345

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Applications

reduction, plasma actuators for flow control, radiation-​driven hypersonic wind tunnels, sonic boom meditation, and enhanced plasma ignition and combustion stability (Bletzinger et  al. 2005). For wave drag reduction, the counter-​flowing plasma injection and remote energy deposition using focused lasers ahead of the flight path of a blunt body were considered; an astonishing amount of drag reduction has been observed from basic experimental research. However, for the counter-​flowing plasma injection, the favorable interaction is actually acquired from the degeneration of the bow shock wave into multiple shock wave structures rather than from the injected plasma. Using plasma actuators for flow control creates a huge number of research activities, because the aerodynamic control is generated from simple surface discharges without parasitic effects when deactivated, and can be actuated in microseconds. The basic mechanisms are derived from the Joule heating and periodic electrostatic forces. It has been found that the most viable applications are limited to a low Reynolds number environment or to the aerodynamic bifurcations of dynamic stall of lifting devices or possibly to the control of laminar-​turbulent transition. The most recent research activity for aerospace engineering application is concentrated on enhanced plasma ignition and combustion stability. The premise rests on improvements to the diffusion and mixing process using plasma to overcome the actuated energy barrier, leading to chemical reaction. Extensive research in engineering applicability by using plasma, however, has revealed that additional and accurate evaluations of the pertinent physics are necessary for modeling and analyzing these innovative concepts to reach a conclusive assessment. From lessons learned, most recent research activities need to refocus on basic and simpler aerodynamic–​electromagnetic interactions. The electromagnetic force and energy indeed have provided an expanded physical dimension for aerospace engineering applications. However, the ionized gas also creates challenges such as the communication blackout phenomenon that the telecommunication by electromagnetic waves has either been reflected at the medium interface or rapidly attenuated in partially ionized air (Smoot and Underwood 1966; Fredrick, Blevins, and Coleman 1995). Magneto–​aerodynamics interaction is truly an interdisciplinary endeavor; the interacting physical phenomena require the interplay of aerodynamics, electromagnetics, chemical physics, and quantum physics to describe ionized gas flow in the presence of electromagnetic fields (Shang 2016). This interdisciplinary endeavor not only presents extremely complex science issues but also demands a significant knowledge base that was not completely available at that time or even today. The plasma dynamics impacts extensive sectors of aerospace engineering and shows remarkable potential for innovations that are limited only by our imagination. The progress in developing viable engineering applications is unabated, thus viable applications are discovered nearly daily and interlocked in a chain of events. Therefore, in our following discussions, only the most documented achievements that have infused our knowledge are highlighted, which should not be biased by our limited exposures.

345

10.1  Ion Thrusters

10.1

345

Ion Thrusters Thrust generated by rocket or jet engines results from the impulse or change of momentum across a control volume defined by the engine and the ambient environment. The thrust is a normal reaction force, according to Newton’s third law, on the control surface that intersects the engine exhaust plane. The complete equation for thrust is just the projection of the conservation momentum equation in integral form onto the outward normal unit vector nˆ of the engine exit plane: T = ∫∫∫



∂ρu dv + ∫∫ nˆ ⋅ [ u(ρu + p I − τ ) + ρe E + J × B ]dA (10.1a) ∂t

The thrust equation in steady state is traditionally simplified by neglecting the shear stress and electromagnetic forces, and assuming a low fuel mixture ratio to get estimated propulsion efficiency. The resultant propulsion efficient for a flight vehicle involves only the vehicle flight velocity u and the propellant exhaust velocity uex normal the engine exhausting plane to appear as:

ηp =



2 u uex (10.1b) 1 + u uex

The estimated propulsion efficiency is valid only when the flight vehicle has a slower speed than the exhaust propellant speed uex u < uex. Speeds of orbiting satellites and deep space exploitation vehicles are much greater than the convenient crafts powered by air-​breathing propulsive systems. A  higher propulsive efficiency of space travel must be derived from a higher jet exhaust speed. The velocity of electrically charged particles can easily achieve the desired objective. Another measure of rocket engine efficiency is the specific impulse that relates to fuel consumption and the total thrust, thus it has also been referred to as the propellant mass efficiency:  g (10.1c) I sp = T m



where g is the standard gravitational acceleration at sea level. This performance parameter has a physical dimension in seconds, and the thrust of an engine is directly proportional to the specific impulse. A  comparison of the ion thrusters with convenient chemical-​propellant rockets is presented in Table 10.1. The advantage of higher propulsion efficiency by ion thrusters over the convenient propulsive systems for space flight becomes obvious. From the third law of Newton mechanics, the total thrust is proportional to the product of the exhaust Table 10.1  Comparison of performance characteristics of propulsion systems Engine type

Exhaust speed (km/​s)

Specific impulse (s)

Chemical rocket

2.5–​4.4

250–​400

Ion thruster

20.0–​50.0

1,700–​4,200

Impulse magneto-​plasma

42.0–​210.0

3,000–​12,000

347

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velocity vector and the exhausting mass flow rate. The magnitude of the ion thrust in operations is, however, limited to hundreds of mille-​Newton and the output power is around tens of kW. In order to achieve the necessary change to flight velocity, a continuous thrust from an ion thruster often has a lifetime operation of more than 20,000 hours. The ion engine is one of the very few bona fide practicing technologies using electromagnetic forces to accelerate ions for propulsion. The first working ion thruster is an 8 cm diameter cesium contact ion engine. The Space Electric Rocket Test (SERT1) was tested in a suborbital flight in 1964 (Cybulski et al. 1966). Four years later, a two-​cesium contact engine was launched aboard the ATS-​4 spacecraft to complete a successful orbital flight. In 1998, the Deep Space (DS1) launched; it was the first spacecraft using ion propulsion to reach another planetary body. Ever since, ion engines have been routinely used for station keeping of commercial and military communication satellites in geosynchronous orbits; such satellites as European Space Agency (ESA) Artemis and US military AEHF-​1 and AEHF-​ 2 have been used to maintain geosynchronous orbit with a perigee of 10,150 km above the earth. NASA also used the ion thruster Solar Technology Application Readiness (NSTAR) in two successful missions; first for a flyby to asteroids (9969 Braillo and comet Borrelley), second to enter and leave more than one orbit (Gallimore et al. 1994). More recently, a future launch has been planned for a 200 kW Variable Specific Impulse Magnetoplasma Rocket (VASIMR) electromagnetic thruster to be placed and tested on the International Space Station. The VASIMR is essentially an electro-​thermal plasma thruster using either argon or xenon as the propellant. Plasma is generated by an electromagnet and heated by a helical RF antenna, then strong electromagnets are used as a convergent-​divergent nozzle to accelerate the ions and electrons. Through these arrangements, a variable specific impulse ion thruster can achieve an engine exhaust speed up to 50,000 m/​s (Longmier et al. 2009). It is important to realize that the electromagnetic propulsion systems are limited by the thrust per unit area of the engine exhaust, and the avoidable degradation of performance due to space charge accumulation by Child’s law (Jahn 1968). In any event, the specific impulse of a typical ion thruster has a range from 1,000 to 10,000 seconds using propellants of xenon, bismuth, liquid cesium, and argon, and can attain a propulsive efficiency up to 80 percent. There are two basic types of ion thrusters known as the gridded electrostatic ion thruster and the Hall effect thruster, shown in Figures 10.1a and 10.1b. The operational environments for both are identical to a rarified gas domain. The typical operating conditions in an ionization chamber of an ion thruster usually have the maximum electron number density around 1013/​cm3 (1019/​m3), and the electron temperature is on the order of 3 eV or greater. Thus, the Debye length in the high number density region of an ionization chamber is just a few microns and the plasma frequency is estimated to be about 1011/​s. The plasma in the discharge chamber of ion thrusters exists in a state similar to most low-​pressure, direct-​ current plasma actuators for flow control. Note that the gas mixture in the ion

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thruster is always in a rarefied condition and measured by the Knudsen number, K n = λ L , and λ is the mean free path between collisions; L denotes a characteristic length of the chamber. At a xenon neutral particle number density of 1019/​m3 and a collision cross-​section of 5 × 10–20 m2, respectively, the mean free path yields a value about 1.4 m within a typical ionization chamber. In general, the mean free path is several times greater than most characteristic dimensions of a typical ionization chamber; therefore, a computational simulation is beyond the valid range of the governing equations in a continuum regime. Under this condition, the numerical results must be generated by a particle dynamics formulation, usually through a combination of Monte Carlo techniques. Meanwhile, the electron confinement is achieved by setting up a magnetic field long the walls of the ionization chamber. The applied magnetic field for axisymmetric configurations is by utilizing magnet rings that reflect electrons back into the discharge chamber, giving them a much longer accelerating trajectory. For the electrostatic ion thruster, ions are accelerated in the same direction by an applied electric field, and the Hall effect thruster is driven by the Lorentz force. The major components of any ion engine consist of a plasma generator, accelerator grids, and neutralizers. The ionization is achieved by electron impact into cesium, argon, krypton, and, most commonly, xenon, because of their relatively low atomic ionization potentials. A contemporary implementation of an ionization chamber employs a hollow cathode inserted into the ionization chamber, which provides the secondary electron emission. High-​energy electrons from the hollow cathode collide with neutrals in the ionization chamber and produce positively charged ions. The positively charged ions are extracted from the ionization chamber and are accelerated downstream of the engine by a system of grids. The high ion exhaust velocity is generated by electrostatic potential difference across these grids and with minimum charged particle impingement on these grids. The grids serve two major functions:  first, keeping neutral particles in the ionization chamber while allowing ions to be separated, and second, accelerating the separated ions to high velocities. The perforation of the grids also focuses the ion stream into an array of smaller beams with minimum impingement on the solid portion of the grids. The last device of an ion thruster is the neutralizer, which prevents charge accumulation on the vehicle. Again, it is basically another hollow cathode that emits electrons into the positively charged ion beam to prevent the ion engine from accumulating a large number of charges over the space vehicle that would prevent additional ions from leaving the ionization chamber. The schematics for two types of electrostatic ion thruster are depicted in Figures 10.1a and 10.1b. For the electrostatic thruster, plasma is generated by ionizing the propellant by electron impact, and the beam of positively charged ions is separated from the electrons at the downstream end of the ionizing chamber; a collision-​less stream of ions is then accelerated by an electrostatic field between two permeable grids. After an array of ion beams is stabilized, electrons are injected into the beam, downstream of the grids, with a neutralizing cathode to produce a high-​speed exhaust of zero net electrical charge.

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Figure 10.1a  Electrostatic ion thruster.

Figure 10.1b  Schematic Hall effect thruster.

The electromagnetic thruster or the Hall effect thruster, on the other hand, relies on the interaction of conducting the propellant charged motion with a magnetic field to provide the acceleration by the Lorentz force J × B . The applied electric field E and the magnetic flux density B are perpendicular to each other and to the ionized propellant velocity u, as shown in Figure 10.1b. The ionized gas is accelerated by electric potential and a radial magnetic field between a cylindrical anode and cathode. At the center of the thruster, a magnetic field is generated by an electric coil winding over a spike; at the end of the spike, the electrons are trapped by the magnetic field and attracted to the anode. Some electrons spiral toward the anode to become the Hall current and close the electric circuit. The major portion of the propellant is introduced near the anode after ionization; the electrons move toward the cathode, and the ions finally leave the thruster at high speed. The streamwise body force accelerates the propellant along the thruster, and transfers streamwise momentum to neutral particles by collisions and by a microscopic polarized field. As a consequence, the thruster’s charged particle number densities

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are greater than that of the electrostatic ion thruster, but at a cost of much more complex electromagnetic–​aerodynamic interactions. It is important to note that the ionized propellant in an electromagnetic thruster is basically macroscopically neutral; therefore, it is not overly constrained by the space charge limitations as in the electrostatic accelerators. Most current methods of computational electromagnetic-​ aerodynamics in a rarefied gas domain within an ionization chamber adopt the particle-​in-​cell (PIC) approach in combination with a direct simulation Monte Carlo (DSMC) technique (Birdsall and Langdon 1985; Verboncoeur 2005). In a full PIC-​DSMC simulation, all particles, including the neutral and charged species, are tracked by the macro-​ particle assumption. In fact, the PIC-​DSMC scheme simulates the dynamics of the electrical charge–​carried and neutral particles. The basic governing equations consist of the classic Newton-​Lorentz equations for charged particles’ motion and the electric field distribution by the Poisson equation:



dx d 2 xi = q( E + i × B ) dt 2 dt (10.2) e 2 ∇ ϕ = ( n− − n+ ) ε mi

In a full PIC-​DSMC computational simulation, all charged particles are tracked by the macro-​particle assumption. In theory, each macro particle represents billions to trillions of real particles of the rarefied plasma, thus the total number of the tracked particles can easily exceed 108. The required huge computational resources must satisfy the peculiar and stringent computational stability criteria of PIC-​ DSMC algorithms in time and space. The numerical procedure also suffers from the statistical noise of a large number of sampled macro particles. Meanwhile, the physical fidelity of modeling imposes severe resolution limitations on the Debye shielding length in space and time through the charge number density, Equation (1.11b). In short, the computational simulations by the Lagrangian formulation are essential to analyzing the plasma dynamics in the rarefied gas regime, and the computational resources requirement is extremely demanding. The outstanding numerical results for the electrostatic ion thrust (NSTAR) have been generated by Mahalingam (2007). The considered particles consist of xenon and single-​and double-​charged xenon ions (Xe+ and Xe++), as well as electrons. A  total of 1.267  million individual macro particles were carried out on 8,383 unevenly spaced cells. The numerical simulation has taken advantage of the axisymmetric configuration; the numerical analyses were performed only for the top-​ half meridional plane of symmetry. The contour of the singly charged xenon ions is presented in Figure 10.2a with respect to the normalized axial location and at steady state at the TH-​15 operation condition. The charge number density spans a range from 1016 to 1019 per cubic meter (1010 to 1013/​cm3), and the high concentration of the ions is clustered along the axis of symmetry. The lines of magnetic field intensity are also superimposed on the ion contour plot; the three magnet rings function well to confine the charged particles away from the solid chamber surface.

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Figure 10.2a Xe+ distribution.

Figure 10.2b  Secondary electron distribution within the NSTAR electrostatic thruster

(Mahalingam 2007).

The electrons in the numerical simulation consist of the primary and the secondary types due to the inelastic collision processes. The primary electrons convert into the secondary after the inelastic collisions and their energy drops below 4 eV. In Figure 10.2b, the secondary electron number density is depicted, which is similar to the singly charged xenon ions to reflect the globally neutral characteristic of plasma. The primary electrons are generally confined by the magnetic field, and the secondary electrons have higher energy around 3 eV to 5 eV along the axis of the thruster. The maximum current density value is observed for the secondary electron at the magnetic cusp region (at r = 0.15 in nondimensional units) on the back wall surface. The primary electron current density value has similar behavior in the same region. Both primary and secondary electron current density values drop sharply before and after the cusp region. At the region away from the cusp, the secondary

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electron current density is found to be smaller and the primary electron current has completely vanished. This behavior is caused by the strong magnetic field at the back wall surface. In order to process the huge amount of data, the parallel computational technique was implemented. The data structure was organized by a one-​dimensional domain decomposition scheme onto thirty processors with overlapping data array for data communication via the message-​ passing interface (MPI) paradigm. The numerical simulations reveal the detailed xenon ionization process to show a significant number of primary electrons lose their energy through the xenon excitation process and only the primary electrons are responsive to the excitation of X e+ +. The simulations also find that increasing the magnetic field strength by the magnetic ring can improve the ion beam current distribution and discharge current. The converged computational results not only have displayed good agreement with the measured ion beam current and propellant utilization efficiency but also achieved a reasonable agreement with the measured ion beam profile within the ion engine. More important, the computational simulation has demonstrated the capability of accurately duplicating the NSTAR discharge chamber at its full operation condition (TH-​15). Currently, academic research in ion thrusters is sustained worldwide. The areas of research span disciplinary topics ranging from fundamental studies of the microscopic-​scale mechanisms and how electric propulsion plasma is accelerated, to how the exhaust plumes will impact the operational characteristics of large satellite structures. Knowledge of these types is paramount for our basic understanding of aerodynamic–​electromagnetic interactions. Wider variations of innovative electromagnetic thruster configurations, such as the magneto-​plasma-​dynamic thruster, are being pursued. A recent research development for deep space exploration and the international space station is VASIMR; the ionization uses the inductive plasma generator for which we have discussed in Chapter 7.7. The generated plasma is further accelerated by the magnetic field to achieve a greater ion exhaust velocity (Longmier et al. 2009). In a demonstration of this new capacity, the impulse thruster has operated with a total of 200 kW direct current power input to the radio frequency generator and coupled with plasma for a short pulse. VASIMR is one of the very few ion thrusters that can process a great power density up to 6 mW/​m2 for a long lifetime. In short, the ion and electromagnetic thrusters exemplify complex electromagnetics–​aerodynamic interactions; it is very effective, but is severely limited in the resultant force magnitude, mostly due to space charge limitations. Therefore, for now, the ion thrusters are relegated to special purpose aerospace engineering applications.

10.2

Reentry Thermal Protection All the kinetic energy of a reentry vehicle has to be dissipated and converted into thermal energy when it lands. The amount of kinetic energy is enormous depending on the original orbits or trajectories of the vehicles, which have values

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from 6.4 × 107 to 2.3 × 109 J/​kg. A part of the energy is deposited on or absorbed by the surface material of the reentry vehicle, and some is transferred to the passing air stream; another portion radiates into free space. The heat transfer rate varies widely depending on the reentry speed; at a low earth orbital (LEO) reentry speed of 8 km/​s and for a Galileo reentry speed up to 47.7 km/​s, the rate ranges from 2.3×102 to 1.7×105 kW/​m2. The conductive, convective, and radiative heat transfer rates are strongly dependent on the surface material of the reentry vehicle, whether it’s a non-​catalytic or a full-​catalytic material. Regardless, thermal protecting materials are necessary to maintain the structural integrity of space vehicles during the reentry phase and of the missile launching and solid rocket propulsion systems. The ablative surface for thermal protection is widely used for its high-​phase-​changing latent heat to absorb the converted thermal energy from the decelerated flow and for its low thermal conductivity to minimize the conductive heat transfer to the substrate (Chen and Milos 1999). The radiative heat transfer strongly depends on the temperature and composition of ionized gas in the shock layer and becomes an increasingly important mechanism for thermal protection for interplanetary exploration. All physics-​ based computational simulations have been built on the understanding of the ionization process around a reentering space vehicle. The air stream surrounding a reentering vehicle downstream of the enveloping bow shock is subjected to extremely strong compression and high temperature to become ionized. In particular, the ionization process occurs in a very narrow region, just a few mean free paths across the shock wave, and continues in the shock layer with a thickness just a fraction of the nose radius of the vehicle. At the reentering speed of tens of kilometers per second, the duration of a reentry with a strong ionization process also occurs in a very short period, usually around 40 seconds (Cauchon 1967; Park 2007; Johnston, Hollis, and Sutton 2008). The thermodynamic nonequilibrium chemical physics process, including quantum transitions and thus the radiative energy transfer, is also presented. The overall process is purely based on thermal ionization by chemical kinetics, including radiation, and the simulation’s capabilities by either experimental or computational techniques for engineering are highly developed. For computational modeling and simulation, the governing equations and numerical algorithms have been formulated and outlined in Chapter 6.5 and the required numerical methods for radiative heat transfer by different approaches have also been presented in Chapter 9.3 through 9.8. In the early experimental research into radiative heat transfer in shock tubes, it was found that, at a shock speed slower than 10 km/​s, the peak radiation occurs in a narrow region immediately downstream of the bow shock wave. The transient radiation is the result of the excited nitrogen molecules and ionized composition. When the shock speed is increased above 10 km/​s, the radiation is generated by the nonequilibrium processes of vibrational relaxation and electronic excitations. The maximum radiative intensity exhibits an inverse relationship with the distance traveled from the bow shock within the shock layer. As a consequence, the spatial integration of the radiative intensity within the nonequilibrium region displays a

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binary scaling relationship (Sharma 1991). The maximum radiation was confirmed later by other experimental facilities to be around 10 w/​cm2 (100 kW/​m2) in the wavelength range from 3,500 A to 4,500 A. The first flight data were provided by experimental efforts by the Fire reentry probe prior to the Apollo mission (Park 2004). The Fire-​II probe has a nose radius of 75  cm and is equipped with a radiometer and a scanning spectrometer to be operated in the range from 3,000 A to 6,000 A. The collected flight data are widely used as the validating base for computational simulation and are included in our discussion later. The radiative heat flux at the stagnation point of Apollo 4 has shown a peak value of 30.09 w/​cm2 (300.9 kW/​m2). It was estimated that two-​thirds of the stagnation point convective heating rate was due to the absorption of radiation by the boundary layer at the peaking heat loading condition of Apollo 4. In 2006, the Stardust interplanetary vehicle was launched and returned to earth safely after a successful mission collecting cometary material from comet Wild 2.  Its reentering speed became the fastest of manmade objects to attain a speed of 12.9 km/​s or a Mach number of 36 at the initial reentry point. The radiative heat transfer rate is calculated to be 30.9 percent of the combined conductive and convective heat transfer rate at the peak heating condition. In numerical simulations, accurate prescriptions of interface boundary conditions of mass, momentum, and energy balance are required, and are strongly dependent on the ambient condition. In fact, some of the critical chemical reactions of the ablative surface are exclusively determined by the components of the surrounding high-​temperature air mixture (Park 2007; Shang and Surzhikov 2010). For this reason, a detailed knowledge of the gas composition in the nonequilibrium shock layer and physically meaningful interface boundary conditions are paramount in describing the ablative phenomenon. In the following discussion, an assessment is presented of the recent progress in simulating nonequilibrium hypersonic flow, including transport properties of the high-​ temperature gas medium, chemical kinetic and chemical-​physic models, and radiative energy transfer. The pacing items of modeling and simulating of nonequilibrium hypersonic flow are also identified and delineated. In the past decades, impressive progress has been made in a broad spectrum of nonequilibrium hypersonic research both in computational and experimental techniques. Since the hypersonic aerodynamic phenomena are especially based simulations, intense computational challenging to replicate by ground-​ modeling and simulation efforts have been invested in for a sustained period of more than five decades. These numerical procedures are all implemented with the advanced numerical algorithms such as the approximate factored scheme (Beam and Warming 1978; Wright, Chandler, and Bose 1998) and the under/​over relaxation finite-​volume algorithm (Cheatwood and Gnoffo 1996). For shock capturing, the flux-​vector splitting and flux-​difference splitting methods that are based on total variation-​diminishing algorithms are adopted with additional options for parallel computing and multi-​grid techniques. Equally important, a wide range of chemical kinetics models for nonequilibrium high-​temperature gas mixtures

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Figure 10.3  Flow field structures at three reentry stages of Stardust probes.

with ionization and energy transition among internal degrees of freedom has been devised and calibrated. For practical engineering applications, these models and detailed formulations can be found in Chapter  6.4. These numerical simulation capabilities have become the backbone of engineering design and of analysis tools for reentry thermal protection. Computational simulations have been successfully applied to all space vehicle programs from Fire, RAM-​C, and Orion to Stardust for earth reentry. Numerical simulation has also been carried out for entry into other planets such as the Pioneer to Venus, and the Galileo probe to Jupiter as well as to Neptune and Titan. Among all reentry projects, the Stardust reentry probe exemplifies the shape of the reentry vehicle; its configuration consists of a 60-​degree one-​half angle spherical cone with a nose radius of 0.229 m as the forebody. The afterbody is a truncated 30-​degree cone with a base radius of 0.406 m. The corner radius of this configuration at the juncture of the forebody and afterbody is merely 0.02 m. This particular shape has a feature common in all reentry vehicles  –​a blunt forebody to reduce the maximum heat transfer rate in the stagnation region. The field structures in the velocity contours of the Stardust probe with an angle of attack of 8 degrees at three different reentry stages in 42, 48, and 54 seconds after a designated reentry altitude are depicted in Figure 10.3. At these points in time, the probe is traveling at the speeds of 12.42, 12.00, and 11.37 km/​s, which belong to the hypersonic region. Again, the flow topology is shared by all the reentry vehicles; they all have a strong bow enveloping shockwave to contain a relatively thin shock followed by rapid expansions around the forebody and the afterbody, then emerging into the wake region or the aerodynamic shadow. In the base region, the flow is separated from the vehicle at the outer region of its base to form a recirculating flow with the lowest heat transfer rate. All these dominant flow topologies of the Stardust probe

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during earth reentry have been clearly captured and displayed by the computational simulations by Surzhikov and Shang (2012). The nonequilibrium chemical reactions, including endothermic reactions, exothermic reactions, and quantum transitions, occur immediately adjacent to the reentering vehicle, which creates an extreme high-​energy environment that always leads to catastrophic vehicle structure failure. Therefore, it is critically important to understand the basic mechanism and to develop an accurate and efficient predictive method. The high-​temperature gas mixture in the shock layers and in the wakes of reentering space vehicles with ablative species usually contains optically active components such as CO2, H2O, CH4, N2, O2, NO, N2+, C2, CO, and other metastable elements. At a reentry speed around 10 km/​s, the strong absorption by C, CO, NO, and O2 has been detected in vacuum ultraviolet (80–​200 nm) and ultraviolet spectra (Surzhikov and Shang 2012). The emission of cyanide, CN, has also been noted, as well as a low-​intensity absorption by CO2 within the boundary layer. In all, nonequilibrium radiation is responsible for 26 percent of the total radiation flux in the spectra range of 80–​600 nm. Some of these transition processes are not fully understood but become critical mechanisms for radiative heat transfer (Liu et al. 2010). The challenge in analyzing the ablation phenomenon arises from its complex interdisciplinary nature. In spite of extensive applications of ablators for thermal protection systems during the past decades, the physical-​chemistry process of ablating in the reentry environment is still far from fully understood. Nevertheless the contributions to aerospace engineering by Park and Balakrishnan (1985); Chen and Milos (1999) are remarkable. For example, the most widely adopted silicone-​impregnated reusable ceramic (SIRCA) or phenolic impregnated carbon ablator (PICA) are frequently reinforced by secondary impregnation of polymethyl methacrylate (PMMA) to maximize the pyrolysis gas generation so to minimize the char surface recession. The complicate modeling has been attempted and successfully applied for engineering analyses by Chen and Milos (2005). The added physical-​chemistry phenomenon at the media interface always occurs at the molecular/​atomic levels that can be modeled only in a computational simulation. The accuracy of modeling of the ablating medium based on the limited experimental database and under a harsh environment is severely limited by the physical fidelity. The aerodynamic phenomena in which the ablation takes place are equally daunting and are not retained as one of the topics in the present discussion, but remain as a specialized engineering topic. In a hypersonic reentry environment, the radiative heat transfer contributes a substantial amount of energy in addition to the conductive and convective processes. An illustrative example via computational simulation is provided by the Stardust reentry probes (Shang and Surzhikov 2011; Surzhikov and Shang 2012). The numerical methods for simulating radiative flux are the multi-​group spectral models via the ray-​tracing and half-​moment methods for the radiative heat transfer over the heat shield. The optical data developed from quantum mechanics are usually included in the wide-​and narrow-​band spectral models, and a line-​by-​line

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Figure 10.4  Heat transfer rates on the symmetrical plane of the Stardust probe at peak heating condition.

spectral approach is integrated over the investigated spectrum. All computational approaches are actually very similar; a studied spectral range is divided into a finite number of spectrum domains with average optical properties. In the limit, the spectral intensity Equation (9.15) is solved with a frequency-​independent coefficient of emission and absorption in the divided spectrum bands. The radiative energy flux is obtained by integrating over the complete spectrum. The complication of the nonlinear line structure of the spectra along the optically thick path is also bypassed by the averaged linear structure of atomic and molecular spectra. By a reduced dimensional approximation, the half-​moment method has proven the best approach to solve the loosely coupled radiative gas dynamics problem. The multi-​group spectral model has been successfully demonstrated by Surzhikov (2002); and Surzhikov and Shang(2012) for Stardust reentry simulation. Figure 10.4 shows the distribution of all heat fluxes along the distance measured from the stagnation point in the plane of symmetry on the entire surface of the Stardust, including conductive, diffusion (or convective), and radiative components. The displayed results are collected at the peak heating condition of the Stardust reentry at t = 54 sec, ρ∞ = 2.34 × 10 −7 g cm3 , T∞ = 238.5 K, and u∞ = 11.14 km s without ablation. The combined conductive and diffusive heat transfer rates yield a value in excess of 1.19×103 W/​cm2 at the stagnation point of the capsule.

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The total heat transfer rate does not include the active ablation, and is in very good agreement with the results by Olynick, Chen, and Tauber (1999)of 1.2 × 103 W/​cm2 and agrees equally well with the laminar flow result of Park (2007) at 1.189 × 103/​cm2. The radiative heat transfer rate over the forebody also reveals a comparable value of 248 w/​cm2 with results by Olynick and colleagues (1999) and Park. When the ablation is included in the heat loading simulation, the total heat transfer rate is 120 W/​cm2 versus the convective-​conductive heat transfer rate of 839 W/​cm2. Again, all the predicted numerical results yield a comparable magnitude summing over all components of heat transfer mechanisms. A major source of predictive discrepancy emerges from the nonequilibrium chemical kinetic modeling. This observation is easily made by examining the detailed temperature distributions of chemical species of air (N2, O2, and NO) in vibration internal degrees of freedom, the translational temperature along the stagnation streamline in the shock layer, and the important stand-​off distance of the bow shock wave (Surzhikov and Shang 2015). The computational simulated condition is duplicating the flight test program of Fire-​II from the recorded stages from t = 1,634 to t = 1,648 seconds (Cauchon 1967; Johnston et al. 2008). The comparison of the kinetics models by Dunn and Kang (1973) and Park (2001) is displayed in Figure 10.5a at the reentry stage t = 1634 s for the Fire II capsule at ρ∞ = 0.372 × 10 −7 g cm 3 , T∞ = 254.0 K, and u∞ = 10.97 km s. From the flight data, the total heat transfer rate has a range from 445 to 609 W/​cm2. For a series of calculations, the local thermal equilibrium (LTE) assumption is applied to both the coupled interdisciplinary gas dynamics and radiation equations. From the selected computational results, Figure  10.5a, the difference in temperature profiles generated by the two models along the stagnation streamline is quite significant and has the maximum discrepancy for all cases examined. The stand-​off distance from the model by Dunn and Kang is slightly greater than that of Park by a maximum ratio of 5 percent. On the other hand, the predicted equilibrated temperatures from the model of Park are 1,000 K to 1,200 K lower during the earlier stages of reentry at t  =  1644 s.  However, the predicted temperatures in the shock layer are essentially identical during the later stages of reentry. In short, the difference in predicted results between the two chemical kinetics models is of the same order of magnitude as the difference produced by the models for nonequilibrium vibration models. However, it should be stressed that the influence of the chemical kinetics and physical kinetics models (nonequilibrium dissociation) by a temperature difference of 1,000 K is quite significant for radiative heat computations. One of the uncertainties of using these models is vibrational relaxation in the presence of dissociation. To address this issue, two models of vibrational degrees of freedom are implemented in the computational procedure. In the first case, the molecular distributions of vibrational states are assumed to be Maxwell-​ Boltzmann; the reaction rates of dissociation are calculated by the LTE model. In the second case, the Treanor-​Marrone model of nonequilibrium dissociation is used (Surzhikov and Shang 2015). A group of comparative studies at eight reentering stages from t = 1,634 s to t = 1,644 s is performed, but only the result at t = 1,644 s

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Figure 10.5a  Comparison of Dunn and Kang and Park nonequilibrium chemical-​physical

kinetics models.

Figure 10.5b  Effects of dissociation biased to vibrational relaxation.

is presented in Figure 10.5b. The Dunn and Kang kinetics model was used in this series of computational simulations. The biased dissociation with vibration relaxation using the kinetics model of Dunn and Kang reveals a consistently greater stand-​off distance of the shock layer than the LTE assumption. Independent from the different assumptions used for the simulations, the maximum stand-​off distance of 5.25 cm is observed at t = 1,634 s. The minimum counterpart is recorded at 3.6 cm at the last stage of the Fire II trajectory simulated, t = 1,648 s. As a consequence, the equilibrated condition between the vibrational and translational temperatures is reached sooner in the shock layer

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at the later stages of reentry. In general, the maximum translational temperature in the immediate post shock region is higher with the Treanor-​Marrone model than that of LTE, which is understandable because the former is based on a realistic energy transport process (Surzhikov and Shang 2015). During the fully nonequilibrium part of the reentering trajectory, for the stages at a time elapse less than 1,637 s, the models of nonequilibrium dissociation significantly alter the vibrational temperatures through the consequence of changing radiative emission. The state of the art at present for the radiation model is highlighted by a distinctive feature, including the contributions of atomic and ion spectral lines to various models of physical and chemical kinetics for radiation heat transfer to a super-​ orbital reentry vehicle. The adopted numerical procedure for radiation transfer in the shock layer over the front shield of the space vehicle is based on the half-​ moment method for solving the radiation heat transfer equation with a line-​by-​line spectral evaluation. By this approach, the spectral optical property computations adopt the ab initio technique, including parameters of atomic line strengths and half-​ widths. This approach allows parametric computations of approximately 4,000 atomic lines for the atoms and ions of N, N+, O, O+, C, and C+. The line-​by-​ line spectral data calculation is carried out through a special procedure by creating an inhomogeneous distribution of optical properties with respect to wave number. As a consequence, it allows a significantly reduced number to the required line-​by-​ line spectral data. It has been shown that the new line-​by-​line approach can achieve high accuracy with less than 80,000 nodes of data (Surzhikov 2000; Surzhikov and Shang 2015). Figure  10.6 presents the contrast between the total heat transfer rates at the stagnation point with/​without the contribution by using the atomic spectral lines calculations for the Fire-​II probe. For these computations, the chemical kinetics model of Dunn and Kang is adopted. The standards for comparison are established by the databases of Cauchon (1967) and the correlated results by Johnston and colleagues (2008). The experimental data indicate heat transfer rates of 150 W/​cm2 at the initial descending stage and a peak heating load of 1140 W/​cm2 at 1644.5 s; the data by Cauchon are designated by the filled black square symbols. The total heat transfer rates reduce to a value less than 320 W/​cm2 at the last measured stage, t = 1,652 ss. The computational results using the line-​by-​line approach agree very well with data collected by Cauchon over the entire trajectory. The conclusion holds independently from the two possible nonequilibrium dissociation models for radiation computations. Equally important, the difference between the total heat transfer rates with/​without the atomic spectral lines is 150 W/​cm2 at the peak heating load; radiative heating without atomic lines can be estimated from this figure for t = 1,644.5 s. Finally, the predicted heat transfer rates with and without the contribution of the atomic spectral lines bracket the results from the LAURA and GIANT codes, presented by Olynick and colleagues (1994). The radiative heat fluxes by both the LTE and the Treanor-​Marrone models are included, and there is a smaller difference in the numerical results than from the predictions omitting the atomic spectral lines. As a reference for determining

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Figure 10.6  Comparison of the line-​by-​line radiation models for simulation of Fire-​II flight test data with bench marks.

the relative magnitudes of total heat flux to the nonequilibrium radiative flux, the correlated data by Johnston, designated by the yellow filled square symbols, are included. The radiative heating calculations at the seven trajectory points by Olynick and colleagues are also inserted and denoted by the black circle symbols. The ratio of the peak radiative and total heat fluxes at the different stages of reentry attains a value of 28 percent to show a significant contribution by radiative heat transfer for space vehicle thermal protection. In summary, the translational  temperature in the enveloping shock layer bounded over the earth reentry vehicles has been predicted and verified to be consistently over 10,000 K. Under the environment, air is fully dissociated and partially ionized, and radiation will be produced by quantum transitions from oxygen and nitrogen atoms as well as the complex abated compounds. The extremely high heat loading on earth reentry space vehicle surfaces is generated in part by the recombination of ionized gas through diffusion from the convective heat transfer process. And then the dissipated kinetic energy converts into thermal form and conducts through the heat shield coated with ablating material. Therefore plasma plays a controlling role in thermal protection of the reentering space vehicle not only by the direct radiative heating but also by the absorption in the shear layer and recombination on the vehicle surface. The computational electromagnetic-​aerodynamics technology integrating quantum chemical kinetics, radiation,

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electromagnetics, and aerodynamics has provided the ability of thermal protection on outer space explorations.

10.3

Plasma Actuators for Flow Control Aerodynamic control using plasma actuators attracts a huge amount of research interest because its implementation on an aerospace vehicle does not require machined components like aileron or strake as control surface, and when deactivated it will not degrade the aerodynamic performance. Furthermore, the actuation is accomplished in microseconds without using any mechanical servos for weight saving. The innovative flow control is possible because plasma is an electrically conducting work medium that increases the physical mechanism through electromagnetic effects. The basic mechanisms for flow control using electron impact ionization are provided by Joule heating, electrostatic force due to charged separation, and Lorentz force by an externally applied magnetic field. However, it is critical to recognize that the relative magnitude of the electromagnetic force and energy can be generated by electronic impact is at the most at the perturbation level. The energy generated by Joule heating via a single direct electrical discharge (DCD) is around a few mW/​cm2, but the heat transfer rate of an aerospace vehicle in flight is at least tens of kW/​cm2. Similarly, the electrostatic force generated by an alternative current discharge (DBD) from charge separation is on the order of magnitude of few Newton/​m3 and the typical volumetric aerodynamic force in subsonic or transonic regions is around tens of thousands Newton/​m3. In practical applications, the applied magnetic flux density B is limited to a fraction of a Tesla and the discharge electric current density J of a discharge is merely in mA; thus the Lorentz force, J × B , is also severely limited. Therefore, the applications of a plasma actuator for flow control can be the most effective at the aerodynamic bifurcation points such as the onset of boundary layer separation, and dynamic stall of the lifting surface, and the possible laminar-​turbulent transition. The plasma actuation in applications is generally separated into two categories, the thermal effect and the non-​thermal type. The former is simply based on the Joule heating in low-​density environments, which has been demonstrated; when actuated, it can generate a local heating spot in the plasma sheath or in the inner region of the boundary layer. The sudden local density by heating increases the displacement of the boundary layer thickness. In any supersonic stream, any sudden growth of the displacement thickness will deflect the flow outward from the surface and cause it to interact with the external flow, resulting in a series of compression waves, and to coalesce into a compression shock. It is a classic viscous–​inviscid flow interaction and is most pronounced at a sharp leading edge of hypersonic flow known as the pressure or Mach wave interaction. The Mach wave interaction occurs mostly at the sharp leading edge of a flat surface in hypersonic flow involving sonic waves through flow deflection and elevates local surface pressure. In hypersonic flow, the boundary layer is no longer

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negligible to the oncoming free stream. This physical phenomenon has been routinely observed in hypersonic flow over a sharp leading edge even if the surface is immersed at a zero incidence to the oncoming stream, but the Mach wave interaction is negligible in low supersonic or subsonic flows. According to classic boundary layer theory, the equivalent body to the external flow is the original body inclination plus the displacement thickness of an attached boundary layer. The local hypersonic similarity parameter is then κ = M ∞ ( α b + d δ* dx ). For the case of a flat plate, α b = 0, the laminar boundary layer thickness near the leading edge can be scaled as:

δ* x  δ x  ( γ − 1) 2 (M ∞2 c∞ Re x ) (10.3a) At the sharp leading edge of a flat plate at zero incidences, the scaled boundary layer thickness has a singular behavior at the leading edge. The phenomenon is confirmed by experimental observations to appear as a blast wave originating at the sharp leading edge; however, the induced surface pressure decays rapidly toward downstream. Meanwhile, the hypersonic similarity parameter of the small perturbation theory becomes:



κ = M ∞ ( δ x ) = M ∞3 c∞ Re x = χ (10.3b) The induced surface pressure on the plate by the viscous–​inviscid flow interaction can be determined by substituting the hypersonic similarity parameter χ into the tangent wedge approximation (a simplified oblique shock equation), and by expanding the approximated shock relationship in terms of the surface-​to-​free-​stream-​pressure ratio ( p w p∞ ) according to the large and small values of χ. The induced pressure is traditionally classified into strong and weak interactions according to whether the value of χ is greater or smaller than the value of three. The resultant strong and weak pressure interaction distributions at the leading edge of a cold flat surface are expressible, respectively, as (Hayes and Probstein 1959):



pw p∞ = 1.0 + 0.5 χ Strong interaction (10.3c) pw p∞ = 1.0 + 0.078 χ Weakinteractio on Under the strong interaction environment, the surface pressure at the sharp leading edge will be elevated by a factor at least 2.5 times the oncoming free stream. The Mach wave interaction induced by a Joule heating plasma actuator generates a surface pressure rise that is inversely proportional to the square root of the distance from leading edge, Equation (10.3b). It is an important illustration that a small local thermal perturbation to the hypersonic surface shear layer growth can substantially modify the flow field structure through the viscous–​inviscid interaction. Figure 10.7 displays an induced oblique shock generated by a simple DCD actuator on a sharp leading plate with a low power input of 164 Watts in a Mach number 5.15 hypersonic flow (Shang et  al. 2005). The second shock downstream of the leading edge induced by the Mach wave interaction is much weaker, but is still captured by the schlieren photograph and the density contours of the accompanying computational simulation. On the electrode, Joule heating at the cathode creates a

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Figure 10.7  Oblique shock generated by DCD over a sharp leading edge plate, M = 5.15.

sudden outward growth of the boundary layer displacement thickness that leads to the Mach wave interaction. In turn, the induced family of surface compression waves coalesces into an oblique shock. This phenomenon is captured by both the schlieren photograph and computational simulation in a Mach 5 plasma channel. The Joule heating from a direct current discharge (DCD) is generated by the applied voltages varying from 0.5 kV to 1.1 kV and at electric currents from 50 mA to 200 mA. The family of compression waves initiated at the leading edge of the cathode produces an induced surface pressure rise that constitutes an effective hypersonic flow control technique. The Langmuir probe measurements show that the electron number density at a given height above the plate can vary by orders of magnitude between the cathode and the anode, and a similar variation is observed in the vertical direction above the cathode within the discharge domain. Nevertheless, this is a bona fide aerodynamics–​electromagnetic interaction, and the DCD-​induced Mach wave or pressure interaction has been applied for flow control to appear as a virtual leading edge strake (Shang et al. 2005) and virtual variable geometric inlet (Shang, Chang, and Surzhikov 2007). The aerodynamic–​electromagnetic interaction was also verified by Borghi and colleagues (2006) on a wedge in hypersonic flows. When the DCD is adopted as a virtual leading strake with a power input from 50 W to 350 W, it generates a pressure rise on a sharp-​leading flat plate equivalent to when the plate has executed a deflection angle up to 5 degrees in a Mach number of 5.16 plasma channel. The induced surface pressure plateau can be further enhanced by 33  percent with an applied transverse of a magnetic flux density of 0.5 Tesla (Surzhikov 2013; Shang 2015). However, the control effectiveness is limited to an ambient pressure of 10 torr, which belongs to the upper atmospheric flight condition. The DCD has been applied for flow control to appear as a virtual leading edge strake (Shang et al. 2005) and can equally apply as a virtual variable geometric inlet (Shang et al. 2007).

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Figure 10.8a  Temperature profiles of a virtual rectangular variable geometric inlet.

When a DCD is applied as a virtually variable geometric inlet by activating the DCD near the entrance of inlets, the mechanism is the identical Mach wave interaction for the virtual leading edge stake. Again, the Joule heating of the discharge suddenly increases the boundary layer’s displacement thickness on the sidewalls of the inlet, which induces flow deflection inward as a compression surface of a variable geometric inlet. A small (10.1 cm × 3.81 cm × 3.11 cm) rectangular inlet model has been tested in a Mach 5.16 plasma channel with a characteristic Reynolds number of 2.57×105, and accompanied by numerical simulation (Shang et al. 2007). Figure  10.8a presents the computed temperature profiles in an x-​y plane over the centerline of the cathode, x/​L = 0.124, and near the inlet exit, x/​L = 0.875, and includes an insert to show the electrodes’ placement of the experimental model. In this presentation, the actuated DCD and the unperturbed simulations are grouped together to accentuate the thermal perturbation to the flow field by the surface gas discharge. The numerical simulation captures the physics in that the Joule heating and the convective electrode heating release a significant amount of thermal energy into the air stream. The elevated temperature in the inner region of the shear layer reduces the value of density locally, and in turn significantly increases the displacement thickness of the shear layer. In addition, the total Joule heating over the cathode is on the same order of magnitude as the convective electrode heating, 7.4 Watts versus 6.6 Watts. The magnitude of the computed electrostatic force is merely 430 dyne/​cm3 ( 4.30 × 10 −3 N m3 ), and this force is exerted mostly downward toward the cathode. In a shear layer over a flat surface, this force is not supported by the shear stress and is directly transmitted to the solid surface. Therefore, the computed result substantiates the fact that the thermal effect of the DCD is dominant over that of the electrostatic force for the ensuing viscous–​inviscid interaction. Far downstream of the discharging electrodes, the Joule heating diminishes and the convective heat transfer persists. The energy-​dissipating shear layer actually heats the model surface to reveal a higher model surface temperature. Despite the

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Figure 10.8b  Comparison of computed Pitot pressure distributions along virtual variable

inlet with data, M = 6.15, Rey = 2.57×105, P = 62.0 W.

high electron temperature, the nonequilibrium ion temperature is retained at the energy state of ambient condition. From experimental measurement, the vibration temperature of electrons is around 5,000 K and is the result of energy cascading from the electron impact ionization. From the viewpoint of plasma generation, this behavior is extremely important, but the vibrational internal degree of freedom contributes very little to the interacting flow field structure. The experiments for creating a virtual variable cross-​section inlet are carried out with a small amount of energy input by two pairs of DCD mounted flush on the sidewalls by an applied DC current of 775 V and with a current of 80 mA. When the DC discharge is activated, the oblique shocks that originate from the leading edge coalesce with the DCD-​induced compression waves, leading to a steepened oblique shock angle. Downstream from the shocks’ intersection, the Pitot pressure results generally indicate a higher level at the wave front than the unperturbed counterpart. The shock waves are clustered toward the middle portion of the rectangular inlet. However, the basic flow field structure remains unaltered. Figure 10.8b depicts a comparison of experimental and computed Pitot pressures along the centerline of the rectangular inlet when the DCD is either actuated or deactivated. Both results, generated at the stagnation pressure of 580 Torr, capture the interacting oblique shocks within the inlet. When the DCD is actuated, the induced oblique shock becomes steeper and moves the incipient point of the shock

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waves upstream. The ensured expansion behind the strengthened shock and a lower static pressure along the centerline produce a lower Pitot pressure downstream. The actuated DCD leads to a higher peak Pitot pressure and an upstream movement, in contrast to its deactivated counterpart. The computed results reach very good agreement with the experimental observations. The computations under predict the peak Pitot pressure by 2 percent and also over predict the uniform entrance condition by 1.2 percent. This small discrepancy in magnitude is directly attributable to the fluctuation of the DC discharge and the uncertain Mach number at the entrance of the inlet due to model blockage. The alternating current (AC) plasma actuator or the dielectric barrier discharge (DBD) eliminates the restriction of electron impact, that it must be operated in a low-​pressure environment by virtue of its self-​limiting ability from transition to arc (Elisson and Kogelschatz 1991; Enloe et al. 2004). In DBD operation, an exposed electrode is placed atop a thin film of dielectrics with few millimeters thickness, and the other electrode is embedded within the dielectric. In the positive polarity of the AC cycle when discharge is ignited, charge separation occurs and all positively charged ions are expelled from the exposed anode and attach to the dielectric surface; the accumulated ions reduce the electric potential across the electrodes, thus helping to prevent the discharge transition to arc. In the negative polarity of the AC cycle, the exposed electrode acts as a cathode; the surface charge accumulation just reverses. During the movement of separated charge, the heavy ions collide with neutral particles to create a wall jet-​like flow field over the dielectric; the velocity of the induced wall jet has a magnitude of less than 10 meter per second (Corke, Post, and Orlov 2007). Although the periodic electrostatic forces during each AC cycle are opposite to each other in microwave frequency, due to the different electric permittivity of electrodes, a net gas motion results from the exposed electrode toward the dielectric, and the magnitude of the so-​called electric wind becomes the main mechanism for using DBD for flow control. The force and momentum of DBD are extremely difficult to measure because the discharge is characterized by multiple random micro discharges or streamers in space and time. The net force of DBD is time-​averaged information over a period of microseconds due to charge separation. There are no concrete experimental data for the periodic electrostatic force, and the required direct numerical resolution is beyond the reach of present computational simulation capability. Nevertheless, based on inelastic collision models, independent computations have been performed by Surzhikov (2002) and Boeuf and Pitchford (2005), as well as Shang, Roveda, and Huang (2011). Figure 10.9 presents the computational results of the x-​component force distributions within the charge separation domain at different heights above the dielectric surface. The numerical results show the maximum force is within the plasma sheath, and, equally important, that the periodic electrostatic force is a push-​and-​pull during the complete AC cycle of the DBD to negate each other. However, the greater magnitude in the positive polarity of the AC cycle than the negative phase counterpart exerts the net force on the charged particles, especially on the ions, and becomes the net balance of the counteracting

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Figure 10.9  X-​component force at different heights above the dielectric surface. ϕ = 8.0 kV, ω = 5.5 kHz, electric permittivity ε = 2.7.

force. The net time-​averaged force over an AC cycle is therefore much less to show a value of just a few Newtown per cubic meters. The first remarkable flow visualization using DBD for flow control is recorded by Post (2004) and presented in Figure 10.10. It illustrates the effectiveness of the DBD actuator in controlling leading edge separation on a generic NACA 663–​018 airfoil by energizing the flow in the stagnation region to overcome the adverse pressure gradient. These images are generated at relatively low free-​stream speeds from 10 m/​s to 30 m/​s, which give a low Reynolds number based on chord length from 7.70 × 104 to 4.60 × 105. The left panel of the photograph image displays smoke traces of a flow field formation without an activated plasma actuator. The flow separated from the airfoil at the leading edge covers the entire leeward side of the airfoil, which is clearly visible and is extending further into the far wake region. A drastic flow field modification takes place when the DBD actuator is actuated, which is depicted on the right side panel; the plasma actuators keeping the flow attached to the airfoil at the angle of attack exceed the stall limit by a value of 8 degrees. The boundary layer over the airfoil remains attached from the leading edge and extends toward the trailing edge and beyond. The laminar structure of the flow from the unperturbed upstream region is maintained until the far wake region. DBD actuators have also been applied for controlling dynamic stall with airfoil leading edge separation at high angles of attack. Numerical and experimental investigations have been conducted for NACA 0009, 0012, 0015, 0021, 663–​018, and HS3412 airfoils. Experimental studies have been conducted for Reynolds numbers based on the chord length covering a range from 1.7 × 104 to 4.6 × 105 and encompassing free-​stream speeds from 10 m/​s to 30 m/​s. The effectiveness of DBD flow control is also confirmed by experimental observations for the NACA 0015 airfoil. The DBD is generated by a symmetrical electrodes configuration and the

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Figure 10.10  Leading edge separation control by DBD actuator, M. Post PhD Dissertation, University of Notre Dame, 2004.

dielectrics material is made of a 5  mm-​thick Kapton film covering the full span of the airfoils. The operational frequency of the AC voltage is 5 kHz at a voltage of 11 kV and an estimated power around 20 Watts per foot span. From the smoke streak-​line patterns, the leading edge stall leads to a full leading-​edge separation and the separated flow region covers the entire suction surface of the airfoil (Corke et  al. 2007). The DBD actuator extends the onset stall angle of this airfoil from approximately 14 degrees to 16 degrees. Again, it is a clear indication that the wall jet generated by the DBD at a possible magnitude up to 5 m/​s can be significant to overcome the local surface adverse-​pressure gradient. This experimental result demonstrates that the wall jet of an activated DBD postpones an airfoil’s stall to high angles of attack. A modeled numerical simulation for an NACA 0021 airfoil at a free stream speed of 35 m/​s and a chord Reynolds number of 8.0 × 105 also shows that the leading edge stall has been delayed from the angle of attack from 16 degrees until beyond 23 degrees. Another striking experimental demonstration of a DBD for flow control is the suppression of the Von Karman street downstream of a circular cylinder (Thomas, Kozlov, and Corke 2008). The classic oscillating wake structure of a circular cylinder is known to take place at a Reynolds number as low as 36 and persisting into tens of millions. In the experimental observation and by a direct contrast of two particle-​image-​velocimetry images for flow at a Reynolds number of 3.3 × 104, four activated DBD actuators keep flow attached on the leeside of a cylinder. The DBD actuators are placed at 90, 135, 225, and 270 degrees measured clockwise from the forward stagnation point of the cylinder. After the activation of the DBD actuator, the alternating vortexes shedding from the cylinder are completely suppressed by the induced velocity on the downstream side of the cylinder. The flow in the wake region rejoins on the centerline of the symmetric flow field without a separation region. From these basic research results using DBD for flow control, a few definitive conclusions may be drawn; namely, at the relatively low Reynolds condition where the wall jet velocity or the electric wind of DBD is comparable with the flow field

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to be a control, the flow control is the most effective. In general, flow control based on electron impact by a single pair of electrodes is severely limited by their power output. Therefore an alternative may be achieved by applying a bank of plasma actuators or augmented by the externally applied magnetic field to bring in the Lorentz acceleration. Using amplifications by consequential viscous–​ inviscid aerodynamic interaction has also proved effective. In principle, flow control using DBD is achieved by the induced wall jet, which is the most common mechanism for hydrodynamic stability, laminar-​turbulent transition, onset of boundary layer separation, and general bifurcation points of aerodynamics. However, there is not a single experimental demonstration that DBD has been successfully applied to control hydrodynamic instability leading to laminar-​turbulent transition. It is fully understood that flow control for aerodynamic bifurcation is complex and that the wall jet of DBD can appear as a large amplitude disturbance, in which a laminar flow may lead to turbulent spots without amplification through the normal mode of the Tollmein-​Schlicting wave. This possibility has been called high-​intensity bypass by Morkovin, and is not an eigenvalue problem (Reshoko 1976). However, the idea of using a strong magnetic field to restrain the random small eddy motion for plasma has not been explored at all (Mitchner and Kruger 1973). Additional research in this area is warranted, which requires extraordinary precision, but the result would be highly valued.

10.4

Remote Energy Deposition Aerodynamic drag reduction using plasma or hot-​gas injection has been reported by Ganiev and colleagues (2000). A  nearly 60  percent drag reduction has been observed experimentally over a Mach number range from 0.5 to 4.0. The mechanisms of this drastic drag reduction by counter-​flow plasma injection can be broken down into the effects of counter-​flow jet–​shock interaction and possible nonequilibrium energy exchange with plasma. A systematic side-​by-​side computational and experimental investigation of counter-​flowing plasma injection has reached its conclusion (Shang 2002). Experimental observations have verified that most wave drag reduction in aerodynamic applications by either a spike or a jet spike on a blunt-​nosed body is derived from splitting a single bow shock into multiple shock wave structures. There is also a bifurcation in the counter-​flowing jet–​ shock interaction that changes from an unsteady shock structure to a steady shock wave system (Shang 2002). In both dynamic states, the additional shock waves associated with a jet spike consist of a ring shock and a Mach disk that terminates the forward motion of the counter-​flowing jet. After the counter-​flow jet reverses its direction by a Mach disk, it appears as a free-​shear layer. The shear layer flows downstream over the recirculating zone, and eventually reattaches to the blunt body, effectively forming a new slender body shape through fluid displacement. The compression waves associated with the reattaching shear layer coalesce into the ring shock.

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Figure 10.11  Shock wave bifurcation by counter-​flow jet injection.

In Figure 10.11, the transition sequence of shock bifurcations of a jet spike over a hemispherical cylinder at a Mach number of 5.85 is depicted. Only six schlieren photographs are included at a wind tunnel stagnation temperature, To = 610 K and pressure po = 0.689×103 kPa. Each photograph represents different jet stagnation pressures normalized by the wind tunnel stagnation pressure pj/​po at the same jet temperature of 295 K.  The flow field becomes unsteady even at the lowest normalized injection pressure of 0.148. As the jet stagnation pressure increases, the length of jet upstream penetration increases accordingly. Due to the unsteady aerodynamic motion, the image of the jet-​induced shock in the schlieren photograph becomes blurred by multiple exposures. Meanwhile, the amplitude of oscillatory motion is also increased until it reaches a maximum around pj/​po = 0.9. In the subcritical region, the shock structure is later identified as the long-​penetration mode (LPM). At the critical point, a bifurcation of shock wave structure takes place, and further increases in jet stagnation pressure do not alter the complex shock structure significantly. The magnitude of oscillation also reduces to an undetectable level, in the supercritical region, the structure is known as the short-​penetration mode (SPM). Identical shock bifurcation was also observed for several other wind tunnel stagnation pressures tested. The measured drag data normalized with respect to the no-​injection case are given in Figure  10.12. The plasma jet is generated by a plasma arc torch with vibrational and electronic temperatures of 4,400 K and 20,000 K and an electron number density of 3×1012 /​cm3. The maximum power input to the torch is rated at 72.8 kW, but in the experiment, the torch is operated only at the starting mode,

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Figure 10.12  Aerodynamics drag reduction by counter-​flow jet injection.

thus the plasma generation power is much lower than the rate value. The flow fields are generated at four wind tunnel stagnation pressures of 0.345×103, 0.689×103, 1.379×103, and 2.068×103 kPa, all at a stagnation temperature of 610 K. The measured data represent the total axial force exerted on the entire hemispherical cylinder, including the wave drag, skin friction drag, base drag, and reverse thrust by the counter-​flow jet. In general, the critical points of bifurcation are confined in the range 0.8 ≤ pj/​po ≤ 1.05, and the value at the critical point increases as the tunnel stagnation pressure decreases. The overall behavior of drag reduction with counter-​ flow plasma jets at different wind tunnel stagnation pressures is similar. The measured data with the counter-​flowing jet are always lower than the no-​injection case, even at the lowest jet injection pressure. The drag reduction due to shock modification overwhelms completely the reverse thrust by the counter-​flow jet. Although the drag distributions vary at different test conditions, the maximum reduction in drag, which occurs at the bifurcation point, is as high as 60  percent from the baseline case. At the highest counter-​flow jet stagnation pressure tested, the drag reduction is still 30 percent lower than the blunt body flow without the jet injection. However, most drag reduction is derived from the single bow shock wave degeneration into a multiple-​shock structure induced by the counter-​flow injection. Only a small portion, much less than 10 percent of the reduced aerodynamics, is contributed by the high-​energy plasma interaction. The energy remote deposition by a pulsating laser upstream of a blunt body is an entirely different phenomenon from the counter-​flow plasma injection for aerodynamic drag reduction. The process is accurately a relatively new ionization

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procedure by optical or radiative means that was discovered in the mid-​1960s, as discussed in Chapter 7.8. The energy is deposited into a gas by a focused microwave of a single frequency; the ionization is initiated through seeding electrons with multiphoton ionization, and the process is accelerated by the cascading release of electrons by the reverse bremsstrahlung radiation. The sudden release of laser energy to the gas stream leads to a blast wave formation that expands and heats up the gas in the laser focus area. The elevating gas temperature increases the local speed of sound; in turn, it lowers the fight Mach number and thus reduces the wave drag when the discharged gas flows over the laser-​emitting aerodynamic body. However, there is also an unintended consequence that the configuration will also be subjected to extra heat and aerodynamic load for the pulsated energy deposition. In general for the remote energy deposition process, the aerodynamic process occurs on the order of tens of microseconds and the blast wave interaction for the duration of microseconds. Based on this time scale, the data acquisition frequencies for pressure and temperature collections are required to be in the 100 kHz range. Meanwhile using a typical pulsed Nd:  YAG laser system, the energy deposition can be consistently maintained from 13 mJ/​pulse to 283 mJ/​pulse. The option that the remote energy can be easily deposited to any desired location becomes a very effective flow control device. The pulsed laser energy deposition has been successfully applied to control the embedded supersonic jet after a shock-​on-​shock impinging interaction, or the Edney IV interaction (Adelgren et  al. 2005). An oblique shock wave intersects the bow shock wave over the blunt body that induces the well-​known discontinuous shock structure (the double image of the incident shock is caused by the turbulent oscillation from the fluid motion). The different density across the segmented shock waves produces parallel slipstreams downstream of the shock interception point; the resulting high-​energy jet that originates from interacting shocks eventually impinges on the body surface to create a high-​heating region. From the experiment, the laser energy is deposited upstream of a hemispherical blunt body in a supersonic Mach number of 3.45 to show a profound modification to the embedded jet stream that originates from the interaction of an oblique incident wave and the blow shock wave. A  selected sequence of two schlieren photos is presented in Figure 10.13 at the instances of 20 microseconds and 60 microseconds after laser pulse with a duration of 10 nanoseconds. The impinging jet is known to cause a high localized heat transfer region on the blunt body. These surface thermal and pressure stresses can be as high as twenty to thirty times greater than the stagnation condition, thus they can lead to catastrophic structure failure. From experimental data, the pulsed remote laser deposition has been demonstrated to reduce the pressure on the surface of the spherical body by 40 percent to 30 percent at the lower Mach number hypersonic flow conditions. The reduction in momentum of the resulting impinging jet is by modifying the shock-​on-​shock interaction through the laser energy depositions to the shock structure upstream of the body.

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10.5  Scramjet MHD Energy Bypass

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Figure 10.13  Laser energy deposition on Edney type IV shock–​shock interaction (Adelgren

et al. 2005).

10.5

Scramjet MHD Energy Bypass Scramjet is a hypersonic air-​ breathing propulsion system with no moving components and only consists of an inlet cowl, isolator, combustor, and exhaust nozzle. This simply constructed system has an upper bound dynamic pressure for its operational limit of around 95.8 kPa (2,000 psf) by the thermal and structure consideration. The lower bound of the operational limit is constrained by the combustion instability, generally considered to be around 24.0 kPa (500 psf). In the high-​speed operation range, it can operate from Mach numbers from 10.0 to 15.0 at an altitude up to 15,240 m (50,000 ft.). According to Currant and Murthyn (2002), the propulsion efficiency measured by the specific impulse is as high as 3,300 secs. The scramjet flow path is subjected to an adverse pressure gradient condition by the compression inlet, which is designed to be stabilized by the isolator, but is followed immediately by thermal perturbations from the combustor, thus the scramjet operation is suppressible to inlet unstart. Nevertheless, the scramjet has become the mainstay of hypersonic flight, thus the new scramjet MHD bypass concept for improved performance has created significant interest. The MHD bypass engine is proposed as a part of the AJAX vehicle concept (Gurijanov and Harsha 1996; Fraishtadt et  al. 1998). The fundamental idea is using MHD energy processes to extract and bypass a portion of the aerodynamic kinetic energy in the form of electricity from the inlet upstream to the combustor, thereby permitting a higher speed operation condition for the scramjet. Meanwhile, the entrance Mach number at the combustor can be controlled to a specific value even at a greater flight Mach number. Then the extracted energy in the form of electric power is reintroduced to the scramjet downstream flow path to accelerate the exhaust gas. A schematic of the MHD bypass engine with added

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Figure 10.14  Schematic of MHD bypass scramjet.

components is presented in Figure 10.14. The MHD electric generator is embedded between the inlet compression ramp and the isolator and the combustor. The MHD accelerator is combined with the thruster to generate a greater jet exhaust speed. The technology for the MHD electric generator, as we have discussed in Chapter 7.4, has reached sufficient engineering maturity in recent years to achieve the efficiency needed for practical applications. Two major parameters for an effective MHD bypass engine are the electrical conductivity σ and the Hall parameter β. The magnitude of the electrical conductivity of ionized gas is important for the aerodynamic and electromagnetic interaction because it interacts with the applied magnetic field by the magnets that impact the engine weight directly. However, the Hall parameter will determine the optimal design configuration for the MHD bypass scramjet engine. In general, the Faraday field associated with the bulk motion of the charged particles for the Lorentz acceleration is given by J × B . The Hall field on the other hand is associated with the electromotive force by the drift velocity of the charged particles u × B . These two induced fields are orthogonal to each other and both perpendicular to the externally applied magnetic field. It may not be surprising that the Hall parameter finally determines the appropriate operational configuration of the MHD bypass engine. It shall be pointed out that the MHD electric generator actually slows down the incoming air flow, thus associating it with an adverse pressure gradient along the inlet ramp compression; the adverse pressure perturbation will propagate upstream through the subsonic portion of the boundary layer as the upstream feeding to increase the workload for the isolator. On the contrary, the MHD accelerator in the nozzle creates a favorable pressure gradient that accelerates the combusted gas to the exhaust jet stream. Unfortunately, the development for the MHD accelerator is lagging behind that of the MHD electrical generator. The rule of thumb for the upper limit of the axial electric field is no more than 100 V/​m, which is imposed on any diagonal conducting wall of a generator that operates on a high electrical resistance medium. The maximum allowable axial electric field intensity will limit the accelerator to a low power density operation.

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An assessment by Bityurin and colleagues (2000) of the MHD bypass scramjet is worth noting. They conclude that the thermodynamic analysis reveals some realizable engineering benefits of the new concept; in principle, the innovation also shows the basic scientific feasibility of implementing the MHD bypass for propulsive efficiency improvements. However, they also caution that the overall system performance is extremely sensitive to any non-​isentropic losses in the flow path of the propulsion system through which the favorable operational characteristic will vanish if these losses become significant. A detailed evaluation has been conducted by Gaitonde (2006) using the full three-​ dimensional interdisciplinary MHD equations supplemented with electromagnetic source terms to the compressible Navier-​Stokes equations and the Poisson equation of plasma dynamics for the consistent electric field and electric current. The governing system is the essential low magnetic Reynolds magnetohydrodynamics formulation. The electric generator and accelerator are simulated by four pairs of Faraday coarsely segmented electrodes mounted on both sides of constant-​area isolator/​combustor components. The computational simulation reveals separate flow regions and a vortical structure with an eddy to interfere with the mainstream electric current, thus the electromotive force field yields complex aerodynamic–​electromagnetic interactions. The separated flow in the electric generator has a major impact on the electric current pattern downstream. One of the more pronounced effects is the formation of eddy current patterns to the flow path. Nevertheless, the numerical results indicate that the electric generator can reduce inlet length and decrease the combustor entrance velocity and the total temperature from the inlet. However, the computational results also show that the accelerator is characterized by significant nonuniformity of the Joule heating in the plasma sheath and the accelerator is not very efficient. Gaitonde’s conclusion for the performance of MHD scramjet bypass components is consistent with nearly all known assessments. The margin of improvement is extremely narrow when evaluating the merits of conventional and the scramjet with MHD energy bypass; Gaitonde recommends a more accurate analysis utilizing physically realistic partially ionized gas and turbulent flow modeling.

10.6

Plasma-​Assisted Ignition and Combustion The chemical kinetics of combustion is characterized by the intrinsically nonlinear and non-​monotonic response to the temperature, pressure, and mixture composition of a chemical-​reacting species. The large number of species and reactions are linked by vastly different time scales and chemical kinetic processes. The classic description of a typical chemical reaction is given by the Damkoler number, which is a dimensionless parameter that defines the ratio between the chemical reacting rate and the mass transfer diffusion rate (Kuo 2005). The ratio is directly proportional to the kinetic reaction constant, species concentration versus diffusion coefficient, and the interfacial area of reacting elements. For most stable combustion

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Applications

processes, the reaction rate is slightly faster than or comparable to the diffusion speed. However, the relative time scales show multiple extremes and increase drastically at the flame front or the media interface to reflect that the chemical reaction rate is faster than the diffusion process. The plasma or the ionized medium has an additional and significant forced diffusion mechanism over the ordinary diffusion through the electrostatic forces; in theory, plasma can enhance combustion stability due to the added diffusion mechanism. Overall, the ignition or rather the delay of ignition is caused by the energy accumulation required to achieve a sufficient amount of actuating energy to start a chemical reaction. For example, the well-​ known triple pressure temperature hydrogen–​oxygen explosion limits in a homogeneous mixture; the lower explosion regime is controlled by the strong chain-​branching cycle involving the radicals H, O, and OH, as represented by the two-​body reaction H+O2→O+OH. However, as the mixture crosses the second limit with increasing pressure, the three-​body termination reaction H+O2+M→HO2+M becomes important because of the increased collision frequency between the molecules at higher pressures. As the air temperature is progressively increased, ignition will occur at the lower ignition turning point. These results have been experimentally observed to demonstrate the richness of the ignition phenomena because of the intricacy of the reaction pathways (Law 2012). Nevertheless, the plasma carries an additional source of energy by the internal degrees of excitations, which can cascade or, via a recombination reaction,  and suddenly release a large amount of electronic energy. At certain conditions, the ionization process can also be achieved by radiation to generate a highly focused thermal energy location using the unique remote energy deposition process or by the random microwave discharge. These additional energy sources will inevitably contribute to overcome the actuate energy barrier to start a chemical reaction. For a simple direct current discharge (DCD), the temperature of the discharge column is known and can be calculated; the electronic temperature distribution can also be determined by the Townsend-​Bailey formula in terms of the reduced electric intensity E/​N with a physical dimension of V/​cm2 (Petrusev, Surzhikov, and Shang 2006). In Figure  10.15, the electron temperature distribution corresponds to a simple DCD across a parallel electrodes configuration with a gap distance of 2.0 cm, at an ambient pressure of 5.0 Torr, and driven by an electric field potential of 2.0 kV. The maximum electron temperature has been determined by numerous experimental observations to be limited to around 3.0 eV, but the input energy is still notable. For the direct current discharge, the electric current occupies only over a part of the electrodes and the high electron temperature is concentrated over the sheath within the cathode layer, but the electronic energy distribution in the positive column connecting between the cathode and the anode is still significant. During the past decades, considerable progress has been made by studying nonequilibrium plasma-​assisted combustion, especially in the low-​temperature plasma generated by electron impact with nanosecond pulses (Adamovich et al. 2009). By this practice, the reduced electric field intensity E/​N can be much higher than most

37

10.6  Plasma-Assisted Ignition and Combustion

377

Figure 10.15  Electron temperature distribution over direct current discharge between parallel electrodes.

other nonequilibrium plasma-​generation procedures measured by hundreds of Townsends (Td). The reduced electric field by Townsends is given as one Td = 10–17 Vcm2 in the cgs system. The plasma generated by the pulsed procedure is also more stable than other means, due to the very low duty cycle less than 10–3, and in a duration of nanoseconds to avoid discharge instability. A series of discharge configurations experiments have been conducted to compare ignition times for different nonequilibrium plasma with a range of the reduced electric field: from the direct current discharge (10 Td < E/​N < 30 Td), a periodic pulse electrode transverse discharge (30 Td < E/​N < 70 Td), freely localize microwave discharge (70 Td < E/​N < 120 Td), to a surface microwave discharge (100 Td < E/​N < 200 Td). The discharge with higher reduced electric field intensity produces the shortest induction time for combustion (Starikovskaia 2006). Most recent experimental investigations using nanosecond pulse discharges for study ignition delay have shown a reduced ignition delay time and improved lame stabilization, as well as sustaining uniform ionization for generating metastable and radical species in supersonic flows (Adamovich et al. 2009). It is definitely confirmed that the initiation of chemical reaction chains can be achieved through low-​temperature plasma via electron impact. For maintaining combustion stability, a basic understanding of the intricate pathways and nonlinear nature of fuel oxidation chemistry must be developed for any hydrocarbon-​based fuel. This remains a formidable challenge for a typical fuel, because the combustion process consists of hundreds of species and thousands of chemical reactions to be recognized as a specialized scientific discipline. For the

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Applications

present purpose, we must restrict our discussion to the kinetics of air plasma. A total of forty-​one species are generated in air plasma of which twenty-​two species are the electrically charged components, and ten species are the excited or metastable species. The dominant charged species during the ionization within 10-​7 second are electrons, positively charged molecular nitrogen N 2+ , and negatively charged atomic oxygen O −. After the time elapse from 10–7 to 10–4 second, the negatively charged oxygen molecular O2− formed by the three-​body electron attachment becomes the dominant negative ion species. These chemical species –​the electron, N 2+ , O −, and O2−  –​are important to incorporate into the chemical kinetic model. At the same time, the important neutral species in excited electronic states of nitrogen molecule N 2 ( A3 Σ ), N 2 ( B 3 Π ), N 2 (C 3 Π ), N 2 ( a ′1Σ ),atomic O, and O3 must be also included in the chemical kinetic formulation. Most electronically excited species exist in the low-​temperature plasma; the reaction rate and the quenching rates are known only for a limited number of the metastable species. On the other hand, the efficiency of electronic state energy for plasma chemistry is strongly dependent on the ratio between the generation and depletion of the excited states. Despite these limitations, ionized gas and rapid plasma recombination has provided an important and innovative channel for developing energy conversion mechanisms. The repetitive pulsed plasma can be manipulated to control the instabilities through the amplification of temperature and pressure perturbations. Some combined theoretical and experimental results have exhibited evidence that the on-​thermal nature of low-​temperature plasma can assist ignition. In the present stage in basic research mode, these results are observed mostly at the perturbation levels and show promising and positive trends. Recently, a special study of ignition processes using hydrocarbon jet fuel and involving the testing of ethylene and JP 7 with a plasma ignitor has made some impressive progress (Jacobsen et al. 2003). In essence, the recent innovation of an igniter is a plasma torch using a central tungsten electrode, which can be alternatively operated as an anode and a cathode for arc generation. The tested plasma ignitors have produced high-​temperature pockets of highly excited gas with peak temperatures above 5,000 K using only 2 kW total power input, with a thermal efficiency exceeding 90 percent. The plasma ignitor thus can produce the necessary condition for a practical scramjet over the auto ignition limits of a combined fuel and combustor configuration. The research into plasma-​assisted ignition and enhanced combustion stability is still an open issue. However, the unique characteristics of plasma of the added force diffusion and metastable elements are favorable facts that enhance combustion stability and ignition; it would be a terrible mistake to overlook a possible new avenue to gain knowledge in the last few scientific frontiers. References Adamovich, I.V., Choi, I., Jiang, N., Kim, J.-​H., Keshav, S., Lempert, W.R., Mntusov, E., Nishihara, M., Samimy, M., and Uddi, M., Plasma assisted ignition and high-​speed flow control:  Non-​thermal and thermal effects, Plasma Source Sci. Technol., Vol. 18, 2009, pp. 1–​13.

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38

Appendix: Physical Constants and Dimensions

Fundamental Constants Atm = 1.0133 × 105 Pa σ a = 8.7974 × 10 −17 cm2 m0 = 1.6605 × 10 −24 g N A = 6.0221 × 1023 1 mole (at T = 273.15 K, p = 1 atm) a0 = 5.2918 × 10 −9 cm k = 1.3807 × 10 −23 J K = 1.3807 × 10 −16 erg K e = 4.8032 × 10 −10 g1/ 2 × cm 3 / 2 / s Electron charge = 1.6022 × 10 −19 Coulomb me = 9.10954 × 10 −28 g Electron mass CF = 6.6485 × 10 4 C mole Faraday constant g = 9.8067 m s 2 Gravitational acceleration Loschmidt’s number N 0 = 2.6868 × 1019 cm −3 (at T = 273.15 K, p = 1 atm) Planck’s constant h = 6.6261 × 10 −27 erg × s Pressure of 1 mm Hg Torr = 1.3332 × 102 Pa m p = 1.6726 × 10 −24  g Proton mass Proton/​electron mass ratio m p me = 1.8362 × 103 Speed of light c = 2.9979 × 1010 cm/​s σ = 5.6705 × 10 −8 W m2 K 4 Stefan-​Boltzmann’s constant = 5.6705 × 10 −5 erg cm2 K 4 s σT = 6.6525 × 1025 cm2 Thomson cross-​section Universal gas constant R0 = 8.3145 J mole K = 8.3145 × 107 erg mole K Atmospheric pressure Atomic cross-​section Atomic mass unit Avogadro number Bohr radius Boltzmann’s constant

Electricity and Magnetism Absolute dielectric permeability of vacuum

ε 0 = 1 µ 0 c 2 = 8.8542 × 10 −12 Farad m Absolute magnetic permeability of vacuum

µ 0 = 4 π × 10 −7 Henry m = 12.566 × 10 −6 Henry m Capacitance 1Farad = 9 × 1011 SGSE units

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Charge 1C = 3 × 109 SGSE units = 9 × 1011 V × cm = 6.25 × 1018 electron charges Conductivity

σ [1 ohm ⋅ cm ] = 9 × 1011 σ( s −1 )

Current 1 A = 1C s = 3 × 109 SGSE units = 9 × 1011 V ⋅ cm s = 6.25 × 1018 electron ch arg e s Magnetic field strength 1 Oersted = 1SGSE units Magnetic inductance 1Henry = 109 SGSE units (cm) Magnetic induction 1 tesla = 10 4 gauss Reduce electric potential E E E V cm ⋅ Torr ] = 3.3 × 1016 [ V ⋅ cm 2 ] = 0.33 [ Td ] [ p N N 1 Td = 10 −17 V ⋅ cm 2 Resistance Ohm = SGSE units 9 × 1011 Voltage 1 V = SGSE units 300 E [ V cm ] = 300 E [SGSE units ] The energy associated with 1 eV ε = 1 eV = 1.6022 × 10–​12 erg The temperature associated with 1 eV T = ε/​K = 1.1604 × 104 K The frequency associated with 1 eV f = ε/​h = 2.418 × 1014 s–1 The wavelength associated with 1 eV λ = hc/​ε = 1.2398 × 10–​4 cm = 12.398 Å The wave number associated with 1 eV ω = ε/​hc = 8.0655 × 103 cm–1

385

Index

Ambipolar diffusion, 31–​34, 136–​137 Ampere’s circuit law, 73–​74, 77 Ampere’s law, 74 Arc plasmatron, 246–​248 Arrhenius formula, 199, 201, 203 Average charged particle speed, 10, 14 Black body radiation, 308 Biot-​Savart law, 6 Boltzmann equation, 56, 108, 112, 116, 119 Enskogs’ infinite expansion, 115 Fokker-​Plank approximation, 115–​116 Lorentz approximation, 115 Vlasov approximation, 117–​118 Charge conservation law, 76, 149 Charge number density, 75 Charge particle mobility, 32 Chemical equilibrium condition, 186 Chemical reaction constant, 198–​199 Backward reaction, 198 Forward reaction, 198 Chernyi et al. Ionization model, 206 Collision Cross section, 108–​109, 113, 115, 117, 151, 190, 251, 255, 306 Complex wave number, 290 Constitutive relationship for electromagnetic variables, 77–​78 Coulomb logarithm kinetic model, 206 Coulomb’s law, 4, 5 Counter-​flowing plasma jet injection, 369–​371 Long penetration monde, 370 Short penetration mode, 370 Cut-​off frequency of wave guide, 48 Cyclotron/​Larmor frequency, 8 Cyclotron/​Larmor radius, 9 Debye shielding length, 2, 12 Diamagnetic dipole, 62 Dielectric barrier discharge (DBD), 198, 216–​219, 234–​240 Direct current discharge (DCD), 56–​61, 213, 227–​234

Drift-​diffusion plasma formulation, 59, 212, 227–​230 Drift velocity of electric-​magnetic field, 9–​10, 30–​31, 39–​40 Einstein relationship between mobility and diffusion, 32, 136, 212–​213 Electric conductivity, 27–​31, 149 Electric currents, 5–​6, 27 Conductive current, 5, 153 Convective current, 5, 150 Displacement current, 44, 74 Electric field, 4–​5 Dielectric constant, 5 Electric field intensity, 5, 73, 77 Electric permittivity, 4 Electromagnetic forces, 4, 6–​8 Electrostatic force, 4–​5, 7 Elementary electric charge, 4, 12 Electromagnetic wave, 18, 78 Alfven wave, 21, 43 Electrostatic wave, 18 Evanescent wave, 51 Hydro-​magnetic/​hydro-​acoustic wave, 21 Transverse electric wave, 44, 45, 47, 285 Transverse electromagnetic wave, 47–​49 Transverse magnetic wave, 44, 46 Electrostatic force, 4–​5, 7 Emission spectroscopy, 276–​285 Faraday’s induction law, 71, 73–​74, 77 Fick’s second law of diffusion, 34 Flow control by electric wind, 366–​368 Fresnel-​Snell law, 337 Gasdynamic radiation equation, 325–​328 Gauss’s law for electric displacement, 75–​77 Gauss’s law for magnetic flux density, 28, 29, 75–​77 Generalized Ohm’s Law, 28–​29, 152–​153 Hall effect, 10, 39–​41 Hall current, 40 Hall parameter, 38, 40

387

386

Index

Hartmann Flow, 178–​183 Hartmann analytic solution, 179–​181 Hartmann number, 179 Ion Thrusters, 345–​351 Electrostatic ion engine, 346 Hall effect ion engine, 347–​348 Magneto-​plasma-​dynamic engine, 350 Induction plasma generator, 248–​252 Intrinsic impendence, 286 Ionization mechanisms, 188–​193 Charge exchange, 190 Dielectric recombination, 192 Dissociation recombination, 191, 215 Electron attachment, 192, 215 Electron impact, 189, 228 Inelastic collision, 188, 210–​216 Penning process, 190 Radiation interaction, 189 Three-​body recombination, 192 Ionization potential, 228 Jacobian of coordinate transformation, 162 Joule heating, 24, 156 Landau damping, 22–​23 Angular frequency, 22 Landau damping factor, 22 Landau-​Teller model, 206 Langmuir Probe, 268–​272 Completely collect current, 271 Double probe, 270 Electron saturate current, 271 Ion saturate current, 271 Single probe, 269 Law of mass action, 195 Light Hill dissociation model, 196 Local Thermodynamic Equilibrium, 309 Lorentz coordinates transformation, 80 Lorentz force/​acceleration, 7 Low Magnetic Reynolds number plasma formulation 127–​129 Magnetic field, 6–​7 Magnetic field intensity, 6, 73 Magnetic field strength, 6, 77 Magnetic flux density, 6, 77 Magnetic induction, 6 Magnetic permeability, 3, 6 Magnetic mirror, 61–​65 Magnetic pressure, 42, 155 Magnetic Reynolds number, 38, 147 Magnetic solenoid, 260–​263 Magnetic stress tensor, 155, 157 Magnetic Transport equation, 171–​172 Magnetohydrodynamic Equation, 170 Eigenvalues of ideal MHD equation, 163

Full MHD equation, 167–​170, 173 Ideal MHD equation, 154, 156 MHD electric generator, 243–​246 Magnetohydrodynamic waves, 18–​21 Alfven wave, 21, 43–​44 Electrostatic wave, 18 Hydro-​magnetic/​hydro-​acoustic wave, 21 Master equation of quantum jump, 207 Maxwell distribution, 10, 23, 25, 113 Maxwell equations, 76–​77, 115 Boundary condition for Maxwell equation, moving frame, 82–​83 Boundary condition for Maxwell equation, stationery frame, 83–​86 Characteristic variables, 92 Eigenvalues and Eigenvectors, 89–​90, 94 Maxwell tensile stress, 21 Microwave dispersion, 289–​292 Microwave plasmatron, 252–​254 Microwave attenuation, 285–​289 Attenuation index, 290–​291 Intrinsic impendence, 286 Refractive index, 290–​291 Millikan-​White kinetic model, 206 Monte Carlo method, 336–​339 Multi-​fluid and temperature plasma model, 118–​120, 216 Multi-​flux formulation for radiation, 313–​325 Discrete ordinate method, 321–​325 Half moment method, 313–​316 Spherical Harmonic method, 316–​321 Nearest neighbor search, 334 No-​reflection boundary condition, 98–​99 Numerical algorithms, 100–​105, 137–​144 Alternating direction implicit scheme, 142 Compact differencing scheme, 101 Gauss’s quadrature, 102–​104 Leap frog scheme, 139 Message passing interface (MPI), 104–​105 Particle in cell method, 137–​138 Polynomial local refinement, 102 Residual diminishing delta formulation, 139–​140 Successive line relaxation scheme, 143–​144 Optic thickness, 309, 312–​313, 319–​320 Partition function, 186, 194 Paschen curve, 228 Penetration depth, 288, 290, 291 Perfectly match layer, 97 Permanent magnets, 260 Planck function, 308 Plasma assisted ignition and combustion 375–​378 Plasma flow control actuator, 361–​369 Virtual leading edge flap, 362–​363 Virtual variable area inlet, 364

387

Index

Plasma frequency, 17, 149 Plasma penetration depth, 52 Plasma pinch, 65–​69 Plasma by radiation, 255–​258 Plasma sheath, 6–​14 Poisson equation of Plasmadynamics, 11, 81 Poynting vector, 19, 49, 286 Proximity search, 334–​336

Saha Equation of ionization, 197 Scramjet MHD Bypass, 373–​375 Shock tube, 240–​243 Spectral radiation energy density, 308 Spectral radiation intensity, 307 Spectral scattering indicatrix, 310 Steward Number, 173 Stotetov constant, 228

Radiation plasmatron, 255–​258 Radiation rate equations, 309–​311, 314, 318, 329 Rankine-​Hugoniot condition, 174–​178 Gasdynamics, 177 Modified by magnetic field, 176–​177 Oblique shock jump conditions, 175–​176 Ray tracing technique, 328–​336 Rayleigh-​Jean law, 308 Reentry thermal protection, 351–​361 Remote laser energy deposition, 372–​373 Retarding potential analyzer, 300

Thomson scattering cross section, 306 Townsend similarity law, 214 Transport property of plasma, 131–​137 Collision Integral and cross sections, 133 Coulomb potential, 134 Exponential repulsion potential, 134 Lenard-​Jones potential, 134 Wilke’s mixing rule, 132 Treanor-​Marrone kinetic model, 206 Wein displacement law, 309

387