COMPUTATIONAL ELECTROMAGNETIC- AERODYNAMICS · 1 Plasma Fundamentals 1 Introduction, 1 1.1 ElectromagneticField, 3 1.2 DebyeLength, 7 1.3 PlasmaFrequency, 10 ... applications. It

IEEE Press Series on RF and Microwave Technology

JOSEPH J.S. SHANG

COMPUTATIONALELECTROMAGNETIC-AERODYNAMICS


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Technical Reviewer

Frank Lu, University of Texas at Arlington


JOSEPH J.S. SHANGEmeritus Research Professor of Wright State UniversityEmeritus Scientist of US Air Force Research Laboratory

IEEE Series on RF and Microwave Technology

Copyright © 2016 by The Institute of Electrical and Electronics Engineers, Inc.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reservedPublished simultaneously in Canada

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CONTENTS

Preface ix

1 Plasma Fundamentals 1

Introduction, 11.1 Electromagnetic Field, 31.2 Debye Length, 71.3 Plasma Frequency, 101.4 Poisson Equation of Plasmadynamics, 121.5 Electric Conductivity, 131.6 Generalized Ohm’s Law, 161.7 Maxwell’s Equations, 191.8 Waves in Plasma, 201.9 Electromagnetic Waves Propagation, 231.10 Joule Heating, 271.11 Transport Properties, 291.12 Ambipolar Diffusion, 32References, 34

2 Ionization Processes 36

Introduction, 362.1 Microscopic Description of Gas, 382.2 Macroscopic Description of Gas, 432.3 Chemical Reactions and Equilibrium, 472.4 Saha Equation of Ionization, 50

v

vi CONTENTS

2.5 Ionization Mechanisms, 512.6 Photoionization, 542.7 Thermal Ionization, 562.8 Electron Impact Ionization, 61References, 67

3 Magnetohydrodynamics Formulation 69

Introduction, 693.1 Basic Assumptions of MHD, 713.2 Ideal MHD Equations, 743.3 Eigenvalues of Ideal MHD Equation, 803.4 Full MHD Equations, 863.5 Shock Jump Condition in Plasma, 913.6 Solutions of MHD Equations, 95References, 100

4 Computational Electromagnetics 102

Introduction, 1024.1 Time-Dependent Maxwell Equations, 1054.2 Characteristic-Based Formulation, 1084.3 Governing Equations on Curvilinear Coordinates, 1124.4 Far Field Boundary Conditions, 1214.5 Finite-Difference Approximation, 1234.6 Finite-Volume Approximation, 1314.7 High-Resolution Algorithms, 137References, 143

5 Electromagnetic Wave Propagation and Scattering 146

Introduction, 1465.1 Plane Electromagnetic Waves, 1475.2 Motion in Waveguide, 1505.3 Wave Passes through Plasma Sheet, 1555.4 Pyramidal Horn Antenna, 1605.5 Wave Reflection and Scattering, 1685.6 Radar Signature Reduction, 1775.7 A Prospective of CEM in the Time Domain, 180References, 189

6 Computational Fluid Dynamics 192

Introduction, 1926.1 Governing Equations, 1946.2 Viscous–Inviscid Interactions, 199

CONTENTS vii

6.3 Self-Sustained Oscillations, 2096.4 Vortical Dynamics, 2216.5 Laminar–Turbulent Transition, 228References, 232

7 Computational Electromagnetic-Aerodynamics 237

Introduction, 2377.1 Multifluid Plasma Model, 2397.2 Governing Equations of CEA, 2427.3 Chemical Kinetics for Thermal Ionization, 2507.4 Chemical Kinetics for Electron Impact Ionization, 2587.5 Transport Properties, 2627.6 Numerical Algorithms, 268References, 276

8 Modeling Electron Impact Ionization 279

Introduction, 2798.1 Transport Property via Drift-Diffusion Theory, 2828.2 Drift-Diffusion Theory in Transverse Magnetic Field, 2898.3 Boundary Conditions on Electrodes, 2928.4 Quantum Chemical Kinetics, 3008.5 Numerical Algorithms, 3058.6 Innovative Numerical Procedures, 311References, 322

9 Joule-Heating Actuators 325

Introduction, 3259.1 Features of Direct Current Discharge, 3279.2 Virtual Leading Edge Strake, 3339.3 Magnetic Field Amplification, 3419.4 Virtual Variable Geometry Cowl, 3499.5 Trailing Edge of Airfoil, 3589.6 Hydrodynamic Stability and Self-Oscillation, 362References, 366

10 Lorentz-Force Actuator 369

Introduction, 36910.1 Remote Energy Deposition, 37110.2 Stagnation Point Heat Transfer Mitigation, 37610.3 Features of Dielectric Barrier Discharge, 37910.4 Periodic Electrostatic Force, 39010.5 DBD Flow Control Actuator, 402

viii CONTENTS

10.6 Laminar–Turbulent Transition, 40610.7 Ion Thrusters for Space Exploration, 40810.8 Plasma Micro Jet, 413References, 415

Index 419

PREFACE

Tremendous amounts of research results have been accumulated to the informationpool for plasmadynamics and plasma, which is the most common state of matterin the universe by either mass or volume. The underlying physics occurs mostlyat the molecular and atomic scales and often under an extremely high enthalpycondition or in a strong electromagnetic field; these phenomena must be analyzed byphysics-based modeling. The research progress faces formidable challenges becausethe compositions of non–equilibrium-ionized gas are always in transient quantumstates where the fundamental chemical-physics processes are the least understood.Furthermore, the validating experimental data are also very sparse.

In spite of these limitations, thousands of conference papers and articles on thissubject are still persistently releasing to the open literature annually. Meanwhile, theelectromagnetics is increasingly recognized to broadening physical dimensions forinterdisciplinary science and technology to meet engineering requirements. For therapidly advancing scientific field, outstanding and fundamental technical referencesare published mostly in the period from the middle 1960s to later 1980s. Althoughall these illuminating accomplishments have withheld the test of time but have notbeen updated by converting information into knowledge for practical engineeringapplications. It is appropriate and timely that summarization of these impressivescientific accomplishments and a systematic review on the progress of this scientificdiscipline are needed for future development.

The inherent nature of interdisciplinary computational electromagnetic-aerodynamics links five major scientific disciplines from aerodynamics, electro-magnetics, chemical-physics kinetics, plasmadynamics, to computational modelingand simulations. The knowledge of each individual discipline has been developedover centuries and numerous treatises on these subject areas have been published

ix

x PREFACE

and firmly established for understanding. There are also some path-finding articlesin classic magnetohydrodynamics that detail the magnetic field–dominated astro-physics around 50 years ago, but none have been documented for computationalelectromagnetic-aerodynamics to reach a comprehensive status in providing a prac-tical working knowledge. For this reason, an attempt is made here to integrate thesepertaining disciplines to supply a working knowledge for engineering and to suggestthe best practice in applications. Hopefully this effort can be a viable reference forprofessional development and graduate studies.

The text is organized in 10 chapters, with the first three chapters consisting ofintroductions to fundamental knowledge of plasmadynamics, chemical-physics ofionization, formulations of classical magnetohydrodynamics, and their extension tolow magnetic Reynolds number electromagnetic-aerodynamics conditions. In fact,most plasma-based flow control actuators, high-speed flows of interplanetary reentry,and ion thrusters in space exploration are operating under this environment.

For fundamental knowledge, a brief review of unique features of the plasmamedium is presented in terms of the Debye shielding length, Coulomb and Lorentzforces, Joule heating, plasma frequency, and plasma waves. All these unique char-acteristics of ionized gas in an electromagnetic field are intermediately related tooutstanding engineering applications of computational electromagnetic aerodynam-ics. Then the statistical thermodynamics that bridges the microscopic to macroscopicthermodynamics is introduced for the ionization processes. The mechanisms of pho-toionization, thermal excitation, and electron impact ionization are discussed on thebasis of the most recent research findings.

The hierarchal relationship among fundamental formulations for plasmadynamicsand their valid domains of approximation is outlined and reviewed next. The basicassumptions for the classic magnetohydrodynamics formulation are articulated, anddetailed derivations from ideal to full magnetohydrodynamics equation, as well as thelow magnetic Reynolds number equations system, are delineated. These governingequations, in fact, define the limitations and physical fidelity of all computationalsimulations. From the eigenvalues and eigenvectors analyses of the magnetohydro-dynamics equations, correlations are revealed between propagating speeds of Alfven,entropy, contact surface, slow and fast plasma waves.

A peculiar property of plasma in an electromagnetic field is that it does not existas an unrealistic one-dimensional phenomenon. Therefore, the shock jump conditionmust be formulated, at least by tangential and normal components across a discretewave. The simplest shock in the electromagnetic field is essentially a two-dimensionaloblique wave. When the macroscopic conservative equation is integrated across theshock, and it requires that the normal components of the magnetic flux density andthe tangential component of the electric field intensity are continuously across anyinterface. As a consequence, the Rankine–Hugoniot relationship of gas dynamicsacross a shock is significantly modified in an ionized gas to become a unique featureof the plasma medium.

The next three chapters of the book provide in-depth and specific descriptionsof numerical algorithms and procedures for solving Maxwell’s equation in the timedomain for computational electromagnetics, plasma wave propagation, and fluid

PREFACE xi

dynamics. All these detailed numerical algorithms and computational procedures arethe shared knowledge for the interdisciplinary computational modeling and simu-lation science and technology. The highlights are especially placed on overlappingareas of technical issues and realizable application opportunities.

Two major foci of the present objective are to integrate the interlinking computa-tional modeling and simulation techniques of aerodynamics and electromagnetics. Tomake the procedure tractable, the computational approach is further restricted to thecontinuum regime so the approach is built on the frameworks of the time-dependentMaxwell equations and the compressible Navier–Stokes equations. It should be recog-nized that the Maxwell equations constitute the hyperbolic partial differential systemand belong to a class of bondless initial-boundary-value problems. This unique fea-ture poses a fundamental dilemma for all numerical analyses that must be carried outin a finite-size computational domain. However, the characteristic-based formulationcan alleviate this constraint by achieving an invariant dependent variables grouping inthe domain of dependence. This formulation permits all propagating electromagneticwaves exit through numerical boundary unimpeded. The computational electromag-netics technique is applying to the microwave propagation in waveguide, plasma wavediffraction and refraction, as well as the plasma diagnostic technique by microwaveattenuation beyond Langmuir probes.

The last four chapters are focused on the modeling and simulation techniques forcomputational electromagnetic-aerodynamics. The most current progress in computa-tional electromagnetic-aerodynamics is summarized for multifluid models, includingchemical kinetics by nonequilibrium thermal excitations, chemical physics by elec-tron impact ionization, transport property by gas kinetics, and numerical algorithms.The emphases are on the physical fidelity of computational models that describe thebehavior of the direct current charge, dielectric barrier charges, microdischarge, andtheir interactions with aerodynamics through the mechanisms of Joule heating andLorentz forces. Transport property by drift motions and diffusions of the chargedparticle is also described by the classical drift-diffusion theory. The interface bound-ary conditions between the chemically reacting media and solid surface are derivedtogether with the requirements on electrodes in an externally applied electric field.The latter are described in detail in conjunction with the discharging electric circuitequation. On this frame of analysis, the secondary emission of electrons from the cath-ode, together for the self-limiting feature of dielectric barrier discharge by preventingtransit to arc, is unambiguously illustrated. For the thermal ionizing model, the mod-els of internal degrees of freedom for vibration relaxation and electron excitationsare also described at practicing level. Some of the innovative numerical proceduresare also included as bench marks for future progression.

It is important to acknowledge that the electromagnetic effect in most flow controlapplications appears only as a small perturbation. In general, the relative magnitudeof aerodynamic inertia overwhelms the electromagnetic force by a thousand fold.In terms of magnetic Reynolds number for an electromagnetics–aerodynamics fieldthat is measured by the ratio of inertia and the product of electric conductivity andmagnetic permittivity Rm = ul∕(𝜎𝜇m)−1, its magnitude is much smaller than unity.The plasma-based actuators are therefore observed to be the mostly effectiveness

xii PREFACE

for flow control at a critical point of a bistable aerodynamic state or to the high-frequency, low-amplitude oscillations. Thus, aerodynamic phenomena are singlingout for possibly enhancements by infusing electromagnetic effects. For this purpose,the aerodynamic bifurcations of viscous–inviscid interaction, including flow sepa-ration, self-sustained oscillation, vortical dynamics, and hydrodynamic instabilityleading to turbulent flow, are specifically included. In this approach, the rudimentarynumerical algorithms of computational electromagnetics and computational fluiddynamics are introduced and the best practices for computational electromagneticaerodynamics are also encompassed.

In the last two chapters, the presentations of plasma-based actuators for flow con-trol are divided into two major categories according to the mechanisms, whether viaJoule heating or by Lorentz force. For the former, the experimentally validated appli-cations in virtual leading-edge strake and a virtual variable inlet cowl are described.Meanwhile, viable amplifications to electromagnetic–aerodynamic interaction byviscous–inviscid interaction and externally applied magnetic field are demonstrated.The actuator by Lorentz force for stagnation point heat transfer mitigation, remoteenergy deposition, and flow control via dielectric barrier discharge for lifting sur-face enhancement are elaborated. A quantification for the flow control effectivenessby dielectric barrier charge is given, and the potential application to hydrodynamicstability is revisited. The successful applications of ion engine or ion thruster byelectrostatic force and plasma micro jet for enhancing combustion stability are alsoincluded to illustrate the possible future explorations.

Here, I would like to express my personal gratitude for lifelong association in thevast arena of science with Professors Robert W. MacCormack of Stanford University,David I. Gottlieb (deceased) of Brown University, and Joseph L. Steger (deceased)of University of California at Davis. I am equally indebted to the interactions andcollaborations with Professor Sergey T. Surzhikov of Russian Academy of Sciencesand my colleagues; Drs. Wilbur L. Hankey, Miguel R. Visbal, and Datta V. Gaitondeof the Center of Excellence for Computational Simulation, United States Air ForceResearch Laboratory, as well as Professor George P. Huang of Wright State Universityand Professor Hong Yan of North Western Polytechnic University of China.

I have gratefully received editorial supports from Professor Frank K. Lu of Uni-versity of Texas at Arlington and James A. Menart of Wright State University. Theirgenerous act of friendship and meticulous efforts are truly a humbling experience tome. At last, but by no mean the least, I earnestly acknowledge the unfailing devotionfrom my wife Karen. I dedicate this book to her.

Joseph J.S. Shang

1PLASMA FUNDAMENTALS

INTRODUCTION

Plasmas are the most common state of matter both by mass and by volume, althoughmost of them are rarefied and intergalactic; nevertheless, plasma is one of the fourfundamental states of matter. Langmuir (1928) first coined the word plasma in 1928.Plasma is an electrically conducting medium and has the basic tendency toward globalelectrical neutrality. The most fundamental definition of this state of matter is thatthe pairs of closely positioned positively and negatively charges in the plasma shieldeach other from an externally applied electric potential. This characteristic chargeseparation distance is the Debye shielding length and is the smallest dimensioncompared with all other macroscopic scales. Within this shielding sphere, the pairedcharged particles are never free from each other but interact permanently.

Plasma does exist not only in gas and liquid but also in a solid conductor. Theelectrons are bound but still can move within atomic or molecular structures betweencollisions. Although mass motion of the charges does not generally take place, whenan external electromagnetic field is applied, the dynamic effect can always be observedin conduction, Hall Effect, and polarization.

Plasmas, like gases, have a negligible shear stress and therefore do not have adefinite shape or volume. However, the electric-conducting feature makes it respondto electromagnetic fields, so plasma can be confined either by a solid container or bya magnetic field. Plasma also appears in a wide range of formations such as filaments,microdischarges, multiple layers in the presence of shock waves, and cellular structure

Computational Electromagnetic-Aerodynamics, First Edition. Joseph J.S. Shang.© 2016 The Institute of Electrical and Electronics Engineers, Inc. Published 2016 by John Wiley & Sons, Inc.

1

2 PLASMA FUNDAMENTALS

1010

109

108

107

106

105

104

103

102

101

1. 3. 5. 7. 9. 11. 13.

Number density in exponents per CC

Tem

pera

ture

(K

)

Insignificant ionization

Gas discharges

Solar wind

Solar corona

Magnetic fusion

Relativistic plasma

Inertial fusion

Ideal classical plasma

15. 17. 19. 21. 23. 25.

DegenerateQuantumPlasma

EF = e2 / n–1/3

EF = KBT

n λD3 = 1

27. 29.

FIGURE 1.1 Classification for types of plasma.

in space. The spontaneous formations of unique spatial features on a wide range oflength scales manifest the complexity of plasma. Because the electric conductivity isdependent on its charged particle number density and thermodynamic condition, theclassification of plasma is usually by the electron number density and temperature inelectron volts or the static temperature (Figure 1.1).

The topics of degenerated quantum plasma and relativistic plasma are beyond thescope of computational magneto-aerodynamics. For this reason, the present discus-sion is focused on the plasma domain that exists around atmospheric pressure, and onmixed thermal and nonthermal plasma. According to convention, the thermal condi-tion of plasma is based on the relative temperatures of the electron, ion, and neutralcomponents. By nonthermal plasma is meant that the ions and neutral particles havea much lower temperature than the electrons. For most gas discharges generated byelectric currents, the electron temperature is at most 3 eV (around 105 K), while theions and neutrals are at near room temperature. When ionization is achieved by astrong shock compression in high temperature or earth reentry, the temperatures ofthe plasma composition are on the same order, around 10,000 K or higher, but arenot always equilibrated so as to remain in thermodynamic nonequilibrium (Jackson,1975). Therefore, the plasma of interest is generally limited to an electron numberdensity up to 1016/cm3 and an overall temperature lower than 106 K.

From the discussion given earlier, plasma is an electric conducting medium but isglobally neutral and characterized by a minimum macroscopic length scale betweencharges of opposite polarity known as the Debye shielding length. A strong electro-static force exists between the paired charges; any small perturbation to the charges

ELECTROMAGNETIC FIELD 3

separation distance will trigger a high-frequency oscillation by the restoring force.This oscillatory motion is referred to as the plasma frequency that is distinguishedfrom the lower frequency oscillation involving mass motion. In the presence of a suf-ficient number of charged particles, the oscillation also leads to a number of uniquecollective behaviors that are exemplified by the strong response to electromagneticfields. Plasmas do not naturally conform to their surroundings but will respond tolocal and distant conditions by the Coulomb force and Lorentz acceleration. Theother unique feature of quasi neutrality is that plasma stores the inductive energyand contributes to resistance and inductance when the current circuit forms a closedloop. Finally, the disturbance communication in plasma not only is through collisionprocesses but also conveys by transverse wave with phase velocities greater than thespeed of sound (Spitzer, 1956; Celexoix, 1960).

All physical phenomena of plasma are governing by the Maxwell and Navier-Stokes equations to become an interdisciplinary subject in macroscopic scales. Oncethe plasma is treated as a continuum, only the global thermodynamic and kinematicproperties are needed to describe its behavior through a link between microscopic andmacroscopic dynamics. This approach is accurate when the microscopic structurecan be linked approximately to macroscopic motion by the Maxwell–Boltzmanndistribution. However, the detailed wave–particle interaction will be unresolved.

1.1 ELECTROMAGNETIC FIELD

An electromagnetic field of plasma always consists of coexisted electric and magneticfields. Charged particles interacting among themselves and with externally appliedfields create induced components; the total field is a sum of applied and inducedcomponents. The magnitude of induced components can vary greatly in comparisonwith applied fields.

According to Coulomb’s law, the electrostatic force between charged particles isa collinear force along the unit space vector between two charges separated by adistance rij,

Fij =1

2𝜋𝜀o

qiqj

r2ij

r̂ij (1.1)

The magnitude of the force is directly proportional to the product of the twocharges and inversely proportional to the square of the distance between them. InSI units, the electric permittivity 𝜀o has a value of 8.85 × 10−12 Farad/m (Krause,1953). The charges qi and qj are measured in coulombs, which attract to each otherof opposite polarity but repulse if the same. The resulting force per unit charge isdefined as the electric field intensity E. Through this definition, the electrostatic forcecan be given as

Fi = Eqi (1.2)


The SI unit of electric field intensity is the newton per coulomb. However, inpractical application, an equivalent unit is often given in volt per meter. In the caseof multiple point charges, the principle of superposition determines the force on aparticular charge. The entire electric field of a charged space is the vector sum of thefield from all the individual source charges,

Ei =1

4𝜋𝜀oΣ

qi

r2ij

r̂ij (1.3)

In an electrically conducting medium, the charge particle movement induces cur-rent and exerts additional force on each other. In an isolator, the molecules of thedielectric material will polarize to reduce the net local field intensity. This effect isdescribed by a dielectric constant 𝜅 and Coulomb’s law becomes

Fij =1

4𝜋𝜀o𝜅

qiqj

r2ij

r̂ij (1.4)

Electric charge in motion constitutes a conductive electric current. In metallicconductors, the charge is carried by electrons with an elementary negative chargeof 1.61 × 10−19 coulombs. In liquid conductors, such as electrolytes, the charge iscarried by both positive and negative ions. The electric field compels the free chargesinto continuous motion and results in an electric current, which can be defined interms of the electric flux vector per unit area.

The conductive current density considered in here is different from the convectiveand displacement current density. The convective current does not involve conductorsand consequently does not satisfy Ohm’s law. The current flows through an isolatingmedium such as the liquid, rarified gas, or vacuum, and the best example is an electronbeam within a vacuum tube (Jahn, 1968). The displacement current arises from thetime-varying electric field and is introduced by Maxwell to account for the generationof a magnetic field when the conductive current is zero. Without the concept of thedisplacement of current, the electromagnetic wave propagation would be impossible.

The conductive electric current density J has a physical dimension of coulomb persecond or ampere,

J = Σniqiui = Σ𝜌iui (1.5)

where ni denotes the charge particle number density, ui is the averaged velocity ofthe charged particles, and the current density 𝜌i is simply the sum of the total electriccharges per unit volume of species i.

A magnetic field is generated by an electric current and the orientation of this fieldis defined by the right-hand rule. The current may be due to electromagnetic fields bya permanent magnet, an electron beam, or a conductive current in current-carryingwire. The differential magnetic field intensity is governed by the Biot–Savart law formagnetostatics, which is the counter part of Coulomb’s law for electrostatics. Thefield intensity is proportional to the product of a differential current element Idl and

ELECTROMAGNETIC FIELD 5

J

B

J

FIGURE 1.2 Induced magnetic field by electric current.

sine of the angle between the element and the line connecting the point of interest andis inversely proportional to the square of the mutual distance. The induced magneticfield line is continuous. In vector form, the Biot–Savart’s law gives (Figure 1.2)

dH =Idl × r̂ij

4𝜋r2ij

(1.6a)

A current-carrying wire produces a magnetic field perpendicular to the electriccurrent, and the magnetic flux density B has the physical unit of newton per amp-meter. In analogy to electrostatics, the intermediary B field is the magnetic flux densityor is also called the magnetic induction. The connection between the magnetic fieldstrength H and the magnetic flux density B in an isotropic medium is the constituterelation, B = 𝜇B, and 𝜇 is referred to as the magnetic permeability. In free space, ithas the dimensional value of 𝜇 = 4𝜋 × 10−1 hennery/meter.

If two current-carrying wires are brought into the vicinity of each other, eachwire is surrounded by two individual magnetic fields leading to a force that acts onthe wires. When the wires are carrying current in the same direction, the wires areattracted to each other, while wires carrying current move in opposite directions, thewires are repelled. The incremental force existing between the elements of the currentpath Li and Lj is

dFij =𝜇

4𝜋

JiJj

r2ij

[dli × (dlj × r̂ij)] (1.6b)


By definition of the magnetic induction is

dBij = (𝜇Jj∕4𝜋)(dlj × r̂ij∕r2

ij

)(1.6c)

Analogous to that of electrostatics, the force on a current element becomes

dFij = Jidli × dBij (1.7)

Basically, there are but two modes of the electromagnetic body force in an elec-trically charged medium that can exert on the charged particles. Figure 1.4 is theinteraction of the electric field with the free charge density of the medium Fe = 𝜌eE,which is known as the electrostatic force. The interaction of an externally appliedmagnetic field with the electric current that is driven by a force within the medium isFm = J × B and it is the well-known Lorentz force.

As a consequence of a moving charge q in a steady and uniform magnetic fieldwith an applied magnetic flux density B will be pushed by a force normal to B,namely,

F = qu × B (1.8)

Under the circumstance, the normal velocity component of the charge particles isalways restricted in the plane perpendicular to a constant and steady magnetic B. Thevelocity component u∥, which is parallel to B, will not be affected by the magneticfield but the velocity component u normal to B will be generated. Therefore, thecharge particle motion in the normal plane to the magnetic field must follow acircular path and the charged particles and will execute a spiral motion. The gyroradius can be determined easily by the balance of the centrifugal acceleration andthe electromagnetic force exerted on a singly charged particle, eu

⊥B = mu2

⊥

/rb. The

radius of the circular motion of a group of charged particles in the plane perpendicularthe magnetic field is rb = mu

⊥∕qB, with an angular gyro velocity 𝜔b = qB∕m. The

gyro radius is often referred to as the Larmor or cyclotron radius and can be evaluatedon the basis of a single electron, rb = mu

⊥∕eB. Similarly, the angular gyro velocity

has also been widely referred to as the Larmor or cyclotron frequency. The chargeparticle trajectory in three-dimensional electromagnetic field is then a helix with itsaxis parallel to B and a pitch of 2𝜋u∥

/𝜔B.

The gyro frequency follows directly from its single-particle motion and can begiven as

𝜔b = eBm

= 1.76 × 1011B (rad∕s) (1.9)

where the magnetic flux density B needs to be in the SI unit of webers/m2.The equation of motion for a charged particle in a uniform electromagnetic field

is then

F = q(E + u × B) = mdudt

(1.10)

DEBYE LENGTH 7

B & E = 0 B E

FIGURE 1.3 Trajectory of charge in electromagnetic field.

The electromagnetic field of moving charged particles therefore is always associ-ated with a combination of electric and magnetic fields. A rectilinear acceleration isaligned with the externally applied electric field and a gyration intrinsically revolvesaround an applied magnetic field vector. In the absence of an electric field, the angularacceleration is restricted in the plane perpendicular to the magnetic field. When theelectric field vector is applied in addition to the magnetic field, the charged particleswill execute a persisted spiral trajectory as shown in Figure 1.3. The orientation ofthe resultant helical trajectory is dictated by the direction of the applied electric fieldintensity.

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 ofplasma in the Debye shielding length and the plasma frequency. While the directcurrent produces magnetostatic field, the dynamics of the electromagnetic field mustmore often be accompanied by a time-varying electric current density. In short,within a static electromagnetic field, the electric and magnetic fields are indepen-dent from each other whereas in dynamic electromagnetic fields the two fields areinterdependent.

1.2 DEBYE LENGTH

To understand and to characterize the macroscopic behavior of a collection of chargedparticles, two fundamental parameters that associate with the electric properties of theionized gas, namely the Debye length and plasma frequency, are paramount. The basicproperty of plasma is its tendency in maintaining electric neutrality (Langmiur, 1929).


This particular state requires an enormously large electrostatic force between anelectron and an ion. A quantitative estimate of this dimension over which a deviationfrom charge neutrality may occur can be obtained by the Gauss law for electricfield ∇ ⋅ D = 𝜌. The electric displacement is related to the electric field intensityby a constitutive relationship D = 𝜀E, where 𝜀 is the electric permittivity by thefact that the electric field intensity is inversely proportional to the gradient of theelectric potential E = −∇𝜑. The resultant equation is known as the Poisson equationof plasmadynamics

𝜀∇2𝜑(r) = −𝜌e = −e(ni − ne) (1.11a)

In a thermodynamic equilibrium condition, the number densities of electrons andions are given by the Boltzmann distributions,

ne = nee𝜑(r)∕𝜅T and ni = ne−e𝜑(r)∕𝜅T (1.11b)

Substituting these expressions into the Poisson equation and reducing the equationto one dimension in space yield

𝜀d2𝜑

dx2= en∞[e(e𝜑∕𝜅Te) − 1] (1.12)

Expanding the exponential function in Taylor series in ascending order of e𝜑∕𝜅Tyields

𝜀d2𝜑

dx2= en∞

[e𝜑𝜅Te

+ 12

(e𝜑𝜅Te

)2

+ ⋅ ⋅ ⋅

](1.13)

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

𝜀d2𝜑

dx2=

e2n∞𝜅Te

𝜑 (1.14)

This leads to the definition of the Debye shielding length

𝜆d =[𝜀𝜅Te

e2n

]1∕2

(1.15)

The electric potential has been found to exhibit an exponentially decay behavioras a function of distance between charged particles,

𝜑 = 𝜑o exp(−|x|/𝜆d) (1.16a)

DEBYE LENGTH 9

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

𝜆d = 69(T∕n)1∕2m, T(K); 𝜆d = 7430(𝜅T∕n)1∕2m, T(ev) (1.16b)

These simplified results for the Debye length are evaluated by the following con-stants: electron volt (ev) 1.1604 × 104 K, Boltzmann constant 1.3807 × 10−16 erg/K,elementary charge 4.8032 × 10−10 coulombs, and electric permittivity of free space8.8542 × 10−12 farad/m.

The typical value of the Debye length in a magnetohydrodynamics generatoris in the order of magnitude of the electron mean free path O(10−6–10−7 m) at atemperature of 2500 K and charge number density of 1020/m3. In plasma generatedby electronic impact at standard atmospheric conditions, the Debye length is 1.14 ×10−7 m.

Another way to estimate the separation distance between charged particles Δx isby recognizing that this position is the neutral state of the balanced average kineticenergy of the electron and the electric potential generated by the charged particles.The averaged separation distance can be obtained by integrating the one-dimensionalPoisson equation of plasmadynamics

d2𝜑

dx2= −

q

𝜀=

nee

𝜀(1.17)

Integrate consecutively with respect to x and set the farfield electric field to 0(Eo = 0) to yield

d𝜑dx

=nee

𝜀x + Eo

Δ𝜑 =nee

2𝜀Δx2

(1.18)

The average energy needed to separate the charged particles by a distance Δxmust overcome the total thermal potential. The mean separation distance between thecharged particles is then

𝜅Te

2=

nee

2𝜀Δx

Δx =[𝜀𝜅Te

nee2

]1∕2 (1.19)

This charge separation distance is exactly the Debye length derived previously.The temperature of the free-electron component of the gas Te is not necessarilyequilibrated with the ions or neutrals. The Debye length has always been adopted asan index of the typical charge separation distance in plasmas that are sustained by therandom thermal energy of the electrons. For this reason, the Debye length is often


FIGURE 1.4 Cathode layer in a direct current discharge.

interpreted as a physical dimension for shielding the plasma from the effects of anexternally applied electric field.

A most illustrative phenomenon of Debye shielding is the plasma sheath. Thecharge neutrality of plasma does not prevail in the immediate adjacent region to asolid surface, especially over electrodes of an applied electric field, and is the so-called plasma sheath. The Debye length gives the order of magnitude of the thicknessof the actual sheath that separates the mass of the neutral plasma from the solid wall.This physical characteristic is easily identifiable in the cathode and anode layers ofa glow discharge that is generated by a strong electric field (Figure 1.4). A majorcontributing factor to this behavior is the recombination mechanisms of chemicalkinetics by depleting the charges of ionized gas when striking the surface. The electronabsorption and emission properties of the solid surface also play an important role.This phenomenon is closely related to the catalyticity of the surface material.

1.3 PLASMA FREQUENCY

Since plasma has a tendency to be macroscopically neutral, it will always tend toreturn to its neural equilibrium state after a local perturbation. Unavoidably in thisprocess, the inertia of electrons will overshoot and oscillate around the equilibriumcondition with a characteristic rate known as the plasma frequency. The plasmafrequency is a measure of the time scale of restoring forces that exerts on a displacedelectron returning within Debye length to an ion and keep plasma to an electrically

PLASMA FREQUENCY 11

neutral state. In the absence of magnetic field and without externally applied thermalperturbation, the coupled ion is considered in a fixed location in uniform plasma thatextends to infinite (Mitchner et al., 1973). The conservation equations of electronmotion are

𝜕ne

𝜕t+

d(neue)

dx= 0

mne

[𝜕ue

𝜕t+ ue

𝜕ue

𝜕x

]= −eneE

(1.20)

The compatible electric field E can be determined by the classic Poisson equationof plasma dynamics

𝜀𝜕E𝜕x

= e(ni − ne) (1.21)

The system of equations can be solved easily with a linearized approach bysetting the known initial conditions and considering the displaced electron as a smallperturbation,

n = no + 𝜀n1, ue = ue + 𝜀u1, E = Eo + 𝜀E1, 𝜀 ≪ 1 (1.22a)

The resultant linearized governing equations to O(𝜀) become

𝜕n1

𝜕t+ u1

dn1

dx= 0

m𝜕u1

𝜕t+ no

𝜕u1

𝜕x= 0

𝜀

𝜕E1

𝜕x= −en1

(1.22b)

The perturbed oscillating variables are naturally definable by simple harmonicfunctions

u1 = u1ei(𝜅x−𝜔t), n1 = n1ei(𝜅x−𝜔t), E1 = E1ei(𝜅x−𝜔t) (1.22c)

Substituting these into the governing equations and eliminating n1 and E1 yields

− im𝜔u1 = −inoe2

𝜀𝜔(1.23)


From which 𝜔2 = noe2

m𝜀. Thus the plasma frequency is

𝜔p =[

noe2

m𝜀

]1∕2

rad∕s (1.24)

The practical approximate formula of the corresponding circular frequency in SIunits is

𝜈p =𝜔p

2𝜋≈ 8.97

√ne (rad∕s) (1.25)

The pertinent parameters for the plasma frequency calculation are the electricpermittivity in free space of 8.8542 × 10−12 farad/m, the elementary charge e =1.6022× 10−19 coulombs, and the electron mass m of 9.1095× 10−25 g (Raizer, 1991).

As an example, the electron number density in an electron-impact discharge hasa value of around 1018/m3 or 1012/cm3. The plasma frequency yields a value of8.7 × 109 Hz, which is within the H and X bands of the microwave frequencywith the corresponding wavelength range between 2.4 and 4.25 cm. The frequency,however, is lower than the typical average electron collision frequency within aplasma generator of 2 × 1011/s.

1.4 POISSON EQUATION OF PLASMADYNAMICS

The electrons are in thermodynamic equilibrium and, according to the statisticmechanics, have the Boltzmann distribution, ne = nee𝜑(r)∕𝜅T . The potential energy ofan electron at a position r is −e𝜑(r). Similarly, under known equilibrium condition,the ions retain the Boltzmann distribution, ni = ne−e𝜑(r)∕𝜅T . The potential energydistribution for plasma is governed in general by the Poisson equation. Therefore,this equation of plasmadynamics is a fundamental formula in describing the collec-tive behavior of plasma in a continuum. In this formulation, the individual chargedparticles are no longer recognized but to be described by the charge number density𝜌e to have a physical unit of coulomb per cubic meter. The sum of charges in anelementary volume can be given as

𝜌e =lim

Δv → 𝜀

∑i

ze

Δv= e

∑i

nizi (1.26)

where ni is the number density of the charge species and zi denotes the number ofelementary charges carried by the particles.

ELECTRIC CONDUCTIVITY 13

The total electric field generated by a group of charges can be evaluated by theprinciple of superposition. The integrated surface electric field over control surfacemust be balanced by the total number of charges within the control volume. Thus

∬i

E(r)ids = 1𝜀 ∭ 𝜌edv (1.27)

By means of Stokes’ divergence theorem, we have

∭ ∇ ⋅ Edv = 1𝜀 ∭ 𝜌edv (1.28)

This equality shall be held to an infinitesimal volume; in fact, it is one of the twoGauss laws of electromagnetics.

∇ ⋅ E = 1𝜀𝜌e (1.29)

If the electric field intensity can be derived from the potential function E = −∇𝜑,which is the most prevailing condition in free space away from any solid surface,then the following equation becomes the well-known Poisson equation of plasmadynamics:

∇2𝜑 = −

𝜌e

𝜀(1.30)

This equation can also be easily derived from Gauss’ law, ∇ ⋅ D = 𝜌e for the electricfield of the Maxwell equations. The relationship between electric displacement D andelectric field intensity E is given by a constitution coefficient 𝜀 known as the electricpermittivity. The electric field intensity just equals to the negative gradient of electricpotential 𝜑.

1.5 ELECTRIC CONDUCTIVITY

The electric and magnetic fields in plasma always produce an electric current that isrelated to the diffusion velocities of the various charged particle species. Therefore, itis necessary to evaluate the electrical conductivity as microscopic properties togetherwith the thermodynamic state of the plasma. Between collisions, the individual par-ticles move in accordance with the E × B drift and diffusion velocities that are thedriving mechanisms for the charged particle motions. Now the electrical resistivity is


the resultant phenomenon of collision process. By introducing a mean or drift motionof the electrons in the gas, the electric current is

J = nqu =nq2

m𝜈eE (1.31)

In this equation, m is the mass of the electron and 𝜈c is the average electron collisionfrequency that is related to the atomic scale momentum transfer cross section andthe random thermal motions of the individual particles. The current density is relatedempirically to the applied electric and magnetic fields by a bulk electric conductivityJ = 𝜎(E + u × B). The proportional constant is known as the electrical conductivity,

𝜎 =nq2

m𝜈c(1.32)

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 conductivityis expressed in reciprocal of ohm-meter or mho. For example, the fused quartzhas the conductivity of 10−17, the glass of 10−12, sea water of 4, and silver of6.1 × 107 mho/m. Note that Ohm’s law at a point is J = 𝜎E and that J and E have thesame direction in an isotropic medium and in the absence of a magnetic field. Underthis special circumstance, electrical conductivity is a scalar quantity.

In the presence of a magnetic field, collisions play a similar role in determiningthe drift velocity of a charged particle. The magnetic field through the Lorentzacceleration generates a gyro motion of the charged particle with a gyro radiusproportional to the velocity component perpendicular to and inversely proportionalto the magnetic flux density B. The angular velocity of the gyro motion has beenintroduced, which has earlier been known as the gyro frequency. It may be clearthe that helical trajectory of the charged particles has a wide and varying range oforientations in collision; the electrical conductivity is no longer a scalar quantity butis a tensor of rank 2.

The trajectory of the drift motion of a charged particle in an electromagnetic fieldis fascinating and complex. In the presence of both electric and magnetic fields, thecharged particle in any inertial frame will execute a helical motion. The electric forceusually is decomposed to components parallel and perpendicular to the magneticfield. The relationship between the transformed (reference) frame moving with themean velocity of plasma and the laboratory (fixed) frame is Etr = E + u × B. Inthe nonrelastivistic approximation, the transformed magnetic flux density is Btr =B − u × E

/c2. For the case when coordinates system is moving with a velocity

perpendicular to both the electric and magnetic field or on a laboratory frame, thehelix trajectory will become a prolate cycloid with loops or the curtate cycloid(Jahn, 1968).

The ratio of the gyro frequency to the collision frequency of the charged particleis known as the Hall parameter 𝜔b

/𝜈c. This parameter characterizes the charged

particle’s motion as it responds to the applied E and B fields. The E × B component of

ELECTRIC CONDUCTIVITY 15

the current is the so-called Hall current that defines the gyro motion. When 𝜔b

/𝜈c ≫

1, the charged particle rarely can execute one cycle of its motion before collisionand hence cannot move in the helical trajectory. Thus, the primary component ofthe electric current is parallel to E. On the opposite limit, 𝜔b

/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. Finallyunder the condition𝜔b

/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

value for various charged species. A singly charged ion and an electron have sub-stantially different masses by a ratio of around 1836 to 1. This disparity complicatesthe formulation for the overall electrical conductivity. In practical applications in thepresence of an applied magnetic field, the electrical conductivity has been formulateddirectly by using the plasma frequency, or by describing the modified electric field inparallel and normal component, equation (1.14) with respect to an applied magneticflux density Bo. The derivation for the nonscalar electric conductivity is tedious. Forpurpose of illustration that the electric conductivity is really a tensor, an abbreviatedderivation by Jahn is presented for the electric conductivity in a steady magneticfield (Jahn, 1968). Three distinct current components have been identified by thecomponent, which lies along the electric field E and follows the drift E × Bo, and thethird component aligns with Bo but in proportional to the component of E parallelto Bo.

If Bo is aligned in the z coordinate of a Cartesian system, the conductivity appearsas a tensor of rank 2,

𝜎

𝜀= 𝜔

2p

⎡⎢⎢⎢⎣bxx bxy 0

byx byy 0

0 0 bzz

⎤⎥⎥⎥⎦ (1.33)

In this matrix formulation,

bxx = byy →vc

w2b + 𝜈

2c

bzz →1𝜈c

bxy = −byx →𝜔b

𝜔2b + 𝜈

2c

(1.34)

The gyro frequency is a rotational vector that requires a sign to distinguish theclockwise and counterclockwise rotations. Again, because of the substantial differ-ence in the mass of an electron and an ion, the electrical conductivity tensors arequalitatively different from each other.

In partially ionized plasma, the electric conductivity is complicated by the hugedisparity in mass of the electrons and ions. Therefore, a systematic approximation


must be applied in the derivation. From the basic characterization of an electromag-netic field, it can be recognized that the B field plays no role in determining thecomponent of the motion of particles parallel to it. Therefore, if one lets the magneticfield to be aligned with the Z coordinate, the particle motion can be decomposed intocomponents parallel and perpendicular to the magnetic field. The effective electronconductivity may be written accordingly as

𝜎e,⊥ =𝜎e

1 + 𝛽2e

, 𝜎e,∥ =𝛽e𝜎e

1 + 𝛽2e

(1.35)

Introducing the slip factor s = 𝛽e𝛽i = 𝜇e𝜇iB2 for weakly ionized plasma, the com-

ponents of the electrical conductivity can be approximated as (Mitchner et al., 1973)

𝜎⊥≈ 𝜎(1 + s)

(1 + s)2 + 𝛽2e

, 𝜎∥ ≈𝛽e𝜎

(1 + s)2 + 𝛽2e

(1.36)

The electric conductivity in Cartesian tensor form appears as

𝜎 =⎡⎢⎢⎢⎣𝜎⊥−𝜎∥ 0

𝜎∥ 𝜎⊥

0

0 0 𝜎⊥

⎤⎥⎥⎥⎦ (1.37)

Ohm’s law subjected to constant driving fields E and B can be constructed byassigning a simplified scalar electrical conductivity as

E = 1𝜎

[J + 𝛽e(J × B) + sB × (J × B)] (1.38)

In this equation, the second term is immediately identifiable as the Hall current andthe third term is often referred to as the slipping current unique to weakly ionizedplasma.

1.6 GENERALIZED OHM’S LAW

A simplified relationship between electric current and electric field intensity thathas been widely used in classic magnetohydrodynamics is the generalized Ohm’slaw. This approximation has the distinct advantage in compact formulation to bypassthe complex and detailed description of electrical conductivity. The derivation andapproximations of the law are based on the single-fluid, quasi-neutral plasma with

COMPUTATIONAL ELECTROMAGNETIC- AERODYNAMICS · 1 Plasma Fundamentals 1 Introduction, 1 1.1 ElectromagneticField, 3 1.2 DebyeLength, 7 1.3 PlasmaFrequency, 10 ... applications. It

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