Computational Physics of Electric Discharges in Gas Flows 9783110270419, 9783110270334

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De Gruyter Studies in Mathematical Physics 7 Editors Michael Efroimsky, Bethesda, USA Leonard Gamberg, Reading, USA Dmitry Gitman, São Paulo, Brasil Alexander Lazarian, Madison, USA Boris Smirnov, Moscow, Russia

Sergey T. Surzhikov

Computational Physics of Electric Discharges in Gas Flows

De Gruyter

Physics and Astronomy Classification Scheme 2010: 52.30.-q, 52.30.Cv, 52.80.-s, 02.70.-c.

ISBN 978-3-11-027033-4 e-ISBN 978-3-11-027041-9 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: P TP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

This book is dedicated basically to numerical modeling of gas-discharge phenomena in gas flows. To define the specificity of computing models of the discharge phenomena presented in the book it was necessary to give some classification of computational models of discharges of different kinds. In the book, the frame of problems arising in the numerical modeling of gas-discharge physical mechanics processes that was considered was far from complete. The major frame of problems of the physical and chemical kinetics of gas-discharge plasma is beyond the scope of this review. To become acquainted with the problems and issues of numerical modeling of physical and chemical kinetics processes some studies [13] and also the comprehensive bibliography referred to in them are recommended to the reader. Hybrid, kinetic and stochastic models of gas-discharge processes are also not featured. To become acquainted with these models, some studies [3, 5, 50] can be recommended. The reason for such a restriction of the frame of problems considered is that to date problems of electrodynamics and physical mechanics of discharges have, as a rule, been covered separately from problems of physicochemical kinetics. Undoubtedly, it is a principal question of the modern state of computer physics and physical mechanics of gas discharges. It is obvious that the problem of joining the two specified concepts is extremely topical. The author hopes that the models of joining discharge and gas-dynamic processes explained in the book will promote their further joining with models of physical and chemical kinetics. When discussing the problems of joining computational models of a different class, it is necessary to mention the studies devoted to numerical modeling of electro-discharge and gas-dynamic lasers [83]. However, modern trends in the development of the aerophysics of electro-discharge processes demand reviewing many similar problems from other points of view. Interest in studying electro-discharge effects in subsonic, supersonic, and, particularly, hypersonic gas flows has arisen in the last decade in view of attempts at developing hypersonic flying vehicles (HFV). The following lines of research discuss the uses of electric discharges in hypersonic aerodynamics:  

problems of global modification of the aerodynamic characteristics of HFV; control of the flow under conditions of external flow and in internal flows, for example, in the circuits of hypersonic scramjet engines (HSE); and

vi 

Preface raising the efficiency of power plants utilizing plasma chemical methods of combustion at high velocities of flow.

A detailed analysis of the processes of interaction of gas flows with discharges of various types is undertaken in the book, as well as an analysis of the prospects for the application of magneto-plasmadynamic methods in aerophysics. Of the problems of discharge effects in hypersonic aerophysics identified above, a separate class of problems is associated with the use of electro-magnetic actuators for the purpose of local modification and control of rarefied flow, which is of fundamental importance from the standpoint of developing HFV capable of flying at altitudes of 30–50 km. The book as a whole is intended for those who wish to develop without assistance a computing model of physical mechanics of glow (first of all) or other type of discharge. For this reason the book contains typical statements of problems, modes of buildup and testing finite-difference schemes, some methods of solving the grid (finitedifference) equations, algorithms of numerical modeling and a series of test variants. The statement of the given material makes no claim to completeness, but it is quite enough to create one’s own design-theoretical model. The author expresses profound gratitude to professors M. Capitelli (University of Bari, Bari, Italy), S. A. Losev (Moscow State University, Mowcow, Russia), Yu. P. Raizer (Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia), J. S. Shang (Wright State University, Dayton, Ohio, USA) for their support of the studies on making methods and computer processes for solving problems of the physics and mechanics of gas discharges, and also for long-term cooperation in this field. Moscow, July 2012

Sergey T. Surzhikov

Contents

Preface

v

I Elements of the theory of numerical modeling of gas-discharge phenomena 1

Models of gas-discharge physical mechanics 1.1 Models of homogeneous chemically equilibrium plasma . . . . . . . . . . . 1.1.1 Mathematical model of radio-frequency (RF) plasma generator 1.1.2 Mathematical model of electric-arc (EA) plasma generator . . . 1.1.3 Models of micro-wave (MW) plasma generators . . . . . . . . . . . 1.1.4 Models of laser supported plasma generators (LSPG) . . . . . . . 1.1.5 Numerical simulation models of steady-state radiative gas dynamics of RF-, EA-, MW-, and LSW-plasma generators . . . 1.1.6 Method of numerical simulation of non-stationary radiative gas-dynamic processes in subsonic plasma flows. The method of unsteady dynamic variables . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Model of the five-component RF plasma generator . . . . . . . . . 1.2.2 Model of the three-component RF plasma generator . . . . . . . . 1.2.3 Two-temperature model of RF plasma under ionization equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 One-liquid two-temperature model of laser supported plasma .

2

Application of numerical simulation models for the investigation of laser supported waves

3 5 14 19 22 25 33

47 49 54 57 59 61 64

2.1 Air laser supported plasma generator . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.2 Hydrogen laser supported plasma generator . . . . . . . . . . . . . . . . . . . . . 74 2.3 Bifurcation of subsonic gas flows in the vicinity of localized heat release regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Qualitative analysis of the phenomenon . . . . . . . . . . . . . . . . . . 2.3.3 Quantitative results of numerical simulation . . . . . . . . . . . . . .

81 83 84 85

2.4 Laser supported waves in the field of gravity . . . . . . . . . . . . . . . . . . . . 91

viii 3

Contents

Computational models of magnetohydrodynamic processes

104

3.1 General relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Vector form of Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 106 3.3 System of equations of magnetic induction . . . . . . . . . . . . . . . . . . . . . . 107 3.4 Force acting on ionized gas from electric and magnetic fields . . . . . . . 111 3.5 A heat emission caused by action of electromagnetic forces . . . . . . . . 112 3.6 Complete set of the MHD equations in a flux form . . . . . . . . . . . . . . . . 114 3.6.1 The MHD equations in projections . . . . . . . . . . . . . . . . . . . . . . 115 3.6.2 Completely conservative form of the MHD equations . . . . . . . 117 3.7 The flux form of MHD equations in a dimensionless form . . . . . . . . . . 120 3.7.1 Definition of the normalizing parameters . . . . . . . . . . . . . . . . . 120 3.7.2 Nondimension system of the MHD equations in flux form . . . 122 3.8 The MHD equations in the flux form. The use of pressure instead of specific internal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.9 Eigenvectors and eigenvalues of Jacobian matrixes for transformation of the MHD equations from conservative to the quasilinear form. Statement of nonstationary boundary conditions . . . . . . . . . . . . . . . . . 129 3.9.1 Jacobian matrixes of passage from conservative to the quasilinear form of the equations . . . . . . . . . . . . . . . . . . . . . . . 129 3.10 A singularity of Jacobian matrixes for transformation of the equations formulated in the conservative form . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.11 System of the MHD equations without singular transfer matrixes . . . . 140 3.12 Eigenvalues and eigenvectors of nonsingular matrixes of quasilinear system of the MHD equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.1 Matrix AQx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.2 Matrix AQy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.3 Matrix AQz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

144 144 148 151

3.13 A method of splitting for three-dimensional (3D) MHD equations . . . 153 3.14 Application of a splitting method for nonstationary 3D MHD flow field, generated by plasma plume in the ionosphere . . . . . . . . . . . . . . . 161

II Numerical simulation models of glow discharge 4

The physical mechanics of direct current glow discharge

171

4.1 Fundamentals of the physics of direct current glow discharge. The Engel–Steenbeck theory of a cathode layer . . . . . . . . . . . . . . . . . . . . . . 172

Contents

ix

4.2 Drift-diffusion model of glow discharge . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Reduction of governing equations to a form convenient for numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Initial conditions of the boundary value problem for the glow discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Glow discharge with heat of gas . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Estimation of typical time scales of the solved problem . . . . . .

178 178

4.3 Finite-difference methods for the drift-diffusion model . . . . . . . . . . . . 4.3.1 Finite-difference scheme for the Poisson equation . . . . . . . . . . 4.3.2 Finite-difference scheme for the equation of charge motion . . 4.3.3 Conservative properties of the finite-difference scheme for the motion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 The order of accuracy of the finite-difference approximation used. The mesh diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 The finite-difference grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6 Iterative methods for solving systems of linear algebraic equations in canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 An iterative algorithm for the solution of a self-consistent problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.8 Characteristic properties of a solution of a two-dimensional problem about glow discharge in a nonstationary statement . .

194 194 197

181 184 186 187

200 203 207 210 221 222

4.4 Numerical simulation of the one-dimensional glow discharge . . . . . . . 225 4.4.1 Governing equations and boundary conditions . . . . . . . . . . . . . 226 4.4.2 The elementary implicit finite-difference scheme . . . . . . . . . . . 228 4.5 Diffusion of charges along a current line and effective method of grid diffusion elimination in calculations of glow discharges . . . . . . . . . . . 4.5.1 Governing equations for the one-dimensional case . . . . . . . . . 4.5.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Numerical methods for the one-dimensional calculation case . 4.5.4 Results of 1D numerical simulation . . . . . . . . . . . . . . . . . . . . . 4.5.5 Method of fourth order accuracy for the solution of the drift-diffusion model equations . . . . . . . . . . . . . . . . . . . . . . . . .

229 230 230 231 232 235

4.6 Two-dimensional structure of glow discharge regarding neutral gas heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 4.6.1 Statement of two-dimensional axially symmetric problem . . . . 242 4.6.2 Numerical simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 5

Drift-diffusion model of glow discharge in an external magnetic field

261

5.1 Derivation of the equations for calculation model . . . . . . . . . . . . . . . . 261 5.2 Numerical simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

x

Contents

5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Constitutive thermophysic and electrophysic parameters . . . . . 5.3.3 The method of numerical integration . . . . . . . . . . . . . . . . . . . . 5.3.4 The finite-difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 The method of numerical integration of the heat conductive equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Numerical simulation results for glow discharge in a magnetic field in view of heating of gas . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Computational model of glow discharge with cross gas flow . . 5.4.2 Simplified hydrodynamic part of the problem under consideration. The Couette flow . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Glow discharge in neutral gas flow. Numerical simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 280 281 282 283 285 288 296 296 305 305

5.5 Computing model of glow discharge in electronegative gas . . . . . . . . . 314 5.5.1 Computational model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 5.5.2 Numerical simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 5.6 Numerical modeling of glow discharge between electrodes arranged on the same surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The equations of the drift-diffusion model for surface glow discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Boundary conditions for surface discharge . . . . . . . . . . . . . . . . 5.6.3 Initial conditions of numerical modeling . . . . . . . . . . . . . . . . . 5.6.4 Numerical simulation results of surface glow discharge . . . . . .

III 6

331 331 334 335 335

Ambipolar models of direct current discharges Quasi-neutral model of gas discharge in an external magnetic field and in gas flow 345 6.1 The spatial scale of electric field shielding in plasma. The Debye radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.2 The ambipolar diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 6.3 Ambipolar diffusion in an external magnetic field . . . . . . . . . . . . . . . . 350 6.4 Two-dimensional model of ambipolar diffusion in an external magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 6.5 Illustrative results of numerical simulation . . . . . . . . . . . . . . . . . . . . . . 354

Contents

7

xi

Viscous interaction on a flat plate with surface discharge in a magnetic field 360 7.1 Statement of a problem about viscous interaction . . . . . . . . . . . . . . . . . 362 7.2 Boundary conditions of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 7.3 Transfer and electro-physical properties of gas . . . . . . . . . . . . . . . . . . . 366 7.4 The numerical method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 7.5 Numerical simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 The heat-insulated plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Heating electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 The surface discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

367 369 370 370

Hypersonic flow of rarefied gas in a channel with glow discharge in an external magnetic field 378 8.1 Model of gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 8.2 Model of electrodynamics of glow discharge in a magnetic field . . . . 380 8.3 Boundary conditions of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 8.4 Closing relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 8.5 Algorithm of solution of complete set of equations . . . . . . . . . . . . . . . 384 8.6 Numerical simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9

Hypersonic flow of rarefied gas in a curvilinear channel with glow discharge

398

9.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 9.2 Boundary conditions and closing relations . . . . . . . . . . . . . . . . . . . . . . 400 9.3 Numerical simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 A Appendix

411

A.1 Fundamental constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 A.2 Ratios between units of electricity and magnetism . . . . . . . . . . . . . . . . 412 Bibliography

415

Index

423

Part I

Elements of the theory of numerical modeling of gas-discharge phenomena In the first part of this book, computational models of electric discharges and gasdynamic processes used in gas-discharge physics and physical gas dynamics are considered. In the first chapter, a classification of hydrodynamic models of electric discharge plasma is described. The set of equations statements for one-liquid onetemperature radiation magnetic gas dynamics is given. Examples for some computational models are considered: 

radio-frequency inductive plasma generator;



arc discharge plasma generator;



micro-wave plasma generator.

A series of inhomogeneous chemically equilibrium and nonequilibrium electric discharge plasma models are discussed. Significant attention is given to the problem of laser combustion waves as one of the most intricate problems of radiation gas dynamics of optical discharges. The problem of the unsteady subsonic movements of laser plasma is investigated. It is essential that in the first chapter only a few computational models developed for the study of gas-discharge plasma dynamics are given. The principal task of the review is to give a common representation of the typical numerical simulation models and important tasks accompanying the numerical investigation of gas discharge and physical gas dynamics problems. The second chapter gives some examples of the application of the numerical simulation models for the investigation of laser supported waves. The third chapter is devoted to reviewing some important singularities of the magnetic hydrodynamic equations, which should be taken into account in numerical simulation gas-discharge plasma dynamics.

Chapter 1

Models of gas-discharge physical mechanics

A large variety of so-called hydrodynamic models in which the whole plasma or its components are considered as a continuous medium is used in numerical modeling of plasma and gas-discharge physics. The plasma dynamics and its thermodynamic state are characterized by a velocity, pressure, density, temperature, thermal capacities, and also by a set of transport properties (viscosity, thermal conduction and electric conduction) and optical properties (spectral coefficients of absorption and emission). Plasma is characterized essentially by greater number of defining parameters than a liquid or gas, therefore the choice of hydrodynamic model for an adequate description of plasma behavior is the first problem which should be solved before practical numerical realization of a chosen model. Creation or selection of the plasma hydrodynamic model is based on comparison of characteristic scales of space L and time t , dictated by a problem under consideration, with the spatial and time scales describing separate elementary processes. The basic elementary physical and chemical processes which are necessary to consider at formation of such a model are: 

collisional processes of plasma particles, characterized by mean free path lc and collision frequency c ;



shielding of elementary charge located in plasma, which is characterized by Debye screening radius rD and Maxwellian time of a volume charge relaxation tM ;



electronic and ionic plasma oscillations, which are characterized by frequencies !e and !i accordingly;



electronic and ionic cyclotron oscillation in an exterior magnetic field, which are characterized by cyclotron frequencies !B,e and !B,i , accordingly. If velocities of electronic Ve and ionic Vi motions are known, in a magnetic field the major space scales of plasma are thus values of radiuses of particles’ cyclotron rotation trajectories, they are the Larmor radiuses for electrons RB,e and ions RB,i ;



collisional and radiative processes resulting in ionization (recombination), dissociation (association), excitation (depletion) of excited energy levels of particles, and other chemical reactions of various types;



physical and chemical processes resulting in collision energy transfer (electronic heating of atoms; translational, rotational, vibrational and electronic heating of molecules);

4

Chapter 1 Models of gas-discharge physical mechanics



processes of energy and momentum transfer which characterize propagation of sound, viscous dissipation, oscillating and electronic thermal conduction, diffusion and ambipolar diffusion;



magnetohydrodynamic waves (magnetosonic waves, the Alfvén waves).

One can find definitions of all introduced parameters in text books on plasma and discharge physics [2, 17, 64, 83]. Unfortunately, the choice of plasma hydrodynamic model becomes more complicated because a lot of above enumerated parameters can be authentically defined only after a solution of the problem. Classification of plasma hydrodynamic models can be presented as following: I. I.1. I.2. I.2.1. I.2.1.1. I.2.1.2. I.2.1.3. I.2.2. I.2.3. I.2.3.1. I.2.3.2. I.2.4. I.2.5.

One-liquid models Models of collisionless plasma Models of collision plasma (perfectly conducting plasma and plasma with finite conductivity) Single-temperature models: Models of equilibrium (Boltzmann) population of particles at excited states Models of homogeneous chemically equilibrium plasma Models of inhomogeneous chemically equilibrium plasma Models of inhomogeneous chemically nonequilibrium plasma Two-temperature models of homogeneous plasma Two-temperature models of inhomogeneous plasma Models of chemically equilibrium plasma Models of chemically nonequilibrium plasma Multi-temperature models of chemically nonequilibrium plasma with equilibrium (Boltzmann) population of particles at excited states Multi-temperature models of chemically nonequilibrium plasma with nonequilibrium population of particles at excited states

II. Multi-liquid models of low temperature plasma II.1 Models with equilibrium (Boltzmann) population of particles at excited states II.1.1. Two-liquid model of partially ionized media [electrons C (atoms C ions C molecules)] II.1.2. Two-liquid model of fully ionized media [ions C electrons] II.1.3. Three-liquid model [(atoms C molecules) C ions C electrons] II.1.4. Multi-liquid models [atoms C    C molecules C    C ions + electrons] II.2. Models with nonequilibrium population of particles at excited states II.2.1. Two-liquid model of partially ionized media [electrons + (atoms C ions)] II.2.2. Two-liquid model of fully ionized media [ions + electrons]

5

Section 1.1 Models of homogeneous chemically equilibrium plasma

II.2.3. Three-liquid model [atoms C ions C electrons] II.2.4. Multi-liquid models [atoms C    C molecules C    C ions C electrons] This classification which will be considered below in more detail, allows bringing the hydrodynamic models into line with specific peculiarities of plasma dynamics and physics of gas discharges.

1.1 Models of homogeneous chemically equilibrium plasma Such models represent the most widespread class of models applied to research of plasma dynamics and its stationary states when the spatial and time scales considered noticeably surpass all other scales. It is assumed that the plasma particles’ collision frequency is so high that the locally-equilibrium thermodynamic state has formed significantly faster than any considered gas-dynamic function has changed. In these conditions, external electric field strength is not so great that electrons, intensively heated up by it, can be characterized by a temperature that strongly differs from the temperature of heavy particles (atoms, molecules, ions). As used here, the term of the homogeneous chemically equilibrium plasma should be understood so that to solve the problem stated there is no necessity to define concretely a chemical composition of plasma in each point of investigated area. That is, the whole plasma is considered as a single liquid characterized by any two thermodynamic parameters (temperature and pressure, density and pressure, etc.), and for definition of all its properties there is no necessity to specify its chemical composition. Certainly, it is good to bear in mind that thermodynamic, thermal-physic and optical properties of the plasma can vary in the investigated volume, and it is impossible to define the specified properties without calculation of plasma chemical composition. However, the basic singularity of such a model is the possibility to separate the problem solution procedure into independent two constituents. At the preparatory stage, calculation of equilibrium chemical composition and definition of thermodynamic, transfer and optical properties is performed. At the second stage, the solution of the stated plasma-dynamics problem itself is fulfilled. The model of homogeneous chemically equilibrium plasma is used as a solution of the following problems: 

super- and hypersonic flow of space vehicles in reentry mode;



calculation of arc, inductive and optical plasma generator performance.

The set of equations for the homogeneous chemically equilibrium plasma model has the following form: 

The continuity equation: @ C div V D 0. @t

(1.1)

6 

Chapter 1 Models of gas-discharge physical mechanics

The equation of motion: @V R CdivŒ.V/  V D  grad.p C p R /1 0 ŒJ  BCF CF CgCe E. (1.2) @t



The energy conservation equation:      @ B2 p pR V2  V2 C C C C div V " C " C @t 2 20 2   D div. gradT / C divW C A C AR  C .g  V/ C .J  E/. (1.3)



Set of the Maxwell equations: @B , @t divD D e , divB D 0, rotE D 

.1 0 /rotB D J. 

(1.4) (1.5) (1.6) (1.7)

The generalized Ohm law:  1 gradpe . J C PH ŒJ  b D  E C ŒV  B C e ne 



(1.8)

The thermal equation of state: pD

R0 T , M

(1.9)

where t is the time; B, D are the magnetic and electric inductions, related with corresponding strengths by the following expressions: B D 0 H, D D "0 E; 0 , "0 are the plasma magnetic inductivity and permittivity: 0 D 4  107Gs/m, "0 D 8.854  1012F/m; J is the current density; g is the acceleration of gravity; F , FR  are the forces of viscous friction caused by gas-dynamic and radiation processes; A , AR  are the work of viscous friction caused by gas-dynamic and radiation processes; W is the vector of integral radiation flux; PH D jBj=e ne is the Hall parameter (for the case ni D ne /; b D B=jBj D B=B; V is the vector of plasma velocity; , p are the density and pressure; p R is the radiation pressure; is the thermal conductivity coefficient;  is the coefficient of viscosity; M is the molecular weight; R0 is the universal RT gas constant; " D T0 cv dT C"0 is the specific internal energy of plasma; cv is the specific thermal capacity at constant volume; T0 , "0 are the temperature and energy taken for a reference point; e D e.ni  ne / is the charge density; ni , ne are the volume ion and electron densities; e is the electron charge;  is the electrical conduction.

Section 1.1 Models of homogeneous chemically equilibrium plasma

7

In the general case, formulas for the calculation of viscous friction force and its work are represented as follows:   F D .2 C /r  .r  V/   r  .r  V/ n  o C 2.r  r/V C .r  V/  .r  / C .r  /  Œr  V , (1.10) n  o  (1.11) A D r   .r  0.5V 2 / C .V  r/V C V.r  V/ , where in addition to the coefficient of dynamic viscosity , the second viscosity coefficient is also considered, which however is practically never taken into account in a solution of plasma-dynamics problems at moderate pressure. The set of equations (1.1)–(1.11) gives the common representation of the basic model of radiative hydrodynamics in the non-relativistic formulation. For the practical realization of such a model in the form of a computing code it is preferable to rewrite the equations in the conservative flux form. In the case of Cartesian rectangular coordinates, such a set of equations has the following form: @U @Fx @Fy @Fz C C C D G, @t @x @y @z ˇ ˇ ˇ ˇ  ˇ ˇ ˇ ˇ u ˇ ˇ ˇ ˇ v ˇ ˇ ˇ ˇ w ˇ, ˇ UDˇ ˇ ˇ E C pm ˇ ˇ ˇ Bx ˇ ˇ ˇ ˇ By ˇ ˇ ˇ ˇ Bz

FEu x

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(1.12)

(1.13)

NS R MGD C FVMGD , Fx D FEu x C Fx C Fx C Fx x

(1.14)

Fy D

(1.15)

FEu y

C

FNS y

C

FRy

C

FMGD y

C

FVMGD , y

NS R MGD C FVMGD , (1.16) Fz D FEu z C Fz C Fz C Fz z ˇ ˇ ˇ ˇ u ˇ ˇ 0 ˇ ˇ ˇ 2 ˇ u C p ˇ ˇ  xx ˇ ˇ ˇ ˇ uv ˇ ˇ  xy ˇ ˇ ˇ ˇ uw ˇ ˇ  xz ˇ  NS ˇ, ˇ R ˇ , Fx D ˇ p p .u xx C v xy C w xz /qx ˇˇ u E C  C  C Wx ˇ ˇ ˇ ˇ ˇ 0 ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 0 (1.17)

8

Chapter 1 Models of gas-discharge physical mechanics

ˇ ˇ 0 ˇ ˇ R ˇ  xx C p R ˇ ˇ R ˇ  xy ˇ ˇ R  xz ˇ FRx D ˇ

 R R R ˇ  u xx C v

C w

xy xz ˇ ˇ 0 ˇ ˇ ˇ 0 ˇ ˇ ˇ 0

FMGD x

FVMGD x

FEu y

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ 0 ˇ  ˇ 1 2 ˇ  20 .Bx  By2  Bz2 / ˇ ˇ .1=0 /Bx By ˇ ˇ ˇ .1=0/Bx Bz Dˇ ˇ 2upm  .1=0/Bx .uBx C vBy C wBz / ˇ ˇ 0 ˇ ˇ vBx C uBy ˇ ˇ ˇ wBx C uBz

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1 20 

h

 By

ˇ ˇ v ˇ ˇ vu ˇ ˇ v 2 C p ˇ ˇ vw ˇ  R Dˇ p ˇ v E C  C p C Wy ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

@Bx @y



@By @x



0 0 0 0 0

C Bz



@Bx @z



@By @Bx 1 0  @x  @y x z  @B  10  @B @z @x

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

FNS y

(1.18)

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ i ˇ ˇ @Bz  @x ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(1.19)

(1.20)

ˇ ˇ 0 ˇ ˇ  yx ˇ ˇ  yy ˇ ˇ  yz D ˇˇ .u

C v

yx yy C w yz / C qy ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(1.21)

9

Section 1.1 Models of homogeneous chemically equilibrium plasma

ˇ ˇ 0 ˇ ˇ R  yx ˇ ˇ R ˇ  yy C p R ˇ ˇ R  yz ˇ  FRy D ˇ R R R ˇ  u yx C v yy C w yz ˇ ˇ 0 ˇ ˇ ˇ 0 ˇ ˇ 0

FMGD y

ˇ ˇ 0 ˇ ˇ .1=0/By Bx ˇ ˇ ˇ .1=20/.By2  Bx2  Bz2 / ˇ ˇ .1=0/By Bz ˇ Dˇ ˇ 2vpm  .1=0/By .uBx C vBy C wBz / ˇ ˇ uBy C vBx ˇ ˇ ˇ 0 ˇ ˇ wB C vB y

FVMGD y

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1 20 

FEu z

z

0 0 0 0

h  @B Bz @zy 



@B @Bz C Bx @xy @y  @B x  10  @xy  @B @y 1 0 



0 @Bz @y



@By @z



ˇ ˇ w ˇ ˇ wu ˇ ˇ ˇ wv ˇ ˇ w 2 C p ˇ  R Dˇ ˇ v E C p C p C Wz ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ ˇ 0

(1.22)

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ i ˇˇ ˇ, x  @B ˇ @y ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(1.23)

(1.24)

(1.25)

10

Chapter 1 Models of gas-discharge physical mechanics

ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ  zx ˇ ˇ ˇ ˇ ˇ ˇ  zy ˇ ˇ ˇ ˇ  zz NS ˇ ˇ, Fz D ˇ ˇ ˇ .u zx C v zy C w zz / C qz ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ R  zx ˇ ˇ ˇ ˇ R ˇ ˇ  zy ˇ ˇ ˇ ˇ R  zz C pR ˇ ˇ R Fz D ˇ ˇ, R C v R C w R / ˇ ˇ .u zx zy zz ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ 0 ˇ ˇ  ˇ ˇ  10 Bz Bx ˇ ˇ ˇ ˇ  ˇ ˇ ˇ ˇ  10 Bz By ˇ ˇ  ˇ ˇ 1 2 2 2 ˇ ˇ  20 .Bz  Bx  By / MGD ˇ, ˇ  Dˇ Fz ˇ ˇ 2wpm  1 Bz .uBx C vBy C wBz / ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ uB C wB z x ˇ ˇ ˇ ˇ ˇ ˇ vBz C wBy ˇ ˇ ˇ ˇ ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ ˇ 0 ˇ  i ˇ 1 h  @Bz @Bx D FVMGD @By @Bz z ˇ 2 Bx C B   y @x @z ˇ 0  @z @y ˇ @Bx @Bz 1 ˇ  ˇ 0   @z @x ˇ @B @Bz 1 ˇ  0  @y  @zy ˇ ˇ ˇ 0

(1.26)

(1.27)

(1.28)

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(1.29)

Section 1.1 Models of homogeneous chemically equilibrium plasma

ˇ ˇ 0 ˇ ˇ C e Ex f x ˇ ˇ fy C e Ey ˇ ˇ fz C e Ez G D ˇˇ Q C .ug v x C vgy C wgz / ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0 where E D " C

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

11

(1.30)

u2 Cv 2 Cw 2 , 2

q D rT D i

@T @T @T  j  k , @x @y @z

P˛ˇ D pı˛ˇ  ˇ ˛ , R P˛ˇ D p R ı˛ˇ  ˇR˛ ,    @uˇ 2 @u˛  ı˛ˇ divV , C

ˇ ˛ D  @x˛ @xˇ 3

V D iu C jv C kw, x, y, z are the rectangular Cartesian coordinates; P˛ˇ are the components of gasR dynamic pressure tensor; P˛ˇ are the components of radiation pressure tensor; ˇ ˛ are the components of viscous stress tensor; E is the total specific energy; "is the specific internal energy; q is the heat flux vector caused by thermal conduction; f is the mass force acting on unit of volume (for example, in case of only gravity force acting f D g/; Wx , Wy , Wz are the components of integral radiation heat flux vector W; Qv is the power source caused by external heat sources; T is the temperature; Bx , By , Bz are the components of magnetic induction vector B; pm D .Bx2 C By2 C Bz2 /=20 is the magnetic pressure. The set of equations of gas dynamics closes with the caloric equation of state, which can be formulated in general form as follows: " D ".p, T / D "., T /. Using the notation of the equations of radiative magnetohydrodynamics in the form of (1.12)–(1.30), it is possible to obtain any necessary form of the equations for one-, two- and three-dimensional cases in any coordinate system, by supposing uniqueness of transformation of rectangular Cartesian coordinates. To determine the radiative heat flux vector W, the equation for selective thermal radiation transfer is used. This equation is formulated for spectral intensity of heat radiation J .s, , t / along an arbitrary ray s with a unit vector , which was eradiated

12

Chapter 1 Models of gas-discharge physical mechanics Ω

s

dΩ

z

r

k

x

i

j

y

Figure 1.1. System of spatial and angular coordinates for definition of spectral intensity of radiation.

from a spatial point r (Figure 1.1): 1 @J .s, ,t / @J .s, ,t / C C Œ .s/ C  .s/J .s, ,t / c @t @s Z 1 Z 1 D Jem .s, t / C  .s/ .s;  0 , ;  0 , /J 0 .s, 0 ,t /d 0 d 0 , 4  0 D0 0 D4 (1.31) where: J .s, ,t / is the spectral intensity of radiation;  is the frequency of radiation; s is the physical coordinate along a ray;  is the unit vector of radiation transfer direction;  .s/ is the volume absorptivity (the spectral volume absorption coefficient);  .s/ is the spectral volume scattering coefficient; Jem .s, t / is the spectral emissivity of unit volume; .s; 0 , ;  0 , / is the spectral scattering indicatrix by direction and frequency of radiation; 0 is the unit vector for rays specifying a direction of thermal radiation propagation, which falls on elementary physical volume with coordinate s and afterwards is dispersed by it with probability .s; 0 , ;  0 , /; c is the speed of light. It is recommended to use the definitions and descriptions of the functions and coefficients presented in books [69, 75, 99]. In the case of a local thermodynamic equilibrium (LTE): Jem .s, t / D  .s/Jb, ŒT .s, t /.

(1.32)

where Jb, ŒT .s, t / is the spectral intensity radiation of the black body (the Planck function).

Section 1.1 Models of homogeneous chemically equilibrium plasma

13

The radiation heat flux vector used in the set of equations (1.12)–(1.30) is calculated under the formula: Z 1 Z W.r, t / D d J .r, , t /d . (1.33) 4

0

It follows from the equations (1.2), (1.3), that energy transfer by heat radiation is presented in the equations of radiation gas dynamics only in the form of radiation stress tensor and in the form of a divergence of integral radiation heat flux vector. After application of divoperator to the equation (1.31) one can determine the last term under the formula: Z 1 Z 1 Z divW.r, t / D 4 Jem d   .r/ J .r, , t / d d. (1.34) 0

D4

0

As to the components of radiation pressure tensor, they are negligible in comparison with components of gas-dynamic pressure tensor in overwhelming majority of cases which are of interest for physics and dynamics of low temperature plasma. Note that these terms can be significant for radiative gas-dynamic problems issued from astrophysical applications [69]. The comparative estimation for values of gas-dynamic and radiation pressure tensor components is given with radiation pressure parameter: Rr D

T Q 4 , 3p

where Q is the Stefan–Boltzmann constant. At Rr  1 the radiation pressure can be neglected. Several physically justified assumptions are used in solutions of problems of gasdischarge physics and radiative gas dynamics. They allow simplifying the radiative transfer equation. Some of them are: (a) as the velocity of thermal radiation propagation is many times more than the velocities of all other processes, it is possible to calculate the radiative transfer in a stationary approximation; (b) light scattering over directions is essentially a more probable process than scattering over frequencies (this is the so-called approximation of coherent scattering); (c) if there is a lack of condensed scattering centers in plasma it is possible to neglect processes of scattering; (d) approximation of local thermodynamic equilibrium is in common use. The equation of the spectral radiation heat transfer in view of the above-mentioned assumptions gets the following form: @J .s, / C  .s/J .s, / D  .s/Jb, ŒT .s, t /. @s

(1.35)

14

Chapter 1 Models of gas-discharge physical mechanics

Despite the relative simplicity of (1.35), it is not actually used in radiative gas-dynamic models in such a form since, according to the definition of a vector of radiative flux density and its divergence (see (1.33) and (1.34)), besides integration in (1.35) along physical coordinate s it is necessary to perform integration on angular variables and on frequency of radiation. The laboriousness of this procedure is extremely great, therefore usually the approximate methods of the definition of functions W.r, t / and divW.r, t / are used. The computational methods of radiation heat transfer with reference to problems of radiative gas dynamics are viewed in detail in [69, 99]. The combined equations of radiative magnetic gas dynamics (1.1)–(1.9) should be added by corresponding boundary conditions. Examples of such boundary conditions will be considered below. Three model problems of gas-discharge physical mechanics are considered below as examples of models of the homogeneous chemically equilibrium plasma.

1.1.1 Mathematical model of radio-frequency (RF) plasma generator Radio-frequency plasma generators (or inductive plasma generators – IPG) are applied in plasma-chemical and aerophysical investigations, as well as in different plasma technologies. Characteristic parameters of the radio-frequency electromagnetic radiation are the following: RF  3–3 000 m and frequency of f  105–108 Hz. A distinctive feature of the inductive plasma generators is the purity of the generated plasma jets, that has made them one of the most widespread plasma generators used in thermophysic and aerophysical applications. The computing model of the IPG operating in steady conditions is stated in a twodimensional cylindrical coordinate frame r  x (Figure 1.2) [30]: 

Continuity equation: 1 @rv @u C D 0. @x r @r



(1.36)

Equation of motion: u

@v @p @u C v D C Su C Fx  g, @x @r @x

(1.37)

@v @v @p C v D C Sv C Fr , (1.38) @x @r @r      @ 2 1 @ @u @v @ @u Su D   div V C r C C .2 /, (1.39) @x 3 r @r @r @x @x @x        @ @u @v 1 @ @v v @ 2  div V C  C C r 2  2 2 (1.40) Sv D  @r 3 @x @r @x r @r @r r u

15

Section 1.1 Models of homogeneous chemically equilibrium plasma

r

1 x1

r2 r1 xc

3

x

g 2 x2

Figure 1.2. Schematic of IPG: (1) inductor; (2) the gas-discharge channel; (3) the coaxial channel for making a near-wall gas stream (possibly with vortex); r1 , r2 – radiuses of the interior and exterior channel of the plasmatron, xc – the axial size of the discharge chamber, x1, x2 – axial coordinates of the inductor.



energy conservation equation:     @T @T @ @T 1 @ @T C cp v D  C r C QE  QR : cp u @x @ @x @x r @r @r



(1.41)

thermal equation of state: pD

R0 T ; M

(1.42)

where x, r are the axial and radial coordinates; V D .u, v/ is the velocity of a plasma flow and its x and r components, accordingly; , p are the density and pressure; cp is the specific heat capacity at constant pressure; T is the temperature; QE is the Joule thermal emission; QR is the power of volumetric radiative losses; F D c 1 Œj  H D .Fx , Fr / is the average electromagnetic force for a field oscillations period. For the definition of energy release power, QE and the components of the magnetic force system of equations (1.36)–(1.41) are supplemented with the Maxwell equations. Two formulations of the Maxwell equations for calculations of IPG parameters are in common use. The local one-dimensional formulation The main assumption of the model given in [7] is that in each section, perpendicular to symmetry axis, the electromagnetic part of the problem is solved as for the infinite cylinder. In this case, only the azimuth component of electric field E' and the axial

16

Chapter 1 Models of gas-discharge physical mechanics

component of magnetic field Hx are subjected to definition. The following system of equations is considered in the case: dHx D E' cos , dr

(1.43)

E' dE' D  a !Hx sin  dr r

(1.44)

together with the equation for phase shift between the specified field components a !Hx E' d sin   cos , D dr Hx E'

(1.45)

where ! D 2f is the circular frequency of the electromagnetic field;  D H  E is the phase angle between Hx and E' ; E' and Hx are the components of the peak values of the field. The power of a thermal emission in a unit volume and the radial component of electromagnetic force are defined under formulas: QE D E'2 ,

(1.46)

Fr D a j' Hx cos ,

where j' D E' is the azimuth component of current density. For the solution of equations (1.38)–(1.41) the following boundary conditions are used: (1.47) x D 0 : u D V0 .r /, v D 0, T D T0.r /; r D0: r D re :

v D 0,

u D v D 0, x D xc :

@u @T D D 0; @r @r 

@T c .Tw  T /; D @r ıc

@v @T @u D D , @x @x @x

(1.48) (1.49) (1.50)

where V0 .r /, T0.r / are the velocity and temperature in the inlet cross section of the discharge chamber; c , ıc are the thermal conductivity and thickness of the discharge chamber (as a rule, a quartz tube); Tw is the temperature of an exterior surface of a tube. For the solution of equations (1.43) and (1.45) for components of the electromagnetic field, the following boundary conditions are used: r D0:

H D H0 .x/,

E D E' D 0,

 D =2.

(1.51)

Section 1.1 Models of homogeneous chemically equilibrium plasma

17

The magnetic intensity H0 .x/ is calculated by the approximate formulas H0 .x/ D H1 D

x2  x H1 x1  x Œq q , 2 ri2 C .x2  x/2 ri2 C .x1  x/2

(1.52)

I! , x2  x1

(1.53)

where I is the current in the inductor; ri is the radius of the inductor. The axial coordinate dependence of the magnetic intensity is considered here as it is created by the inductor without plasma, while the inductor is considered the homogeneous finite cylinder. In the given model it is necessary to correct H1 value, considering the influence of plasma. For this purpose the full thermal emission in induction plasma is calculated as: Z x 2 Z rc P D 2QE r dr dx, (1.54) x1

0

Then the specified correction could be made p H1 D H1 P0 =P  ,

(1.55)

where P0 is the given value of heat input into the plasma (this value can be measured experimentally). Returned to the discussion of a class of homogeneous chemically equilibrium plasma models, we shall note that in the IPG problem viewed, definitions of all thermodynamic and thermophysical properties are defined by local values of temperature and pressure (or density). Two-dimensional axisymmetric formulations A system of two-dimensional electromagnetic equations can be formulated relating the real E1 and imaginary E2 components of the complex amplitude of an electric field intensity (E D E1 C iE2 /: @ @r @ @r

 

1 @rE1 r @r 1 @rE2 r @r

 C

@2E1 4! D E2 , 2 @x c2

(1.56)

C

4! @2E2 D E1 . 2 @x c2

(1.57)



With the purpose of formulating boundary conditions, a series of simplifying assumptions were made. It was assumed that the inductor consists of Nk ring currents Ii with radius ri . Then the axial-oriented intensity of the magnetic field created by such an

18

Chapter 1 Models of gas-discharge physical mechanics

inductor with discharge chamber radius .rc / is equal to N

k Ii X '.lk /, (1.58) c k   ri2  rc2  .x  xk /2 2   '.lk / D p  K .lk / C E .lk / , .ri  rc /2 C .x  xk /2 .ri C rc /2 C .x  xk /2 (1.59) 4rc ri lk D , (1.60) .ri C rc /2 C .x  xk /2

Hx D

where k is the number of coils with a current; xk is the axial coordinate of the k-th coil; K  .lk /, E  .lk / are the full elliptic integrals of the first and second kind. One more physical assumption was made, namely the inductor was located far enough from input (x D 0/ and output (x D xc / sections of the discharge channel, therefore the axial gradients of an electric field are small. Considering the above-mentioned assumptions, boundary conditions are written in the following form: xD0: r D0:

@E1 @E2 D D 0; @x @x

(1.61)

E1 D E2 D 0;

(1.62)

N

r D rc :

k 1 @ ! X .rE2/ D 2 Ii '.lk /, r @r c

(1.63)

k

1 @ .rE1 / D 0; r @r @E2 @E1 x D xc : D D 0. @x @x

(1.64) (1.65)

Another formulation of the electromagnetic part of the problem was offered in [30]: 1 @rE' @Hx D a , r @r @t

(1.66)

@Hx @Hr  D E' , @x @r

(1.67)

@Hr @E' D a , @x @t

(1.68)

where Hx , Hr are the axial and radial components of the electric field intensity; E' is the azimuth component of the electric field intensity.

Section 1.1 Models of homogeneous chemically equilibrium plasma

19

In the same study the method of the equations’ integration with use of the vector potential was stated. Recommendations were given based on the investigation of the application of approximating dependencies for the magnetic field axial component on the boundary of the discharge channel: Hx .x, r D rc / D     r 0.5.x2  x1 /  x 0.5.x2  x1 / C x 1 ri I! arctg C arctg  x2  x1  r c ri  rc ri C rc  r    0.5.x2  x1/  x rc  1 arctg ri ri  r    0.5.x2  x1/ C x rc  1 arctg , ri ri (1.69) where ri is the radius of the plasma generator inductor.

1.1.2 Mathematical model of electric-arc (EA) plasma generator Electro-arc plasma generators are widely applied in practical aerophysical investigations along with inductive plasma generators, as well as in plasma technologies. The physical mechanics of the low temperature plasma of EA plasma generator are considered in detail in some studies [83, 90]. The electro-arc plasma is usually considered thermally equilibrium. However, at low pressures and currents, and near walls of the plasma generators the use of a local thermodynamic equilibrium approach becomes unjustified. In these cases, two temperature and more complex models of plasma should be used. Electric arcs are characterized by currents of I  100–105 A, densities of a current on the cathode of j  102–107 A/cm2 , and voltage drop between electrodes of V  20  30 V. A typical diagram of an electro-arc generator is shown on Figure 1.3. The equations of the homogeneous chemically equilibrium electro-arc plasma are formulated in the following form: 

Continuity equation: @ C div V D 0; @t



(1.70)

Equation of motion: @V 1 C divŒ.V/  V D grad.p C p R /  Œ J  B C F C FR  C g C e E; @t 0 (1.71)

20

Chapter 1 Models of gas-discharge physical mechanics A

1

3

2 K

Figure 1.3. Schematic of electro-arc generator: K is the cathode; A is the anode; (1) gas stream; (2) electro-arc plasma; (3) plasma jet.



Energy conservation equation: @ @t



 V2 B2 " C C 2 20



   V2 p pR C div V " C C C D 2  

div .grad T / C divW C A C AR  C .g  V/ C . J  E/; (1.72) 

The Maxwell equations: @B , @t div D D e , rot E D 

div B D 0, 1 rot B D J; 0 

(1.74) (1.75) (1.76)

The Ohm law:  1 J C PH ŒJ  b D  E C ŒV  B C grad pe ; e ne



(1.73)

(1.77)

Thermal equation of state: pD

R0 T , M

(1.78)

where B, D are the magnetic and electrical inductions related with corresponding strengths as B D 0 H, D D "0E; 0 , "0 are the magnetic and dielectric permeabilities of the plasma; F , FR  are the forces of viscous friction caused by gas-dynamic and radiative processes; A , AR  are the work of the viscous friction caused by gasdynamic and radiative processes; W is the vector of integrated radiation heat transfer

Section 1.1 Models of homogeneous chemically equilibrium plasma

21

flux; PH D jBj=e ne is the Hall parameter (at ni D ne /; b D B=jBj D B=B; V is the velocity of plasma; , p are the density and pressure; M is the molecular weight; RT R0 is the universal gas constant; " D T0 cv dT C "Q0 ; cv is the specific heat capacity at constant volume; T0 , "Q 0 are the temperature and energy taken for a reference thermodynamic point; e D e.ni  ne / is the density of electric charge; ni , ne are the volume concentrations of ions and electrons; e is the electron charge; pe is the pressure of electronic gas. It is assumed that the homogeneous chemically equilibrium plasma model of arc discharges at atmospheric pressure is well-founded if the following inequality is satisfied:     Te  Ti ƒe eE 2 ƒe e 2 3 mi 3 mi    1, Te 32 me .3=2/kTe 32 me .3=2/kTe where Te , Ti are the temperatures of electrons and ions; me , mi are the masses of electrons and ions; ƒe is the Coulomb logarithm. Typical parameters entering into this inequality for air plasma at atmospheric pressure are: T  10 000 K, ƒe  106 m, and   101 W/(m  K). Therefore, at a typical current in arc discharge of I  10 A one can get: Te  Ti  104 . Te The following assumptions simplifying equations (1.70) –(1.78) are also used: (1) Neglect of a gravity g. However, it is necessary to note that for open-flame arcs of great volumes, located horizontally, a gravity leads to their curvature. (2) Neglect of electrostatic force e E. The estimate of the ratio of electrostatic force to inertial force shows that "0 E 2 e E   108, V 2 LV 2 where typical values for electric force E D 103 V/m, velocity V D 102 m/s, density  D 101 kg/m3, and characteristic scale distance L D 0.1 m are used. (3) Neglect of the Hall effect (the second item in the left-hand part of (1.77)). For a substantiation of this approach it is necessary to consider two possible cases: the self- and the exterior magnetic fields. In the case of the self-magnetic fields, the Hall parameter is estimated as follows: PH D

B E   106 , e ne Ve ne

where ne D 1022 m3 , E D 103 V/m, V D 102 m/s,  D 104 (Ohm  m)1 . It is obvious from the definition of the Hall parameter that the induction of exterior magnetic fields should also be rather high to have an appreciably effect on the dynamics of plasma.

22

Chapter 1 Models of gas-discharge physical mechanics

(4) Neglect of an electronic pressure gradient and the induced electric field in comparison with electric intensity E: 1 kTe rpe   1, e ne E LEe ŒV  B VB   1. E E Assumptions (3) and (4) give grounds to use the Ohm law in the following simplified form: J D E. (1.79) Note that use of the full system of equations (1.70)–(1.72) is extremely rare. The gasdynamic models based on the approach of a boundary layer are currently used considerably.

1.1.3 Models of micro-wave (MW) plasma generators The typical schematic diagram of the MW plasma generator is shown in Figure 1.4.

r

2 1

3 4

5 x

2

Figure 1.4. Schematic diagram of the MW axial type plasma generator: (1) the gas-discharge channel; (2) axially symmetric flow of MW radiation; (3) gas stream (possibly with vortex); (4) discharge plasma; (5) plasma jet.

Calculation models of the MW discharge are similar to models of equilibrium electro-arc and RF plasma generators. The only difference is in the part of the equation that features an energy release in the low temperature plasma. In case of a MW plasmatron model, this is the Maxwell equation formulated in the most convenient form for the geometry explored, subject to features of interaction between electromagnetic

Section 1.1 Models of homogeneous chemically equilibrium plasma

23

microwaves and plasma. For MW discharges, the frequency band of fMW  109 – 1011 Hz and wavelengths MW  0.3–30 cm are typical. In the overwhelming majority of cases under consideration, the conditions in which the electron concentration ne appreciably surpasses the critical electron concentration [71, 83] nc,e D

" 0 me ! 2 , e2

(1.80)

are realized in the MW discharges. Therefore, as a rule the valid requirement is ! < !e ,

p where ! is the circular frequency of electromagnetic field; !e D ne e 2="0me is the plasma frequency. Note that frequency of collisions of electrons with heavy particles  can surpass circular frequency ! or be less than it as well. A distinctive feature of MW discharges is that the wavelength of electromagnetic radiation is usually close to the characteristic size of plasma volume. Therefore, the geometrical optics approximations, which allow the simplification of an electromagnetic formulation of the problem, are in most cases inapplicable. In this sense, the formulation of MW discharge mathematical model represents a greater difficulty than the formulation of such models for RF discharge. Nevertheless, practically explored models of the MW plasmatron allow the application of some simplifications. Some such models are presented below. The elementary one-dimensional computing model of MW discharge The MW discharge is considered in one-dimensional cylindrical geometry. It means that a change of its parameters in a radial direction is many times greater than changes in an axial direction. If gas-dynamic processes are negligible for such a discharge, the following governing equations can be used: 



Energy conservation equation:   dT 1 1 d r D  jEj2  div W; r dr dr 2 Equation for electric field intensity:   dE 1 d  C ! 2 "00 "k E D 0, r dr dr

(1.81)

(1.82)

where "k D " C i.="0!/ is the complex permeability; ",  are the dielectric permeability and electrical conductivity of the medium; 0 is the magnetic permeability of free space; E is the complex electric field intensity.

24

Chapter 1 Models of gas-discharge physical mechanics

Boundary conditions for equations (1.81), (1.82) are formulated along a symmetry axis and on an external radial boundary of the discharge channel: r D0: r DR: RR

@E D 0; @r

(1.83)

Q† D QR C QA ,

(1.84)

@T D 0, @r

T D Tw ,

where QA D  0 jEj2 r dr is the absorbed power of electromagnetic radiation; Q† , QR are the radiation power falling on the plasma and reflected by it. Quantity Q† is the prescribed parameter of the problem; R is the channel radius. Two-dimensional axisymmetric model of MW discharge Several assumptions are used in this model. These are: 

gas flow in the MW plasmatron is stable and axially symmetric;



gas-discharge plasma is in chemical equilibrium;



external electromagnetic energy falls to plasma in radial direction (see Figure 1.4);



change of electromagnetic functions in a radial direction appreciably surpasses their change in an axial direction.

Governing equations of the model are formulated in the following form: 

Continuity equations: @ C div V D 0; @t



(1.85)

Equations of motion: u

@u @v @p C v D C Su C Fx  g, @x @r @x

(1.86)

@v @p @v C v D C Sv C Fr , (1.87) @x @r @r        1 @ @u @v @ @u @ 2  div V C r C C 2 , (1.88) Su D  @x 3 r @r @r @x @x @x        @ 2 @ @u @v 1 @ @v v Sv D   div V C  C C r 2  2 2 ; (1.89) @r 3 @x @r @x r @r @r r u



Energy conservation equations:

  @ @T 1 @ @T @T @T C  cp v D . / C r C QE  QR ;  cp u @x @r @x @x r @r @r

(1.90)

Section 1.1 Models of homogeneous chemically equilibrium plasma 

Thermal equations of state : pD



25

R0 T ; M

(1.91)

The equations of an electromagnetic field: 1 d @Ex !2 .r /C "k Ex D 0, r dr @r " 0 0 i @Ex , !0 @r

(1.93)

i @H' , !"0"k @x

(1.94)

H' D  Er D

(1.92)

where x, r are the axial and radial coordinates; V D .u, v/ is the velocity of the plasma flow and its projections to axes x and r accordingly; , p are the density and pressure; cp is the specific heat capacity at constant pressure; T is the temperature; QE D 0.5jEj2 is the Joule thermal emission; QR is the power of volumetric radiative losses; E D .Er , E' D 0, Ex /, H D .Hr D 0, H' , Hx D 0/ are the intensities of electrical and magnetic field; "k is the complex inductivity. Features of the solution of the gas-dynamic and thermal parts of the problem (the equations (1.85)–(1.90)) will be discussed in Sections 1.1.5 and 1.1.6. A feature of the solution of the electrodynamic part of the problem (equations (1.92)–(1.94)) is that components of electrical and magnetic fields are complex functions. Therefore, these equations should be rewritten in the form of a system of equations concerning the real and imaginary parts, but even so it is still necessary to take into account that the boundary conditions are set for incident radiation energy only on an exterior radial boundary of the channel.

1.1.4 Models of laser supported plasma generators (LSPG) In many respects the schematic diagram of the laser plasma generator considered in the present part (Figure 1.5) is analogous to the classical diagrams of arc and highfrequency plasma generators. The plasma, which is the source of the plasma jet used for various technological and research purposes, is created in a cylindrical channel through which a gas is pumped. In the case of the laser plasma generator, the plasma is formed by absorption of the radiation of a continuous laser, most frequently a CO2 laser with the radiation wavelength  D 10.6 micrometers. The plasma thus obtained is called a continuous (steady-state) optical discharge or simply a steady-state laser plasma and the process of its propagation through a stationary or moving gas is called a laser combustion wave or laser supported wave [83]. The laser supported plasma generator has unique properties which make it possible to regard it as a promising source for solving important practical problems of radiation

26

Chapter 1 Models of gas-discharge physical mechanics r 3

1

2R c

5 0 4

xp

6 7 L x

Figure 1.5. Schematic diagram of a laser plasma generator: cylindrical chamber of the generator (1), focusing lens (2), unfocused laser beam (3), undisturbed gas flow in the inlet cross section of the chamber (4), steady-state laser plasma (5), gas and generated plasma jet mixing layer (6), and plasma jet (7).

gas dynamics: at atmospheric pressure the temperature reaches .1.5–2/  104 K in the laser radiation absorption zone, and the high temperature plasma zone has no contact with the surfaces. This ensures a high-purity plasma flow. In several experimental and theoretical investigations [24, 36, 56, 59, 60, 101, 104, 107, 108, 111] the conditions for steady-state existence of the laser plasma for a stationary or moving gas and focused or unfocused laser beams were considered. In [57, 70, 103] continuous or pulse-periodic laser-plasma generators and laser ramjets were analyzed. In continuous laser ramjets the gas is heated inside the combustion chamber by means of continuous laser radiation. The heated gas flows out through a nozzle as in an ordinary jet engine. The operating principle of the pulse-periodic engine is different. In this case, both pulse-periodic lasers and specially profiled channels are used. They ensure that the optical gas breakdown shock wave pulse is transferred with maximum efficiency to the vehicle which, as it were, is pushed along by the shock waves. Clearly, the power of the laser radiation used and its wavelength are the most important parameters determining the regimes of existence of the laser plasma and the working characteristics of the generator. Accurate focusing of the several kilowatt CO2 -laser radiation at a wavelength of 10.6 microns makes it possible to generate laser plasma with a characteristic space scale of the order of 0.01–0.03cm. At such small dimensions, the laser plasma exists in the so-called heat-conduction regime. This means that the laser radiation energy absorbed in the plasma is mainly transferred to the surrounding gas by heat conduction. The plasma radiation zone is so small that the thermal radiation power is insufficient to heat the surrounding medium up to the single-ionization temperatures at which intensive absorption of the laser radiation begins. Transition from the heat-conduction to the radiation regime of existence of the laser plasma is possible only if the laser radiation power increases significantly. In order for the radiation regime of existence of the laser plasma (or its propagation along an unfocused or weakly focused laser beam) to be realized in atmospheric air it

Section 1.1 Models of homogeneous chemically equilibrium plasma

27

is necessary for the plasma dimensions to be of the order of 1 cm. It is precisely these dimensions of the laser plasma that were observed in 1974 in experiments [60] on plasma propagation along an unfocused CO2 -laser beam and investigated numerically with reference to problems of laser physics [107]. In the radiation regime of existence of the laser plasma the energy of the absorbed laser radiation is transferred to the surrounding gas by thermal radiation whose power is sufficient to heat the adjacent gas layers to the single-ionization temperature. Hence, there follows the vital importance of correctly taking into account the effect of the thermal radiation transfer processes on the formation of the laser plasma and the plasma jet. Clearly, in this case the problem under consideration should be assigned to the class of radiation-gasdynamic problems with strong radiation-gasdynamic interaction. Another basic feature of the radiation regime considered is the selectivity of the thermal radiation. This significantly complicates the theoretical analysis of the phenomena and the numerical realization of the radiation-gasdynamic models since the spectral absorption coefficient varies by orders of magnitude on different spectral ranges and at different points of the laser plasma and the gas flow. In investigating the characteristics of the generator, one significant problem should also be considered. The problem is the radiation heating of the walls of the gas-dynamic channel in which the laser plasma is ignited and from which the plasma jet flows out. Solving this problem is important for two reasons. Firstly, it enables to determine the energy losses, integral over the spectrum, to the generator channel walls. Secondly, there is a risk of thermal damage to the channel walls by the thermal radiation at such high plasma temperatures. This is directly associated with the problem of generator design. Self-oscillations of the laser plasma and bifurcations of the gas-plasma structures accompanying the processes of gas flow past a laser plasma can develop in both the heatconduction and radiation regimes (see Chapter 2). Therefore, some unusual regimes of laser plasma generators and atmospheric-pressure laser combustion waves will be considered below for transition from the heat-conduction to the radiation regime of existence of the laser plasma, as well as to study the effect of the initial free-stream gas flow turbulence and reabsorption of the thermal radiation of the laser plasma on flow stabilization. The radiation-gasdynamic model should be developed and used in the case of LSPG in order to upgrade the calculation of the radiative thermal fluxes on the internal surface of the generator. Summing up this brief analysis of the problem of creating a radiation-gasdynamic model of the laser plasma generator, note that this model (or its elements) can also be used for solving problems of strong radiation-gasdynamic interaction in the shock layer near the surface of space vehicles re-entering the denser layers of the atmosphere at super-orbital velocities. In fact, the unique properties of the laser plasma, namely, its temperature T  .1.5–2/104 K at atmospheric pressure, make it possible to simulate the conditions in the shock layer in the neighborhood of space vehicles re-entering

28

Chapter 1 Models of gas-discharge physical mechanics

the Earth’s atmosphere at velocities higher than 16 km/s. However, this model must be supplemented by taking the physical-chemical processes important for hypersonic flow problems into account. In the current section the problem is formulated in two-dimensional cylindrical geometry corresponding to the conditions of symmetry of the process considered. As the gas medium, let us take air at atmospheric pressure. A continuous CO2 -laser beam with a radiation wavelength of 10.6 micrometers falls on the plasma from the left (Figure 1.5). For describing the thermodynamic state of the low temperature plasma, the local thermodynamic equilibrium approximation is used. As before, it is assumed that particle collision processes predominate in the atmospheric pressure plasma considered and that its thermodynamic state can be completely characterized by the pressure and the temperature. Of course, this is not actually entirely true at such high temperatures and densities of the laser and thermal self-radiation and there are zones in the neighborhood of the laser combustion wave in which the local thermodynamic equilibrium approximation is certainly not satisfied. For example, the powerful ultraviolet radiation from the high temperature zone leads to photo-dissociation and photo-ionization of the relatively cold gas layers, as this takes place ahead of the front of a strongly radiating shock wave (action of “precursor” radiation). Nevertheless, preliminary investigations showed that the nonequilibrium processes have no appreciable effect on the laser combustion wave dynamics or on the parameters of the steady-state laser plasma. In order to solve the problem of the laser plasma dynamics, the following system of continuity, Navier–Stokes, and energy conservation equations and selective thermal and laser radiation transport equations will be used as before: @ C div.V/ D 0 @t

(1.95)

@p @ u C div. uV/ D  C Su @t @x @v @p C div.vV/ D  C Sv @t @r @T C cp VgradT D div.†gradT /  QR C QL @t   Z x 1 rn ! .x 0 , r D 0/dx 0 QL D ! .x, r D 0/PL exp. n / exp  2 RL RL 0 cp

QR D c

Nk X

k .Ub,k  Uk /!k

(1.96) (1.97) (1.98) (1.99)

(1.100)

kD1

  1 div gradUk D k .Ub,k  Uk /, 3k

k D 1, 2, ..., N k

(1.101)

Section 1.1 Models of homogeneous chemically equilibrium plasma

     @u @v @ @u 2 @ 1 @ .† divV/ C r† C C2 † Su D  3 @x r @r @r @x @x @x    @u @v 2 @ @ Sv D  .† d ivV/ C † C 3 @r @x @r @x     @v @ v @ † C 2† . C2 @r @r @r r

29

(1.102)

(1.103)

Here, x and r are the axial and radial coordinates; p, cp , and T are the density, specific heat at constant pressure, and temperature, respectively; u and v are the axial and radial components of the velocity V; p is the pressure, † and † are the effective dynamic viscosity and thermal conductivity coefficients; QR and QL are the volume energy release powers associated with the transport of selective thermal and laser radiation, respectively; , U , and Ub are the volume thermal absorption coefficient and the volume radiation densities for the medium and an absolutely black body; ! is the laser radiation absorption coefficient; PL is the laser power; RL is the radial boundary of the laser beam, and n is the index of the intensity distribution over the laser beam cross section. The subscripts ! and k denote the spectral and group characteristics determined by averaging the corresponding spectral characteristics over each of Nk spectral wave number ranges !k . In (1.98), (1.102), and (1.103) the turbulent viscosity and thermal conductivity coefficients can be calculated using the Boussinesq hypothesis [15, 132], according to which the effective gas flow viscosity can be determined from the formula † D m C t ,

(1.104)

where m is the dynamic viscosity coefficient, which takes into account the molecular collision processes, t is the turbulent viscosity coefficient, which is determined by a turbulent mixing model. In the case under consideration, the k  " model [15, 132] is used. Let us also assume that the turbulent Prandtl number, Prt D t cp =t , is equal to unity. This makes it possible to find the corresponding thermal conductivity coefficient t D cp t . The equations of the k  " model have the form:      @k 1 @ t @k @ t @k C r kv  C ku  D P  ", @t r @r k @r @x k @x

(1.105)

     t @" @ t @" @" 1 @ " C r "v  C "u  D .C1P  C2 "/ , (1.106) @t r @r " @r @x " @x k t D C

k2 , "

(1.107)

30

Chapter 1 Models of gas-discharge physical mechanics

 2  2     @v @v @u v 2 @u 2 C P D t 2 C C C @r @x r @x @r     @u 1 @r v t @ @p @ @p 2 C  2 C  k 3 @x r @r  @r @r @x @x C D 0.09;

(1.108)

C1 D 1.44; C2 D 1.92; k D 1.0; " D 1.3.

In the immediate neighborhood of the surface, different modifications of the turbulent mixing model can be used [15]. The propagation of the laser radiation is described in the geometric optics approximation. The continuum absorption mechanism, which is the inverse of the electron bremsstrahlung mechanism under a local thermodynamic equilibrium conditions, is assumed to be determining for the laser radiation absorption coefficient in equation (1.99): ! D 2.82  1029ne .nC C 4nCC/T 3=2 lg.2.17  103T ne1=3 /,

1=cm

(1.109)

where ne , nC , and nCC are the number densities of electrons and singly and doubly charged ions, and T is the temperature in K. The power of the energy release associated with the thermal radiation transport process can be found by integrating the selective thermal radiation transport equation in the form of the multigroup Pl -approximation of the spherical harmonics method, i. e., as a result of solving the system of Nk equations (1.101), each of which is assigned its own k .T, p/ function. The following boundary conditions can be used for the solution of the problem: for the cooled surface (at a temperature T0 ), no-slip conditions on the surface, axial symmetry of the functions T , u, v, and Uk and absence of flow perturbations at the channel inlet: u D u0 , v D 0, and T D T0. In the exit cross section of the cylindrical channel there is the possibility to use boundary condition fx .x ! L/ D 0 (a) or fxx .x ! L/ D 0 (b), where f D fT , u, v, Uk g. The distance from the exit boundary of the computation domain should be taken so as to reduce as for as possible its effect on the laser plasma gas dynamics. The Neumann boundary condition (a) turned out to be reasonable in computation variants with a high temperature laminar plasma jet. As the initial conditions, a spherical plasma cloud of 0.5 cm radius at a temperature of 20 000 K against the background of an unperturbed gas flow can be used. The thermophysical properties of equilibrium air should be calculated over the entire temperature range from normal conditions to double ionization using, for example, the approximations in [41]. These approximations were obtained over a wide pressure and temperature range using compact formulas, which are based on data on the transport properties in the third approximation of the Chapman–Enskog method [48]. To solve the radiation heat transfer equation, the multigroup spectral optical model can be recommended. The spectral optical model of hot air (thermal radiation absorption coefficient in (1.100), (1.101)) was calculated using ASTEROID software [106].

Section 1.1 Models of homogeneous chemically equilibrium plasma

31

Initially, it was found that the absorption coefficients of high temperature air over the temperature range from 300 K to 20 000 K at a million points along the spectrum. In this case, all the most significant elementary radiation processes which contribute to the total absorption coefficient of air under local thermodynamic equilibrium conditions were taken into account. A nonuniform computation grid in the wavelength was used. This made it possible to describe the contours of about 4 000 atomic lines [109] of nitrogen and oxygen atoms and ions (with resolution of each multiplet structure) at not less than 10 spectral points. This method is usually called the line-by-line calculation. By averaging the results of the line-by-line calculations, it is possible to create any group spectral models. Three group spectral models with 37, 74, and 148 spectral groups are demonstrated here. Figure 1.6 reproduces the line-by-line and group absorption coefficients at temperatures of 4 000, 10 000, and 16 000 K. These temperature points give a representation of the features of the spectral absorption coefficients of hot air for dissociation temperatures (4 000 K), almost complete breakdown of the molecular components (10 000 K), and developed ionization (16 000 K). Let us note the main structural features of the hot air absorption spectrum over the mentioned temperature range investigated. Strong radiation absorption increasing with a decrease in the temperature is observed over the ultraviolet spectrum range, in particular, in its ‘vacuum’ part (! > 80 000 cm1 /. In this range the absorption coefficient is attributable to both atomic photo-ionization from the ground and low-excitation energy states and molecular photo-dissociation and photo-ionization. In this part of the spectrum, the atomic and ionic spectral lines formed in quantum transitions from the ground and low-excitation energy levels radiate and absorb at high temperatures. As the temperature increases, the effect of the atomic lines in the integral transported radiant energy balance also increases. At the centers of the atomic lines the absorption coefficients are appreciably greater than 1 cm1 so that under the conditions considered their radiation is absorbed in the plasma itself without leaving it. From Figure 1.6 (a) one can see that the radiation with the wave number ! > 90 000 cm1 is almost completely absorbed by the plasma layers at a temperature T D 4 000 K since the coefficient ! > 10 cm1 and, consequently, the characteristic free path of the thermal photons lr < 0.1 cm. Colder gas layers absorb the ultraviolet radiation even more strongly. The molecular plasma components, which dissociate as the temperature increases (as a result, the plasma become more “transparent”), absorb appreciable amounts of radiation over the near ultraviolet spectrum range (! D 25 000–60 000 cm1 /. Over the visible (! D 13 160–25 000 cm1 / and near infrared spectrum ranges, the molecular components absorb radiation mainly in electron-vibrational quantum transitions and also in atomic lines formed in quantum transitions from excited energy states. On this spectral range, the continuous absorption spectrum is associated with the interaction between free electrons and plasma ions. Therefore, at low temperatures, i. e., at low degrees of ionization, the heated gas is almost transparent for radiation. The chosen temperature points are important for investigating the laser plasma dynamics since in a laser plasma located in a gas flow the temperature distribution has the

32

Chapter 1 Models of gas-discharge physical mechanics

κ, cm−1 10 000 (a) 1 000 100 10 1 0.1 0.01 0.001 0.0001 1E-005 1E-006 1E-007 1E-008 0 40 000

80 000

−1

10 000 κ, cm (b) 1 000

120 000 ω, cm−1

κ, cm−1 100 10 1 0.1 0.01 0.001 0.0001 1E-005 1E-006 1E-007 1E-008 0 40 000

80 000

120 000 ω, cm−1

80 000

120 000 ω, cm−1

80 000

120 000 ω, cm−1

−1

100 κ, cm 10

100

1

10 1

0.1

0.1

0.01

0.01

0.001

0.001 0.0001

0.0001 1E-005 0

40 000

80 000

−1 1 000 κ, cm (c) 100

120 000 ω, cm−1

1E-005 0

40 000

−1 10 κ, cm

1

10 0.1

1 0.1

0.01

0.01 0.001

0.001 0.0001 0

40 000

80 000

120 000 −1 ω, cm

0.0001 0

40 000

Figure 1.6. Line-by-line (left) and group (right) absorption coefficients of high temperature air for T D 4 000 K (a), 10 000 K (b), T D 16 000 K (c).

Section 1.1 Models of homogeneous chemically equilibrium plasma

33

following characteristic: the temperature reaches maxima of .1.5–2.0/  104 K in the zone of strong laser radiation absorption and then sharply decreases to 104 K, while the resulting plasma jet has a temperature of the order of 5  103 K. Note that if the zone with the maximum temperature is localized in space by a focused laser beam, then the relatively low temperature zones occupy a significant volume and are mainly determined by the gas dynamics of the generator. The need to analyze various group models is due to the following. Ideally, in order to solve radiation gas-dynamic problems one should use spectral line-by-line models. The tendency to use the most detailed spectral models is fully expressed in world practice for solving various radiation gas-dynamic problems as computer power increases and the parallel algorithms are developed. However, in the overwhelming majority of cases the use of these models is not justified, not to mention their extreme computational inefficiency. Therefore, the problem of choosing an adequate multigroup optical model, in particular for analyzing processes admitting the use of the local thermodynamic equilibrium model, remains fairly topical for the majority of radiation gas-dynamic problems. The generator calculations carried out in the present part using various group models demonstrate the possibility of using optical models with approximately 37 groups, within the framework of which it is possible to describe the most important structural features of the absorption spectrum, including the groups of strongest atomic lines. Nevertheless, note that the multigroup models presented here still cannot solve the problems of adequately calculating the radiation transport in atomic and ion lines.

1.1.5 Numerical simulation models of steady-state radiative gas dynamics of RF-, EA-, MW-, and LSW-plasma generators The present section considers computing two-dimensional models of radiative gasdynamic processes in different kinds of plasmatrons. Governing equations of gasdynamic processes in plasma generators were formulated above in two forms. The first one was the formulation of equations (1.36)–(1.42) in the explicitly stationary form. The second one was the formulation of gas-dynamic equations in nonstationary form (see equations (1.70)–(1.72) and (1.95)–(1.98)). Here we will consider numerical simulation approaches for the solution of the equations for steady-state regimes. If the equations do not contain time derivatives, they will be modified by adding artificial time derivatives of corresponding functions. These artificial time derivatives will be used, as in the case of explicit nonstationary formulation, as auxiliary summands for time-marching numerical procedure. The gas-dynamic models of plasma generators considered above are based on the process of the subsonic gas motion through an area of intensive absorption of electromagnetic energy where abrupt changes (by a factor of ten or a hundred!) of a flow rate, temperature, thermodynamic, thermophysic and optical properties take place.

34

Chapter 1 Models of gas-discharge physical mechanics

The energy conservation equation in all models contains a rate of energy deposition caused by electromagnetic radiation of different wavelength. This equation is classified as the elliptic equation in stationary form or as the parabolic equation in nonstationary form. The basic difficulty of its solution is connected with the creation of optimum iterative process between the functions defining a plasma heating degree (temperature, enthalpy, internal energy), and non-linear rates of energy exchange. For example, we should recall that radiant losses are proportional approximately to T 4 , and laser radiation absorption is defined by an exponential function of a temperature. The significant methodological complexity of the problem is represented with a method of the equation of gas-dynamic integration. In the practice of computational fluid dynamics (CFD), two basic approaches for the numerical simulation of subsonic gas flows with any arbitrary changes of density (caused by changing temperature at practically constant pressure in subsonic flows) are well known. These are: the approach based on the natural form of the flow equations, i. e., with the use of velocity components and pressure as unknown functions, and the approach based on the socalled “dynamic variables”. The first of these methods allows to predict the unsteady and stationary gas flows, however it demands special arrangements for the accurate definition of small pressure field disturbances. The use of the second method allows to manage without the definition of a pressure field, but corresponds to stationary flows only. Both these approaches are considered in this chapter. Computing model of the natural form of the flow equations Governing equations of gas-dynamic processes in the channel of a plasma generator in a two-dimensional axisymmetric geometry are formulated in the model as follows: @ C div .V/ D 0, @t @p Su @u C div .uV/ D  C , @t @x Re @p Sv @v C div .vV/ D  C , @t @r Re   1  Q  L2 @H C div .VH / D div grad H C , @t Pr  Re cp  0 T0 Q  D W  QR , Z

!

QR D

(1.110) (1.111) (1.112) (1.113) (1.114)

c .Ub  U / d

0

D

Nk X 1

ck .Ubk  Uk / k , c D 3  1010 cm/s,

(1.115)

Section 1.1 Models of homogeneous chemically equilibrium plasma

  1 grad Uk D k .Uk  Ub,k /, div 3k where:

  2 @ Su D   div V C 3 @x   2 @  div V C Sv D  3 @r

k D 1, 2, : : : , Nk ,

35

(1.116)

     1 @ @u @v @ @u r C C2  , r @r @r @x @x @x      @ @u @v @ @v @v 2v  C C2  C 2  2 , @x @r @x @r @r r @r r

where x, r are the axial and radial variables; , cp , T are the density, specific heat capacity at constant pressure, and temperature; u, v are the axial and radial velocity components V; p is the pressure; ,  are the coefficients of dynamic viscosity and thermal conductivity; QR is the volume power flux related to thermal radiation transfer; , U , Ub are the volume absorption coefficient, volume radiation density of a medium and of a black body; !, k are the indexes of spectral and group characteristics; W is the energy deposition due to absorption of external electromagnetic energy in plasma. The group absorption coefficient k is averaged in each of Nk spectral ranges !k covering the spectral region 2 000–250 000cm1 . All variables in the equations (1.110)–(1.113), namely velocity V D fu, vg, density , pressure p, enthalpy H , viscosity  and thermal conduction t , are normalized by corresponding values in an undisturbed input gas stream with temperature T0 , density 0 and velocity V0. The similarity parameters Pr D 0cp0 =0, Re D 0 V0L=0 are constructed with thermodynamic and thermophysic properties .cp,0 , 0 , 0 / for the same conditions. Inclusion of the typical length scale L also allows to define the time scale t0 D L=V0. Attempts to get a self-consistent solution of equations (1.110)–(1.116) encounter the complexity proper for all systems of stiff equations, where the strong distinction of the characteristic time scales of running processes is observed. Let us make explaining estimates. The characteristic processes for the first two equations are the convective transfer with the velocity V0 and propagation of sound perturbations with the acoustic speed a0 . For a characteristic spatial scale of L D 1 cm we shall estimate a computational mesh spacing with the value h D 0.01 cm. Therefore, supposing V0 D 1 000 cm/s, T0 D 500 K, 0 D 0.00124 g/cm3, p0 D 1 atm D 106 g=.cm  s2 /, one can estimate the characteristic times of convective and sound processes as tc D L=V0  103 s, ts D L=a0  3  106 s (a0 D 3.36  104 cm/s, D cp =cV D 1.4). Typical time of thermo-conductive wave propagation can be estimated under the formula: L2 L2 D , tT D  =cp  where, however, values of functions should be taken at the representative temperature of discharge T D 15 000 K,  D 7.8  106 g/sm3, cp D 21.4 J/.g  K/,  D 3.62  102W=.cm  K/, whence tt Š 4.5  105 s. However, at high temperatures it is

36

Chapter 1 Models of gas-discharge physical mechanics

necessary to consider heat transfer by a radiative heat conductivity, then R Š lR c=3. For ultraviolet photons, a free length is lR Š 102 cm and R Š 108 cm2=s. Therefore, the characteristic time of the radiative heat conductivity is about tR Š 1010 s, that is much less than other characteristic times. Thus, the limiting process for the phenomena under consideration is the radiative heat exchange. A two-stage numerical simulation procedure is used for the solution of equations (1.110)–(1.116). In the first step, the so-called “energy stage”, iterative procedure for self-consistent solution of the energy conservation equation (1.113) and radiation transfer equation (1.116) is used. The implicit finite-difference scheme is preferable here. It is easy to determine by means of numerical experiments that a maximum computational time step for execution of the first stage is significantly less than that necessary for the satisfaction of a stability condition of explicit schemes with reference to gas-dynamic equations. Therefore, the Navier–Stokes equations (1.111), (1.112) together with the continuity equation (1.110) can be solved with use of explicit schemes. The solution of these equations makes the second stage (the so-called “gas-dynamic stage”). Let us assume that at the point in time t pC1 the enthalpy field H pC1 , as well as pC1 pC1 other thermodynamic and thermophysic functions . pC1 , pC1 , ! , t / have pC1 pC1 D fu, vg , p pC1 with the use been calculated. It is necessary to calculate G p of these functions at points in time t , and thermodynamic and thermophysic functions at t pC1 . The splitting method [20] can be used for these purposes. In accordance with this method, the calculation procedure is represented in the form of three sequential steps. The first calculation step. It is supposed in equations (1.111) and (1.112) that @p=@x D @p=@r D 0, then @Gu Su C div Gu V D , @t Re

(1.117)

@Gv Sv C div Gv V D , @t Re

(1.118)

where Gu D u, Gv D v. The required finite-difference scheme is created with the use of the finite-volume method which is applied for elementary grid volume shown in Figure 1.7 (a). The integral operator Z O,ij f g D

Z

pC1

dt p

Z

j C1 j 1

dx

i C1=2

i 1=2

r f gdr,

(1.119)

37

Section 1.1 Models of homogeneous chemically equilibrium plasma υi,j+1/2

(a) r

j

(b) r

υi,j j

pij

ui+1/2,j i

ui,j

pij i

x

x

Figure 1.7. Finite-difference grids.

when applied to the equations (1.117) and (1.118) results in the following relations for preliminary values of mass fluxes:  br Gu,i ,j C1 C .brC  bl /Gu,i ,j  blCGu,i ,j 1 p GQ u,i ,j D Gu,i C

 ,j hj   C ar .r Gu/i C1,j C .ar  al /.r Gu/i ,j  alC .r Gu/i 1,j  i Ri O,i ,j fSug , (1.120) C Re hj i Ri   br Gv,i ,j C1 C .brC  bl /Gv,i ,j  blCGv,i ,j 1 p GQ v,i ,j D Gv,i ,j C   hj

 ar .r Gv /i C1,j C .arC  al /.r Gv /i ,j  alC .r Gv /i 1,j i Ri O,i ,j fSv g C , (1.121) Re hj i Ri



where hj D 0.5.xj C1  xj 1 /, i D 0.5.ri C1  ri 1/; Ri D 0.25.ri 1 C 2ri C ri C1/, br˙ D 0.5.ur ˙ jur j/,

bl˙ D 0.5.ul ˙ jul j/;

ar˙ D 0.5.vr ˙ jvr j/,

al˙ D 0.5.vl ˙ jvl j/;

ur D 0.5.ui ,j C ui ,j C1 /,

ul D 0.5.ui ,j C ui ,j 1 /;

vr D 0.5.vi ,j C vi C1,j /,

vl D 0.5.vi ,j C vi 1,j /.

38

Chapter 1 Models of gas-discharge physical mechanics

The finite-difference approximations of dissipative summands in the Navier–Stokes equations are expressed as follows: O,i ,j fSug D   ui C1,j  ui ,j 0.25.vi ,j C1  vi ,j 1 C vi C1,j C1  vi C1,j 1/

hj i ,j ri C1=2 C ri C1  ri hj   ui ,j  ui 1,j 0.25.vi ,j C1  vi ,j 1 C vi 1,j C1  vi 1,j 1/ C  ri 1=2 ri  ri 1 hj    ui ,j C1  ui ,j ui ,j  ui ,j 1  C Ri i i ,j 2 xj C1  xj xj  xj 1  1 1 C . div V/i ,j 1  . div V/i ,j C1 , (1.122) 3 3 O,i ,j fSv g D 

i Ri i ,j

vi ,j C1  vi ,j vi ,j  vi ,j 1  xj C1  xj xj  xj 1

0.25.ui C1,j C1 C ui 1,j 1  ui 1,j C1  ui C1,j 1 / i



  ri C1=2 vi C1,j  vi ,j ri 1=2 vi ,j  vi 1,j  C 2 hj i ,j ri C1  ri ri  ri 1   ri C1=2 2 hj i ,j vi ,j ln ri 1=2



C

˚  23 hj i ,j 0.5ri C1=2Œ. div V/i C1,j C . div V/i ,j 

  0.5ri 1=2 Œ. div V/i ,j C . div V/i 1,j   . div V/i ,j i . (1.123)

In deriving approximations (1.122) and (1.123) it was supposed that i ˙1=2,j ˙1=2 D i ,j . The second calculation step. Intermediate values of the mass fluxes GQ u and GQ v , obtained at the first calculation step, are used here to find the pressure disturbances. Following to the basic ideas of the splitting method, the initial system of equations of the second step is formulated in the following form: @Gu=@t D @p=@x,

(1.124)

@Gv =@t D @p=@r.

(1.125)

39

Section 1.1 Models of homogeneous chemically equilibrium plasma

Let us act on equation (1.124) with the operator @f g=@x, and also on equation (1.125) with the operator r 1@ fr g=@r, then: @. div G/ D  div . grad p/. @t

(1.126)

Following the splitting scheme and transforming to finite time discretization, one can obtain the Poisson equation relative to pressure div .gradp/ D 

Q div GpC1  div G .

(1.127)

This equation gives birth to two groups of possible calculation models. The first group includes stationary models which correspond to identical conditions at each time step, div GpC1 D 0. The second group includes the nonstationary models, div GpC1 ¤ 0. In the latter case for the determination of the function div GpC1 it is necessary to include an equation of continuity. Let us consider the equation for pressure in the general form (1.127), but take into account specific properties of slow subsonic motion of heated gas in plasma generators. For these purposes, the thermal equation of state p 0 D  0 R0 T 0 =M 0 (all values are dimensional; R0 is the universal gas constant) can be presented in the following form: 0 D

.p00 C pg0 /M 0 R0 T 0

D

.1 C pg0 =p00 /M 0 p00 R0 T 0

D

.1 C pg0 =p00 /M 0 T00 00 M00 T 0

,

(1.128)

where p00 is the averaged pressure, pg0 is the local perturbation of pressure p00 . Considering a02 D R0 T00 =M00 , where D cp,0 =cv,0 , cp,0 , cv,0 are the specific heat capacities at constant pressure and volume at temperature T0 , one can get the dimensionless thermal equation of state: 2  D .1 C M1 p/M =T ,

2 M1 D p0=0,

(1.129)

2 and at requirements M1  1 considered here, it assumes the even simpler form   M=T . In our case, at practically constant pressure, the density can be considered a function of an enthalpy  D .H / only, therefore:

d D

@  @ dH D cp1 dH D  dH . @H @T cp T

(1.130)

Now, combining the continuity equation d=dt D  div V and the energy conservation equation dH =dt D Q† , where Q† is the right-hand side of equation (1.113), one can rewrite equation (1.130) in the following form:  div V D

Q† cp T

or

div .V/ D V grad  C

Q† . cp T

(1.131)

40

Chapter 1 Models of gas-discharge physical mechanics

Taking into account that

@H  @H  C v @r u , V grad  D  @x cp T

the analog of the continuity equation becomes: Q†  G grad H , G D fu, vg. cp T

div .V/ D

(1.132)

Now, the five-point finite-difference scheme for the calculation of pressure by the iterative method can be presented in the form: pC1 pC1 pC1 pC1 N N Ai ,j pipC1 1,j CBi ,j pi C1,j C Ai ,j pi ,j 1 C Bi ,j pi ,j C1 Ci ,j pi ,j CFi ,j D 0, (1.133)

with the following coefficients: ri 1=2 Ai ,j D , i Ri .ri  ri 1/ ANi ,j D Fi ,j D

1 , hj h j

BN i ,j D

1 hj hC j

,

Bi ,j D

ri C1=2 , i Ri .ri C1  ri /

Ci ,j D Ai ,j C Bi ,j C ANi ,j C BN i ,j ,

  1 GQ u,i ,j C1  GQ u,i ,j 1 . div GpC1 /i ,j ri C1GQ v,i C1,j  ri 1GQ v,i 1,j  C .

2

hj i Ri

Note that in the finite-difference approximation of equation (1.127) we used the grid shown in Figure 1.7 (b). It is assumed that the numerical errors generated by the two kinds of grids will be neutralized in the third calculation step. Boundary conditions for pressure were set in the form of @=@x D @p=@r D 0. For the solution of the difference equation (1.133), any known methods [1, 51] can be recommended, namely the successive over relaxation method (SOR), alternating direction implicit (ADI) method, etc. Some of them will be considered below. The third calculation step. The final correction of the components of mass flux is performed here. Calculation relations which are in use at this stage follow from (1.124) and (1.125): @p @p and GvpC1 D GQ v  , GupC1 D GQ u 

@x @r with the following approximation:  pC1 pC1 pC1  pipC1

pi ,j C1  pi ,j ,j  pi ,j 1 pC1 Q Gu,i ,j D Gu,i ,j  , C 2 xj C1  xj xj  xj 1 Gv,i ,j D GQ v,i ,j  pC1

 pC1 pC1 pC1 pC1  pi ,j  pi 1,j

pi C1,j  pi ,j . C 2 ri C1  ri ri  ri 1

(1.134)

Section 1.1 Models of homogeneous chemically equilibrium plasma

41

Note that the use of mass flux vector components as required functions appears more preferable than using density and velocity components separately for the problems of radiative gas dynamics of plasma generators considered. Computing model in dynamic variables Flow equations in the dynamic variables can be obtained by the Curl-operator acting on the original Navier–Stokes equations in the vector form. In the two-dimensional axysimmetric formulation this mathematical procedure is simplified significantly. Let the operator @f g=@r act on the x-projection of the impulse conservation equation (1.111), and let the operator @f g=@x act on the r -projection of the impulse conservation equation (1.112). The system of governing equations in the dynamic variables has the following form: div

1 grad r 2

! D , r

@ D ru, @r

@ D rv, @x

 r! @  ! C div D @t r   r  !  2 @  @  @u 1 @  @u 1 C    div 2 grad r2 r r r @x @x @r @r @x    @ @v @  @v 2v @ 2 @     3 C 2 r r @r @x @r @r @x r @x   1 @ @  u2 C v 2 @ @  u2 C v 2  C , r @r @x 2 @x @r 2 !D



@v @u  , @x @r

  @H t C div VH D div grad H C W  QR , @t cp

Z

1

QR D

c .Ub  U / d D

0

Nk X

c .Ubk  Uk / dk ,

(1.135)

(1.136)

(1.137)

(1.138)

(1.139)

kD1

div

1 grad Uk D k .Uk  Ubk / , 3k

(1.140)

where u, v are the projections of velocity V D fu, vg on axial x and radial r coordinates; , cp , , t are the density, heat capacity at constant pressure, viscosity and RT thermal conduction; H D T1 cp d T is the enthalpy; T , T1 are the temperature in the

42

Chapter 1 Models of gas-discharge physical mechanics

discharge and in the undisturbed input stream; , U , Ub are the volumetric absorption coefficient, volumetric density of heat radiation of heated gas and of the black body; W is the energy deposition due to absorption of external electromagnetic energy in plasma. Equation (1.139) represents the diffusion approach to the solution of the radiation heat transfer equation. The gas stream and gas-discharge plasma, as before, are in the local thermodynamic equilibrium. A small pressure change in the gas stream allows to use the functional relationships of thermodynamic, thermophysic and optical properties in the form of f D f .T /, where f D f, , t , H , k g. Time derivatives in the equations (1.136) and (1.138) are used for the solution of the equations by a time-marching method and in the case under consideration do not have the physical meaning. Actually, all numerical methods suppose the following generalized formulation of boundary conditions: a C bf C c

@f D 0, @n

(1.141)

where n is the normal coordinate to the given boundary; f is the decision function. The concrete definition of boundary conditions (definition of a, b, c coefficients) and the assignment of initial conditions are carried out at the solution of specific problems. The problem can be solved with non-uniform regular meshes in the lines of x and r . Finite-difference implicit scheme can be applied not only for the energy conservation equation (as before), but also for gas flow equations (1.135) and (1.136). Numerical analogs of all solved equations (1.135), (1.136), (1.138), (1.140) are reduced to canonical form of the five-point finite-difference equation: Ai ,j ˆi 1,j CBi ,j ˆi C1,j C ANi ,j ˆi ,j 1 C BN i ,j ˆi ,j C1 Ci ,j ˆi ,j CFi ,j D 0, (1.142) where indexes i D 1, 2, : : : , NI set the nodes of the computational mesh in a radial direction, and indexes j D 1, 2, : : : , NJ are used for the axial direction; ˆ D f , !=r , H , Uk g. Formulas for the calculation of coefficients in (1.142) for each of the equations (in the dimensionless form for (1.135) and (1.136)) are given below. (1) The finite-difference equation for a stream function Ai ,j D ANi ,j D

ri 1=2 ai 1=2,j , p pp O 

ai ,j 1=2 , qq 

BN i ,j D

Bi ,j D

:

ri C1=2ai C1=2,j , p pp O C

ai ,j C1=2 , qq C

Fi ,j D

Ci ,j D Ai ,j C Bi ,j C ANi ,j C BN i ,j ,

!i ,j , pO

43

Section 1.1 Models of homogeneous chemically equilibrium plasma

where p D 0.5.ri C1  ri 1 /;

q D 0.5.xj C1  xj 1/;

q ˙ D ˙.xj ˙1  xj /;

p ˙ D ˙.ri ˙1  ri /; pO D 0.25.ri 1 C 2ri C ri C1/;      1 1 1 ai Cm=2,j Cn=2 D C 2 r 2 i ,j r 2 i Cm,j Cn or ai Cm=2,j Cn=2 D

1



2ri2Cm=2

 1 1 , C ij  i Cm,j Cn

m, n D 0, ˙1.

(2) The finite-difference equation for vortex function !=r : Ai ,j D

ANi ,j D

Ci ,j

Fi ,j D

C C ri21 i 1,j al,i ,j

Re ri 1=2 p  p pO

C C .i ,j 1 =Re q  / bl,i ,j

q

,

Bi ,j D

,

BN i ,j D

ar,i ,j C ri2C1i C1,j Re ri C1=2 p Cp pO

,

br,i ,j C .i ,j C1=Re q C / q

,

  arC,i ,j  al,i brC,i ,j  bl,i i ,j ,j ,j C C D

p pO q 



  1=q C C 1=q  i ,j 2 1= p C ri C1=2 C 1= p  ri 1=2 r C , C Re i p pO q

i ,j !is,j

ri

Ni C1,j  Ni 1,j 1 Mi ,j C1  Mi ,j 1 C C Re pO q p  1 1 2Ni ,j ln.ri C1=2=ri 1=2/ vi ,j .i ,j C1  i ,j 1/.ri C1=2  ri 1=2 / C C p pq   i C1,j  i 1,j Œ.ui ,j C1  ui ,j 1 /.ui ,j 1 C 2ui ,j C ui ,j C1/ C 8qp pO C .vi ,j C1  vi ,j 1 /.vi ,j 1 C 2vi ,j C vi ,j C1/   i ,j C1  i ,j 1 Œ.ui C1,j  ui 1,j /.ui C1,j C 2ui ,j C ui C1,j /  8p pq O C .vi C1,j  vi 1,j /.vi 1,j C 2vi ,j C vi C1,j /, 

44

Chapter 1 Models of gas-discharge physical mechanics

where Re D

u1  L ; 

ri ˙1=2 D

ri C ri ˙1 ; 2

D

tL , u1

and functions M D

@ @u @ @u  ; @x @r @r @x

(1.143)

@ @v @ @v  (1.144) @x @r @r @x are approximated by the central differences; s is the index of the previous time layer. Here density  and viscosity  at the characteristic temperature of the gas discharge (for example, for LSW-plasma generator, T D 18 500 K) together with u1 and L were used as the reference parameters. Functions a D v and b D u were approximated with the use of the simplest approach:  



ar D 0.5 ri ai ,j C ri C1ai C1,j , al D 0.5 ri ai ,j C ri 1ai 1,j ;  



br D 0.5 bi ,j C bi ,j C1 , bl D 0.5 bi ,j C bi ,j 1 ; ˇ ˇ ˇ ˇ



br˙,l D 0.5 br ,l ˙ ˇbr ,l ˇ . ar˙,l D 0.5 ar ,l ˙ ˇar ,l ˇ , N D

(3) The finite-difference equation for enthalpy H :   ri 1=2 aQ i 1=2,j 1 C , aO l,i C Ai ,j D ,j p pO p   ri C1=2aQ i C1=2,j 1 , Bi ,j D aO r,i ,j C p pO pC     aQ i ,j 1=2 aQ i ,j C1=2 1 C 1 C N N , Bi ,j D , Ai ,j D b br ,i ,j C C q l,i ,j q q qC   aO rC,i ,j  aO l,i brC,i ,j  bl,i aQ i ,j 1=2 aQ i ,j C1=2 i ,j ,j ,j C C C C 

p pO q qq qq C ri 1=2aQ i 1=2,j ri C1=2aQ i C1=2,j C C ,  p pp O p pp O C

Ci ,j D

Fi ,j D

i ,j His,j

where aQ D t =cp ; aO D rv; b D u; aQ i Cm=2,j Cn=2 D

C Wi ,j  QR,i ,j ,

aQ i ,j C aQ i Cm,j Cn , m, n D 0, ˙1. 2r 2

(4) The finite-difference equation for group radiative energy density Uk .

Section 1.1 Models of homogeneous chemically equilibrium plasma

45

Equation (1.116) corresponds to the P1-approximation (or to the so-called diffusion approximation) of the spherical harmonic method [75,80]. Let us rewrite this equation in a form acceptable for further derivation of a finite-difference scheme (spectral index k is omitted for simplicity, index p D 0 corresponds to plane geometry, and p D 1 corresponds to axisymmetric geometry):     1 @ r p @U @ 1 @U  p    C U D Ub , (1.145) r @r 3 @r @z 3 @x with the boundary conditions on the symmetry axis: @U D 0, @r and on the exterior boundaries: 2 1 @U   D U , 3  @n or with asymptotic boundary conditions of the form: U .r/js D Ub .r/js . Let the density of heat radiation energy U be determined in a center of elemental calculation volume and projections of the radiation heat flux vector .Wx , Wr / on its sides (see Figure 1.8). With the use of the finite-volume approach one can get the following canonical five-point scheme: Ai ,j Ui 1,j CBi ,j Ui C1,j C ANi ,j Ui ,j 1 C BNi ,j Ui ,j C1 Ci ,j Ui ,j CFi ,j D 0, (1.146)

Wi,x j+1

x i, j+1 Wi,r j

i+1, j+1 x r Ui, j Wi+1, j

i, j

i+1, j Wi,xj

r Figure 1.8. Radiative fluxes in a computational mesh.

46

Chapter 1 Models of gas-discharge physical mechanics p

Ai ,j D

p

ri 1=2

,

ANi ,j D

1 , 3i ,j 1=2 q.xj  xj 1/

BN i ,j D

3i 1=2,j p pN p .ri  ri 1/

Bi ,j D

ri C1=2 3i C1=2,j p pN p .ri C1  ri /

,

1 , 3i ,j C1=2q.xj C1  xj /

p

p

ri 1=2

ri C1=2

C 3i C1=2,j p pN p .ri C1  ri / 3i 1=2,j p pN p .ri  ri 1/ 1 1 C C , 3i ,j C1=2q.xj C1  xj / 3i ,j 1=2q.xj  xj 1/

Ci ,j D i ,j C

Fi ,j D i ,j Ub,i ,j , where: pD i ˙1=2,j D

ri C1  ri 1 , 2

i ˙1,j C i ,j , 2

pN D

ri 1 C 2ri C ri C1 , 4

i ,j ˙1=2 D

i ,j ˙1 C i ,j , 2

xj C1  xj 1 , 2   ri C ri ˙1 p p . ri ˙1=2 D 2

qD

Boundary conditions for computing models in dynamic variables Boundary conditions of the first kind [33] are stated for mesh functions i ,j , Hi ,j at x D 0 and r D Rc (Rc is the radial boundary of the calculation domaine). The function of the velocity vortex !i ,j =ri is defined from undisturbed flow requirements at x D 0 and r D Rc with the first order accuracy (for calculation cases in free space). At r D 0 for all functions conditions of axial symmetry are used. The formulation of the boundary conditions at the exit cross section is traditionally the most difficult problem for subsonic flows with localized heat release regions. Numerical experiments performed in series of papers [104,108,110,111,113–115] show that the following conditions are acceptable: @f D 0 or @x

@2 f D0 @x 2

if the exit cross section is located far from the region of heat release. More accurate boundary conditions can be formulated if special prohibitive conditions to admit any disturbances transferred from outside of the calculation domain region are taken into account.

47

Section 1.1 Models of homogeneous chemically equilibrium plasma

1.1.6 Method of numerical simulation of non-stationary radiative gas-dynamic processes in subsonic plasma flows. The method of unsteady dynamic variables This method will be described with the example of computing model of unsteady laser supported wave considered in Section 1.1.4. Let, as before, at some fixed stage of the p p p p p nonstationary process (t D t p / all desired functions (u , v ,  , T , Uk / and other p constitutive functions cpp , p , p , p ! , kk be known. Here we shall apply the explicit method of consecutive calculation of a system of equations for gas-dynamic and energy stages. In cases of the subsonic flow, the plasma density is defined by its temperature, therefore it is possible to suppose that  pC1 has been defined together with T pC1 (hence, @=@t  . pC1   p /= has been defined also, where is the time step). Let us exclude pressure from the system of gas-dynamic equations as it is usually done at transition from the native to dynamic variables, by cross differentiation on r and x of the equations (1.96) and (1.97) accordingly, with the subsequent subtraction of one from another. As a result one can get the equation: @! @ @ C div .vV/  div .uV/ D @t @x @r



@Sv @Su  @x @r

 Cg

@ , @r

(1.147)

where !D

@v @u  . @x @r

(1.148)

It is obvious that for incompressible flows this equation is equivalent to the usual equation for “vortex” function !=

@v @u  , @x @r

i. e., for the azimuth component of the vector function Curl V. The second equation of the desired system is usually derived with the use of the continuity equation (1.95). There is the possibility to introduce function from the requirement of identical satisfaction to this equation. However, it is well known that the stream function can be entered in such a way only in two special cases when the density is not changed in time. These are: (a) for the incompressible flow, when div V D 0; (b) for the steady motion of compressible gas, when div V D 0. In the plasma model under consideration these conditions are not satisfied and it is not possible to enter the stream function in the usual way.

48

Chapter 1 Models of gas-discharge physical mechanics

Let us enter a new scalar function E by means of equation div .EV/ D

@ , @t

(1.149)

then the continuity equation can be rewritten in the form of div Œ.1 C E/V D 0. Now it is possible to enter an analog of a stream function with the following relations: @‰ D .1 C E/ru, @r

@‰ D .1 C E/rv. @x

(1.150)

The equations (1.150) will serve for the calculation of projections of a mass flux vector on coordinate axes: u D

@‰ 1 , r .1 C E/ @r

v D 

1 @‰ . r .1 C E/ @x

(1.151)

Substituting these relations in (1.151), we shall find the relation between functions ! and ‰:     1 @‰ r @‰ @ 1 @ ! (1.152) C D . 2 2 @x r .1 C E/ @x r @r r .1 C E/ @r r Before formulation of the final system of equations, we shall use one formal replacement of the newly entered function E which, however, will make the computing procedure more effective. Let E D exp."/  1, where " is the unknown scalar function, then instead of equation (1.149) it is possible to derive a new equation (using the continuity equation): @" @" @ u C v D . (1.153) @x @r @t Having made corresponding replacements in the equations (1.151), (1.152) we shall formulate equations for the gas-dynamic stage in the final form:     @! @ @u @u @ @v @v C div .!V/ C v  u C v  u @t @x @x @r @r @x @r   @.v=r / @ @Sv @Su @ .v=r /  u D  C g , (1.154) C v @x @r @x @r @r   ! exp."/ (1.155) grad ‰ D  , div 2 r r V grad " D

@ , @t

(1.156)

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma

49

@‰ @‰ D exp."/ru, D  exp ."/ rv. (1.157) @r @x Thus, by introduction of a new scalar function " (or E/ it is possible to formulate a system of nonstationary equations for viscous compressible fluid in dynamic variables “"  ‰  !”, which have been called nonstationary dynamic variables [117]. Boundary conditions for functions ! and ‰ are formulated as for the usual dynamic variables. The formulation of boundary conditions for " does not cause difficulties: x D 0:

" .x D 0, r / D 0,

(1.158)

r D R:

" .x, r D R/ D 0, (1.159) @" r D 0: D 0, (1.160) @r @" x D X: D 0. (1.161) @x Concerning the derived system of equations, it is possible to make some remarks of a general character. (1) For simplified cases of incompressible fluid or stationary flow of a compressible fluid " 0, and equations (1.154)–(1.157) automatically transfer to equations for the usual dynamic variables. (2) The structure of the new system of equations is similar to those for the usual dynamic variables which allows for the use of numerical methods developed previously, practically without changes , for solving the Navier–Stokes equations in dynamic variables. (3) Numerical integration of the new equation (1.156) does not present fundamental difficulties since it is an equation well investigated in the theory of numerical methods.

1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma The necessity for the development of nonuniform chemically equilibrium plasma models arises from the analysis of the complex chemical composition plasma dynamics when preliminary computation of thermodynamic, thermophysic, and optical properties in a wide range of pressures and temperatures is obviously not possible. Such a case is often observed when the local thermodynamic equilibrium approach is valid, but the equilibrium plasma composition is very complex. As an example, the problems of reentry space vehicles with ablative thermo-protection system can be considered. A distinctive feature of the nonuniform plasma models is the necessity to define its chemical composition for all grid nodes in the whole calculation domain. It also allows

50

Chapter 1 Models of gas-discharge physical mechanics

to calculate thermodynamic, thermophysic and optical plasma characteristics of the same points, which appreciably raises the universality of the computing models. A negative feature of such models is the essential complication of computing procedure so it is necessary to develop special high-performance algorithms. The system of governing equations of the nonuniform chemically reacting plasma has the following form: 

Continuity equation: @ C divV D 0; @t







Continuity equation for individual components (diffusion equations): 

@ni C div .ni V/ C div ni Vd ,i D !P i , i D 1, 2, : : : , Nc ; @t Momentum equation (the impulse conservation equation);  @V C div Œ.V/  V D  grad p C p R C Fg C Fe C F C FR  ; @t The energy conservation equations:

(1.162)

(1.163)

(1.164)

     V2 @ V2 B2 p pR C div V " C " C C C C @t 2 20 2   D div q C div W C Ag C Ae C A C AR 

(1.165)

or      V2 V2 B2 @.p C p R / @ C div V h C h C C D C @t 2 20 2 @t D div q C div W C Ag C Ae C A C AR  ; 

The Maxwell equations: @B , @t div D D e , rot E D 

(1.166) (1.167)

div B D 0,

(1.168)

1 rot B D J; 0

(1.169)

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma 



51

Generalized Ohm law:

 1 J C PH ŒJ  b D  E C ŒV  B C grad pe e ne Thermal equation of state: R0 pD T ; M

(1.170)

(1.171)

where D

Ns X

mi ni ,

pD

i D1

" D

Ns X

Ns X

Z i "i ,

"i D

i D1

h D

Ns X

hi D

Ns X

Fg,i D

i D1

Fe D

i D1

Ag D

T T0

cp,i dT C hi ,0;

Fe,i D

Xi D ni =

i D1

Fg D Ns X

cv,i dT C "i ,0;

T0

Ns Ns X R0 X mi Xi D Mi Xi , k i D1

Ns X

Ns X

ni ;

i D1

T

Z i hi ,

i D1

M D

pi D kT

i D1

Ns X

Ns X

ni ;

i D1

i g D g,

i D1

i qi .E C ŒVi  B/ D e E C ŒJ  B ;

i D1

X

 Fg,i  Vi D  .g  V/ ,

Ae D

X

.Fe,i  Vi / D .J  E/ C e .V  E/ .

Here Fg is the gravity force acting on a unit mass; Fe is the electromagnetic force (the Lorentz force) acting on a unit mass; Ag is the work of gravities; Ae is the work of the Lorentz force; Mi D .R0 =k/mi D M0 m Q i is the molecular weight of the i-th species; M0 D 1.66  1024 g is the atomic mass unit; m Q i is the relative atomic weight of the ith species; mi is the particle mass of the i-th species; ni is the volume concentration of the i-th species; pi is the partial pressure of the i-th species; Vi is the velocity of the i th species; !P i is the volume formation rate of species i, caused by chemical reactions; "i , hi are the specific internal energy and enthalpy of the i-th species; cv,i , cp,i are the specific heat capacities of the i-th species at constant volume and pressure; "i ,0, hi ,0 are the specific internal energy and enthalpy of the i-th species at the temperature T D T0, used for reference; Ns is the number of chemical species. Other designations are as in Section 1.1.

52

Chapter 1 Models of gas-discharge physical mechanics

Using a symbolic formulation of the n-th chemical reaction Ns X

Ns   X   aj ,n Xj D bj ,n Xj , n D 1, 2, : : : , Nr ,

j D1

(1.172)

j D1

the mole rate of formation of species i can be written for the n-th chemical reaction as follows: .dXi =dt /n D kf ,n .bi ,n  ai ,n / D .bi ,n 

Ns Y

a Xj j ,n

 kr ,n.bi ,n  ai ,n /

j D1 n ai ,n /.Sf ,i  Srn,i /

Ns Y

b

Xj j ,n

j D1

(1.173)

where ai ,n , bi ,n are the stoichiometric coefficients of the n-th reaction; ŒXj  is the chemical symbol of reagents and products of reactions; Nr is the number of chemical reactions; kf ,n , kr ,n are the rate constants of the forward and reverse reactions; Sfn ,i , Srn,i are the rates of the forward and reverse reactions. The mole formation rate for species i is written in the following form: Wi D

Nr X

.bi ,n  ai ,n /.Sfn ,i  Srn,i /, mol=.cm3  s/,

(1.174)

nD1

and the volume formation rate of species i can be calculated as follows: !P i D NA Wi ,

1=.cm3  s/

(1.175)

where NA is the Avogadro number. To calculate the rates of formation it is necessary to know the forward and reverse reaction constants for each n-th reaction   Ef .r /,n nf .r/,n kf .r /,n D Af .r /,n T , (1.176) exp  kT where Af .r /,n , nf .r /,n , Ef .r /,n are the approximation coefficients for the forward .f / and reverse .r / reaction rates. Note that the equilibrium constant for each n-th chemical reaction is determined as follows Keq,n D kf ,n =kr ,n,

(1.177)

which are determined by thermodynamic data. The vector of a heat flux density q in the case of the nonuniform chemically active plasma encloses not only a component caused by the molecular thermal conduction

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma

53

(in the form of the Fourier law), but also the components related to a concentration diffusion and a thermo-diffusion: q D  grad T C

Nc X

i hi Vd ,i C p

i D1

Nc X

kiT Vd ,i ,

(1.178)

i D1

where  is the molecular thermal conduction; Vd ,i D Vi  V is the vector of diffusion velocity of the i-th species. The diffusion velocities satisfy to the requirement Nc X

i Vd ,i D 0,

i D1

where kiT is the thermal diffusion coefficient of the i-th species. To define the diffusion velocities of plasma components, the matrix equation of the following kind is necessary to solve [34, 48]: Gi D

Ns X  Xi Xj

Vd ,j  Vd ,i , Di ,j

i D 1, 2, : : : , Ns ,

j D1

where vectors of diffusion forces Gi have been defined as:   grad p grad T X i qi E i Gi D grad Xi   Xi  kiT  ,  p T kT

i D 1, 2, : : : , Ns ,

where coefficients of binary diffusion Di ,j have been defined with use of the molecular-kinetic theory. The formulated equations suppose an essential simplification in two asymptotic cases: “frozen” chemical reactions and “equilibrium” chemical reactions. In the case of “frozen” chemical reactions, the gas-dynamic process rate is many times greater the than chemical reaction rates so no chemical transformations have time to occur, that is !P i D 0. In the case of “equilibrium” chemical reactions, their rates essentially surpass the gas-dynamic process rate so in any spatial point of the gas flow the local thermochemical equilibrium has time to become steady. It allows to formulate an algebraic system of equations for the calculation of species partial pressures j Keq .T / D

Ns Y i

j

b

pi i,j

Ns .Y

a

pi i,j ,

j D 1, 2, : : : , Np ,

i

where Keq .T / is the equilibrium constant of the j-th chemical reaction. At the solution of this system of equations it is possible to take into account the Dalton law and a quasi-neutrality requirement (in the case of charged intermixture of gases).

54

Chapter 1 Models of gas-discharge physical mechanics

In both cases of the relation between rates of chemical reactions and gas-dynamic processes, it is necessary to calculate thermodynamic and thermophysic properties (viscosity, thermal conduction and electrical conductivity) [41]. Examples of the nonuniform nonequilibrium gas-discharge plasma models are given below.

1.2.1 Model of the five-component RF plasma generator Let us consider five-species plasma mixture, consisting of electrons .e/, atoms .a/, diatomic homonuclear molecule .m D 2a/, atomic ions .ai/ and molecular ions .mi/. It can be, for example, the molecular nitrogen plasma which is widely spread in plasma C  investigations: N2 , N, NC 2 , N , e . Kinetic processes which will be taken into account are presented in Table 1.1. Table 1.1. The kinetic scheme of processes in the five-component plasma. Item Type of reaction 1

Name of the forward reaction Name of the reverse reaction

f k1

N C N C P  N2 C P Association k1r

Dissociation

f

2

k2

N C N  NC 2 Ce k2r

Associative ionization

Dissociative recombination

Collisional ionization

Recombination

Collisional ionization

Recombination

Exchange reaction

Exchange reaction

f

k3

3

N C e  N+ C 2e

4

N2 C e  NC 2 C 2e

5

N2 C NC  NC 2 CN

k3r

f

k4

k4r

f

k5

k5r

Note: In the given model the processes of vibrational and electronic excitation are not considered. This is the so-called thermally equilibrium model.

The system of governing equations includes Navier–Stokes equations and a continuity equation for the determination of the mass-averaged gas-dynamic functions, the system of energy conservation equations for individual species, and the system of kinetic equations. To reduce the total number of equations used, it is advisable to suppose that temperatures of atoms and atomic ions Ta D Tai as well as temperatures of molecules and the molecular ions Tm D Tmi are equal. The computational model of the five-component RF plasma generator has the following form:

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma 

Continuity equation: div V D 0.





55

(1.179)

Equations of impulse conservation (with reference to two-dimensional axisymmetric geometry): div .uV/ D 

@p C Fx C Sx C gx , @x

(1.180)

div .vV/ D 

@p C Fr C S. @r

(1.181)

Energy conservation equation of electronic gas: div .Ve He / D div .e grad Te /  Ba .Te  Ta /  Bm .Te  Tm / C div W C E 2 , (1.182) where He D e1.3=2ne kTe C pe C nai Ea C nmi Em / is the enthalpy of electronic gas subject to the ionization energy of atoms and molecules; pe D ne kTe is the pressure of electronic gas; Ea , Em are the ionization energy of atoms and molecules; nai , nmi are the volume concentration of atomic and molecular ions; E is the electric intensity module; Ve is the velocity of electronic gas; Ba , Bm are the efficiency factor for energy exchange at collision of electrons with atoms and molecules.



Equation of motion of electrons: N

me

k @Ve grad pe X D eE C eŒVe  B   ek ek .Ve  Vk /, @t ne

(1.183)

kD1

where Vk is the averaged velocity atoms (k D a/, molecules (k D m/, ions (k D ai/ and molecular ions (k D mi/; ek is the frequency of collisions of electrons with particles of the k-th kind; ek D me mk =.me C mk /  me is the reduced mass. Equation (1.183) supposes an essential simplification of the stationary electronic flow without an external magnetic field. In this case, the left-hand part of this equation can be equal to zero. Let us suppose that the velocities of the atomic and molecular ions are close, as well as their mobilities, then Ve 

grad n  C V, E  Da e ne n

(1.184)

i CDi e where Da D De ; De , Di are the diffusion coefficients of electrons and e Ci ions; e , i are the mobilities of electrons and ions.

56 

Chapter 1 Models of gas-discharge physical mechanics

Energy conservation equation for atoms and ions: div .ai VHai / D div .ai grad Ta /  Ba .Te  Ta / C A,a ,

(1.185)

1 Œ.5=2/kT .n C n / C n E  is the enthalpy of atom-ionic gas where Hai D ai a a i ai d subject to energy of dissociation Ed ; ai is the density of atom-ionic gas; ai is the thermal conduction coefficient of atom-ionic gas; Ta is the temperature of atomionic gas; na , ni are the volume concentrations of atoms and ions. 

Energy conservation equation for the molecular gas: div .m VHm / D div .m grad Tm / C Bm .Te  Tm / C A,m ,

(1.186)

1 Œ.5=2/kTm .nm C nmi / is the enthalpy of the molecular gas (withwhere Hm D m out taking into account the energy reserved in excited vibrational and rotational energy states); m is the density of the molecular gas; m is the thermal conduction coefficient of the molecular gas; Tm is the temperature of the molecular gas; A,m is the work of viscous friction forces in the molecular gas. 

System of the kinetic equations defining a chemical composition of the plasma: f

f

f

f

div .na V/ D k1 na na np  k2 na na  k3 na ne C k5 nm nai C 2k1r nm nP C 2k2r nmi ne C k3r nai ne ne , div .nm V/ div .nai Vai / div .nmi Vmi /

f f f D Ck1 na na np  k4 nm ne  k5 nm nai  k1r nm nP C k4r nmi ne ne C k5r nmi na , f f D Ck3 na ne  k5 nm nai  k3r nai ne ne C f f f D Ck2 na na C k4 nm ne C k5 nm nai  k2r nmi ne  k4r nmi ne ne  k5r nmi na ,

(1.187) (1.188)

k5r nmi na ,

(1.189) (1.190)

f

where ki , kir are the rate constants of the forward and the reverse i-th reactions; p is the index of any third particle (in the case under consideration this particle can be any of five considered species). The rate constants, as a rule, are functions of temperature of heavy particles or temperatures of electrons. However, if electrons are generally heated by electromagnetic fields, as is observed in MW, RF, and optical plasma generators, the rates of ionization by electron impact can depend on electric intensity and local pressure. Basically, it is necessary to formulate the kinetic equation for electron concentration, similar to equations (1.187)–(1.190). However, it is possible to neglect this because one can use an additional approximate condition of quasi-neutrality: ne D nai C nmi .

(1.191)

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma

57

Nevertheless, sometimes, especially at the initial stages of the debugging of the calculation code, it is recommended to solve the kinetic equation for electronic concentrations. Then the condition (1.191) is expedient for checking the precision of the solution of the dynamic equations of atomic and molecular ions: grad nai grad .kTa /  DT , ai C V, (1.192) Vai D ai E  Dai nai kTa grad nmi grad .kTm /  DT , mi C V. (1.193) nmi kTm Note that the equations (1.192) and (1.193) suppose some simplifications similar to those for equation (1.183). Boundary conditions at r D0 are formulated in the usual form: @Ta @Tm @Te D D D 0, @r @r @r Vmi D mi E  Dmi

@na @nai @nm @nmi @ne D D D D D 0. @r @r @r @r @r Similar boundary conditions can be used for all axial velocities, while all the radial velocities are equal to zero on the symmetry axis. Conditions of adhesion are applied to heavy particles on the walls of the plasma generator channel. Sometimes the adhesion condition can be replaced by conditions of gas blowing through the walls. Temperature balance with the channel wall (or a blown stream) also should be taken into account. The formulation of the boundary conditions for electronic gas is less obvious, and it demands the application of the kinetic approach. The following assumptions can be additionally analyzed. (1) For the definition of electronic temperature near to the wall of the plasma generator chamber the assumption is used that the energy of the external electric field near to the wall is spent for heating heavy particles. (2) The model of a collisionless screening layer is used close to the wall where the polarization field has been created by significantly more mobile electrons. This field promotes alignment of the ionic and electronic current towards the wall (it brakes electrons in the wall direction and accelerates ions). (3) Zero total electric current towards the wall is supposed. The requirement of quasineutrality is used and the balance of energy brought by particles to the wall and energy released on the wall at their recombination is calculated.

1.2.2 Model of the three-component RF plasma generator This model is a special case of the model presented above. The essence of the simplification of the previous model consists in assuming that the plasma consists only

58

Chapter 1 Models of gas-discharge physical mechanics

of three components: atoms, electrons and ions. Despite so essential a simplification, the present model is the most well-developed one among other nonequilibrium models because it corresponds to the argon induction plasmatron widely explored in scientific research. The system of governing equations of the model is formulated in the following form: 

Continuity equation: div V D 0;



Momentum equation (the equations of motion with reference to two-dimensional cylindrical geometry): div .uV/ D 

@p C Fx C Sx C gx , @x

@p C Fr C Sr ; @r Energy conservation equation for electronic gas: div .vV/ D 



(1.194)

(1.195) (1.196)

div .Ve He / D div .e grad Te /  Ba .Te  Ta / C div W C E 2 , (1.197)

 where He D 1e 52 ne kTe C ne Ea ; Ea is the ionization energy of atoms; 

Energy conservation equation for atomic-ionic gas: div .VHa / D div .a grad T / C Ba .Te  Ta / ,

(1.198)

where Ha D 1 Π52 kT .na C ni /; 

The equation of ionization equilibrium: div .ne V/ D ki na ne  kr n2e ni ,

(1.199)

where ki , kr are the rate constants of collision atom-electronic ionization and an ion-electronic recombination related by a condition of detailed equilibrium. The system of equations for the electromagnetic part of the problem is the same as in Section 1.1. In the case of the optically thin argon plasma, its energy losses by heat radiation can be calculated under the formula from [131]:       C  p Ei h h div W D 1.19  1040 Te exp  ne ni exp C 4n2i exp , kTe kTe kTe (1.200) where h D 2.85 eV, h C D 8.2 eV; ,  C are the limit frequencies of photo ionization thresholds from the ground states of a neutral argon and its ion (here the second addend in square brackets considers the contribution of double ions to the emissive power).

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma

59

1.2.3 Two-temperature model of RF plasma under ionization equilibrium The model is based on the following assumptions: (1) The Maxwellian distribution of particles with temperatures Te and T accordingly is assumed in electronic and heavy particles gases (atoms, molecules and their ions). (2) Processes of heavy particle and electron transfer in inhomogeneous plasma do not disturb its local ionization-recombination equilibrium. That is, the plasma composition is defined by local values of thermodynamic parameters of electronic gas and heavy particles gas. In other words, velocities of the kinetic processes are supposed to be many times greater than the transfer process rates. (3) It is supposed that processes of ionization of particles and their distributions on the excited electronic states are defined by the temperature of the electronic gas. This allows for the use of the Saha equation of ionization equilibrium for the definition of a plasma composition [2]. For example, for three-component plasma (atoms, ions, electrons) we have:     2gi 2kme Te 3=2 Ea ne ni , (1.201) D exp  na ga h2 kTe where ga , gi are the statistical weights of atoms and ions; Ea is the atomic ionization potential. Together with equation (1.201) it is necessary to use 

The requirement of quasi-neutrality ni D ne ;



(1.202)

The Dalton law .na C ni / kT C ne kTe D p,

(1.203)

where p is the total pressure. Relations (1.201)–(1.203) result in a quadratic equation for electron concentration which is defined now by the two temperatures T and Te , and also by the full pressure. In spite of the introduction of two temperatures T and Te , the given model allows to reduce the labor of the calculation model due to preliminary computations of thermodynamic, thermophysic and transport properties of the low temperature plasma. For argon plasma the following relations can be used: 

The electro conductivity and electronic thermal conduction D

e 2 ne K , P k me e N n Q kD1 k k

e D

ne kTe2 K , P k me e N n Q kD1 k k

(1.204)

60

Chapter 1 Models of gas-discharge physical mechanics

where Nk is the number of particle kinds; K and K are the corrections to the electron and heavy particle temperatures; Qk is the cross section of inter-particle interaction (electron-electron, electron-atom, electron-ion, atom-atom, atom-ion, ion-ion). Some useful approximations are: Qei D

8.78  1010 2.79  104Te ln , m2 , 1=3 Te2 ne Œ1 C .Te =T /1=3

Qea D .3.6  104 Te  0.1/  1020, m2 , 

(1.205) (1.206)

The thermal conduction of atomic-ionic gas is defined under the formula s   na ne 75k nkT , C D 64 Ma na Qaa C ne Qai na Qi a C ne Qi i

(1.207)

which is simplified at T  17 000 K as follows  D 2.5  104T 0.75 

(1.208)

Viscosity of the plasma  D 6  10

25

T

0.5

Nk X kD1



na , W=.m  K/; na C 5ne

PNk

nk

lD1

nl Qlk

, kg=.m  s/;

(1.209)

Heat capacity of heavy particles gas 5 cp D k .na C ni / 2

(1.210)

and of electronic gas cp,e

5 D k ne C 2



5 kTe C Ei 2



@ne @Te

 ,

(1.211)

p

where the sum in parenthesis corresponds to the electrons’ energy taking into account an ionization energy. Now we can formulate a closed system of the governing equations of the two-temperature model: 

Continuity equation div V D 0;

(1.212)

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma 



Momentum equations (with reference to the two-dimensional cylindrical geometry) div .uV/ D 

@p C Fx C Sx C gx , @x

(1.213)

div .vV/ D 

@p C Fr C Sr ; @r

(1.214)

Energy conservation equation of electronic gas div .VHe / D div .e grad Te /  Ba .Te  Ta / C div W C E 2 , where He D



61

1 5 e . 2 ne kTe

(1.215)

C ne Ea /, Ea is the ionization energy of atoms;

Energy conservation equation of atomic-ionic gas div .VHa / D div .a grad T / C Ba .Te  Ta / ,

(1.216)

where Ha D 1 Π52 kT .na C ni /; 

The equations of ionization equilibrium   Te p n2e C ne 1 C Fei  Fei D 0, T kT where Fei D

2gi 2 kme Te 3=2 Ea . h2 / exp. kT /, ga e

(1.217)

p is the total pressure.

Other equations of the model (system of electrodynamic equations, the equation of thermal radiation transfer) are the same as in the previous models.

1.2.4 One-liquid two-temperature model of laser supported plasma This model is formulated in a two-dimensional axisymmetric geometry with reference to the stationary flow of hydrogen plasma. This model includes the Navier–Stokes and a continuity equation for the description of viscous gas dynamics, and also two energy conservation equations for electronic gas (e  / and for heavy particles gas .H2 ,H,HC /: 

Continuity equation 1 @rv @u C D 0; @x r @r



(1.218)

The Navier–Stokes equations @ 1 @ @p .uu/ C .ruv/ D  C Sx  g, @x r @r @x

(1.219)

1 @ @p @ .uv/ C .rvv/ D  C Sr , @x r @r @r

(1.220)

62

Chapter 1 Models of gas-discharge physical mechanics

where

       @u @v 2 @ @ @u 1 @ C Sx D   div V C 2  C r , 3 @x @x @x r @r @r @x        @ @u @v 2 @ @ @v @ v . div V/ C C  C2  C ; Sr D  3 @r @x @r @x @r @r @r r



Energy conservation equation for heavy particles gas      3 1 @ 3 k nu T C r k nv T D 2 r @r 2       @ @T 1 @ @T @u 1 @  C   .p C P / C .r v/ C Ah  QR , (1.221) @x @x r @r @r @x r @r @ @x





Energy conservation equation for electrons       3 3 1 @ @ k ne u Te C r k ne v Te D @x 2 r @r 2     @ @Te @Te 1 @ e C r e  Ae C QL, (1.222) @x @x r @r @r where x, r are the axial and radial coordinates; u, v are the axial and radial components of velocity; Te , T are the temperature of electrons and heavy particles;  is the coefficient of dynamic viscosity; e ,  are the thermal conductivities of electronic gas and gas of heavy particles;  is the total density; p is the dynamic component of the pressure; P is the static pressure defined by the following equation of state: P D .ne kTe C nkT /.1  PcDH /; PcDH   1.5 n C 1 4e 2kc ne D C H 24.ne C n/ k Te T

(1.223)

is the correction of the Debye–Hueckel; kc is the Coulomb constant; QL is the power of a heat release caused by the radiation of laser source; QR is the power of radiative energy losses; Ae , Ah is the work of friction caused by interaction of an electronic gas with a gas of heavy particles: Ae D Ah D 3k.Te  T /ne me

Nh X ej j

mj

;

(1.224)

ej is the frequency of collisions of electrons with a j -th heavy particle; Nh is the number of kinds of heavy particles (in this case Nh D 3 .H2 , H, HC//; ne is the volume concentration of electrons; n is the volume concentration of heavy particles .n D nH2 C nH C nH+ /; me , mj are the masses of an electron and j -th heavy particle; k is the Boltzmann constant.

Section 1.2 Models of nonuniform chemically equilibrium and nonequilibrium plasma

63

For the calculation of a chemical composition of nonequilibrium plasma, the following system of equations is used      .nH /2  mH kT 3=2 QH,ex Te EH2 2 D exp  , nH2 kT h2 T QH2 , rot  QH2 ,vib     nH+ T =Te 2 exp .EH =kTe / 2 mekTe 3=2 ne D , nH QH,ex h2

(1.225)

(1.226)

where EH2 , EH are the potentials of dissociation of H2 and ionization of H; QH,ex, QH2 ,vib, QH2, rot are the partition functions on electronic states of H and of vibrational and rotational states of H2 . Collision frequencies of electrons and heavy particles are estimated under the formula ej D ve nj Qej , (1.227) p where ve D .8k= me /Te is the average thermal velocity of electrons; Qej is the cross section of electron-particle collision. Depending on the type of heavy particle the three types of collisions are defined as: 

The electron-atom collision (in the given case: e  H/;



The electron-molecule collision (e  H2 /;



The electron-ion collision (e  H+ /.

In the study [68] the tabular data gained by [54, 55] were used. For the Coulomb interaction of charged particles the cross section is estimated as follows: Qei D

e 4kc2 ln ƒe

,

(1.228)

k 3 Te2 .  ne

(1.229)

2 .kTe /2 s

where 1.5 ƒe D 3 1.5 e kc

The electronic thermal conduction is calculated under the formula [68] p e D 5  1025 ne Te ,

(1.230)

derived before for argon plasma. Summing up the description of the equilibrium and nonequilibrium computing models of low temperature plasma, we shall note that it is important to provide the models with well-tested transport and optical properties because with the use of incorrect models for the properties, the value of detailed nonequilibrium models of gas-discharge plasma dynamics can be lost.

Chapter 2

Application of numerical simulation models for the investigation of laser supported waves 2.1 Air laser supported plasma generator The schematic diagram of a laser supported plasma generator that will be considered in the chapter is presented in Figure 1.5. The subsonic flow model used in the present section (pressure variations due to small perturbations are neglected) makes it possible to reduce the continuity and Navier–Stokes equations in the form (1.95)–(1.97) to the following system of equations in time-dependent dynamic variable (for details see Section 1.1.6):     @! @ @u @u @ @v @v C div .!V/ C v  u C v  u @t @x @x @r @r @x @r

1

1   @ vr @ vr 1 @Sv @Su  u D  , (2.1) C v @x @r Re @x @r   exp."/ ! div grad‰ D  , (2.2) r2 r @" @ @" C v D , @x @r @t @‰ @‰ D ru exp ."/ , D rv exp ."/ , @r @x @v @u  , !D @x @r and the function " D ln.E C 1/ introduced using the hypothesis u

div.EV/ D

@ @t

(2.3) (2.4) (2.5)

(2.6)

makes it possible to represent the continuity equation in the form (2.3). The use of hypothesis (2.6) is based on the fact that the time variation of the gas density is primarily initiated by internal heat sources (in our case by absorption of the laser radiation energy) under subsonic flow conditions, i. e., this variation can be assumed to be a known function to be determined from the energy conservation equation and the equation of state and not from the gas-dynamic equations. On the other hand, the continuity equation, which nevertheless must unconditionally be satisfied, dictates

Section 2.1 Air laser supported plasma generator

65

the divergent form of the partial derivative of the density with respect to time. In order to conserve the formulation of the problem of finding the unknown vector-function we propose the following hypothesis: let us assume that there is a scalar function E satisfying equation (2.6). Thus, by introducing a new scalar function " (or E) we can formulate the system of equations of a viscous compressible fluid in the dynamic variables "  ‰  !, which makes it possible to solve time-dependent problems of subsonic radiation gas dynamics. The boundary conditions for the functions ! and ‰ are specified as for the ordinary dynamic variables. It is also easy to write the boundary conditions for the function " x D 0,

".r / D 0,

(2.7)

r D Rc , ".x/ D 0,

(2.8)

r D 0,

@" D 0, @r

(2.9)

x D L,

@" D 0. @x

(2.10)

Conditions (2.7) and (2.8) are the consequences of the constant temperatures on the corresponding surfaces (Rc is the radius of the cylindrical channel). Condition (2.9) determines the axial symmetry and condition (2.10) corresponds to the free boundary x D L of the computation domain. In order to determine the thermal radiation transfer to the inner surface of the generator, let us consider the ray-tracing method. In our calculations we imposed the condition of axial symmetry. The algorithm of the ray-tracing method is as follows. In order to find the density of the thermal radiation flux to a surface element we introduce a local spherical coordinate system with the normal n. In this coordinate system, each ray  is determined by two angular coordinates, namely, the latitude angle  2 Œ0, =2 and the azimuthal angle ' 2 Œ0, 2. On the inner surface of the generator the densities of the spectral and integral radiation fluxes can be calculated from the formulas Z

Z

2

W! .rj / D

=2

d' 0

Z

J! .rj , / cos  sin d

0 1

WR .rj / D

(2.11)

W! .rj /d!, 0

where rj D .rj , xj / is the radius vector of the j -th point on the surface in the laboratory coordinate system, rj , xj are the radial and axial coordinates of the j -th point in the laboratory coordinate system, and J! .rj , / is the spectral intensity of the radiation.

66

Chapter 2 Nunerical investigation of laser supported waves

Introducing a computation grid of angular directions makes it possible to integrate the spectral intensity of the radiation on the generator surface over the space of the angular variables and find the spectral density of the flux (2.11) N' 1

 1 X .'mC1  'm / W! r j D 2 mD1



N X

J! .rj , m,n /.sin nC1 cos nC1 C sin n cos n /.nC1  n /

nD1

or N' 1

W! .rj / D 

X

.'mC1  'm /

mD1

NX  1 nD1

1 J! .rj , m,n /.cos2 nC1  cos2 n /. 2 (2.12)

The direction cosines of the vector m,n D .$x /m,n i C .$y /m,n j C .$z /m,n k can be calculated from the formulas .$x /m,n D .$y /m,n

p

1  N 2n cos 'Nm , p D 1  N 2n sin 'N m ,

.$z /m.n D N n , 1  N n D .n C nC1 /, 2

(2.13)

1 'Nn D .'n C 'nC1 /, 2

 D cos .

In order to determine the quantity J! .rj , m,n /, it is necessary to integrate the radiation transport equation along an inhomogeneous optical ray. For these purposes we can use a formal solution of the transport equation of the following form for a nonscattering medium: Z J! .rj , m,n / D

sDL

sD0

 Z Jb,! ŒT .s 0 /! .s 0 / exp 

s0

 ! .s 00 /ds 00 ds’

(2.14)

0

where s D 0 and L are the initial (on the generator surface at r D rj ) and end (on the opposite generator surface along the ray or in the exit cross section) coordinates of the interval of the ray m,n along which integration is performed, and Jb,! .T / is the radiation intensity of an absolutely black body (Planck function). A finite-difference grid in the spatial variable s, by means of which we carry out the numerical integration, can be found for each ray m,n . For this purpose in the laboratory coordinate system we find the coordinates of intersection of the ray m,n with

67

Section 2.1 Air laser supported plasma generator

all the surfaces of the finite-difference grid  D fhi D ri  ri 1, hj D xj  xj 1; i D 1, : : : , NI , j D 1, : : : , NJ g it encounters. Figure 2.1 illustrates this algorithm graphically (ray a). We note that in order to determine the coordinates of intersection it is necessary to use the relations of analytic geometry. However, this algorithm turns out to be uneconomical with respect to curvilinear, in particular nonstructured, computation grids. Therefore, it is recommended to use a highly-efficient algorithm of quasi-random sampling of the computation grid coordinates. (a)

y

(b) sk

j

i

x

Figure 2.1. Methods of calculating selective thermal radiation transport: calculation of the coordinates of intersection of a ray with all the surfaces along it (a) and quasi-random sampling (b).

This algorithm consists of the following. A segment of a ray with the directional vector m,n between the first (s D 0) and the end (s D L) points is divided into Ns identical fragments as shown for a ray in Figure 2.1 (in fact, the requirement of uniformity of the computation grid on the segment s 2 Œ0, L is not mandatory for the algorithm used). Then for each gridpoint along the ray m,n one can find the nearest gridpoint of the spatial computation grid (or the cell to which this gridpoint belongs) and on which the thermophysical and optical properties of the medium are specified. The temperature and the spectral absorption coefficient of the gridpoint (or cell) thus found are assigned to the current gridpoint of the computation grid along the ray. Thus, the temperature and the optical properties of the gas are known at each gridpoint of the computation grid along the ray m,n . We note that this algorithm of the ray-tracing method (2.11)–(2.14) is far from being the most efficient. An alternative is the more efficient discrete ordinates method [35]. However, only the ray-tracing method can be further developed in order to take the specifics of radiation transport in atomic lines into account by means of statistical models of their spectrum. In this case, the algorithm outlined corresponds to the approximation based on the molecular spectrum averaged over the rotational structure and the atomic absorption spectrum averaged over the spectral line structure.

68

Chapter 2 Nunerical investigation of laser supported waves

Now we will consider the results of calculating the radiation gas dynamics of an air generator at atmospheric pressure, which are given for three laser radiation powers: PL D 30, 50, and 200 kW. Single-mode CO2 -laser radiation (see parameter n D 2 in equation (1.99)) with a wavelength of  D 10.6 micrometers was specified. The radiation was focused at a distance xp D 3 cm from the inlet cross section of the cylindrical channel. The divergence of the laser radiation was assumed to be equal to L D 0.02, 0.04, and 0.08 mrad, respectively, for the three increasing powers. On the one hand, this corresponds to the physics of the process and, on the other hand, made it possible to obtain a stable laser plasma in the same gas flows with a velocity u0 D 30 m/s. The minimum radius of the caustic of the focused radiation was calculated using the approximate relation Rc  xp L. Transition from the heat-conduction regime of laser plasma existence (for low power) to the radiation regime (for high power) was simulated by specifying three laser radiation powers. Figures 2.2–2.4 show the temperature fields inside the generator for various laser radiation powers under different assumptions concerning the turbulent mixing and the radiative heat transfer mechanism. The radiative heat transfer calculations were carried out for the 37-group model (see details in Section 1.1.4).

0.5

r, cm (a) 21893

11909

1926

10483 13336

9057

7631

3352

4779 6205

(b)

0.5

1926 21893

17614 10483

3352

7631

4779

6205

(c)

0.5

1926 11909 9057

21893

3352

7631

4779

6205

(d) 0.5

1926 11909 10483

21893

2

4

9057

6

3352

7631

8

4779

6205

10

x, cm

Figure 2.2. Temperature distribution over the generator (T in K) for PL D 30 kW, Rc D 0.06 cm, u0 D 30 m/s, and xp D 3 cm: for laminar initial flow without and with reabsorption of thermal radiation ((a) and (b), respectively) and for turbulent initial flow without and with reabsorption of thermal radiation ((c) and (d), respectively).

The heat-conduction regime of laser plasma existence was realized in the calculations with the minimum laser radiation power PL D 30 kW. In this case, the caustic radius Rc D 0.06 cm. The temperature fields reproduced in Figures 2.2 (a) and 2.2 (b) correspond to the assumption of a laminar flow regime in the inlet cross section of the generator. In the first case (Figure 2.2 (a)) the reabsorption of thermal radiation was

Section 2.1 Air laser supported plasma generator

69

neglected, i. e., we used the volume de-excitation model in accordance with which the thermal radiation leaves the computation domain without absorption in the gas. In the second case (Figure 2.2 (b)), the reabsorption of thermal radiation was taken into account. Clearly, the differences between the temperature fields calculated with and without taking the reabsorption into account are insignificant. This again confirms the initial assumption that in this case the heat-conduction regime was realized. As mentioned above, the principal reason for this is the fact that the volume of the high temperature laser plasma region is so small that the power of the emitted thermal radiation is insufficient for intense heating of the adjacent gas layers, which can be more effectively realized by means of the heat-conduction heating mechanism. The self-oscillations of the gas flow behind the region of heat release associated with the absorption of laser radiation are extremely important for the variant of low laser radiation power in question. In Figures 2.2 (a) and 2.2 (b) we have reproduced the instantaneous structures of the temperature field. In fact, the calculations indicate that in this case there exist un-damped self-oscillations of the flow. A pulsating plasma wake, in which the velocity and the temperature vary with a period of the order of 0.1–0.5 ms, is formed behind the heat-release region. This period corresponds to the characteristic drift time of the vortices formed behind the heat-release region. Taking the turbulent mixing in the gas flow and the initial flow turbulization at the inlet to the generator channel into account qualitatively changes the flow pattern (Figures 2.2 (c) and 2.2 (d)). In the calculations we can observe the complete stabilization of the solution and the laser plasma becomes more stable to various perturbations of the gas flow introduced into the numerical solution for research purposes. In the variants of the calculations presented, the initial turbulence level (boundary conditions for equations (1.105), (1.106)) was simulated by specifying the boundary conditions for the functions k and " in the form: x D 0 and k D " D 0.1. This corresponds to approximately 15–20 % of the intensity of the turbulent pulsations at the inlet to the generator channel. A decrease in the initial turbulence level led to a corresponding decrease in the effect of turbulent dissipation on the gas-dynamic structure of the laser plasma and the gas flowing past it. Hence, we may conclude that in this case it is precisely the initial flow turbulence that has the main effect on the gas flow stabilization. This is attributable to the fact that the self-turbulence of the gas flow manifests itself only at certain distances downstream from the high temperature laser radiation absorption region, where this effect is no longer so significant. The second series of calculations corresponds to the case of transition from the heat-conduction to the radiation regime of plasma existence. In this case, the caustic radius Rc D 0.12 cm. As in the first series of calculations, we have reproduced the temperature fields in the generator obtained for a laminar inlet flow (Figures 2.3 (a) and 2.3 (b)) and with allowance for turbulent mixing (Figures 2.3 (c) and 2.3 (d)). It is noteworthy that in this case the reabsorption of radiation already has an appreciable effect on the gas-dynamic structure of the generator. In particular, in this case

70

Chapter 2 Nunerical investigation of laser supported waves r, cm 1 (a) 0.5

1926 10483 9057

21893

1 (b) 0.5 1 (c)

3352

14762

13336

9057

7631

9057

11909 1926

20467

1

3352

4779

1926 21893

0.5

7631

3352

4779

6205

4779

6205

(d) 1926

0.5 0

7631 6205

11909

21893

2

4

10483

6

9057

3352

4779 7631

8

6205

10

x, cm

Figure 2.3. The same as in Figure 2.2 for PL D 50 kW, Rc D 0.12 cm, u0 D 30 m/s, and xp D 3 cm.

taking the reabsorption of radiation into account leads to gas flow stabilization. This is attributable to the decrease in the temperature gradients in the gas flow as a result of the absorption of thermal radiation by the cold gas layers. The effect of reabsorption can also clearly be seen from the temperature of the plasma jet flowing out of the generator. When the reabsorption is taken into account, the jet temperature increases appreciably, the increase being more marked for laminar flow. These facts indicate a decrease in the total energy losses from the plasma and the plasma wake due to the reabsorption of radiation and the absence of turbulent dissipation. However, in this case it should again be noted that the solution obtained is stabilized when the initial gas flow turbulence is taken into account. In Figure 2.4 we have reproduced the results of the calculations for the radiation regime of existence of the laser plasma in the generator channel. In this case, the caustic radius Rc is equal to 0.24 cm. As in the previous case, the reabsorption of radiation favors stabilization of the gas flow. Attention is drawn to the large length of the high temperature zone of the laser plasma and its appreciably increased transverse dimensions. In Figures 2.4 (a) and 2.4 (d) we can clearly see that the reabsorption of thermal radiation leads to the laser plasma being displaced upstream toward the laser beam by more than 0.2 cm. In the case considered, the reabsorption of radiation is so significant that the initial flow turbulence has almost no effect on the gas-dynamic structure of the laser plasma in the neighborhood of the highest temperature absorption zone, although, of course, in the resulting plasma jet this effect remains strong with respect to flow stabilization. As in the second calculated case, taking the reabsorption of radiation into account leads to an appreciable increase in the plasma jet temperature. Corresponding changes are also observed for the plasma jet velocity. For example, when the reabsorption of radiation was taken into account for an initial turbulent flow, the outlet cross section plasma jet velocity u on the axis of symmetry was 1860 m/s

71

Section 2.1 Air laser supported plasma generator r, cm 1

1926

(a)

0.5

21893

1

11909 17614

(b)

0.5 21893

1

4779 10483

13336

6205

1926 3352 4779 6205 13336 20467 17614 16188 14762

7631

7631

9057

10483

(c) 1926

0.5 21893

1

(d)

3352

3352 13336

4

6205 7631 11909 10483 9057

4779 14762

17614

21893

2

3352 19040

1926

0.5 0

3352

6

6205 13336

7631

8

9057

10483

10

x, cm

Figure 2.4. The same as in Figure 2.2 for PL D 200 kW, Rc D 0.24 cm, u0 D 30 m/s, and xp D 3 cm.

(the Mach number M D 0.58), while in the absence of reabsorption u D 780 m/s (M D 0.32). The spectral and integral radiative heat fluxes on the inner generator surface were calculated by means of the ray-tracing method using computation grids in the angular coordinates N D N' D 11, 21 and 41. The number of grid points Ns along the rays issuing from the surface varied from 10 to 40. The calculations showed that the results are most sensitive to the specified number of grid points Ns . This is easily attributable to the high degree of inhomogeneity of the laser plasma and the dependence of the solutions on the accuracy of the description of the plasma properties along the issuing rays. The error introduced into the calculations due to grid coarseness is well known in computation physics as the “ray effect” [35]. In the case considered, this effect was observed when N and N' were less than 11. Figure 2.5 shows the results of calculating the integral-over-the-spectrum radiative heat fluxes on the inner generator surface for the maximum laser radiation power. In this case, the 37-group optical model was used. These distributions completely reflect the above-mentioned features of radiation reabsorption and initial gas flow turbulence. When only the volume de-excitation is taken into account, the decrease in the dimensions of the high temperature laser plasma region manifests itself in an approximately twofold decrease in the radiative heat fluxes WR on the wall (curves a and c). The maximum fluxes WR are observed in the case of the laminar flow with allowance for the reabsorption of thermal radiation (curve b). The initial gas flow turbulence leads to a regular but small reduction in the radiation fluxes associated with an insignificant decrease in the dimensions of the high temperature laser plasma region. In the case shown in Figure 2.6, the initial flow was assumed to be turbulent and the reabsorption of thermal radiation was taken into account. These data indicate, on the one hand, the ambiguous effect of the choice of group model on the results obtained

72

Chapter 2 Nunerical investigation of laser supported waves

WR, W2 cm (b) 800 (d) (a) 400 (c)

0

4

8

x, cm

Figure 2.5. Integral radiation flux distribution along the inner generator surface for the 37group model and PL D 200 kW, Rc D 0.24 cm, u0 D 30 m/s, and xp D 3 cm: (a)–(d) the same as in Figure 2.2.

1000

WR, W2 cm

1

3

800 2 600 400 200 0

2

4

6

8

10 x, cm

Figure 2.6. The same as in Figure 2.5 for different group models: curve 1 corresponds to 37 groups, curve 2 to 74 groups, and curve 3 to 148 groups; the input data are the same as in Figure 2.5.

and, on the other, the acceptable accuracy of the recommended 37-group model as compared with other group models. In this case we note that different group models were used not only for calculating the radiative heat fluxes but also for solving the entire problem.

73

Section 2.1 Air laser supported plasma generator

In conclusion, we will consider the group radiation flux distributions at three points on the inner generator surface obtained for three group models. Figure 2.7 shows the radiation fluxes, integral over the spectral groups, which reach the inner generator surface. We note a qualitative similarity between the radiation flux spectra predicted by the three group models at each of the chosen surface points (x D 0.4, 4.0, and 10.7 cm). Even the roughest of these group models makes it possible to obtain a fairly complete representation of the structure of the radiation flux spectrum. Naturally, the models with a larger number of groups make it possible to obtain a more detailed description of these structural features. 103 2

10

WR, W2 cm

(a) 1 2 3

1

10

100 10–1 10–2 10–3 2

10

WR, W2 cm

(b) 1 2 3

1

10

100 10–1 10–2 10–3

50 000

100 000

ω, cm–1

Figure 2.7. Radiation fluxes integral over the spectral groups to three parts of the generator surface: curves 1–3 correspond to x D 0.4, 4.0, and 10.7 cm for 37 (a) and 148 (b) group models; turbulent initial flow with reabsorption of thermal radiation, PL D 200 kW, Rc D 0.24 cm, u0 D 30 m/s, and xp D 3 cm.

An analysis of these group fluxes suggests that a significant fraction of the radiation emitted by the laser plasma in the ultraviolet spectrum range, primarily, by intense atomic lines, reaches the inner generator surface. The radiation of the visible and near infrared spectral regions reaching the surface is also due to atomic lines formed in quantum transitions from excited states. Therefore, as the group models solve the

74

Chapter 2 Nunerical investigation of laser supported waves

problem of radiation transport in atomic lines very approximately, we note the need for the further development of the corresponding models. So, the role of reabsorption of thermal radiation in the formation of the gas-dynamic structure of laser plasma generators in the radiation regime is demonstrated in the section by comparing the thermo-gasdynamic parameters of generators operating in the heat-conduction and radiation regimes. Numerical experiments show that there are generator operating regimes with laminar and oscillatory gas and plasma flow regimes associated with self-oscillations of the gas flow behind the heat-release region localized in the laser plasma. It is shown that initial free-stream turbulence at the working chamber inlet and the reabsorption of thermal radiation suppress the large-scale flow fluctuations induced by flow past the local heat-release region. A comparative analysis of various group hot air models with reference to radiationgasdynamic generator calculations showed that the 37-group optical model can be used for these purposes.

2.2 Hydrogen laser supported plasma generator Radiative gas dynamics of a hydrogen laser plasma generator (see Figure 1.5) is considered in this section for the following initial conditions: the power of CW CO2 -laser is PL D 200 kW, the focal length of the focusing lens is xf D 3 cm, the divergence of the laser radiation is  D 0.1 mradian, the initial radius of the laser beam is Rb D 1.0 cm, the pressure in the LSPG channel is p0 D 1 atm, the gas velocities in the input section were changed from u0 D 2 to 60 m/s, and the length and radius of the LSPG channel are Lc D 11 cm and Rc D 1.3 cm. As mentioned above, the investigated process develops at approximately constant pressure that causes significant problems for a numerical solution of the task, because available disturbances of the pressure are hundreds of times less than background pressure. However, in this case it is quite sufficient to take into account only the temperature dependencies of the thermophysical and optical properties which are included in the mathematical model (1.95)–(1.101). Temperature dependencies of the mentioned properties , cp , , , ! , and g can be calculated only for the background pressure and for the main calculations an interpolation procedure for temperature can be used. The thermophysical and transport properties of low temperature hydrogen plasma were borrowed from [14]. Spectral and group (Figure 2.8) absorption coefficients of hydrogen plasma were calculated using the computing program ASTEROID [106]. The group optical model was created by averaging the spectral absorption coefficients over 37 spectral groups. This number of spectral groups was determined by performing numerical experiments with different numbers of spectral groups (from 2 up to 300). There is a problem of optimization of the number: too low a number provides physically unrealistic results

75

Section 2.2 Hydrogen laser supported plasma generator

104

Absorption coefficient, 1/cm

103 102 101 3

100

2

10–1 1

10–2 10–3 10–4

50 000

100 000 Wavenumber, 1/cm

150 000

Figure 2.8. Spectral (solid line) and group absorption coefficients of a hydrogen plasma at p D 1 atm, T D 20 000 K (1), 16 000 K (2), and 12 000 K (3).

(for example, one can observe extinction of laser supported waves for some calculation cases with a low number of spectral groups), while too large a number of spectral groups results in large computational times. Therefore, from one side the boundaries of the spectral groups (actually the spectral grid) must take into account the general peculiarity of the spectra of heat emissivity (processes of photo recombination, electronic bands of diatomic molecules, strong atomic lines, continuous radiation processes), from the other side the spectral group model with a low number of spectral groups is favored over the spectral group model with a larger number of spectral groups. It was shown in the numerical experiments that the 37-group model is quite acceptable for the problem under consideration. Just this group model was used for the calculation of radiative gas dynamics and radiation fluxes on an internal surface of the LSPG. The numerical simulation results for succession nonhomogeneous finite-difference meshes with grid condensation around the high temperature zone of the LSW and near to the wall of the cylindrical channel will be considered below. Numerical simulation results are presented for the following number of nodes along the axial and radial directions respectively: 60  30, 80  40, 120  50, 240  100. Let us consider the results of the following two series of calculations. The first one was carried out for the radiation emission model (i. e., the optically thin approximation was used in this case, or the so-called “volume emission approach”). For an entrance velocity of u0 D 2 m/s the laser plasma was moved towards the laser and its forward front was fixed at a distance of  2.3 cm from the channel entrance (Figures 2.9 (a),

76

Chapter 2 Nunerical investigation of laser supported waves Level

1

2

3

4

5

6

7

8

9

10

11

T: 2 000 2 564 3 288 4 215 5 404 6 928 8 883 11 388 14 601 18 719 24 000

r, cm

(a) 1

1

11

0

r, cm

–1 (b) 1

2 4 7 5 8 6

1

3

2 3

0

11

7 5 8 6

4

–1 0

2

4

6

8

10 x, cm

Figure 2.9. Temperature distributions in hydrogen LSPG at u0 D 2 m/s (a) and u0 D 5 m/s (b). The volume emission approach.

2.10 (a)). In these cases, the distributions of all thermal and gas-dynamic functions quickly stabilized and did not vary with time, i. e., the steady-state solution for the task was obtained. A temperature inside the LSW achieved  24 500 K, but the plasma jet temperature was less than  3 200 K. The temperature behavior inside the laser plasma can be explained by using the volume emission approach. Practically full absorption of the laser radiation by the plasma was obtained. The next case studied was performed for the entrance velocity of u0 D 5 m/s. Actually, this is an extreme velocity case at which the LSW can exist in a volume emission mode. A gas flow with u0 D 6 m/s blows off the LSW and the laser plasma cannot exist as the numerical solution predicts. Figures 2.9 (b) and 2.10 (b) show that in this case the high temperature plasma is more localized than in the previous case. In this case, the maximum temperature in the LSW is also about 24 000 K and the temperature in the laser plasma jet is about 3 000 K. The numerical simulation results show extremely high cooling of the laser plasma in the case of the volume emission approach. In this case, a configuration of the numerically calculated LSW follows the laser beam form near to its focusing point. It is worth noting that the volume emission approach predicts extinction of the LSW at low entrance velocities. The second series of calculations was performed using the full radiation heat transfer model, i. e., all radiation characteristics were calculated by solving the full system of equations (1.95)–(1.101) by using of the 37-group model. In this series, the entrance velocity was varied for significantly wide regions. Figures 2.11 and 2.12 show distributions of temperature and axial velocity for entrance velocities of u0 D 5 and 50 m/s. Comparing the data presented in Figures 2.9–2.12 one can conclude that radiation heat transfer processes in the given type of LSPG are very significant, especially

77

Section 2.2 Hydrogen laser supported plasma generator Level

1

2

3

4

VX: 1.0

1.6

2.7

4.4

r, cm

(a) 1

1 11

0

3

2

6

7

4

9

10

11

85.4 140.0

5

7

3

4

5 6

7

9

8

31.8 52.1

6

10 9 8

11

0

–1 (b) 1 r, cm

2

1

5

7.2 11.8 19.4

–1 0

2

4

6

8

10 x, cm

Figure 2.10. Axial velocity (VX D u=u0 ) distributions in hydrogen LSPG at u0 D 2 m/s (a) and u0 D 5 m/s (b). The volume emission approach.

Level

1

2

3

4

5

6

7

8

9

10

11

T: 2 000 2 564 3 288 4 215 5 404 6 928 8 883 11 388 14 601 18 719 24 000

(a) 1

2

1

r, cm

8

0

9 10 11

r, cm

–1 (b) 1

1

0

11

3

6

7

5

2

10 9 8

4

3 7

4

5

6

–1 0

2

4

6

8

10 x, cm

Figure 2.11. Temperature (T in K) distributions in hydrogen LSPG at (a) u0 D 5 m/s and (b) 50 m/s. Full radiation heat transfer approach.

for the forming of the high temperature region and for prediction of the plasma jet velocity. For the full radiation heat transfer mode, the high temperature region of the LSW has a noticeably larger dimension. Its front is located at distance  1.5 cm from the entrance cross section of the LSPG channel for u0 D 5 m/s. The greatest temperature inside the LSW runs up to  24 000 K. This temperature is slightly lower than that obtained for the volume emission mode; but due to an appreciably larger dimension of the heated region and due to reduction of total energy losses, the temperature of the plasma jet and its velocity are essentially larger. For u0 D 5 m/s the plasma jet

78

Chapter 2 Nunerical investigation of laser supported waves Level VX:

(a) 1

1

2

1

2

3

4

5

3

4

r, cm

7

5

8

9 10 11

0

11

–1 (b) 1 0

7

11

2

1 9

6 8

9

10

r, cm

6

1.0 9.9 18.8 27.7 36.6 45.5 54.4 63.3 72.2 81.1 90.0

10

3

5

7

9 11

11

–1 0

2

4

6

8

10 x, cm

Figure 2.12. Axial velocity (VX D u=u0 ) distributions in hydrogen LSPG at (a) u0 D 5 m/s and (b) u0 D 50 m/s. Full radiation heat transfer approach.

temperature at the exit section of the LSPG channel reaches a value of 10 800 K and the axial velocity is 250 m/s. The maximal temperature in the plasma jet slightly decreases up to  9 000 K with increasing entrance velocity (see Figure 2.12). For maximum entrance velocity u0 D 50 m/s, at which the LSW exists in the conditions under consideration, the plasma jet velocity achieves the value of 1 075 m/s. At velocity u0 D 60 m/s, the LSW is blown off by entrance hydrogen flow. Results of methodological CFD study of the finite-difference method accuracy are shown in Figure 2.13, where two temperature fields obtained for two kinds of grids are presented. As mentioned above, to obtain the steady-state solution for the temperature field, the k  " turbulent model was used for predicting effective viscosity of gas flow inside LSPG chamber. One can see that varying the calculation grids within a wide region results in some change of temperature profiles, but the general peculiarities of the thermal structure of LSW were left unchanged. Investigations of the effect of spectral group models on numerical simulation results have shown that two spectral group models (Ng D 37 and 77) provide acceptable agreement between numerical simulation results. At a lower number of spectral groups, large differences were observed in the temperature fields. It is worth noting that the spectral group model has an especially large influence on the temperature distribution for the calculation cases in which the heat radiation plays a dominant role in the energy balance inside the laser supported waves. Sometimes, low group optical models result in the extinction of the LSW. Calculations of radiative heating of the internal surface of a LSPG were performed for each calculation case. The ray-tracing method was used with an angular mesh of N D N' D 21 and Ns D 20, where Ns is the number of nodes along a ray. Preliminary calculations have confirmed that the radiative heat fluxes predicted by the RTM

79

Section 2.2 Hydrogen laser supported plasma generator Level

1

2

3

4

5

6

7

8

9

10

11

T: 2 000 4 200 6 400 8 600 10 800 13 000 15 200 17 400 19 600 21 800 24 000

1

1

(a)

2

r, cm

3

0 –1 1

9

1

(b)

4

6

8

11

5

7

2

r, cm

3

0

8

11

6

9

4

5

7

–1 0

2

4

6

8

10

x, cm

Figure 2.13. Temperature (in K) distribution with an entrance velocity of 30 m/s; (a) grid 60  30, (b) grid 120  50.

strongly depend on the parameters of the spatial finite-difference mesh. Numerical simulation results presented in Figures 2.14–2.16 correspond to the minimal acceptable mesh. The angular mesh was established with numerical experiments in which the number of nodes in each angular direction was varied over a wide region (10–100). Figure 2.14 illustrates the dependence of the integral radiation flux distribution on the internal surface obtained by using different numbers of spectral groups. One can see Integral radiative flux, W/cm**2 600 37 13

77

400

200

0

1

2

3

4

5 6 x, cm

7

8

9

10

Figure 2.14. Integral radiation heat flux along internal cylindrical surface of the hydrogen LSPG, for different spectral group models (numbers near to the curves designate number of spectral groups).

80

Chapter 2 Nunerical investigation of laser supported waves Spectral radiation flux, W*cm/cm**2 103 102 101 2 1

0

10

10–1 10–2 50 000

100 000

ω, cm–1

Figure 2.15. Spectral group heat flux at x D 0 (1), and x D 3.6 cm (2) for Ng D 13, 37, and 77 groups.

1 000 Radiation Heat Flux W/cm2

900

U = 20 m/s U = 40 m/s U = 60 m/s U = 70 m/s

800 700 600 500 400 300 200 100 0

5 x, cm

10

Figure 2.16. Integral radiation heat flux along internal cylindrical surface of the hydrogen LSPG, W/cm2 , at different entrance velocities. The ray-tracing method.

that there is satisfactory convergence of the radiation flux distributions depending on the number of spectral groups. Not only the number of spectral group is significant here; the distribution of the wave numbers of the spectral grid is also important. Figure 2.15 gives a representation of the spectral grid used. This figure presents spectral group radiation heat fluxes at two axial points along the internal surface of the LSPG

Section 2.3 Bifurcation of subsonic gas flows

81

chamber at x D 0, and at x D 3.6 cm. The spectral distribution shows that the maximum radiative heating lies in the range of the visible spectrum around  20 000 cm1 . The tendency of decreasing radiative heating of the internal surface with increasing entrance velocity is demonstrated in Figure 2.16. The high temperature region of the LSW in the plasma generator channel, as well as the relatively low temperature region, becomes smaller for increasing entrance velocities. One can also see that the region of heated gas is constricted (compare, for example, Figures 2.11 and 2.12). These two facts lead to a decrease of the radiative heating of the internal surface. The numerical simulation results were also obtained for a power of the CW CO2 -laser of PL D 100 kW and a radius of the LSPG channel of Rc D 2.0 cm. Temperature distributions were obtained by solving the radiative gas-dynamic equations at values of entrance velocities of hydrogen flow of u0 D 20–70 m/s. Note that for the larger channel radii the laser supported wave exists for higher input velocities. This can be explained by the larger mass flow which is blowing out through the region of higher temperature of the laser supported wave in the case of a decreased radius of the channel. It is obvious that in this case the conditions of the laser plasma cooling are much better. Therefore, the limit of the existence of the laser plasma in variables of laser power and input velocity is shifted in the direction of smaller velocities. So, the computational model presented in this section allows the study of hydrogen LSPG functioning in the radiative mode, i. e., when there are two general mechanisms of energy exchange: absorption of laser radiation in hydrogen plasma and radiation heat transfer of the high temperature plasma region with the ambient gas. A numerical study has been performed for regimes of strong radiative-convection interaction. It has been shown that the distribution of the thermic and gas-dynamic parameters is generally formed by the heat radiation transfer.

2.3 Bifurcation of subsonic gas flows in the vicinity of localized heat release regions This section is dedicated to a little-studied phenomenon of subsonic gas flows. The necessity for studying the gas-dynamic structure of subsonic gas flows through localized heat release areas arises in connection with research on laser supported waves (LSW). Certain kinds of energy devices are based on the specified phenomenon: the laser supported plasma generators (LSPG) [111] and the laser supported rocket engines (LSRE) [70, 103]. The interest in problems of gas flow interactions with localized heated areas has increased noticeably in physical gas dynamics recently, first of all in connection with the probable applications in practical aerodynamics. This study concentrates basically on physical regularities of subsonic gas motion through localized heat release areas. The problem considered in the following is characterized by a number of distinctive features:

82

Chapter 2 Nunerical investigation of laser supported waves

(1) The study of unsteady subsonic motion of viscous heat-conducting and radiating gas in area of heat release is based on the full system of the Navier–Stokes equations. Thus, specified nonsteady-state modes can be caused not only by external reasons, but also by the properties of the gas flows through the heat release area. (2) Temperature in the heat release areas can achieve  5 000–20 000K, that is, the gas becomes completely dissociated or ionized. Differences in gas densities in the region under consideration may reach  200 times, because the pressure in this area slightly differs from atmospheric pressure. (3) The radiative heat transfer, real thermophysical and transport properties are also taken into account. A heat release area is considered in the present section as being fixed in space. This means that any changes in the gas-dynamic parameters do not influence the heat release capacity. In reality when the gas is heated by the LSW, the situation is slightly different: the heat release capacity depends rather strongly on the distribution of gasdynamic parameters. However, in the latter case it is difficult to specify the reason for the occurrence of unsteady movements: whether on account of its internal properties, whether on account of periodic changes in the heat release configuration. In other words, the fixed heat release area allows for the exclusion of the influence of variability in the heat release area and, thus, to study the regularity of the occurrence of the unsteadiness caused by the properties of the gas flow.

r u0 R0

xp

Figure 2.17. Schematic of the problem. The heat release region has a spherical shape. Typical velocities are u0 D 10–200 m/s; typical temperatures inside the heat release region are T D 1 000–20 000 K.

83

Section 2.3 Bifurcation of subsonic gas flows

2.3.1 Statement of the problem The problem schematic is shown in Figure 2.17. On the symmetry axis (r D 0) at a point xp there is a hot area with the given distribution of heat release capacity   4    r x  xp 4 3Q0 exp   QV D , R0 R0 4 R03

(2.15)

where R0 is the radius of the heat release region, xp is the axial coordinate of its center, Q0 is the heat release power. Note that such a form practically coincides with the heat release distributions obtained from the self-consistent model of LSW (see, for example, Sections 2.1 and 2.2). Again, it should be stressed that in the case under consideration the distribution of heat release capacity (equation (2.15)) is determined only by spatial variables and depends neither on time, nor on gas-dynamic processes. Parameters of undisturbed gas were used at the entrance of the area under consideration: velocity u0 and temperature T0 . For the numerical simulation of subsonic gas flows through heat release regions the following system of the Navier–Stokes equations, mass and energy conservation equations, and also the equation of radiative heat transfer in the form of the P1 -multigroup approximation is used @ C div. V/ D 0, (2.16) @t @ u C div. uV/ D @t      @p 2 @ 1 @ @u @v @ @u   . divV/ C r C C2  , (2.17) @x 3 @x r @r @r @x @x @x @v C div.vV/ D @t        @p 2 @ @ @u @v @ @v @ v   . divV/ C  C C2  C , @r 3 @r @x @r @x @r @r @r r (2.18) cp

@T C cp VgradT D div. gradT /  QHR C QV , @t QHR D c

Ng X

 kg Ub,g  Ug !g ,

(2.19) (2.20)

gD1

div .3kg1 grad Ug / D kg .Ub,g  Ug /,

g D 1, 2, : : : , Ng ,

(2.21)

84

Chapter 2 Nunerical investigation of laser supported waves

where x, r are the radial and axial coordinates; , cp , T are the density, specific heat capacity at constant pressure and temperature; u, v are the axial and radial components of the flow velocity V; p is the pressure; ,  are the coefficients of viscosity and thermal conductivity; QHR is the volume capacity due to radiation heat transfer; k, U, Ub are the absorption coefficient, radiation volume density of the medium and absolutely block body. Subscript g indicates group properties as obtained by averaging the appropriate spectral characteristics within each of Ng spectral ranges of wave numbers !g . Validity of the local thermodynamic equilibrium (LTE) is assumed. The gas composition (air in this case) is considered at chemical equilibrium in each point of the calculation area at given temperature and pressure. Because of small speeds of the gas, the energy conservation equation does not contain the term representing heat release due to gas compressibility. Temperature dependence of the thermophysical (, cp ), transport (, ) and optical (kg ) properties of air are used only at atmospheric pressure, as their changes dependent on pressure are insignificant. At the initial time instant a Gaussian temperature distribution with maximal temperature 2 000 K is set. The following boundary conditions are used:    

at x D 0: V D .u D u0 , v D 0/, T D T0; at x D L.x ! 1/: @f =@x D 0 or @2 f =@x 2 D 0; at r D 0: @f =@r D 0; at r D Rc .r ! 1/: f D f0 or @f =@r D 0;

where f D fu, v, T , Ug g. The values xp , L, Rc are chosen in numerical experiments from the conditions of weak influence of the boundaries site on the calculation results in the vicinity to the heat release area. We need not formulate boundary conditions for pressure, because one is excluded from the consideration. The method of the Unsteady Dynamical Variables was applied to the solution of the problem (see Section 1.1.6). Thermophysical and group optical properties of low temperature air plasma were calculated using the ASTEROID computing system [106].

2.3.2 Qualitative analysis of the phenomenon Essentially, the problem is as follows: if at the fixed undisturbed gas velocity, for example at u0 D 100 m/sec, we gradually increase heat release power QV , each time using the just obtained solution for thermo-gasdynamic functions, then at some value QV a vortical (steady-state or unsteady) gas motion may arise behind the heat release area. If then at some value QV > QV we start to reduce QV gradually, then one can find that the vortical gas-dynamic structure is preserved at QV < QV down to a

Section 2.3 Bifurcation of subsonic gas flows

85

certain value QV < QV , at which the gas flow once again becomes laminar. In other words, we can fix such well-known phenomenon as hysteresis. But, from the point of view of computational fluid dynamics, one can fix the very significant fact that there is a certain range QV < QV < QV , where two qualitatively different gas-dynamic structures correspond to the same given data (u0 , QV , and other invariable entrance data). The specified gas-dynamic structures with vortical movement can be observed in a wide range of velocities u0  30–200 m/sec. One more fact may also be of interest. One of the two solutions obtained, namely the one with recoverable vortical motion, is steadier. The numerical experiments have shown that if the laminar solution is chosen as the initial one, and some indignation is introduced into the flow (for example, change of QV by a value QV within the limits ŒQV , QV /, then the following two results may be obtained: 



At small values QV the solution does not leave the initial branch of the solutions, that is, the flow remains laminar; At large enough QV the solution always converges to the alternative configuration, that is, becomes vortical.

If the vortical solution is chosen as the initial one, then indignations of gas-dynamic functions do not result in a change of the gas-dynamic pattern. It is also necessary to take into account three additional features of the obtained results: (1) At speeds of u0  20 m/sec the bifurcation of the gas-dynamic structure was not revealed. However, it should be stressed that it is impossible to interpret this conclusion as an absolute, because the calculations were carried out at fixed entrance parameters (pressure, geometry, size of the heat release region, etc.). (2) Near to the border of the conditional stability, that is, at QV  QV (boundary of the transition “laminar flow–vortical flow”) the unsteady solutions are observed as a rule. (3) With increasing QV , reduction of the range ŒQV , QV  in computing experiments is observed.

2.3.3 Quantitative results of numerical simulation Calculations were carried out for the following entrance speeds u0 D 10–200 m/s, and for fixed radial and axial coordinates of the heat release region: R0 D 0.4 cm, xp D 3 cm. A maximal value of the heat release power QV was varied in the range 2–20 kW/cm3. The greatest value QV was limited by maximal temperature inside the heated area (T  25 000 K).

86

Chapter 2 Nunerical investigation of laser supported waves

Firstly, we will consider numerical simulation results obtained from a step-by-step increase in the entrance velocity u0 . The computed results obtained for conditions corresponding to stability boundaries of the gas flows will then be considered in detail. Temperature and axial velocity distributions at u0 D 10 m/s and QV D 14.6 kW/cm3 are shown in Figure 2.18. r, cm 2

(a)

1.5 1

1 2

0.5 0

3

5 9 6 4

(b)

1.5

1

1

2 3

0.5

5 9 6

0 0

2

4

4

6

8

10

12

9 8 7 6 5 4 3 2 1

22 755 20 228 17 702 15 176 12 650 10 124 7 598 5 072 2 545

9 8 7 6 5 4 3 2 1

8.03 7.16 6.30 5.43 4.57 3.70 2.83 1.97 1.10

14 x, cm

Figure 2.18. Temperature in K (a) and axial velocity u=u0 (b) distributions at u0 D 10 m/s and QV D 14.6 kW/cm3 . Set of calculations with increasing heat release.

As mentioned above, transition from a laminar to a vortical mode of gas-dynamic structure is not detected in this case. A maximal temperature inside the heated region achieves approximately 24 000 K. It should be emphasized that the numerical solution in this case is not completely steady-state, and small periodic fluctuations of the velocity and temperature fields are observed. The steady-state solution was obtained at essentially smaller heat release power QV  2–5 kW/cm3. Beyond entrance velocity u0 D 30 m/s the phenomenon of bifurcation in the flow was detected, but solutions obtained at this velocity were not stable. A detailed discussion of these results will be presented slightly later. First, we will consider the calculated data obtained at velocities u0 D 40–200 m/s, because these data are stable enough. Temperature and velocity fields for the case of stationary laminar (QV D 2.5 kW/cm3 ) and quasi-steady-state vortical flow (QV D 3.0 kW/cm3) at u0 D 40 m/s are shown in Figures 2.19 and 2.20 respectively. The name “quasi-steady-state” is used here to emphasize that the solution obtained is not perfectly steady-state, but rather contains moderate velocity oscillations. The specified solutions were obtained in the set of calculations where QV was increased with steps of QV D C0.5 kW/cm3 . Maximal temperature inside the heated area achieves 5 200 K.

87

Section 2.3 Bifurcation of subsonic gas flows r, cm 2

(a)

1.5 1 9

0

2 1 6 5

3 7

0.5 8

(b)

1.5 1

0

3

3

0.5

2 1

0

8

9

7 1

2

4

6

6 4

8

2

5 3

10

12

9 8 7 6 5 4 3 2 1

3 589 3 224 2 858 2 493 2 127 1 762 1 396 1 031 665

9 8 7 6 5 4 3 2 1

1.81 1.67 1.52 1.37 1.22 1.07 0.93 0.78 0.63

14 x, cm

Figure 2.19. Temperature in K (a) and axial velocity u=u0 (b) distributions at u0 D 40 m/s and QV D 2.5 kW/cm3 . Set of calculations with increasing heat release.

r, cm 2 (a)

1.5 1 0.5

1 2 3

5 4 9

0 (b)

1.5 1

9 8

0.5 6

0 0

2

8

7

7 24 5 6 1

4

6

8

10

12

9 8 7 6 5 4 3 2 1

5 101 4 567 4 034 3 500 2 967 2 434 1 900 1 367 833

9 8 7 6 5 4 3 2 1

2.07 1.62 1.16 0.71 0.25 –0.21 –0.66 –1.12 –1.58

14 x, cm

Figure 2.20. Temperature in K (a) and axial velocity distributions u=u0 (b) at u0 D 40 m/s and QV D 3.0 kW/cm3 . Set of calculations with increasing heat release.

Figure 2.21 shows the temperature and velocity distribution at u0 D 40 m/sec and QV D 21.0 kW/cm3 . In this case, the maximal temperature amounts to 23 000 K. The numerical simulation results shown in Figures 2.22 and 2.23 were obtained for the case with a gradual reduction of QV with the steps of QV D 0.5 kW/cm3, when the vortical solution was taken as the initial one. Figure 2.22 shows temperature and velocity distributions at QV D 2.25 kW/cm3 , and Figure 2.23 shows the data at

88

Chapter 2 Nunerical investigation of laser supported waves r, cm 2 (a)

1.5 1 0.5

4

1

2

9

0 (b)

1.5 1

5

9 7

0.5 4

0 0

21

2

6

6 4 3 5

4

6

8

10

12

9 8 7 6 5 4 3 2 1

22 332 19 884 17 436 14 988 12 540 10 092 7 644 5 196 2 748

9 8 7 6 5 4 3 2 1

5.54 4.48 3.43 2.37 1.31 0.26 –0.80 –1.86 –2.91

14 x, cm

Figure 2.21. Temperature in K (a) and axial velocity distributions u=u0 (b) at u0 D 40 m/s and QV D 21.0 kW/cm3 . Set of calculations with increasing heat release.

r, cm 2 (a)

1.5 1 0.5

5 9

0

1

2

3

4

(b)

1.5 1 0.5

9 6 8

0 0

2

234 5 1

4

7

8 6

6

7

8

10

12

9 8 7 6 5 4 3 2 1

4 191 3 758 3 326 2 894 2 461 2 029 1 597 1 165 732

9 8 7 6 5 4 3 2 1

1.00 0.56 0.13 –0.31 –0.75 –1.19 –1.63 –2.06 –2.50

14 x, cm

Figure 2.22. Temperature in K (a) and axial velocity distributions u=u0 (b) at u0 D 40 m/s and QV D 2.25 kW/cm3 . Set of calculations with decreasing heat release.

QV D 2.0 kW/cm3. Note that in the set of calculations with increasing QV , the laminar solution was obtained at QV D 2.5 kW/cm3 . Thus, with the conditions considered it is possible to specify the range of values of QV inside which there is a bifurcation of the solution: u0 D 40 m/s, QV D 2.25 < QV < QV D 2.5 kW/cm3. At other speeds these bifurcational ranges are found to be:

89

Section 2.3 Bifurcation of subsonic gas flows

(a) u0 D 50 m/sec, QV D 2 < QV < QV D 3 kW/cm3 ; (b) u0 D 100 m/sec, QV D 3.5 < QV < QV D 5.5 kW/cm3; (c) u0 D 150 m/sec, QV D 5.5 < QV < QV D 7 kW/cm3 ; (d) u0 D 200 m/sec, QV D 7.5 < QV < QV D 9 kW/cm3 . r, cm 2 (a)

1.5 1 0.5

9

0

2 1 6

3 7

8

4

5

(b)

1.5 1

3 3

0.5

2 1

0 0

2

7

8

9

1

4

6

2 3

8

4

10

5

6

6

12

54

3

9 8 7 6 5 4 3 2 1

2 929 2 637 2 345 2 053 1 760 1 468 1 176 884 592

9 8 7 6 5 4 3 2 1

0.72 0.59 0.47 0.35 0.22 0.10 –0.02 –0.15 –0.27

14 x, cm

Figure 2.23. Temperature in K (a) and axial velocity distributions u=u0 (b) at u0 D 40 m/s and QV D 2.0 kW/cm3 . Set of calculations with decreasing heat release.

Numerical simulation results for the set of calculations with u0 D 30 m/sec are of particular interest for the analysis of unsteady gas-dynamic structures. Remember that at lower speeds the phenomenon of a flow bifurcation was not revealed. The specified speed u0 D 30 m/sec is near the bottom border of the range of speeds within which the bifurcation was found. This case is characterized by significant instability in the calculated data. Therefore, the calculations at this specified speed were performed with different values of QV . In the first set of calculations, the step for increasing the heat release power was QV D 0.5 kW/cm3 , in the second set QVII D 0.1 kW/cm3, and in the third set QVIII D 0.025 kW/cm3 . The transition from laminar to vortical motion was observed in the range QV D 1.5–2 kW/cm3 in the first case, and in the range QV D 3.7–3.8 kW/cm3 in the second case. In the third case, the transition from laminar to vortical movement was found in the range QV D 5.925–5.95kW/cm3. Figure 2.24 shows temperature and velocity distributions for laminar mode at u0 D 30 m/s and QV D 3.7 kW/cm3 , and Figure 2.25 shows the same data for vortical mode at u0 D 30 m/s and QV D 3.8 kW/cm3 .

90

Chapter 2 Nunerical investigation of laser supported waves r, cm 2 (a)

1.5 1 0.5

6

3 5 4

2 1

9 8 7 6 5 4 3 2 1

9 723 8 676 7 629 6 582 5 535 4 488 3 441 2 394 1 347

9 8 7 6 5 4 3 2 1

1.69 1.40 1.11 0.82 0.52 0.23 –0.06 –0.36 –0.65

987

0 (b)

1.5

3 3

1

4

3

0.5

8

9 2

1

0 0

2

4

6

7 4

6 2 3

6

5

8

10

12

14 x, cm

Figure 2.24. Temperature in K (a) and axial velocity distributions u=u0 (b) at u0 D 30 m/s and QV D 5.925 kW/cm3. Set of calculations with increasing heat release (QV D C0.025 kW/cm3).

r, cm 2 (a)

1.5 1

1

3

0.5

2

9

0 (b)

1.5 1

6

0.5 5

0 0

2

9 8

7

6

7 24 1 35 6

4

6

8

10

12

9 8 7 6 5 4 3 2 1

11 100 9 900 8 700 7 500 6 300 5 100 3 900 2 700 1 500

9 8 7 6 5 4 3 2 1

1.97 1.34 0.70 0.07 –0.57 –0.21 –0.84 –2.4 –3.11

14 x, cm

Figure 2.25. Temperature in K (a) and axial velocity distributions at u=u0 (b) u0 D 30 m/s and QV D 5.95 kW/cm3. Set of calculations with increasing heat release (QV D C0.025 kW/cm3).

Figure 2.26 shows the axial velocity distribution for laminar and vortical solutions obtained with increasing heat release power for this case. It indicates that reducing the level of heat release disturbance results in essentially larger values of the heat release

91

Section 2.4 Laser supported waves in the field of gravity

capacity at the transition from laminar to vortical flow. It is reasonable that the gas is heated up to the higher temperatures. For example, at the greatest heat release capacity QV D 5.925 kW/cm3 at which the laminar solution was observed, the temperature inside the heated area achieved T D 10 000 K. The distributions of temperature and axial velocity u for this case are shown in Figure 2.24. Appropriate distributions of the functions after transition to vortical motion (QV D 5.95 kW/cm3) are shown in Figure 2.25. The maximal temperature in this case reaches 12 000 K. u / u0 3 2 1 0 –1 –2 –3 0

5

10

x, cm

15

Figure 2.26. Axial velocity distributions at r D 0; u0 D 30 m/s and QV D 5.925 (solid line) and 5.95 (dashed line) kW/cm3. Set of calculations with increasing heat release.

The case of the calculations discussed here is also remarkable since the steady-state laminar solution does not exist at QV D 5.925 kW/cm3 . A self-oscillatory process with periodic variations of velocity components and temperature is observed in this case. Partly it can be seen from the distributions of temperature and speed (see Figure 2.24). Actually an instant photo is shown here. So, the numerical investigation of subsonic flows with localized heat release regions showed that at certain conditions in gas flows it is possible to detect a bifurcation of gas-dynamic structure. Namely, two different quasi-steady-state gas-dynamic configurations with the same initial data do exist.

2.4 Laser supported waves in the field of gravity This section deals with the late stage of optical breakdown of gas by laser radiation in the vicinity of a solid surface, where the breakdown-induced shock wave departed

92

Chapter 2 Nunerical investigation of laser supported waves r=R x=0

Laser breakdown region

(A)

x g

r

LSW

(B)

Laser beam x=L Direction of LSW motion

g

Vortex motion of gas

Direction LSW of LSW motion x=L Laser beam

Laser breakdown region

x

Figure 2.27. Schematic of laser supported waves in a gravity field.

from the region of almost instantaneous energy through a significant distance where a heated plasma region remained with an almost level atmospheric pressure. The scheme of the problem is given in Figure 2.27. It is assumed that the gas is quiescent at the site of optical breakdown. The approximation of local thermodynamic equilibrium (LTE) is applied, because the process develops at atmospheric pressure, and the characteristic times of development of the process (milliseconds) significantly exceed the times of relaxation to local equilibrium. The gas is taken to be viscous, heat-conducting, and selectively emitting and absorbing the radiation. As discussed before in Section 1.1, the LTE approximation enables one to pre-tabulate the data on the thermal, transport, and spectral optical properties of laser plasma in the entire temperature range of interest to us (300–20 000 K). The gas is assumed to be under normal conditions and quiescent at a significant distance from the site of formation of initially heated plasma cloud. Numerical analysis of regularities of propagation or steady existence of laser supported combustion waves for the experimental conditions of [60] was performed in [56, 107, 108, 111, 114]. The distinctive element of the problem being solved in the present study is the inclusion of the direction of gravitational acceleration with respect to the direction of the laser beam. Because the problem is solved in a two-dimensional axisymmetric formulation, only two combinations shown in Figure 2.27 are possible, namely, the vector of gravitational acceleration and the laser beam have the same or opposite directions. It is their relative orientation that will define the special features of convective motion arising behind the laser supported combustion wave, including gas-dynamic instabilities (conventionally shown in Figure 2.27).

93

Section 2.4 Laser supported waves in the field of gravity

The following radiation-gasdynamic model is employed for solving the problem of motion of slow laser supported combustion waves: @ C div.V/ D 0, @t

(2.22)

   @ u @p 2 @ 1 @ @u @v C div. uV/ D   .divV/ C r C @t @x 3 @x r @r @r @x     @u @ @u @  C2  C g, C2 @x @x @x @x    @ v @p 2 @ @ @u @v C div.vV/ D   .divV/ C  C @t @r 3 @r @x @r @x   @v 2 @v v @  C  2 2 , C2 @r @r r @r r   t @ H C div. VH / D div grad H C QL  QR , @t cp Z 1 Ng X k! .Ub,!  U! /d!  c kg .Ub,g  Ug /!g QR D c 

0

1 gradUg 3kg



(2.23)

(2.24) (2.25) (2.26)

gD1

D kg .Ug  Ub,g /,

g D 1, 2, : : : , Ng ,

(2.27)

  Z x   r2 1 kL,! dx , QL D kL,! PL exp  2 exp  2 RL  RL 0

(2.28)

div

where u and v are projections of the velocity vector V onto the x- and r -axes of cylindrical coordinates; p is the pressure;  and cp denote the density and specific heat capacity at constant pressure, respectively;  is the dynamic viscosity; t is the thermal conductivity coefficient; k, U , and Ub denote the volume absorption coefficient, the energy density of the medium, and the energy density of the black body, respectively; the subscript ! indicates the spectral characteristics, and the subscript g indicates the group spectral characteristics obtained by the averaging of the spectral optical properties in each one of Ng D 37 spectral ranges !g into which the spectral range of thermal radiation 1 000–500 000Rcm1 is divided; kL,! is the volume absorption coefficient of laser radiation; H D cp dt is the enthalpy; T is the temperature; and g is the vector of acceleration of gravity. The presence of a Gaussian distribution of radiation intensity along the beam radius RL is assumed in equation (2.28). The following relation PL WL D 2 RL will characterize, in what follows, the flux density of laser radiation of power PL .

94

Chapter 2 Nunerical investigation of laser supported waves

Because the approximation of local thermodynamic equilibrium in laser supported combustion waves is assumed to be valid, it is thereby assumed that the zone of nonequilibrium ionization, which exists in the immediate vicinity of the high temperature region because of intense ultraviolet radiation of the LSW proper, does not make a strong impact on the regularities of development of gas-dynamic processes. This assumption may be partly validated by the fact that the previous calculations of the velocity of propagation of laser supported combustion waves in the LTE approximation were indicative of the adequate description of experimental data. Nevertheless, this assumption calls for thorough verification in the future. The boundary conditions are formulated as follows: on the free boundaries of the computational domain, the gas is taken to be immovable under atmospheric conditions (in what follows, the parameters under normal conditions are marked by subscript 0); the conditions of attachment and constant temperature are used on the surfaces marked by hatching (see Figure 2.27); the conditions of symmetry are employed at r D 0. Boundary conditions which introduce no disturbances in the flow field are applied downstream from the computational domain. Special computational experiments were performed to confirm this. The mathematical formulation of the boundary conditions is as follows: @u D 0, @r

v D 0,

@ D 0, @r

r !1:

u D 0,

v D 0,

 D 0 ,

xD0:

u D v D 0,

r D0:

xDL:

@T D 0; @r T D T0 ;

T D T0 ;

@u @v @T D D D 0. @x @x @x

Atmospheric air is considered. The thermal, transport, and group optical properties of high temperature air (, cp , , t , kg , and kL,! ) are given in tabulated form at atmospheric pressure in the temperature range from 300 to 20 000 K. The ASTEROID computer code [106] is used for the calculation of spectral optical characteristics. The thermodynamic and transport properties are borrowed from [41]. It should be emphasized that the modes of motion of slow LSW under consideration correspond to strong radiation-gasdynamic interaction, and the thermal radiation of laser plasma has the decisive impact on the propagation of LSW. This is the so-called radiation mode of LSW motion. The calculations were performed for a CO2 -laser with wavelength  D 10.6 m. In this case, the principal mechanism of absorption of laser radiation is that of transferring the energy of electromagnetic radiation to electrons. This heating turns out to be very significant, so that the electrons transmit the energy to heavy particles by colliding with the latter and heating them up to dissociation and ionization, at which new electrons are produced. In accordance with the mechanism inverse to bremsstrahlung, the spectral

Section 2.4 Laser supported waves in the field of gravity

95

coefficient of absorption of laser radiation at wavelength  D 10.6 m is calculated by the formula ! D 6.435pe2.T =104 /7=2  lnŒ27.T =104/4=3 pe1=3  with electron pressure pe (in atm). For calculating this pressure (within the employed assumptions of local thermodynamic equilibrium), one must use the Saha equation     xe2 gi I pe T 5=2 D 6700 exp  , xe D , 4 1  xe ge p 10 kT p where xe is the relative mole concentration of electrons, gi and ge denote the statistical weight of ions and electrons, respectively, I is the ionization potential, and p is the gas pressure in atm. p The following numerical values were used: u0 D 981L D 31.3cm/s, L D 1.0cm, 0 D 1.177  103 g/cm3, 0 D 1.983  104 W/(cmK), cp,0 D 0=0 . In the present section, numerical simulation is used to investigate the regularities of the emergence of gas motion in the region of motion of laser supported combustion waves, which are defined by the presence of the force of gravity. Two versions of the relative orientation of the laser beam and the acceleration of gravity, which were realized in the calculations, demonstrate a fundamentally different gas-dynamic structure of LSW. But the main feature is that this phenomenon is observed only at certain values of density of laser radiation. The calculations were performed in the power density range of laser radiation WL D 0.06–0.029 MW/cm2 with a fixed radius of laser beam RL D 1.05 cm. This is the maximal transverse dimension of the beam in the experiments [60], which guarantees the radiation mode of propagation of LSW. In the case of the flux density of laser radiation WL < 0.0155 MW/cm2, the laser plasma in the calculations was extinguished. At WL > 0.29 MW/cm2, the plasma temperature exceeded 20 000 K, and the computational model employed became incorrect (first of all because of the need for the inclusion of double ionization of gas and for the correction of the thermal and optical properties employed). We will consider the results of the calculation of the dynamics of LSW in a laser beam of radius RL D 1.05 cm for different values of the power of laser radiation PL D 0.06, 0.08, 0.12, 0.207, 0.346, 0.5, and 1.0 MW. Only two of the these values (PL D 0.207 and 0.346 MW) correspond to the region investigated in the experiments [60]. The calculation data are analyzed in the following sequence. First, the results of calculation by scheme A (see Figure 2.27) are given for the entire set of values of the power of laser radiation, i. e., for the case where the LSW moves downward toward the laser beam. Then the results are given of calculations for all values of the power by scheme B, where the LSW moves upward (in the opposite direction to vector g). We will start analyzing the calculations results with the version corresponding to the least power of laser radiation PL D 0.06 MW, which corresponds to the flux density of laser radiation WL D 0.017 MW/cm2 . For a lower flux density of radiation, the calculation revealed the decay of LSW.

96

Chapter 2 Nunerical investigation of laser supported waves x, cm 0 1

Level

5 2

10

T

10 13 000 9 11 611 8 10 222 7 8 833 6 7 444 5 6 065 4 4 667 3 3 278 2 1 889 1 500

15 3

20 4

25

30

5 6

35

8 10

40 0

5

10

15

20 r, cm

Figure 2.28. The temperature field in LSW at PL D 0.06 MW at the instant in time t D 0.54 s. Computational scheme A.

Figure 2.28 gives the temperature field in LSW at time t D 0.54 s in calculations by scheme A. The LSW moves downward at an average velocity hV i D 0.69 m/s and leaves behind a heated region in which the gas moves at velocity about u D 7.8 m/s. The distribution of axial velocity at the instant in time t D 0.54 s is given in Figure 2.29. Here and in what follows, the axial velocity of gas is related to the characteristic velocity of convective motion, where u0 D 0.313 m/s is the characteristic velocity of convective motion. The negative values of axial velocity in Figure 2.29 indicate that the gas in LSW moves in the opposite direction relative to the direction of LSW motion (gas moves upward, and LSW downward). The average velocity of LSW motion increases with the flux density of laser radiation: at PL D 0.08 MW, it is hV i D 2.44 m/s, and at PL D 0.12 MW nonmonotonic with respect to two regions of maximal velocity, namely, in the high temperature region of LSW and in the vicinity of the surface (Figure 2.29 (b)).

97

Section 2.4 Laser supported waves in the field of gravity Vx 0 (a) –5 –10 –15 –20 0 (b) –5 –10 –15 –20 –25 –30

0

5

10

15

20

25

30

35 40 x, cm

Figure 2.29. The axial velocity of gas in LSW at PL D 0.06 MW (a) and at PL D 0.12 MW (b).

The subsequent two versions (PL D 0.21 and PL D 0.35 MW) correspond to the region of flux density of laser radiation, which was studied in the experiments [60]. The average velocity of LSW at PL D 0.21 MW (Figure 2.30) increased to hV i D 13.6 m/s, and the maximal velocity of gas within the LSW remained unchanged: u D 8.74 m/s. Therefore, the LSW in this case moves faster than the gas is accelerated within the heated region owing to the Archimedes force. Note further that the maximal temperature in the LSW likewise increases with the flux density of laser radiation. The degree of heating of plasma in the LSW and the velocity of its motion make a regular impact on the gas-dynamic structure. On the one hand, the heating of plasma to a higher temperature (with increasing PL / causes an increase in the rate of thermal gravitational convection. On the other hand, the velocity of motion of the leading front of LSW increases so that the heated gas does not have enough time to accelerate while

98

Chapter 2 Nunerical investigation of laser supported waves x, cm 0

Level T 10 9 8 7 6 5 4 3 2 1

1

5

10

19 822 17 869 15 917 13 964 12 012 10 059 8 107 6 154 4 202 2 249

15 2

20

25

30

3 4 5

35

10

0

5

10

15

20 r, cm

Figure 2.30. The temperature in LSW (in K) at PL D 0.21 MW at the instant in time t D 0.025 s. Computational scheme A. The modulus of gas velocity in the vicinity of the symmetry axis is 8.74 m/s.

the high temperature region is located at the site of the initial acceleration of the gas. The gas in the wake of LSW has enough time for cooling down and is accelerated to a lower velocity. The latter effect shows up at PL D 0.35 MW, where the average velocity of LSW motion continues to increase (hV i D 19.6 m/s), and the maximal velocity of gas decreases (u D 5.9 m/s). With further increases in the power of laser radiation, this tendency persists. The average velocity of LSW motion for PL D 0.5 MW and PL D 1.0 MW is hV i D 25.8 m/s and 38.9 m/s, and the highest axial velocity of gas decreases from u D 4.7 m/s to u D 3.8 m/s. In summing up the results of the analysis of the regularity of LSW motion in the direction of the vector of acceleration of gravity (downward), we make the following inferences:

Section 2.4 Laser supported waves in the field of gravity 

 







99

given the flux density of laser radiation, the LSW moves at an approximately unchanged velocity; the average velocity of LSW motion increases with flux density of radiation; when WL increases in the range from 0.017 to 0.29 MW/cm2 , the average velocity of LSW increases from hV i D 0.69 m/s to 38.9 m/s; the dependence of the maximal velocity of gas within the LSW with increasing flux density of laser radiation is nonlinear. For relatively low values of flux density of radiation (WL < 0.08 MW/cm2/, the maximal velocity increases somewhat in magnitude from u D 7.8 m/s to u D 8.7 m/s. With further increase in WL , the maximal velocity of gas within the LSW decreases to u D 3.8 m/s; the radial dimensions of LSW decrease somewhat with increasing flux density of radiation; and no gas-dynamic instabilities are present in the LSW wake: the gas flow is laminar.

The second series of calculations were performed for computational scheme B (see Figure 2.27). In this case, the LSW moves upward, i. e., in the opposition direction to the vector of acceleration of gravity. The gas heated by LSW likewise moves upward under the effect of the Archimedes force. The results of calculations with the power of laser radiation PL D 0.06 MW (WL D 0.0173 MW/cm2) are given in Figures 2.31 (a) and 2.31 (b). The average velocity of LSW motion is in this case hV i D 1.9 m/s. The velocity of gas within the LSW reaches a value of u  4.8 m/s. However, note must be made of the significant nonmonotonicity of axial distribution of velocity in the LSW. The two-dimensional distributions of temperature and velocity in the LSW, given in Figure 2.31, are indicative of the formation of vortices in the heated gas region during the LSW motion along the beam. The data given in these figures suggest the periodic emergence of vortex zones. Zones of vortex motion periodically arise on the boundary of the heated region. With developing vortex motion, conditions are developed on the boundary which promote the emergence of heated gas jets in the axial region directed toward the LSW motion. This pattern recurs with fixed periodicity, so that the velocity of gas periodically changes its direction from the stand point of an observer located in the frame of reference associated with the leading front of LSW. With the power of laser radiation increased to PL D 0.08 MW (WL D 0.023 MW/cm2/, the gas-dynamic structure of the LSW wake becomes much smoother, although some nonmonotonicity in the distribution of axial velocity is still observed. The average velocity of LSW motion up the beam is in this case hV i D 5.97 m/s. The axial distribution of hot gas velocity within the LSW wake has a maximum at 19 cm away from the surface. The maximal value of velocity is u D 6.24 m/s; one can see that this is somewhat higher than the velocity of motion of the LSW front. For the power PL D 0.12 MW, the average velocity of LSW is hV i D 15.7 m/s, and the maximal velocity of hot gas within the LSW wake is only u D 3 m/s. The vector

100

Chapter 2 Nunerical investigation of laser supported waves x, cm 40

x, cm LevelT T Level 1010 13000 13 000 9 9 11611 11 611 8 8 10222 10 222 7 7 8833 8 833 6 6 7444 7 444 5 5 6065 6 056 4 4 4667 4 667 3 3 3278 3 278 2 2 1889 1 889 1 1 500 500

(a)

35 10

30

(b)

8 5

25

4

20

15

3

10 2

5 1

0 0

5

10

15

20 r, cm

5

10

15

20 r, cm

Figure 2.31. The temperature (a) and the vector velocity (b) fields in LSW at PL D 0.06 MW at t D 0.0186 s. Computational scheme B.

velocity field (Figure 2.32 (b)) and the temperature distribution (Figure 2.32 (a)) point to the laminar pattern of flow of gas within the LSW and its wake. The subsequent four series of calculations, performed at PL D 0.21, 0.35, 0.5, and 1.0 MW, fully confirm these regularities. At PL D 0.21 MW (WL D 0.06 MW/cm2 ), the average velocity of LSW motion is hV i D 28.5 m/s, and the maximal velocity of gas within the LSW wake is u D 2.1 m/s; at PL D 0.35 MW (WL D 0.1 MW/cm2 ), we have hV i D 40.0 m/s and u D 1.9 m/s; at PL D 0.5 MW (WL D 0.145 MW/cm2 ), the velocity of LSW increases to hV i D 52.0 m/s; and, at PL D 1.0 MW (WL D 0.289 MW/cm2 ), the LSW velocity is hV i D 76.0 m/s. In the latter two cases, the maximal velocity of gas was u D 1.7 m/s and 1.56 m/s, respectively; in so doing, the maximal values of velocity in the front and tail of the wake become virtually the same.

101

Section 2.4 Laser supported waves in the field of gravity x, cm

x, cm

(a) 10

35 8 6

30

Level T 10 13 000 9 11 611 8 10 222 7 8 833 6 7 444 5 6 065 4 4 667 3 3 278 2 1 889 1 500

(b)

25 5

20

15

4

3

10

2

5 1

0 0

5

10

15

20 r, cm

5

10

15

20 r, cm

Figure 2.32. The temperature (a) and the vector velocity (b) fields in LCW at PL D 0.12 MW at t D 0.023 s. Computational scheme B.

In summing up the results of the analysis of regularities of upward LSW motion in the field of gravity, one can make the following conclusions: 

at fixed flux density of laser radiation, the LSW moves approximately uniformly, with the possible exception of the case of extremely low values of flux density of laser radiation;



the average velocity of LSW motion increases with increasing flux density of laser radiation. With WL increasing in the range from 0.017 to 0.29 MW/cm2, the average velocity increases from hV i D 19 m/s to 76.4 m/s; at relatively low values of flux density of laser radiation (WL < 0.020 MW/cm2/, the velocity of gas within the LSW exceeds the velocity of motion of LSW proper.



102

Chapter 2 Nunerical investigation of laser supported waves 〈V 〉, m/s 80 1 60

3

40 2 20

0

0

10

15

20

5

15 10 W, MW/cm2

Figure 2.33. The average velocity of LCW motion: (1) in opposition to the force of gravity (upward), (2) in the direction of the force of gravity (downward), (3) disregarding the thermal gravitational convection.

In these cases, the emergence of vortex motion is observed: the heated gas under the effect of the Archimedes force develops a higher pressure on the LSW front, which is the principal reason for vortex formation; 



at high values of flux density of laser radiation (WL 0.030 MW/cm2 /, the velocity of convective motion of gas within the LSW is lower than the velocity of motion of the LSW front. In this case, the convective motion of gas within the LSW is laminar, because the floating up gas falls behind the leading front of LSW motion; and the vortex motion of gas, which develops at low velocity of LSW motion, causes a periodic variation of the velocity of gas in the LSW front.

So, as a result of studying the LSW motion in the field of gravity, it was found that the thermal gravitational convection makes an appreciable impact on the velocity of this motion. The LSW moves upward (in opposition to the field of gravity) approximately twice as fast as it moves downward. The calculated dependencies of the velocity of LSW motion on the flux density of laser radiation are given in Figure 2.33. Also given in this figure is the predicted velocity of LSW motion disregarding the thermal gravitational convection. Two mechanisms of emergence of convection are identified during the upward motion of LSW. In the case of low velocity of movement of the LSW front, a vortex turbulent motion of hot gas arises in the LSW wake. In so doing, a periodic variation of the direction of gas velocity is observed in the LSW front. In fact, one can speak

Section 2.4 Laser supported waves in the field of gravity

103

of a new hybrid form of thermal-convective instability of the gas-dynamic structure of flow. This instability is characterized by classical gravitational convection and by the velocity of movement in space of the source of heating the gas (in this case, laser supported combustion wave). At relatively high velocities of movement of the LSW front, a laminar flow of hot gas is formed in the LSW wake.

Chapter 3

Computational models of magnetohydrodynamic processes The calculation models of homogeneous chemically equilibrium plasma considered in Section 1.1 include as the important constituent the account of a magnetic field action on plasma flows. The one-liquid one-temperature magnetohydrodynamic (MHD) model was considered. Such computing models are widely applied in plasma dynamics and in gas-discharge physics. The set of MHD equations is also an object of interest for many basic directions of aerospace sciences, as well as computational fluid dynamics (CFD). However, at numerical realization of the MHD equations there is a series of problems which are discussed in the given chapter. Singularity of the CFD problems, which relates to plurality of a statement of physically equivalent equations expressing conservation laws of mass, impulse and energy [1], appears in the MHD to an even greater degree. In MHD theory, the fluid dynamic equations are added not only by the Maxwellian equations, but also the momentum conservation equation and energy conservation equations have to be modified. The outcome is the complication of the complete system of the equations and an increasing number of variants of the equations statements. From the physical point of view, these various statements of the equations are anything other than equivalent entries of the same conservation laws. From the point of view of the computing analysis, the various statements of the equations lead to consequences of basic character. In this chapter the following systems of variables will be considered: ˇ ˇT ˇ ˇ U D ˇ   u  v  w  E Bx By Bz ˇ , (3.1) ˇ ˇ W e D ˇ  u v w e Bx By Bz ˇ ˇ W p D ˇ  u v w p Bx By Bz

ˇT ˇ ˇ ,

(3.2)

ˇT ˇ ˇ ,

(3.3)

where , p are the density and pressure; u, v, w are the projections of a mass flux vector to coordinate axes x, y, z; u, v, w are the velocity projections to coordinate axes; E is the total energy of volume unit; e is the specific internal energy (note, that unlike Chapter 1, the symbol “e” is chosen here for a specific internal energy); Bx , By , Bz are the projections of a magnetic induction vector to coordinate axes. Variables (3.1) are known as conservative variables, while variables (3.2) and (3.3) are referred to as natural variables. In addition, a vector of characteristic variables will be formulated below with corresponding comments and explanations.

105

Section 3.1 General relations

Conversion from one type of variables to another, as a rule, means a modification of the type of the combined equations. However, it is necessary to note that it is not always possible to pass from one type of notation to another without using some additional relations. The problems of the use of such relations with reference to three-dimensional MHD problems is the general goal of the given chapter.

3.1 General relations Initially we shall choose the system of the MHD equations formulated and substantiated in the form of conservation laws of continuum mechanics together with Maxwellian set of equations in the works [1, 65]: @ C divV D 0 1 , @t

(3.4)

@ V C div.V  V/ D grad PO C ŒJ  B, @t

(3.5)

@e C div.e  V/ D divq  PO : divV C .E0  J/, @t @B D rot E, @t 1 rotB, JD 0

(3.6) (3.7) (3.8)

divB D 0,

(3.9)

divJ D 0,

(3.10)

where , V are the density and velocity with components along x, y, z coordinate axes designated accordingly u, v, w; e is the specific internal energy; PO is the stress tensor; J, B, E are the current density, induction of a magnetic field, electric field strength; q is the vector of a thermal flux; 0 is the magnetic permeability of free space (further, the system of units SI is used, therefore 0 D 4  107 kg  m  C2 or H/m). In the energy conservation equation (3.6) the electric field strength E0 is calculated in a plasma moving with an average velocity V, i. e., it is related to an electric field strength in a laboratory coordinate system with the following relation: E0 D E C ŒV  B.

(3.11)

Components of a stress tensor can be written as follows: P˛ˇ D pı˛ˇ  ˇ ˛ , 1

Furthermore, in equations and formulas the letter symbols for differential operators and also inverted delta operator r, equivalent for the field theory, will be used in an equal measure.

106

Chapter 3 Computational models of magnetohydrodynamic processes

where p is the pressure; ı˛ˇ is the Kronecker delta; ˇ ˛ are the components of viscous stress tensor    @ uˇ 2 @ u˛

ˇ ˛ D   ıˇ ˛ div V , C (3.12) @ x˛ @ xˇ 3 where  is the coefficient of dynamic viscosity.

3.2 Vector form of Navier–Stokes equations The compressible Navier–Stokes equations in Cartesian coordinates can be written as: 

@U E  FE Eu C F E NS D Q, Cr @t

(3.13)

E Eu is the vector function of where U is the vector function of conservative variables; F NS flow in the set of Euler equations; FE is the vector function, which includes components of a viscous stress tensor and a thermal conduction flux: E FE Eu D Ei f C jEg C kh, ˇ ˇ ˇ ˇ ˇ U D ˇˇ ˇ ˇ ˇ

 u v w E

ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ

E NS , E NS D Ei fNS C jEgNS C kh F

ˇ ˇ ˇ ˇ ˇ ˇ v u ˇ ˇ ˇ 2 ˇ ˇ ˇ vu C p u ˇ ˇ ˇ ˇ, g D ˇ uv v 2 C p f D ˇˇ ˇ ˇ ˇ ˇ ˇ uw vw ˇ ˇ ˇ ˇ vŒE C .p=/ ˇ uŒE C .p=/ ˇ ˇ ˇ ˇ ˇ w ˇ ˇ ˇ ˇ wu ˇ ˇ ˇ wv h D ˇˇ ˇ 2 ˇ ˇ w C p ˇ ˇ ˇ wŒE C .p=/ ˇ

ˇ ˇ 0 ˇ ˇ

xx ˇ

xy fNS D  ˇˇ ˇ

xz ˇ ˇ u C v C w C  @T xx xy xz @x ˇ ˇ 0 ˇ ˇ

yx ˇ NS ˇ

yy g D ˇ ˇ

yz ˇ ˇ u yx C v yy C w yz C  @T @y

ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ

(3.14)

107

Section 3.3 System of equations of magnetic induction

hNS

ˇ ˇ 0 ˇ ˇ

zx ˇ

zy D  ˇˇ ˇ

zz ˇ ˇ u C v C w C  @T zx zy zz @z

ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ

(3.15)

E D e C .V2 =2/ is the total specific energy, T is the temperature;  is the thermal conductivity coefficient; Q is the source of mass, impulse or energy in a full set of the Navier–Stokes equations. Viscous stress tensor components have the following form:

xx D 2

@u 2   divV, @x 3

@v 2   divV, @y 3   @v @u C , D yx D  @y @x   @w @v C , D zy D  @z @y

yy D 2

@v 2   divV; @z 3   @u @w C , D zx D  @z @x

zz D 2

xy

xz

yz

where  is the coefficient of dynamic viscosity. The perfect gas equation of state will be used further in the chapter: p eD ,  .  1/

(3.16)

where D cp =cv is the ratio of specific thermal capacities at constant pressure and volume. We shall also give some useful relations which will be used further: e D cv T ,

p Q D cv .  1/T , D RT  RQ D cv .  1/,

cp D

RQ , 1

cv D

RQ , 1

p D .E  0.5VE 2 /.  1/.

(3.17) Under air normal conditions (p D 10 Pa, T D 293 K) one can get: D 1.4, RQ D 287 J/(kg  K). 5

ı

3.3 System of equations of magnetic induction The basis of the equations of perfect gas magnetohydrodynamics (further referred to as the MHD equations) are the Faraday and the Ampere laws, which are formulated in the following form [2, 64, 71, 98]: Z Z @B Edl D  dS, (3.18) S @t  Z Z  @D dS, (3.19) JC Hdl D @t S

108

Chapter 3 Computational models of magnetohydrodynamic processes

The Faraday law relates an electric field strength along a contour l enveloping an area S to time change of a magnetic flux through this area. The Ampere law describes the scalar magnetic potential induced in the contour l due to currents through this area, enveloped by this contour. Differential consequences of these conservation laws (neglecting displacement currents) enter into a complete set of MHD equations in the form of @B D rot E, @t rotB D 0J.

(3.20) (3.21)

The set of equations (3.20) and (3.21) is not closed. To integrate the equations there is a need to define additional closing conditions. The best known technique is the use of the generalized Ohm law J D fE C ŒV  Bg. (3.22) where  is the electroconductivity. Basically, the three equations presented here are enough to describe all variety of electromagnetic processes within the limits of an ideal magnetohydrodynamics. However, the set of Maxwellian equations contains one more equation E  B/ D 0, .r

(3.23)

which should be fulfilled in the whole investigated area (including initial conditions). Besides, the above-mentioned assumption about neglecting displacement currents imposes the following condition on total current E  J/ D 0, .r

(3.24)

which also should be fulfilled in the whole investigated area. It has been established [2, 53, 64, 71] that the optimal procedure to complete the set of MHD equations is adding to equations of Euler or Navier–Stokes the equations for components of a magnetic induction vector obtained by a combination of the equations (3.20)–(3.22). However, before starting on the realization of this procedure, it is necessary to solve a basic problem, namely how to use additional, but the unconditional equations (3.23) and (3.24). We will consider two methods of obtaining the required equations. In the first, during transformations of the initial set of equations (3.20)–(3.22) to the final form the equation (3.23) will be used, and in the second case it will not. Note that from equation (3.20) the divergence of a magnetic field at any moment automatically follows zero if it was equal to zero in initial. However, from the mathematical point of view, the set of equations obtained will possess various properties, and these differences will be rather significant for the subsequent numerical realization.

109

Section 3.3 System of equations of magnetic induction

First of all, let us derive the equation for electric field strength with the use of equations (3.21), (3.22): ED

1 1 E J  ŒV  B D Œr  B  ŒV  B.  0

(3.25)

Then, from equation (3.20) @B E  1 J C Œr E  ŒV  B D Œr @t 

  1 E E Œr  B  ŒV  B D r 0 D

(3.26)

1 E  B/  V2 Bg C V.r E  B/ C .B  r/ E V  B .r E  V/  .V  r/ E B. f.r 0

The first part of equation (3.26) can be presented in the form of the sum of two functions: @B D Fmd C FB , (3.27) @t where 1 E 1 E E  B D  1 fr. E r E  B/  .r E  r/Bg, E  J D  Œr  Œr Fmd D  Œr  0 0 E  ŒVE  B D V.r E  B/ C .B  r/V E  B.r E  V/  .V  r/B. E FB D Œr The first of the introduced functions Fmd corresponds to a change of a magnetic field induction as a result of currents in a medium with finite conductivity, and the second function FB corresponds to a change of a magnetic induction due to generation electric fields in a moving medium. Note that the approximation of infinite electrical conductivity . ! 1/, which is rather widespread in MHD applications, gives the following inequality Fmd  FB . Now one can write projections of vector functions FB and Fmd with and without E  B/ D 0. including the above-mentioned absolute condition .r E  B/ D 0: (1) Components of function FB taking account of the equation .r   @u @u @w @Bx @Bx @Bx @v C C By C Bz u v w , FxB D Bx @y @z @y @z @x @y @z   @v @v @By @By @By @u @w B  By C C Bz u v w , Fy D Bx @x @x @z @z @x @y @z   @w @w @Bz @Bz @Bz @u @v C By  Bz C u v w . FzB D Bx @x @y @x @y @x @y @z

(3.28)

110

Chapter 3 Computational models of magnetohydrodynamic processes

E  B/ D 0: (2) Components of function FB without taking into account the equation .r FxB D

@ @ .uBy  vBx / C .uBz  wBx /, @y @z

(3.29)

  @

@

vBx  uBy C vBz  wBy , @x @z  @ @

.vBx  uBz / C wBy  vBz . D @x @y

FyB D FzB

E  B/ D 0 the Let us pay attention to the fact that when taking account of the equation .r set of equations (3.28) is obtained for the definition of magnetic field induction components in the quasilinear form, in which differential operators are applied only to natural E  B/ D 0 has not been taken into consideration, variables (u, v, w, Bx , By , B/. If .r the set of equations (3.29) is formulated in a conservative form. The consequences of this fact will be discussed below. E  B/ D 0: (3) Components of function Fmd taking account of the equation .r Fxmd

1 D 0

Fymd D

1 0

Fzmd D

1 0

  

@ 2 Bx @ 2 Bx @ 2 Bx C C @x 2 @y 2 @z 2 @ 2 By @ 2 By @ 2 By C C 2 2 @x @y @z 2 @ 2 Bz @ 2 Bz @ 2 Bz C C @x 2 @y 2 @z 2

 ,

(3.30)

 ,  .

E  B/ D 0: (4) Components of function Fmd without taking into account the equation .r Fxmd

1 D 0

Fymd

1 D 0

Fzmd



@ 2 Bx @ 2 Bx @ 2 Bx C C @x 2 @y 2 @z 2



 

@ 2 Bx @ 2 By @ 2 Bz C C @x 2 @x@y @x@z

 , (3.31)

 @ 2 By @ 2 Bx @ 2 Bz C ,  C @x@y @y 2 @y@z  2   2  @ Bz @ Bx @ 2 By @ 2 Bz 1 @ 2 Bz @ 2 Bz  . C C D C C 0 @x 2 @y 2 @z 2 @x@z @y@z @z 2 

@ 2 By @ 2 By @ 2 By C C @x 2 @y 2 @z 2





So, two physically equivalent forms of a set of magnetic induction equations have been obtained.

111

Section 3.4 Force acting on ionized gas from electric and magnetic fields

In summarizing the section we shall also give component-wise expressions of Fmd with projections of a current density 

Fxmd

@Jz @Jy  D @y @z Fzmd





@Jx @Jz  D , @z @x   @Jy @Jx  D . @x @y Fymd

 ,

(3.32)

3.4 Force acting on ionized gas from electric and magnetic fields The full force acting on gas volume unit is: Fem D c E0 C ΠJ  B ,

(3.33)

where c is the density of charges. In the theory of ideal magnetohydrodynamics c D 0, hence, Fem D ŒJ  B .

(3.34)

This force is also presented in the right-hand side of the equation (3.5). To write expressions for projections of this force to coordinate axes we shall substitute in (3.34) an expression for a current density in the form of (3.21)    1  E  B D  1 r E .B  B/  .B  r/B E B  Œr . 0 0

(3.35)

Fxem

  1 @B 2 1 @Bx @Bx @Bx C Bx C By C Bz ,  D 0 2 @x @x @y @z

(3.36)

Fyem

  1 @B 2 @By @By @By 1  C Bx C By C Bz , D 0 2 @y @x @y @z

(3.37)

Fzem

  @Bz @Bz @Bz 1 @B 2 1 C Bx C By C Bz ,  D 0 2 @z @x @y @z

(3.38)

Fem D Then

where B 2 D Bx2 C By2 C Bz2 . Equations (3.36)–(3.38) are written in the hybrid conservative and nonconservative form. To rewrite these equations in the homogeneous form we shall use some formal transformations.

112

Chapter 3 Computational models of magnetohydrodynamic processes

Let us add and subtract from the right-hand side of formula (3.36) an item of , then

1 B @Bx 0 x @x

Fxem D    @Bx Bx @Bx By @Bx Bz @By @Bz 1 @B 2 @Bx 1 C C C  Bx C C .  0 2 @x @x @y @z @x @y @z (3.39) E  B/ D 0, Or, with the use of .r   @B B @Bx Bz 1 1 @  2 x y em 2 2 Bx  By  Bz C C . Fx D 0 2 @x @y @z

(3.40)

By analogy

  @B B @By Bz 1 1 @  2 x y By  Bx2  Bz2 C C , (3.41) 0 2 @y @x @z   @B B @By Bz 1 1 @  2 x z em 2 2 Fz D Bz  Bx  By C C . (3.42) 0 2 @z @x @y Thus, identical transformations of formulas (3.36)–(3.38) with the use of the equation E  B/ D 0 allow to obtain the locally-conservative form (3.40)–(3.42). .r Otherwise, the nonconservation form for the force projections can be obtained:   @Bx @Bx 1 @  2 1 Fxem D By C Bz2 C By C Bz , (3.43)  0 2 @x @y @z   1 @  2 1 @By @By  Fyem D Bx C Bz2 C Bx C Bz , (3.44) 0 2 @y @x @z   @Bz @Bz 1 1 @  2 em 2 C By Fz D  Bx C By C Bx . (3.45) 0 2 @z @x @y It will be shown below that exactly these formulas are most convenient for deriving the quasilinear form of the MHD equations. Fyem D

3.5 A heat emission caused by action of electromagnetic forces Let us consider transformations of Joule heat .E0  J/ entering into the energy conservation equation (3.6). According to the MHD theory, the primed electric field strength E0 considers its difference in a moving continuum from an electric field strength E in the same point of space of a laboratory frame [2, 53, 64, 71]: E0 D E C ŒV  B.

(3.46)

113

Section 3.5 A heat emission caused by action of electromagnetic forces

Thus, Joule heat can be written in the following form QJ D .E C ŒV  B/ D

1 E 1 E  B C ŒV  B  Œr E  Bg Œr  B D fE  Œr 0 0

1 E  E  r E  ŒE  Bg, fB  Œr 0

(3.47)

where the following vectorial identity was used: E  H D H  Œr E  E  r E  ŒE  H , E  Œr E  B  Œr E  B is equal to zero as a scalar product of parallel vectors. and the item Œr Designed expression for Joule heat can be derived from (3.47) with the use of the Maxwell equation (3.7) after calculation of a vector product ŒE  B divergence: 1 @B 2 20 @t   @ @ 1 @ .Ey Bz  Ez By /C .Ez Bx  Ex Bz / C .Ex By  Ey Bx / .  0 @x @y @z (3.48)

QJ D 

Projections of electric field strength E are expressed by projections of current density, velocity and induction of a magnetic field to coordinate axes by using the generalized Ohm law J D  fE C ŒV  Bg (3.49) as follows: Ex D

1 1 Jx  .vBz  wBy / D  0



@Bz @By  @y @z

  .vBz  wBy /,

(3.50)

  @Bx @Bz 1 1 Jy  .wBx  uBz / D   .wBx  uBz /, (3.51)  0 @z @x   @By @Bx 1 1   .uBy  vBx /. (3.52) Ez D Jz  .uBy  vBx / D  0 @x @y Further transformations of derivatives in (3.48) are identical, therefore we shall consider only the first one (let us pay attention to adding a term which is equal to zero): Ey D

1 @ .Ey Bz  Ez By / 0 @x      By @By @Bx 1 @ Bz @Bx @Bz   wBx Bz C uBz Bz    0 @x 0 @z @x 0 @x @y  C uBy By  vBx By C uBx Bx  uBx Bx

QJx D 

D QJx , C QJx ,B ,

(3.53)

114

Chapter 3 Computational models of magnetohydrodynamic processes

where QJx ,

     @Bx @Bz @By @Bx 1 @ D 2  Bz   By  , @z @x @x @y 0  @x

QJx ,B D 

1 @ ŒuB 2  Bx .uBx C vBy C wBz /.  0 @x

Making similar calculations with two other derivatives and introducing the magnetic pressure pm D B 2 =20, we obtain an expression for Joule thermal emission   @ @pm 1  2upm  Bx .uBx C vBy C wBz / @t @x 0   1 @ 2vpm  By .uBx C vBy C wBz /  @y 0   1 @ 2vpm  By .uBx C vBy C wBz /  @z 0    

@Bx @Bz @ @Bx @By 1 By  C Bz   2  @y @x @z @x 0  @x      @By @Bz @By @Bx @ 1 Bz  C Bx   2  @z @y @x @y 0  @y      @ @Bz @Bx @Bz @By 1 Bx  C By  .  2  @x @z @y @z 0  @z

QJ D 

(3.54)

This expression is written in a locally-conservative form which will be applied further. The time derivative from magnetic pressure is usually included in a time derivative from total volume energy, so that the energy conservation equation can be solved concerning the sum of a total energy and magnetic pressure @ .E C pm / D : : : . @t

3.6 Complete set of the MHD equations in a flux form The generalized form of the MHD equations in three-dimensional rectangular Cartesian coordinates is presented as follows: @U E F E † D Q or Cr @t

@f @g @h @U C C C D Q, @t @x @y @z

(3.55)

115

Section 3.6 Complete set of the MHD equations in a flux form

where E Eu C F E NS C F E MGD C FE VMGD , FE † D F f D fEu C fNS C fMGD C fVMGD ,

E † D Ei f C jE g C kE h, F

g D gEu C gNS C gMGD C gVMGD ,

h D hEu C hNS C hMGD C hVMGD , E † to coordinate axes. f, g, h are the projections of vector function of flows F

3.6.1 The MHD equations in projections First we shall give the flux form of the MHD equations at which the electric field strength and a current density is present in explicit form2 : ˇ ˇ ˇ ˇ ˇ ˇ u ˇ ˇ  ˇ ˇ ˇ ˇ 2 ˇ u C p ˇ ˇ ˇ u ˇ ˇ ˇ ˇ ˇ ˇ uv ˇ ˇ v ˇ ˇ ˇ ˇ ˇ ˇ uw ˇ ˇ w ˇ ˇ,  ˇ , fEu D ˇ U D ˇˇ (3.56) ˇ p ˇ ˇ u E C  ˇ ˇ E C pm ˇ ˇ ˇ ˇ ˇ Bx ˇ ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ By 0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Bz 0

gEu

2

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ  xx ˇ ˇ ˇ ˇ  xy ˇ ˇ ˇ ˇ 

xz ˇ ˇ  fNS D ˇ ˇ, @e ˇ  u xx C v xy C w xz C cv @x ˇ ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ 0 ˇ v ˇ ˇ ˇ 

ˇ yx vu ˇ ˇ ˇ  yy v 2 C p ˇˇ ˇ ˇ  yz ˇ

vw p  ˇ , gNS D ˇˇ  v E C  ˇˇ ˇ  u yx C v yy C w yz C ˇ ˇ ˇ 0 0 ˇ ˇ ˇ ˇ 0 0 ˇ ˇ ˇ ˇ 0 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, @e ˇ cv @y ˇˇ ˇ ˇ ˇ ˇ ˇ

For this reason it is impossible to divide here components of flows related only with a magnetic induction and electrical conduction.

116

Chapter 3 Computational models of magnetohydrodynamic processes

hEu

ˇ ˇ w ˇ ˇ wu ˇ ˇ wv ˇ ˇ w 2 C p ˇ  Dˇ ˇ w E C p ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

hNS

fMGD C fVMGD

gMGD C gVMGD

hMGD C hVMGD

ˇ ˇ 0 ˇ ˇ 

zx ˇ ˇ 

zy ˇ ˇ 

zz ˇ Dˇ  ˇ  u zx C v zy C w zz C ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ 1 1 2 2 ˇ 0 . 2 B  Bx / ˇ ˇ ˇ  10 Bx By ˇ ˇ 1  0 Bx Bz ˇ ˇ, 1 ˇ 0 .Ey Bz  Ez By / ˇ ˇ ˇ 0 ˇ CuBy  vBx C 1 Jz ˇˇ ˇ CuBz  wBx  1 Jy ˇ ˇ ˇ 0 ˇ ˇ 1 ˇ  0 By Bx ˇ ˇ 1 1 2 2 . B  B / ˇ y 0 2 ˇ ˇ 1  0 By Bz ˇ ˇ, 1 ˇ 0 .Ez Bx  Ex Bz / ˇ ˇ vBx  uBy  1 Jz ˇ ˇ ˇ 0 ˇ ˇ 1 vBz  wBy C  Jx ˇ ˇ ˇ 0 ˇ ˇ ˇ  10 Bz Bx ˇ ˇ 1 ˇ  0 Bz By ˇ ˇ 1 1 2 2 . B  B / ˇ z 0 2 ˇ, 1 .Ex By  Ey Bx / ˇˇ 0 ˇ wBx  uBz C 1 Jy ˇ ˇ wBy  vBz  1 Jx ˇˇ ˇ ˇ 0 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, @e ˇ cv @v ˇˇ ˇ ˇ ˇ ˇ ˇ

117

Section 3.6 Complete set of the MHD equations in a flux form

where Ex D  1 Jx  .vBz  wBy /, Ez D 

1



Jz  uBy  vBx ;

  1 @Bx @Bz Jy D  , 0 @z @x

Ey D  1 Jy  .vBx  uBz /;   1 @Bz @By Jx D  ; 0 @y @z   1 @By @Bx Jz D  . 0 @x @y

3.6.2 Completely conservative form of the MHD equations In this section we rewrite equations (3.56) in the form containing results of Sections 3.3 and 3.4. ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ u  ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ ˇ u C p u ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ v uv ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ uw w Eu ˇ ˇ ˇ ˇ D UDˇ (3.57) , f ˇ u .E C p=/ ˇ , ˇ ˇ ˇ ˇ E C pm ˇ ˇ ˇ ˇ ˇ 0 Bx ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ By 0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 0 Bz

fNS

gEu

ˇ ˇ 0 ˇ ˇ  xx ˇ ˇ  xy ˇ ˇ  xz D ˇˇ

 u

C v

xx xy C w xz C ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ v ˇ ˇ vu ˇ ˇ v 2 C p ˇ ˇ

vw p  D ˇˇ v EC ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

gNS

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ,  @e ˇ cv @x ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ 0 ˇ ˇ  yx ˇ ˇ  yy ˇ ˇ  yz D ˇˇ

 u

C v

yx yy C w yz C ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ,  @e ˇ cv @v ˇ ˇ ˇ ˇ ˇ ˇ

118

Chapter 3 Computational models of magnetohydrodynamic processes

hEu

ˇ ˇ w ˇ ˇ wu ˇ ˇ wv ˇ ˇ w 2 C p

D ˇˇ p ˇ w E C  ˇ 0 ˇ ˇ 0 ˇ ˇ 0

fMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

hNS

ˇ ˇ 0 ˇ ˇ  zx ˇ ˇ  zy ˇ ˇ  zz D ˇˇ

 u

C v

zx zy C w zz C ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ 1 1 2 2 ˇ . B  B / x 0 2 ˇ ˇ 1  0 Bx By ˇ ˇ ˇ  10 Bx Bz ˇ 1 1 2 uB  0 Bx .uBx C vBy C wBz / ˇˇ 0 ˇ ˇ 0 ˇ ˇ vBx C uBy ˇ ˇ ˇ wBx C uBz 0

ˇ ˇ 0 ˇ ˇ 1 2 ˇ  20 .Bx  By2  Bz2 / ˇ ˇ  10 Bx By ˇ ˇ ˇ  10 Bx Bz D ˇˇ 1 ˇ 2upm  0 Bx .uBx C vBy C wBz / ˇ ˇ 0 ˇ ˇ vBx C uBy ˇ ˇ ˇ wBx C uBz

fVMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1 20 

 @Bx By @y 

0 0 0 0 @By  @x

0

C Bz

@Bx @z

@By @Bx  1 0  @x  @y

x @Bz   @x  10  @B @z

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ,  @Bz ˇ  @x ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ,  @e ˇ cv @v ˇ ˇ ˇ ˇ ˇ ˇ

119

Section 3.6 Complete set of the MHD equations in a flux form

gMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ  10 By Bz ˇ ˇ 1 1 2 0 vB   0 By .uBx C vBy C wBz / ˇ ˇ ˇ uBy C vBx ˇ ˇ 0 ˇ ˇ ˇ wBy C vBz

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

0 1  0 By Bx

 1 1 2 B  By2 0 2

ˇ ˇ 0 ˇ ˇ ˇ .1=0 /By Bx ˇ 1 ˇ  20 .By2  Bx2  Bz2 / ˇ ˇ ˇ  10 By Bz D ˇˇ 1 ˇ 2vpm  0 By .uBx C vBy C wBz / ˇ ˇ uBy C vBx ˇ ˇ 0 ˇ ˇ ˇ wBy C vBz

gVMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

hMGD

1 20 

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

 @By Bz @z 

0 0 0 0  @Bz

C Bx

@y

@B 1  0  @xy





0

@Bz 1 0  @y



@By

@Bx @y

@x



@By  @z

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ  ˇ, @Bx ˇ  @y ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

0 .1=0/Bz Bx .1=0 /Bz By .1=0/. 12 B 2  Bz2 / 1 wB 2  10 Bz .uBx C vBy C wBz / 0 uBz C wBx vBz C wBy 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

120

Chapter 3 Computational models of magnetohydrodynamic processes

ˇ ˇ 0 ˇ ˇ .1=0 /Bz Bx ˇ ˇ ˇ .1=0 /Bz By ˇ ˇ .1=20 /.Bz2  Bx2  By2 / ˇ Dˇ ˇ 2wpm  1 Bz .uBx C vBy C wBz / 0 ˇ ˇ uBz C wBx ˇ ˇ ˇ vBz C wBy ˇ ˇ 0

hVMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

0 0 0  @Bz 1 Bx @x  2  0

0  @Bx @z



C By

@Bz @y



@Bx @Bz 1 0  @z  @x

@B  z  10  @B  @zy @y

0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ. @By  ˇ  @z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

The set of equations (3.55)–(3.57) can be computed with use of any computing flux method. However, for the sake of convenience it is recommended to normalize the system. Possible variants of such a renormalization are discussed in the following section.

3.7 The flux form of MHD equations in a dimensionless form 3.7.1 Definition of the normalizing parameters Let us choose the following normalizing parameters in an undisturbed medium:  ,

cp  ,

cv  ,

 ,

 ,

 ,

B ,

L ,

where  is the density of the undisturbed medium; cp  , cv  are the specific thermal capacities at constant pressure and volume;  is the thermal conduction;  is the electroconductivity;  is the coefficient of dynamic viscosity; B is the induction of magnetic field; L is the characteristic size dictated by physical formulation of the problem.

121

Section 3.7 The flux form of MHD equations in a dimensionless form

One can see that pressure p is not used in an explicit form as the normalizing parameter. However, the definition of this value is necessary for the definition of characteristic velocity and an estimation of the temperature of the undisturbed medium. So, p the choice of density and pressure allows to introduce a characteristic velocity V D p = , which is the sound velocity in the medium, where D cp =cv . For example, in Table 3.1 the typical parameters of the Earth ionosphere at different heights are presented. Table 3.1. Parameters of the Earth ionosphere.

H , km

 , kg/m3

p , Pa 4

4.49  10 2.53  105 2.72  106 1.45  106 3.02  107

150 200 300 400 500

V , m/s

9

2.00  10 2.50  1010 1.90  1011 2.80  1012 5.20  1013

560 691 802 851 902

The velocity of a sound V has been calculated at D 1.4. The magnetic induction in a terrestrial ionosphere varies over a wide range, but it is possible to choose characteristic value B D 3  105–5  105T. If all dimensional parameters and functions are marked with a tilde, and the system of MHD equations is rewritten in a dimensionless form, there will be the following values in the system (dimensionless parameters and functions are not marked in any way): .x, y, z/ D ED

.x, Q y, Q zQ / , L

D

Q , 

pD

pQ , V2

.u, v, w/ D

.u, Q v, Q w/ Q , V

Q  Q Q cQp cQv , D , D , cp D , cv D ,    cp cv .BQ x , BQ y , BQ z / .EQ x , EQ y , EQ z / .Bx , By , Bz / D , .Ex , Ey , Ez / D , B E .JQx , JQy , JQz / 0 B .Jx , Jy , Jz / D , J D , E D VB . J L

EQ , V2

eD

eQ , V2

D

The renormalization procedure appears to follow well-known dimensionless complexes. These are: Re = V L =

is the Reynolds number;

Pr =  cp =

is the Prandtl number;

=  D cp =cv ReM = 0 VL

is the adiabatic index; is the magnetic Reynolds number;

122

Chapter 3 Computational models of magnetohydrodynamic processes

V p V    D B VA, p VA = B =  

MA =

is the Alfvén Mach number; is the Alfvén velocity.

If there is a need to introduce the Mach number as one of the dimensionless complexes, it is necessary to use a characteristic velocity of investigated process (for example,V0) instead of a velocity V . In this case, instead of criteria Re , ReM  , MA it is necessary to write Re M1 , ReM M1 , MA M1 , where Re D  V0 L = , ReM D   V0 L, p MA D .V0 =B /  , M D V0=V is the Mach number. And, finally, if there is a need to introduce dimensionless pressure, related to any characteristic pressure, for example, p , a factor of 1= will be added before pressure gradient components in momentum equations.

3.7.2 Nondimension system of the MHD equations in flux form

gEu

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ

@U @ f @g @h C C C D 0, @t @x @y @z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ u  ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ ˇ C p u u ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ v uv ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ uw w ˇ, ˇ , fEu D ˇ U D ˇˇ ˇ ˇ ˇ ˇ u .E C p=/ ˇ ˇ E C pm ˇ ˇ ˇ ˇ ˇ 0 Bx ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ By 0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 0 Bz ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ  xx ˇ ˇ ˇ ˇ  xy ˇ ˇ ˇ ˇ 

1 xz NS ˇ ˇ f D  @e ˇ , ˇ .u

C v

C w

C / Re ˇ xx xy xz Pr  cv @v ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 v ˇ ˇ ˇ ˇ  yx vu ˇ ˇ ˇ  yy v 2 C p ˇˇ ˇ ˇ ˇ  yz 1 vw ˇ , gNS D ˇ

v .E C p / ˇˇ Re ˇˇ  u yx C v yy C w yz C ˇ ˇ 0 0 ˇ ˇ ˇ ˇ 0 0 ˇ ˇ ˇ ˇ 0 0

(3.58)

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ   @e ˇ , Pr cv @v ˇ ˇ ˇ ˇ ˇ ˇ

123

Section 3.7 The flux form of MHD equations in a dimensionless form

hEu

ˇ ˇ w ˇ ˇ wu ˇ ˇ wv ˇ ˇ w 2 C p D ˇˇ p ˇ w .E C  / ˇ 0 ˇ ˇ 0 ˇ ˇ 0

fMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

hNS

ˇ ˇ 0 ˇ ˇ  zx ˇ ˇ  zy ˇ  zz 1 ˇˇ

D ˇ  u

C v

Re ˇ zx zy C w zz C ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ 0 ˇ ˇ 2 .1=MA/.0.5B 2  Bx2 / ˇ ˇ ˇ .1=M2A /Bx By ˇ ˇ .1=M2A /Bx Bz ˇ Dˇ ˇ .1=M2A /ŒuB 2  Bx .uBx C vBy C wBz / ˇ ˇ 0 ˇ ˇ ˇ vBx C uBy ˇ ˇ wB C uB

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ 0 ˇ ˇ 2 .1=2MA /.Bx2  By2  Bz2 / ˇ ˇ ˇ .1=M2A /Bx By ˇ ˇ .1=M2A/Bx Bz ˇ Dˇ ˇ 2upm  .1=M2A /Bx .uBx C vBy C wBz / ˇ ˇ 0 ˇ ˇ ˇ vBx C uBy ˇ ˇ wB C uB

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

x

z

x

fVMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1 ReM M2A 

 @Bx By @y  1

z

0 0 0 0

@By  @x

0

@By

ReM  @x

@B 1 x  ReM  @z

C Bz

@Bx



@Bx @y  z  @B @x



@z

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ   @e ˇ , Pr cv @z ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ,  z ˇ  @B @x ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

124

Chapter 3 Computational models of magnetohydrodynamic processes

gMGD

ˇ ˇ 0 ˇ ˇ 2 ˇ .1=MA /By Bx ˇ ˇ ˇ .1=M2A /.0.5B 2  By2 / ˇ ˇ .1=M2A /By Bz ˇ Dˇ ˇ .1=M2 /ŒvB 2  By .uBx C vBy C wBz / A ˇ ˇ uBy C vBx ˇ ˇ ˇ 0 ˇ ˇ ˇ wBy C vBz

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ 0 ˇ ˇ 2 .1=MA /By Bx ˇ ˇ ˇ .1=M2A /.By2  Bx2  Bz2 / ˇ ˇ .1=M2A /By Bz ˇ Dˇ ˇ 2vpm  .1=M2A/By .uBx C vBy C wBz / ˇ ˇ uBy C vBx ˇ ˇ ˇ 0 ˇ ˇ wB C vB

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

y

gVMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

hMGD

1 ReM M2A 

h



z

0 0 0 0

 @By z C B  @B x  @y @x @By 1 x  ReM  @B  @x @y

Bz

@By @z

1

ReM 

 0

@Bz @y



@By @z



ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ i ˇˇ x ˇ,  @B ˇ @y ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ 0 ˇ ˇ 2 .1=MA /Bz Bx ˇ ˇ ˇ .1=M2A /Bz By ˇ ˇ .1=M2A /.0.5B 2  Bz2 / ˇ Dˇ 2 ˇ .1=MA /ŒwB 2  Bz .uBx C vBy C wBz / ˇ ˇ uBz C wBx ˇ ˇ ˇ vBz C wBy ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

125

Section 3.7 The flux form of MHD equations in a dimensionless form

ˇ ˇ 0 ˇ ˇ 2 .1=MA/Bz Bx ˇ ˇ ˇ .1=M2A /Bz By ˇ ˇ .1=M2A /.Bz2  Bx2  By2 / ˇ Dˇ ˇ 2wpm  .1=M2A /Bz .uBx C vBy C wBz / ˇ ˇ uBz C wBx ˇ ˇ ˇ vBz C wBy ˇ ˇ 0

hVMGD

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

0 0 0 1 ReM M2A 

0 h   @Bx @Bz z Bx @B C B  y @x  @z @y @Bx @Bz 1 ReM   @z  @x @B @Bz 1  @zy  ReM  @y 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ i ˇˇ @B ˇ.  @zy ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

In the presented equations dimensionless magnetic pressure is calculated under the formula pm D B 2 =2M2A , where B 2 is the squared dimensionless magnetic induction. Let us consider further modification of vectors fVMGD , gVMGD and hVMGD with the purpose of deriving more convenient calculation relations. Only the vector components corresponding to the energy conservation equation (the 5th line of each vector) are subject to transformation. Writing

fVMGD mod

ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0 D fVMGD C ˇˇ E  B/ E ˇ Bx .r ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

gVMGD mod

ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0 ˇ ˇ 0 D gVMGD C ˇˇ E  B/ E ˇ By .r ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇT ˇ ˇ ˇ VMGD E E D h C hVMGD ˇ , ˇ 0 0 0  . 0 0 0 0 B z r B/ mod

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

126

Chapter 3 Computational models of magnetohydrodynamic processes

one can receive

fVMGD mod

gVMGD mod

hVMGD mod

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1 ReM M2A 

h

@ @x



0 0 0 0 Bx2 By2 Bz2 2

 0

@By ReM   @x @Bz 1 ReM  @x 1

1 ReM M2A 



C

 

@Bx By @y



@Bx @y @Bx @z

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ i ˇ ˇ @ Bx Bz C @z ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(3.59)

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ h i 2 B 2 B 2 B ˇ @Bx By @By Bz y x z @ C C ˇ, 2 @x @y @z ˇ  ˇ @By @Bx 1 ˇ  @x ReM  @y ˇ ˇ 0 ˇ  ˇ @B @Bz 1 y ˇ  Re 

(3.60)

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ h  2 2 2 i ˇ Bz Bx By ˇ @By Bz @Bx Bz @ C @y C @z ˇ. 2 @x ˇ  ˇ @Bx @Bz 1 ˇ  ReM   @z @x ˇ ˇ @By @Bz 1 ˇ  ReM  @z @y ˇ ˇ 0

(3.61)

0 0 0 0 

M

@y

@z

0 0 0 0

1 ReM M2A 

3.8 The MHD equations in the flux form. The use of pressure instead of specific internal energy One effective computational method for integrating the MHD equations is based on the quasilinear form of the equations. Transfer from one form of the equations to another is carried out with the introduction of the Jacobian transformation from the conservative flux form @U @ f @g @h C C C D0 (3.62) @t @x @y @z

Section 3.8 The MHD equations formulated relative to pressure

127

to the quasilinear form @W @W @W @W C Ax C Ay C Az D 0, @t @x @y @z

(3.63)

where W is the vector of physical primitive variables, for example:

or

ˇT ˇ W D ˇ  u v w e Bx By Bz ˇ ,

(3.64)

ˇT ˇ W D ˇ  u v w p Bx By Bz ˇ ,

(3.65)

where Ax , Ay , Az are the Jacobian matrixes, generated at transfer from a set of equations (3.62) to (3.63). Below, in Section 3.9, such transfer procedure will be considered explicitly. Here we shall pay attention to the fact which will be used in the following: transfer from the equations .3.62/ to the equations .3.63/ is essentially fulfilled more easily if, as a base variable of the energy conservation equation, not the specific internal energy is used, but the pressure. However, unfortunately, it considerably restricts a class of problems subject to a solution with this method. For example, problems of viscous heat-conducting flow (the nonisentropic process) with thermal emission sources cannot be solved yet in such a way (without the introduction of any special methods of linearization or the frozen factors type, etc.). At the same time, there is a wide class of problems of magnetohydrodynamics that satisfies this condition, therefore this basic possibility of simplification of a solution should be considered. Let us rewrite the MHD equations to the conservative form concerning the pressure: @U @ f @g @h C C C D E, @t @x @y @z where

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ UDˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

p 1

C

V 2 2

 u v w .B 2 CB 2 CB 2 / C x 2M2y z A

Bx By Bz

(3.66)

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇD ˇ ˇ ˇ ˇ ˇ ˇ ˇ

128

Chapter 3 Computational models of magnetohydrodynamic processes

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

fEu

gEu

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 2 2 2

 .B CB CB / ˇ ; C 2 u2 C v 2 C w 2 C x 2M2y z ˇ ˇ A ˇ Bx ˇ ˇ By ˇ ˇ B  u v w

p 1

z

ˇ ˇ u ˇ ˇ u2 C p ˇ ˇ uv ˇ ˇ uw

2  D ˇˇ p 1 2 C w2 u u C u C v 2 ˇ 1 ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ v ˇ ˇ vu ˇ 2Cp ˇ v ˇ ˇ vw

2  D ˇˇ p 1 u C v2 C w2 v C v 2 ˇ 1 ˇ 0 ˇ ˇ 0 ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ; ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ w ˇ ˇ ˇ ˇ wu ˇ ˇ ˇ ˇ wv ˇ ˇ 2 ˇ ˇ w ˇ ; hEu D ˇ p

C2 p 2  1 ˇ ˇw u C v C w2 C w 2 ˇ ˇ 1 ˇ ˇ 0 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ. ˇ ˇ ˇ ˇ ˇ ˇ ˇ

where E is the vector function of the source terms which are not considered in further transformation operations, and functions fVMGD , gVMGD and hVMGD remain invariable. They can be used in the form of (3.58). As to functions fNS , gNS and hNS , since they break an isentropic condition they should be either excluded from reviewing, or included in a right-hand side of the equations as some source terms (as it was done in (3.66)). Such an approach to a solution of the Navier–Stokes equations demands special research for each specific case, as on the whole it is incorrect. However, it is admissible far from bounding surfaces and also outside of areas of a sharp changing of functions.

Section 3.9 Transormation of MHD equations from conservative to the quasilinear form 129

3.9 Eigenvectors and eigenvalues of Jacobian matrixes for transformation of the MHD equations from conservative to the quasilinear form. Statement of nonstationary boundary conditions The system of the MHD equations in the flux form formulated in the Section 3.7 possesses the property of conservatism and is rather effective for numerical solutions with use of a wide class of finite-difference schemes. First of all, among such schemes it is necessary to note the numerical schemes which are based on characteristic properties of the equations [65]. Effectiveness of such schemes is defined by simplicity of calculation of eigenvalues and eigenvectors of Jacobian matrixes for the transformation of the system of equations from the conservative form to a quasilinear one, and then to the characteristic form of the equations. Unfortunately, the system of equations in the conservative form has the serious shortage connected with the definition of boundary conditions on a free surface, in particular for subsonic flow conditions. In a series of publications [82, 102, 125–127] devoted to this problem, the possibility to use the characteristic form of boundary conditions with subsequent formulation of corresponding conditions for conservative variables is investigated. Thus, the problem of determination of eigenvectors and eigenvalues of Jacobian matrixes for transformation from a conservative form to a quasilinear one is important for the further development of methods of MHD equations solution.

3.9.1 Jacobian matrixes of passage from conservative to the quasilinear form of the equations Let us take for a basis the conservative flux form of MHD equations @f @g @h @U C C C D E, @t @x @y @z

(3.67)

where vector functions of conservative variables U, f, g, h are defined in Sections 3.6 and 3.7. Vectorial function E contains the variables, which are breaking hyperbolical properties of the equations and are considered further as source terms of the equations. Our first problem will be the reformulation of the set of equations (3.67) in the quasilinear form. As a vector of required functions we shall choose W D Wp (see (3.3)). Then the equation concerning variables W will have the following preliminary appearance: @W @W @W @W P C Qx C Qy C Qz D E, (3.68) @t @x @y @z

130

Chapter 3 Computational models of magnetohydrodynamic processes

where P , Qx , Qy , Qz are the Jacobian matrixes of transformation P D

@U , @W

Qx D

@f , @W

Qy D

@g , @W

Qz D

@h , @W

(3.69)

which components are defined under formulas Pij D

@Ui @fi @gi @hi , .Qx /ij D , .Qy /ij D , .Qz /ij D , @Wj @Wj @Wj @Wj

(3.70)

where i is the index of the matrixes line (and simultaneously the serial number of the line in U, f, g, h vectors); j is the number of the column (corresponds to the serial number of the line in a vector W/. Numbers of lines in vectors U and W are established under the following scheme: ˇ ˇ ˇ ˇ ˇ  ˇ U1 ˇ ˇˇ ˇ ˇ ˇ ˇ ˇ u ˇ U2 ˇ ˇ ˇ ˇ ˇ ˇ ˇ v ˇ U3 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ U4 ˇ ˇ w ˇ ˇ ˇ p  2 UDˇ D 1 ˇ 2 2 2 2 2 ˇ ˇ 1 C 2 .u C v C w / C 2 .Bx C By C Bz / ˇˇ , U 5 2MA ˇ ˇ ˇ ˇ U6 ˇ ˇˇ ˇ Bx ˇ ˇ ˇ ˇ ˇ U7 ˇ ˇ ˇ By ˇ ˇ ˇ ˇ ˇU ˇ ˇ ˇ 8 Bz ˇT ˇ W D ˇ W1 W2 W3 W4 W5 W6 W7 W8 ˇ ˇ ˇT D ˇ  u v w p Bx By Bz ˇ .

(3.71)

Further, we shall name the set of variables (3.71) as primitive variables. For deriving the final form of the equation concerning vector function W it is necessary to multiply equation (3.68) at the left by a matrix P 1 , being inverse in relation to matrix P @W @W @W @W C Ax C Ay C Az D .P 1 E/, @t @x @y @z where

Ax D P 1 Qx ,

Ay D P 1 Qy

, Az D P 1 Qz .

(3.72)

(3.73)

Now the problem is a determination of eigenvectors and eigenvalues of matrixes Ax , Ay , Az . We shall consider a method of searching for them with an example of a matrix Ax . The equation (3.72) we shall rewrite in the form of @W @W C Ax C C D .P 1 E/, @t @x

C D Ay

@W @W C Az . @y @z

(3.74)

Section 3.9 Transormation of MHD equations from conservative to the quasilinear form 131

Let us define the left eigenvectors of a matrix Ax under the formula: lEiT Ax D  ix lEiT ,

i D 1, 2, : : : , 8,

(3.75)

where lEiT D .li ,1 , li ,2 , li ,3 , : : : , li ,8 / is the transposed matrix of the left vectors, li ,j is the j -th eigenvector corresponding to the i-th eigenvalue. Then eigenvalues i of a matrix Ax will be defined from the solution of the following equation: det .Ax  x I / D 0, (3.76) where I is the unit matrix. All the eigenvalues are real due to the hyperbolic properties of the system of equations. So the problem of the calculation of eigenvalues of a Jacobian matrix has been solved. It allows to formulate effective finite-difference schemes of Roe’s type [89]. However, as already mentioned above, having defined eigenvalues of transformation matrixes, it is possible to construct nonstationary boundary conditions using characteristic equations. For this purpose the eigenvectors of a matrix should be determined that allow to define the following two matrixes: 1 0 l1,1 l1,2 : : : : : : : : : l1,8 0 1 B l2,1 l2,2 : : : : : : : : : l2,8 C l1,1 l2,1 : : : li ,1 : : : l8,1 C B B .. .. C B l1,2 l2,2 : : : li ,2 : : : l8,2 C B . . C B C C. Sx D B . , Sx1 D B C . C B .. A @ .. B li ,1 li ,2 : : : : : : : : : li ,8 C B .. .. C l1,8 l2,8 : : : li ,8 : : : l8,8 @ . . A l8,1 l8,2 : : : : : : : : : l8,8 (3.77) Rows of matrix Sx1 are the left eigenvectors, and the matrix Sx has the columns which are the left eigenvectors. Introducing these matrixes allows to fulfill sufficiently easily the procedure of Ax matrix diagonalization, i. e., a determination of the diagonal matrix ƒx corresponding to Ax : ƒx D Sx1 Ax Sx , where diagonal components of a matrix ƒx are eigenvalues of a matrix Ax : 0 x 1 1 0 : : : 0 B 0 x : : : 0 C B 2 C ƒx D B . .. C . @ .. . A 0 0 : : : x8

(3.78)

(3.79)

The further transformations are rather formal, but laborious. We shall multiply (3.74) by a matrix Sx1 at the left Sx1

@W @W C Sx1 Ax C Sx1 C D Sx1 .P 1 E/, @t @x

(3.80)

132

Chapter 3 Computational models of magnetohydrodynamic processes

that in a component-wise notation looks as follows: @W @ W ET C j lEjT C lj C D lEjT .P 1 E/. lEjT @t @x

(3.81)

Let us introduce the label @W Lxj D j lEjT @x

or Lx D Sx1 Ax

@W . @x

(3.82)

Let us substitute .3.82/ in (3.80) and then multiply all terms of the equation from the left side by Sx : @W C Sx Lx C C D .P 1 E/. (3.83) @t Using designation dx D Sx Lx we can finally write @W C dx C C D .P 1 E/. @t

(3.84)

To use a vector dx in the given equation it is necessary to determine its components by means of solution of the following set of equations: Sx1 dx D Lx .

(3.85)

Vectors dy and dz are introduced similarly, so we have obtained the following set of equations for time derivatives from primitive variables @W D dx  dy  dz  C C .P 1 E/. @t

(3.86)

These equations are also used for the determination of the boundary conditions for conservative variables. A method of calculating functions dx , dy , dz for supersonic and subsonic flows on boundaries is considered in [125]. It uses the fact that in the case of a supersonic flow on a boundary all perturbations which have been generated in the calculation domain are taken out from it. But in the case of subsonic flow on a boundary, there will be at least one perturbation coming to the calculated area outside. Presence of the waves, bringing perturbations from the outside, can render an appreciable influence on a solution of the problem. To exclude their influence on a solution inside of the area, Thompson [125, 126] has suggested to calculate vectors Lx , Ly , Lz as follows: Lj D j lj Lj D 0

@W @n

for outgoing waves;

(3.87)

for entering wavesj D 1, 2, : : : , 8,

where n is one of the coordinates. As the j -th eigenvalue has physical sense of velocity of a perturbation propagation, then if on the right boundary nR of the calculated

Section 3.10 A singularity of transformation Jacobian matrixes

133

domain the value j > 0, the perturbation is taken out from the domain, and if j < 0 it is taken in. And, conversely, if on the left boundary nL : j < 0, the perturbation is taken out, and at j > 0 it is taken in. So, the procedure of determining eigenvalues of Jacobian matrixes for transfer from one system of functions to another has been considered in the given section. It has been shown how with use of these vectors it would be possible to construct nonstationary boundary conditions for primitive variables. However, it is not always possible to realize this procedure in practice. First of all, we shall indicate the significant calculation problems connected with the singularity of the Jacobian matrixes of transformation. This problem will be considered below.

3.10 A singularity of Jacobian matrixes for transformation of the equations formulated in the conservative form Let us consider transformation of the set of equations concerning conservative variables @f @g @h @U C C C D E, (3.88) @t @x @y @z where ˇ ˇ ˇ ˇ  ˇ ˇ ˇ ˇ u ˇ ˇ ˇ ˇ ˇ ˇ v ˇ ˇ ˇ ˇ w ˇ ˇ UDˇ p (3.89)  2 1 2 2 2 2 2 ˇ, ˇ 1 C 2 .u C v C w / C 2m .Bx C By C Bz / ˇ ˇ ˇ Bx ˇ ˇ ˇ ˇ ˇ ˇ By ˇ ˇ ˇ ˇ Bz ˇ ˇ ˇ ˇ u 0 ˇ ˇ ˇ ˇ 1 2 2 2 2 ˇ ˇ  2m .Bx  By  Bz / u C p ˇ ˇ ˇ ˇ 1 uv  m Bx By ˇ ˇ ˇ ˇ 1 ˇ ˇ uw  B B x z m ˇ f D ˇˇ p u2 Cv 2 Cw 2 1 2 2 2 C m Œu.Bx C By C Bz /  Bx .uBx C vBy C wBz / ˇˇ ˇ u 1 C u 2 ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ C uB vB x y ˇ ˇ ˇ ˇ ˇ ˇ wBx C uBz (3.90)

134

Chapter 3 Computational models of magnetohydrodynamic processes

ˇ ˇ ˇ ˇ v 0 ˇ ˇ ˇ ˇ 1 ˇ ˇ vu  m By Bx ˇ ˇ 1 2 2 2 2 ˇ ˇ v C p  2m .By  Bx  Bz / ˇ ˇ ˇ ˇ 1 ˇ ˇ By Bz vw m ˇ, g D ˇˇ p 2 2 2 (3.91) 2 2 2 Œv.B CB CB /B .uB CvB CwB / ˇ y x y z x y z ˇ v 1 C v u Cv2 Cw C ˇ m ˇ ˇ ˇ ˇ uBy C vBx ˇ ˇ ˇ ˇ 0 ˇ ˇ ˇ ˇ ˇ ˇ wBy C vBz ˇ ˇ ˇ ˇ w 0 ˇ ˇ ˇ ˇ 1 ˇ ˇ B B wu  m z x ˇ ˇ ˇ ˇ 1 wv  m Bz By ˇ ˇ ˇ ˇ 1 ˇ ˇ 2 2 2 2 w C p  2m .Bz  Bx  By / ˇ ˇ h D ˇ p ˇ. 2 2 2 u Cv Cw 1 2 2 2 ˇw C m Œw.Bx C By C Bz /  Bz .uBx C vBy C wBz / ˇˇ 2 ˇ 1 C  w ˇ ˇ ˇ ˇ uBz C wBx ˇ ˇ ˇ ˇ vBz C wBy ˇ ˇ ˇ ˇ ˇ ˇ 0 (3.92) here m D M2A , and vector functions fNS , gNS , hNS and fVMGD , gVMGD, hVMGD are taken in a right-hand side of the equation and included in a vector E. At transfer from the initial set of equations (3.88) to the system of an intermediate form: @W @W @W @W C Qx C Qy C Qz D E, (3.93) P @t @x @y @z Jacobian matrixes of the following form are turned out: ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ P D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1 u v w u2 Cv 2 Cw 2 2

0 0 0

0 0 0  0 0 0  0 0 0  u v w 0 0 0

0 0 0

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 1

Bx m

By m

0 0 0

1 0 0

0 1 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Bz ˇ , m ˇ ˇ 0 ˇˇ 0 ˇˇ ˇ 1 ˇ 0 0 0 0

(3.94)

135

Section 3.10 A singularity of transformation Jacobian matrixes

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Qx D ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

u



0

0

0

0

0

0

u2

2u

0

0

1

 Bmx

Bz m

uv

v

u

0

0



By m  Bmx

uw

w

0

u

0

0

 Bmx

u .u2 Cv 2 Cw 2 / 2

x x x Q5,2 Q5,3 Q5,4

u 1

By m Bz m x Q5,6

0

x x Q5,7 Q5,8

0

0

0

0

0

0

0

0

0

By

Bx

0

0

v

u

0

0

Bz

0

Bx

0

w

0

u

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(3.95)

where x D Q5,2 x Q5,3 D uv 

By2 C Bz2 1 p C .u2 C v 2 C w 2/ C u2 C ; 1 2 m Bx By ; m

x Q5,7 D

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Qy D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

x Q5,4 D uw 

2uBy  vBx ; m

Bx Bz ; m

x Q5,8 D

v uv v2 vw

0 v 0 0

 u 2v w

0 0 0 v

0 0 1 0

v .u2 Cv 2 Cw 2 / 2

y Q5,2

y Q5,3

y Q5,4

v 1

0 0 0

By 0 0

Bx 0 Bz

0 0 By

0 0 0

x Q5,6 D

vBy C wBz ; m

2uBz  wBx ; m 0 By m Bx m



0 y

0

0 0

 Bmx B  my Bmz B  Bmz  my y

y

Q5,6 Q5,7 Q5,8 v 0 0

u 0 w

0 0 v

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(3.96)

where y D  uv  Q5,2 y

Q5,3 D

Bx By ; m

1 p B 2 C Bz2 C  .u2 C v 2 C w 2/ C v 2 C x ; 1 2 m y

Q5,6 D

2vBx  uBy ; m

y

Q5,7 D

uBx C wBz ; m

y

Q5,4 D vw 

x Q5,8 D

By Bz ; m

2vBz  wBy ; m

136

Chapter 3 Computational models of magnetohydrodynamic processes

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Qz D ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

w

0



0

0

0

0

uw

w

0

u

0

 Bmz

0

vw

0

w

v

0

0

 Bmz

w2

0

0

2w

1 w 1

Bx m z Q5,6

By m z Q5,7

w .u2 Cv 2 Cw 2 / 2

z z z Q5,2 Q5,3 Q5,4

0

Bz

0

Bx

0

w

0

0

0

Bz

By

0

0

w

0

0

0

0

0

0

0

ˇ ˇ ˇ ˇ  Bmx ˇˇ B ˇ  my ˇ ˇ  Bmz ˇˇ ˇ, z ˇ Q5,8 ˇ ˇ u ˇ ˇ v ˇˇ ˇ 0 ˇ 0

(3.97)

where Bx Bz By Bz z ; Q5,3 ; D vw  m m 

Bx2 C By2  u2 C v 2 C w 2 p C C w 2 C ; D 1 2 m

z D  uw  Q5,2 z Q5,4 z D Q5,6

2wBx  uBz ; m

z Q5,7 D

2wBy  vBz ; m

z Q5,8 D

uBx C vBy . m

The analysis of the presented Jacobian matrixes allows to draw a conclusion on presence in matrixes Qx , Qy and Qz of the zero lines corresponding to components of a magnetic induction Bx , By , and Bz accordingly. In other words, Jacobian matrixes are degenerated in relation to corresponding projections of a magnetic induction. To obtain the system of the MHD equations in a quasilinear form @W @W @W @W Q C Ax C Ay C Az D E, @t @x @y @z

(3.98)

it is necessary to multiply the equation (3.93) from the left side by a matrix P 1 , which is the inverse matrix to P , then Ax D P 1 Qx , Ay D P 1 Qy , Az D P 1 Qz , Q D P 1 E. A structure of the matrix P 1 looks like: E ˇ ˇ ˇ ˇ 1 0 0 0 0 0 0 0 ˇ ˇ ˇ ˇ 1 0 0 0 0 0 0  u ˇ ˇ  ˇ ˇ 1 ˇ ˇ  v 0 0 0 0 0 0  ˇ ˇ w 1 ˇ ˇ 0 0 0 0 0 0  1  ˇ ˇ P D ˇ 1

 B .1/ B .1/ B .1/ ˇ 2 2 2 u u .  1/ v .  1/ w .  1/ .  1/  C v C w   ˇ 2 ˇ m m m ˇ ˇ 0 0 0 0 0 1 0 0 ˇ ˇ ˇ ˇ ˇ ˇ 0 0 0 0 0 0 1 0 ˇ ˇ ˇ ˇ 0 0 0 0 0 0 0 1 x

y

z

137

Section 3.10 A singularity of transformation Jacobian matrixes

The matrix Ax is obtained by multiplication of a matrix Qx from the left side by a matrix P 1 : ˇ ˇ ˇ ˇ 0 0 0 0 0 0 ˇ ˇu  ˇ ˇ ˇ ˇ By Bz Bx 1  0 0 ˇ ˇ 0 u  m m m ˇ ˇ ˇ ˇ By Bx u 0 0  m  m 0 ˇ ˇ 0 0 ˇ ˇ ˇ Bz Bx ˇ ˇ ˇ 0 0 0  m 0 u 0  m ˇ . (3.99) Ax D ˇˇ ˇ .1/.uBx CvBy CwBz / ˇ ˇ 0 p 0 0 u 0 0 m ˇ ˇ ˇ ˇ ˇ ˇ 0 0 0 0 0 0 0 0 ˇ ˇ ˇ ˇ ˇ 0 By Bx 0 0 v u 0 ˇˇ ˇ ˇ ˇ ˇ 0 Bz 0 Bx 0 w 0 u ˇ Eigenvalues of the given matrix have the following appearance: 1 D 0,

2 D u,

3 D u C VA,x ,

6 D u  Cf ,

4 D u  VA,x ,

7 D u C Cs ,

5 D u C Cf ,

8 D u  Cs ,

(3.100)

r

where

q 1 2 2 Œa C VA2 C .a2 C VA2 /2  4a2 VA,x ; 2 s   q 1 2 2 a C VA2  .a2 C VA2 /2  4a2 VA,x ; Cs D 2 Cf D

VA,x D

(3.101)

(3.102)

pBx m

is the Alfvén velocity of disturbances propagation along x-th component p 2 2 2 Bx CBy CBz p of a magnetic induction; VA D is the Alfvén velocity; Cs , Cf are the m velocities of the slow and the fast magnetosonic waves. It is seen that the singularity of matrix Qx follows to zero row in matrix Ax . It leads to certain difficulties in the calculation of eigenvectors of matrix Ax . It is necessary to underline that the determined eigenvalues completely correspond to available representations about presence of the two Alfvén waves, four magnetosonic waves and two simple waves within the bounds of the MHD theory. However, from the received data it is not possible to tell for certain concerning one of those waves. Let us consider a procedure of eigenvector definition for a matrix Ax . For this purpose we shall take the equation (3.76), having rewritten it in the following form: l1 .u  / D 0,

l1  C l2 .u  / C l5 p C l7 By C l8 Bz D 0,

(3.103)

138

Chapter 3 Computational models of magnetohydrodynamic processes

1 C l5 .u  / D 0,  Bx By Bz uBz C vBy C wBz  l3  l4 C l5 .  1/  l7 v  l8 w D 0, l2 m m m m By Bx Bz Bx  l3 C l7 .u  / D 0, l2  l4 C l8 .u  / D 0. l2 m m m m Due to a degeneracy of a matrix Ax , the set of equations (3.103) is redefined. If one of the equations were omitted, namely the sixth, it is possible to find a matrix of the left eigenvectors for the set of eigenvalues (3.103). As the normalization of a missing eigenvector .lE1 / can be arbitrary, and simultaneously it should be independent linearly in relation to other vectors, it can be presented in the following form: l1 D .0, 0, 0, 0, 0, 0, 1, 0, 0/. As a result one can see obtain a matrix Sx1 whose rows are the left eigenvectors of a matrix Ax : ˇ ˇ ˇ ˇ ˇ0 ˇ 0 0 0 0 1 0 0 ˇ ˇ ˇ ˇ ˇ ˇ 1 0 0 0 0 0  a2 0 ˇ1 ˇ ˇ ˇ ˇ ˇ By ˇ B z ˇ0 0 Bz By 0 0 p m  pm ˇ ˇ ˇ ˇ ˇ ˇ By Bz p p 0 0 B B 0 0  ˇ ˇ z y m m ˇ 1 ˇ Sx D ˇ , 2 2 Cf By Cf Bz ˇ ˇ 0 Cf  Cf Bx By  Cf Bx Bz ˇ 1 0 ˇ ˇ zf zf zf zf ˇ ˇ Cf2 By Cf2 Bz ˇ ˇ Cf Bx By Cf Bx Bz 1 0 ˇ 0 Cf ˇ zf zf zf zf ˇ ˇ ˇ Cs2 By Cs2 Bz ˇˇ ˇ 0 Cs  Cs Bx By  Cs Bx Bz 1 0 zs zs zs zs ˇ ˇ ˇ ˇ ˇ Cs2 By Cs2 Bz ˇ Cs Bx By Cs Bx Bz 1 0 ˇ 0 Cs ˇ zs zs zs zs (3.104) where zf D mCf2  Bx2 ; zs D mCs2  Bx2 ; a2 D p. We shall pay attention once again that serial numbers of rows in a matrix (3.104) correspond to numbers of eigenvalues (3.100). For simplification of the further calculations we shall introduce following designations: Bx By Bx By By By af D Cf , as D Cs , bf D Cf2 , bs D Cs2 , zf zs zf zs Bz Bz , ps D Cs2 , gf D Cf , gs D Cs , pf D Cf2 zf zs Bz Bx Bz Bx Bz By , hs D Cs . e D p , f D p , hf D Cf m m zf zs l3 .u  /  l7 Bz D 0,

l4 .u  /  l8 Bx D 0,

l2

139

Section 3.10 A singularity of transformation Jacobian matrixes

Then the set of equations for determination of the vector function dx used in boundary conditions, looks like the following 1

0 d1

B B B 1 Sx  B B @

d2 .. .

0 C B C B C C DB C @ A

d8 where

Sx1

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

x

Lx1 Lx2 .. .

1 C C C, A

(3.105)

Lx8

0

0

0

0

0

1

0

0

0

1=a2

0

0

By

Bz

0

0

0

By

Bz

0

0

gf

af hf

1

0

gs

as

hs

1

0 gf

af

hf

1

0 gs

as

hs

1

ˇ 0 ˇˇ ˇ 0 0 0 ˇˇ ˇ 0 e f ˇ ˇ 0 e f ˇˇ ˇ. 0 bf pf ˇˇ ˇ 0 bs ps ˇ ˇ 0 bf pf ˇˇ ˇ 0 bs ps ˇ

1

0

(3.106)

The set of equations (3.105) with the matrix .3.106/ can be solved fairly simply owing to a special kind of the matrix. However, we shall not give this solution here, as it is not further used. Similarly, components of vector functions dy and dz are determined. For reference, we shall give the structure of transfer matrixes Ay and Az : ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Ay D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

v

0



0

0

0

v

0

0

y 0  m

0

0

0

B

Bx  m

0

Bx m

Bz m By  m

0

0

v

0

1 

0

0

0

v

0

0

0

0

p

0

v

0

By  m Bz  m .1/.uBx CvBy CwBz / m

0 By Bx

0

0

v

u

0

0

0

0

0

0

0

0

0

0

Bz By 0

0

w

v

0

0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(3.107)

140 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Az D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

Chapter 3 Computational models of magnetohydrodynamic processes

w

0

0



0

0

w

0

0

Bz 0  m

0

0

w

0

0

0

0

0

w

1 

0

0

0 0

0 Bz 0

0

0

0

0

0

0

x B m

0

Bz  m

y  m

Bx m

By m

Bz  m

p w

0

0

.1/.uBx CvBy CwBz / m

Bx 0

w

0

u

Bz By 0

0

w

v

0

0

0

0

0

0

0

B

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ. ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(3.108)

Thus, it is possible to say that the problem of determination of eigenvectors and eigenvalues for Jacobian transfer matrixes has been solved. But we did not yet consider the significant problems which have arisen in connection with the singular character of these matrixes. In connection with the problem of singularity of matrixes we shall mention two works which immediately concern this problem. In the work [82], an attempt to bypass this problem was made by adding to the MHD equations system the vector function E  B/, E where F 0 is a vector of functions corresponding to each of a special kind F0 .r equation of the complete set ˇ ˇT FE 0 D ˇ 0 Bx By Bz .V  B/ u v w ˇ . It was shown in [82] that such reception eliminates a singularity of transfer matrixes, not breaking their spectral characteristics. In the work [102], the problem of nonstationary boundary conditions for MHD problems was solved on the basis of the use of transferred matrixes which have no singularity. How singular matrixes have been obtained on the basis of completely conservative MHD equations, has been shown in the given section. In the following section, the system of MHD equations which do not lead to a singularity of transfer matrixes will be formulated. Besides, the use of transformations will be shown that has helped to obtain the transfer matrixes used in [102].

3.11 System of the MHD equations without singular transfer matrixes Transformation of the MHD equations from the completely conservative form (3.88) to the quasilinear form (3.98) has shown that singularities in matrixes Ax , Ay , Az are caused by singularities in Jacobian transfer matrixes in those rows which correspond

Section 3.11 System of the MHD equations without singular transfer matrixes

141

to the equations of a magnetic induction. Thus, the reason for the specified singularity is clear: in the equation for x-th component of a magnetic induction there is no association with a change of magnetic field components along an axis x, and it is similar for two other components. Its physical reason is also clear: in the ideal magnetogasdynamics the following vectorial connection between components of a magnetic field and a velocity of flow is postulated:   @B E  ŒV  B . D r @t From this equation it follows that the variation in time of any magnetic field component depends only on a spatial variation of magnetic field components perpendicular to the given direction. If the magnetic induction equations are formulated in the conservative form, integral conservation laws are satisfied unconditionally, which naturally results in lack of association of the given magnetic field induction component from a variation of the field in the given direction. If the equations are to be formulated so that there is an association on all directions in them, it is necessary to superimpose an additional connection on components of the E  B/ D 0, which should magnetic field. In this case, this connection is the equation .r certainly be satisfied. The given equation is used for deriving a set of equations of a magnetic induction in the nonconservative form:   @v @Bx @u @u @w @Bx @Bx @Bx D Bx C C By C Bz u v w , (3.109) @t @y @z @y @z @x @y @z   @u @w @v @v @By @By @By @By D Bx  By C C Bz u v w , (3.110) @t @x @x @z @z @x @y @z   @u @v @w @w @Bz @Bz @Bz @Bz D Bx C By C Bz C u v w . (3.111) @t @x @y @x @y @x @y @z It is enough to formulate the system of MHD equations in the quasilinear form where the matrixes Ax , Ay , Az do not contain singularities (further we shall mark these matrixes with tildes to distinguish them from singular matrixes): ˇ ˇ ˇ ˇu  0 0 0 0 0 0 ˇ ˇ ˇ0 u 0 0 1= 0 By =m Bz =m ˇˇ ˇ ˇ ˇ0 0 u 0 0 By =m Bx =m 0 ˇ ˇ ˇ0 0 0 Bx =m ˇˇ 0 u 0 Bz =m ˇ Q Ax D ˇ ˇ, 0 u 0 0 0 ˇ ˇ 0 p 0 ˇ ˇ0 0 0 0 0 u 0 0 ˇ ˇ ˇ ˇ 0 By Bx 0 0 0 u 0 ˇ ˇ ˇ ˇ0 B 0 Bx 0 0 0 u z

142

Chapter 3 Computational models of magnetohydrodynamic processes

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ AQy D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ

v 0  0 0 0 0 0 0 0 v 0 0 0 By =m Bx =m 0 0 v 0 1= Bx =m 0 Bz =m 0 0 0 v 0 0 Bz =m By =m 0 0 p 0 v 0 0 0 0 By Bx 0 0 v 0 0 0 0 0 0 0 0 v 0 0 0 v 0 0 Bz By 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ AQz D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ

ˇ ˇ w 0 0  0 0 0 0 ˇ 0 Bx =m ˇˇ 0 w 0 0 0 Bz =m 0 0 w 0 0 0 Bz =m By =m ˇˇ ˇ 0 0 0 w 1= Bx =m By =m 0 ˇ. ˇ 0 0 0 p w 0 0 0 ˇ ˇ 0 Bx 0 w 0 0 0 Bz ˇ ˇ 0 0 Bz By 0 0 w 0 ˇ ˇ 0 0 0 0 0 0 0 w

(3.112)

In the representation of matrixes AQx , AQy , AQz in the form of (3.112) some transformations of the MHD equations have been made: (1) In the energy conservation equation the term connected with Joule heat has been transferred to the right-hand part as a source term. It does not change the spectral properties of the matrixes; (2) In the momentum conservation equations the structure of the terms connected with the magnetic field has been changed slightly. Until now in the momentum conservation equations the following terms corresponding to the magnetic field were used (see system (3.88); those unnecessary for the transformation are omitted below): 1 @ 1 @Bz Bx 1 @By Bx @u C   .Bx2  By2  Bz2 /   D 0, @t 2m @x m @y m @z

(3.113)

@v 1 @Bx By 1 @ 1 @Bz By C    .B 2  Bx2  Bz2 /  D 0, @t m @x 2m @y y m @z @w 1 @Bx Bz 1 @By Bz 1 @ C     .B 2  Bx2  By2 / D 0. @t m @x m @y 2m @z z Let us select from parentheses in (3.113) the derivative of squared components of magnetic field on corresponding coordinates, and the remaining two terms of each equation we shall differentiate by parts. After that we shall take the identical condition

Section 3.11 System of the MHD equations without singular transfer matrixes

143

E  B/ D 0. As a result we shall receive: .r 1 @ 1 @u 1 @Bx @Bx C  .By2 C Bz2 /  By  Bz D 0, @t 2m @x m @y m @z

(3.114)

@v @By @By 1 1 @ 1 C     Bx  .Bx2 C Bz2 /  Bz D 0, @t m @x 2m @y m @z @w @Bz @Bz 1 1 1 @ C     Bx  By  .B 2 C By2 / D 0. @t m @x m @y 2m @z x However, the set of equations (3.114) and, accordingly, matrixes (3.112) can be simplified even more regarding the terms considering force action of a magnetic field. For this purpose it is enough to pass to the quasilinear form of the equations by such a transformation, at which there will be only required functions under a differential sign. After differentiation of the terms in parentheses by parts, we receive the following set of equations: @u 1 1 1 1 @By @Bz @Bx @Bx C : : : C By C Bz  By  Bz D 0, @t m @x m @x m @y m @z

(3.115)

@v @By @Bx @Bz @By 1 1 1 1 C : : :  Bx C Bx C Bz  Bz D 0, @t m @x m @y m @y m @z @w @Bz @Bz @Bx @By 1 1 1 1 C : : :  Bx  By C Bx C By D 0. @t m @x m @y m @z m @z In view of relations (3.115) the matrixes in the equations of the quasilinear form take the following form: ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ AQx D ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

u  0 0 0 0 0 0 0 u 0 0 1= 0 By =m Bz =m 0 0 0 u 0 0 0 Bx =m 0 0 0 u 0 0 0 Bx =m 0 p 0 0 u 0 0 0 0 0 0 0 0 u 0 0 0 0 0 u 0 0 By Bx 0 Bz 0 Bx 0 0 0 u

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(3.116)

144

Chapter 3 Computational models of magnetohydrodynamic processes

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ AQy D ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

v 0  0 0 0 0 v 0 0 0 By =m 0 0 v 0 1= Bx =m 0 0 0 v 0 0 0 0 p 0 v 0 0 0 v 0 By Bx 0 0 0 0 0 0 0 0 0 Bz By 0

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ AQz D ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

w 0 0  0 0 0 0 0 w 0 0 0 Bz =m 0 0 w 0 0 0 Bz =m 0 0 0 w 1= Bx =m By =m 0 0 0 p w 0 0 0 Bx 0 w 0 0 Bz 0 w 0 0 Bz By 0 0 0 0 0 0 0 0

0 0 0 0 0 Bz =m 0 By =m 0 0 0 0 v 0 0 v 0 0 0 0 0 0 0 w

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ. ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

It is remarkable that such MHD equations were used in the work [82] exactly in the same form. Thus, in the given section the MHD equations have been stated in the form that is physically equivalent to the set of equations (3.88), however not containing singular matrixes in the quasilinear form. In the following section, spectral characteristics of matrixes AQx , AQy and AQz will be discovered, and also relations for the formulation of nonstationary boundary conditions will be obtained.

3.12 Eigenvalues and eigenvectors of nonsingular matrixes of quasilinear system of the MHD equations Let investigate spectral properties of matrixes AQx , AQy and AQy .

3.12.1 Matrix AQ x Eigenvalues of this matrix have the following expression: 1 D u,

2 D u,

5 D u C Cf ,

3 D u C VAx ,

6 D u  Cf ,

4 D u C VAx ,

7 D u C Cs ,

8 D u  Cs ,

(3.117)

145

Section 3.12 Nonsingular matrixes of quasilinear MHD system

where Bx Bx ; Dp VA,x D p m MA VA2 D

Bx2 C By2 C Bz2 m

D

a2 D

p ; 

Bx2 C By2 C Bz2 M2A

;

s   q 1 2 2 Cf D ; a C VA2 C .a2 C VA2 /2  4a2 VA,x 2 s   q 1 2 2 a C VA2  .a2 C VA2 /2  4a2VA,x . Cs D 2 For determination of the left eigenvectors it is necessary to solve the set of equations (3.76). These equations can be written, in view of a matrix AQx structure, as: l1 .u  / D 0,

l1  C l2 .u  / C l5 p C l7 By C l8 Bz D 0,

l3 .u  /  l7 Bx D 0,

l4 .u  /  l8 Bx D 0,

l6 .u  / D 0, l2

l2

(3.118)

l2  1 C l5 .u  / D 0,

By Bx  l3 C l7 .u  / D 0, m m

Bz Bx  l4 C l8 .u  / D 0. m m

Procedure of determination of left eigenvectors consists of four stages. The first stage. Let us suppose .u  / D 0. This means that l1,1 is any number, including 0;  D 1 D u. Let l1,1 ¤ 0, also we shall suppose l1,1 D 1, then from the system (3.118) it follows l1,2 D 0, l1,3 D 0, l1,4 D 0, l1,5 D .1=a2 /, l1,6 is number, l1,7 D 0, l1,8 D 0. It is necessary to suppose l1,6 D 0, because having chosen a normalization for l1,1 D 1, we have no condition for a normalization of the l1,6 value. So   El T D 1, 0, 0, 0,  1 , 0, 0, 0 . (3.119) 1 a2 The second stage. Let us suppose .u  / D 0,  D 2 D u and l2,1 D 0. Then we obtain all elements l2,i D 0, except for l2,6 . Normalizing l2,6 per unit, we receive (3.120) lE2T D .0, 0, 0, 0, 0, 1, 0, 0/ . Let us notice that the case of l2,6 D 0 is excluded, as the zero vector will turn out.

146

Chapter 3 Computational models of magnetohydrodynamic processes

The third stage. Let us suppose that .u  / ¤ 0,  D 3,4 D u ˙ VAx , then 2  2 D .u  3,4/2 D VAx ,  D ˙VAx . From the set of equations (3.118) l2 D l5 D 0, and remaining components of the vectors are connected among themselves by relations Bx Bz l3 D , l8 By

l4 Bx , D l8 

l7 Bz D . l8 By

Supposing l8 D 1, we receive   Bx Bz Bx Bz T D 0, 0,  , lE3,4 , 0, 0,  , 1 By  By Or, choosing a more obvious normalization, we have: at 3 D u C VAx ,  D VAx   By Bz , lE3T D 0, 0, Bz , By , 0, 0, C p ,  p m m and at 4 D u  VAx ,  D CVAx   Bz By lE4T D 0, 0, Bz , By , 0, 0,  p , C p . m m

(3.121)

(3.122)

2 , k D 5, 6, 7, 8. The fourth stage. Let us suppose .u  k / ¤ 0,  2 ¤ VAx From the set of equations (3.118) the following relations can be received:

lk,1 D 0, lk,3 D lk,4 D

By Bx ,  m 2

Bx2

Bz Bx ,  m 2

Bx2

lk,5 D  lk,7 D lk,8 D

lk,2 D 1 (here we use unconditioned normalization),

1 , 

By , Bx2  m 2 Bx2

Bz .  m 2

Choosing a more obvious normalization, we receive   Bx By Bx Bz  2 By  2 Bz . (3.123) , , 1, 0, , lEkT D 0, , m 2  Bx2 m 2  Bx2 m 2  Bx2 m 2  Bx2

147

Section 3.12 Nonsingular matrixes of quasilinear MHD system

In view of the received formulas (3.119)–(3.123) it is possible to generate the matrix Sx1 .5 D u C Cf , 6 D u  Cf , 7 D u C Cs , 8 D u  Cs /: ˇ ˇ ˇ ˇ 1 ˇ1 ˇ 0 0 0 0 0 0  2 ˇ ˇ a ˇ ˇ ˇ ˇ 0 0 0 0 1 0 0 ˇ0 ˇ ˇ ˇ ˇ ˇ By ˇ B ˇ0 z 0 Bz By 0 0 pm  pm ˇ ˇ ˇ ˇ ˇ ˇ By Bz ˇ0 ˇ p p 0 Bz By 0 0  m ˇ m ˇ 1 Sx D ˇ ˇ, Cf2 By Cf2 Bz ˇ ˇ ˇ 0  Cf  CfzBx By  CfzBx Bz ˇ 1 0 zf zf f f ˇ ˇ ˇ ˇ Cf2 By Cf2 By ˇ Cf Bx By Cf Bx Bz ˇ 0  C 1 0 ˇ ˇ f zf zf zf zf ˇ ˇ ˇ 2 2 Cs By Cs By ˇˇ ˇ 0  Cs  Cs Bx By  Cs Bx Bz 1 0 ˇ ˇ zs zs zs zs ˇ ˇ 2B 2B ˇ ˇ C C Cs Bx By Cs Bx Bz y y s s ˇ 0  Cs ˇ 1 0 z z z z s

s

s

s

(3.124) where zf D mCf2  Bx2 , zs D mCs2  Bx2 . For determination of vector function dx it is necessary to solve the set of equations 0 0 x 1 1 d1 L1 B d2 C B Lx C B B 2 C C Sx1  B .. C D B .. C , (3.125) @ . A @ . A d8

x

Lx8

where Lxj D j lEjT .@W=@x/, j D 1, 2, : : : , 8, where j is the number of the eigenvalue; lEjT is the left eigenvector corresponding to the j -th eigenvalue. Using the procedure of Gaussian elimination, the following triangular matrix can be obtained: ˇ ˇ ˇ ˇ ˇ1 0 0 0 0 ˇ 0 0 1=a2 ˇ ˇ ˇ ˇ f f 1 0 Eyf Ezf ˇ ˇ 0 Cf Exy Exz ˇ ˇ ˇ0 0 ˇ Q Q B B 0 0 B  B z y z y ˇ ˇ ˇ ˇ Cf CCs ˇ0 0 ˇ 0 S  0 S S 44 47 48 ˇ ˇ Cf SQx1 D ˇ (3.126) ˇ, f f ˇ ˇ0 0 0 0 2 0 2Fy 2Fz ˇ ˇ ˇ ˇ ˇ0 0 ˇ 0 0 0 1 0 0 ˇ ˇ ˇ ˇ Q z 2BQ y ˇ ˇ0 0 0 0 0 0 2 B ˇ ˇ ˇ ˇ ˇ0 0 0 0 0 0 0 S88 ˇ

148

Chapter 3 Computational models of magnetohydrodynamic processes

where Fyf D Fzs D

Cf2 By zf

Cs2Bz ; zs

Fzf D

;

Cf2 Bz

Bz BQ z D p ; m

Fys D

f Fxy D

Cf Bx By ; zf

s Fxy D

zf

By ; BQ y D p m f D Fxz

;

Cf Bx Bz ; zf

s Fxz D

zf D mCf2  Bx2 ;

Cs Bx By ; zs

Cs Bx Bz ; zs

zs D mCs2  Bx2 ;

f

S44 D

Cs2By ; zs

f

s s C Cs Exz / C By .Cf Exy C Cs Exy / Bz .Cf Exz

C f Bz

S47 D  S48 D 

f f s Cf  Exy Cs / Bz .Fys Cf C Fy Cs / C BQ z .Exy

C f Bz f f s Cf  Exy Cs / Bz .Fzs Cf C Fz Cs /  BQ y .Exy

S88

C f Bz

;

;

;

f f Fzs BQ z  Fs BQ z C BQ y Fys  BQ y Fy D2 . BQ z

3.12.2 Matrix AQ y Eigenvalues of this matrix have the following form: 1 D v,

2 D v,

5 D v C Cf ,

3 D v C VAy ,

6 D v  Cf ,

4 D v C VAy ,

7 D v C Cs ,

(3.127)

8 D v  Cs ,

where VAy

By By Dp ; Dp m MA r Cs D

r Cf D

q 1 2 2 ; Œa C VA2 C .a2 C VA2 /2  4a2 VAy 2

q 1 2 2 Œa C VA2  .a2 C VA2 /2  4a2VAy ; 2 VA2 D

Bx2 C By2 C Bz2 m

D

Bx2 C By2 C Bz2 M2A

a2 D

.

p ; 

149

Section 3.12 Nonsingular matrixes of quasilinear MHD system

The set of equations for the determination of left eigenvector components of a matrix AQy has the following form: l1 .v  / D 0,

l2 .v  /  l6 By D 0,

l1  C l2 .v  / C l5 p C l6 Bx C l8 Bz D 0, l3

1 C l5 .v  / D 0,  l7 .v  / D 0,

l4 .v  /  l8 By D 0,

By Bx C l3 C l6 .v  / D 0, m m

l2

l3

(3.128)

Bz By  l4 C l8 .v  / D 0. m m

The procedure of determination of the left eigenvectors is the same as for the matrix AQx . We shall give the final result in the form of matrix Sy1 :

Sy1

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ D ˇˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1

0

0

0

 a12

0

0

0

0

0

0

1

0

Bz

0

Bx

0

pBz m

0

0

Bz

0

Bx

0

0  0

Cf Bx By zf

Cf Bx By zf

0  0

Cs Bx By zs

Cs Bx By zs

Cf



1

Cf Bz By zf

Cf Cs

Cf Bz By zf



1

Cs Bz By zs

Cs Bz By zs

Cs

0

0

z  pBm 0

Cf2 Bx zf Cf2 Bx zf

0 0

1

Cs2 Bx zs

0

1

Cs2 Bx zs

0

ˇ ˇ ˇ 0 ˇ ˇ ˇ 0 ˇ ˇ ˇ x ˇ  pBm ˇ ˇ ˇ pBx ˇ m ˇ , 2 Cf Bz ˇ ˇ ˇ zf ˇ Cf2 Bz ˇ ˇ zf ˇ 2 Cs Bz ˇˇ zs ˇ ˇ Cs2 Bz ˇ ˇ zs (3.129)

where zf D mCf2  By2 ; zs D mCs2  By2 . For the determination of vector function dy it is necessary to solve the set of the following equations 0 0 y 1 1 d1 L1 B d2 C B Ly C B B 2 C C Sy1  B .. C D B .. C , (3.130) @ . A @ . A d8

y

x

L8

where L yj D j lEjT .@ W=@y/, j D 1, 2, : : : , 8, where j is the number of the eigenvalue; lET is the left eigenvector corresponding to the j -th eigenvalue. j

150

Chapter 3 Computational models of magnetohydrodynamic processes

Triangular matrix SQy1 can be obtained using the procedure of Gaussian elimination:

SQy1

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Dˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

1

0

0

0

f

f

0 Exy Cf Eyz

 a12

0

1

Ex

0 Ez

S36

0

S38

S46

0

S48

0

f

0 f

0

0

S33

S34

0

0

0

S44

Bz f Exy Cf CCs Cf

0

0

0

0

2

2Fxf

0

0

0

0

0

2BQ z 0 2BQ x

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

S88

0 2Fzf

ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ, ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ

(3.131)

where Fxf D

Fzs D

Cf2 Bx zf

Cs2Bz ; zs f Fyz D

Cf2 Bz

Cf By Bz ; zf

f

;

Exy

Bz BQ z D p ; m

Fxs D

f Fxy D

Cf Bx By ; zf

s Fxy D

s Fyz D

f

Bx Exy C Bz Eyz

;

zf

Bx BQ x D p ; m

f

S34 D

Fzf D

;

S36 D

Cs By Bz ; zs

f f BQ z Exy C Bz Fx f

Cs Bx By ; zs

Bz Cf

S38 D 

;

Exy f

S44 D

S33 D 

Cs2Bx ; zs

f f BQ x Exy  Bz Fz f

Exy

f

s s C Cs Exy Bx C Cs By Eyz C Exy C f Bx Bz Cf Eyz

Bz C f s Bz Cf Fxs C Cs Exy BQ z C Cs Bz Fx  Exy Cf BQ z f

S46 D

;

f f s Bz Cf Fzs C Cs Exy BQ x  Cs Bz Fz  Exy Cf BQ x

S88 D 2

;

f

Bz C f

S48 D 

;

f

Exy

Bz C f Fzs BQ z C Fzf BQ z  BQ x Fxs C BQ x Fxf . BQ z

;

;

151

Section 3.12 Nonsingular matrixes of quasilinear MHD system

3.12.3 Matrix AQ z Eigenvalues of this matrix have the following form: 1 D w, 5 D w C Cf , where VAz

2 D w,

3 D w C VAz ,

6 D w  Cf ,

4 D w  VAz ,

7 D w C Cs ,

8 D w  Cs

(3.132)

s   q Bz Bz 1 2 2 Dp a C VA2 C .a2 C VA2 /2  4a2 VAz ; Dp ; Cf D m MA 2 s   q 1 2 p 2 2 2 2 2 2 Cs D a C VA  .a C VA /  4a VAz ; a2 D ; 2  VA2 D

Bx2 C By2 C Bz2

D

Bx2 C By2 C Bz2

. M2A The set of equations for the determination of the left eigenvector components of the matrix AQz has the following form: l1 .w  / D 0,

m

l2 .w  /  l3 Bz D 0,

l3 .w  /  l7 Bz D 0,

(3.133)

1

l1  C l4 .w  / C l5 p C l6 Bx C l7 By D 0, l4  C l5 .w  / D 0, Bz Bx Bz By C l4 C l6 .w  / D 0, l3 C l4 C l7 .w  / D 0,  l2 m m m m l8 .w  / D 0. Using the same procedure for the determination of components of eigenvectors as in case of the matrix Sx1 , one can obtain the matrix Sz1 : ˇ ˇ ˇ ˇ 1 ˇ1 0 0 0 ˇ 0 0 0  a2 ˇ ˇ ˇ ˇ ˇ ˇ0 0 0 0 0 0 0 1 ˇ ˇ ˇ ˇ ˇ ˇ B Bx y ˇ0 p p Bx 0 0  m 0 ˇˇ By m ˇ ˇ ˇ ˇ ˇ By Bx p p B 0 0  0 0 B ˇ ˇ y x m m 1 ˇ, ˇ Sz D ˇ 2 2 ˇ Cf Bx Cf By ˇ 0  Cf Bx Bz  Cf By Bz  C 1 0 ˇˇ f ˇ zf zf zf zf ˇ ˇ Cf2 Bx Cf2 By ˇ ˇ Cf By Bz ˇ ˇ 0 CfzBx Bz  C 1 0 f zf zf zf f ˇ ˇ ˇ ˇ Cs2 Bx Cs2 By ˇ ˇ 0  Cs Bx Bz  Cs By Bz  C 1 0 s ˇ ˇ zs zs zs zs ˇ ˇ 2 2 ˇ ˇ C B B C B C B C B B s y z s x z s x s y ˇ0  Cs 1 0 ˇ zs zs zs zs (3.134) 2 2 2 2 where zf D mCf  Bz , zs D mCs  Bz .

152

Chapter 3 Computational models of magnetohydrodynamic processes

For determination of vector function dz it is necessary to solve the set of equations 0 0 z 1 1 d1 L1 B d2 C B Lz C B B 2 C C Sz1  B . C D B . C , (3.135) @ .. A @ .. A d8 x Lz8 where Lzj D j lEjT .@ W=@z/, j D 1, 2, : : : , 8, where j is the number of an eigenvalue, lEjT is the characteristic left vector corresponding to j -th eigenvalue. Triangular matrix SQz1 is obtained with the use of the Gaussian elimination procedure: ˇ ˇ ˇ ˇ 1 0 0 0 1 0 0 0  ˇ ˇ a2 ˇ ˇ f f ˇ 0 E f E f C 1 Ex Ey 0 ˇˇ xz yz ˇ f ˇ ˇ By ˇ0 0 S36 S37 0 ˇˇ S33 S34 f ˇ Exz ˇ ˇ ˇ ˇ0 0 0 S S S S 0 ˇ ˇ 44 45 46 47 1 SQz D ˇ (3.136) ˇ, f f ˇ0 0 0 0 2 2Fx 2Fy 0 ˇˇ ˇ ˇ ˇ ˇ0 0 0 0 0 2BQ y 2BQ x 0 ˇˇ ˇ ˇ ˇ ˇ0 0 0 0 0 0 S77 0 ˇˇ ˇ ˇ ˇ ˇ0 0 0 0 0 0 0 1ˇ where Fxf D

Cf2 Bx zf

; Fyf D

By BQ y D p ; m s D Fyz

f Fyz D

Cf2 By zf

Cf By Bz ; zf

Cs By Bz ; zs

S36 D

S46 D

s Fxz D

f

f

S37 D 



s C B Es Bx Exz y yz f

f

Bx Exz C By Eyz

/;

s s Bx Exz C By Eyz f

;

Exz ;

f

f Fyz D

f

Bx Exz C By Eyz

Exz

Fxf

Cs Bx Bz ; zs

f

S33 D

f f BQ y Exz C By Fx

S44 D .Cs  Cf Fxs

Bx Cs2Bx Cs2By ; BQ x D p ; Fxs D ; Fys D ; m zs zs

Bx Exz C By Eyz

S34 D

Cf By f

;

Exz

f f BQ x Exz  By Fy f

;

Exz

S45 D 1  C BQ y

Cf By Bz ; zf

s C B Es Bx Exz y yz f

f

Bx Exz C By Eyz f

f

s s Exz Eyz  Exz Eyz f

f

Bx Exz C By Eyz

;

;

153

Section 3.13 A method of splitting for three-dimensional (3D) MHD equations

S47 D

Fys

C Fyf

s s Bx Exz C By Eyz f

f

Bx Exz C By Eyz "

S77 D 2

Fys



Fyf

C BQ x

f

f

s s Exz Eyz  Eyz Exz f

f

Bx Exz C By Eyz

;

# BQ x s f C .Fx  Fx / . BQ y

In conclusion of the given section we shall pay attention to the fact that spectral characteristics of a matrix AQx , namely, eigenvalues (3.117) and left eigenvectors (3.124), practically completely coincide with spectral characteristics of complete (without simplifications) matrix Ax (3.99), which has one singular line. The unique difference is the equality to zero of the first eigenvalue of the specified singular matrix (see (3.100)).

3.13 A method of splitting for three-dimensional (3D) MHD equations In the previous sections it has been shown how to formulate nonstationary boundary conditions for three-dimensional (3D) MHD equations [102, 125–127]. More simple nonstationary boundary conditions for 3D MHD problems are considered in the given section. This approach is based on a method of splitting [116]. As before, we shall consider the system of MHD equations in the quasilinear form @W @W @W @W C AQx C AQy C AQz D G, @t @x @y @z

(3.137)

where matrixes AQx , AQy , AQz are taken in the form of (3.116), and vector G includes components of the complete set of the equations, corresponding to viscous dissipation and Joule heat. Let us rewrite matrixes AQx , AQy , AQz , having added the separation of these matrixes on blocks 1 0 u  0 0 0 0 0 0 C B C B =m B =m 0 u 0 0 1= 0 B C B y z C B C B 0 0 0 u 0 0 By =m Bx =m C B C B B 0 0 0 Bx =m C 0 u 0 Bz =m C B AQx D B C , (3.138) C B 0 p 0 0 u 0 0 0 C B C B C B 0 0 0 0 0 u 0 0 C B C B C B 0 By Bx 0 0 0 u 0 A @ 0 Bx 0 0 0 u 0 By

154

Chapter 3 Computational models of magnetohydrodynamic processes

0 B B B B B AQy D B B B B B @ 0 B B B B B AQz D B B B B B @

v 0  0 0 0 0 0 0 0 v 0 0 0 By =m Bx =m 0 0 v 0 1= Bx =m 0 Bz =m 0 0 0 v 0 0 Bz =m By =m 0 0 p 0 v 0 0 0 0 By Bx 0 0 v 0 0 0 0 0 0 0 0 v 0 0 0 v 0 0 Bz By 0

1 C C C C C C, C C C C A

1 w 0 0  0 0 0 0 0 Bx =m C 0 w 0 0 0 Bz =m C 0 0 w 0 0 0 Bz =m By =m C C C 0 0 0 0 w 1= Bx =m By =m C. C 0 0 0 p w 0 0 0 C C 0 Bz 0 Bx 0 w 0 0 C A 0 0 Bz By 0 0 w 0 0 0 0 0 0 0 0 w

The matrixes have been divided into four blocks in such a manner that gas-dynamic and magnetic parts of the problem are explicitly sorted. The essence of the method of decomposition is that for the solution of the set of equations (3.137) on the current time step the two sequential stages will be used. At the first stage, the following set of gas-dynamic equations will be solved (further we shall suppose G D 0/: @ W1 @ W1 @ W1 @ W1 C AQG C AQG C AQG D T, x y z @t @x @y @z 0

where

B B W1 D B B @ 0 B B B AQG D x B @ 0 B B G Q Ay D B B @

v 0 0 0 0

0  v 0 0 v 0 0 0 p

u  0 u 0 0 0 0 0 p 1

0 0 0 0 C C 0 1= C C; v 0 A 0 v

 u v w p 0 0 u 0 0

AQG z

(3.139)

1 C C C; C A 1 0 0 0 1= C C 0 0 C C; u 0 A 0 u 0 w 0 B 0 w B DB B 0 0 @ 0 0 0 0

(3.140)

0  0 0 0 0 w 0 0 0 w 1= 0 p w

1 C C C; C A

155

Section 3.13 A method of splitting for three-dimensional (3D) MHD equations

1

0 0 B B TDB B @

T Tu Tv Tw Tp

0

1

B B By @By By @Bx Bz @Bz Bx @By Bz @Bx Bx @Bz C B m @x C m @x  m @y  m @y  m @y  m @z C B By @Bx By @Bz Bx @By Bx @Bx Bz @Bz Bz @By CDB  m  m C m C m  m  m C B @x @x @y @y @y @z B A B B @B By @Bz By @By Bx @Bz Bz @By Bx @Bx B  mz @xx  m   C C m @y m @y m @z m @z @x @ 0

C C C C C C. C C C A

(3.141) Components of the vector W1 can be calculated at the first intermediate step after QG QG diagonalization of matrixes AQG x , Ay , Az . We shall mark the determined values with index .p C 1/ unlike all other functions known on the p-th time layer. At the second stage, the set of equations for magnetic induction is to be solved @ W2 @ W2 @ W2 @ W2 C AQB C AQB C AQB D K, x y z @t @x @y @z where

0

0

0 1 Bx u 0 @ 0 u W2 D @ By A ; AQB D x 0 0 Bz 1 0 0 v 0 0 w @ 0 v 0 A ; AQB @ 0 AQB y D z D 0 0 v 0 pC1

By @u@y

B pC1 B K D  B By @u@x @ pC1 Bz @u@x

pC1

Bz @u@z

Bx @v@x

pC1

pC1 Bx @w@x

CBx @v@y

(3.142)

1 0 0 A; u

(3.143)

1 0 0 w 0 A; 0 w

pC1

CBx @w@z

pC1

Bz @v@z

pC1

CBy @w@z

pC1 CBz @v@y

pC1 By @w@y

pC1

1 C C C. A

(3.144)

The results of the solution of this system will be functions Bx , By , Bz at the upper time layer .p C 1/. Let us note the principal properties of the method. QG QG Q Q (1) Matrixes AQG x , Ay , Az are formed from the left upper blocks of matrixes Ax , Ay , AQz accordingly, while the right upper blocks form the vector of right-hand side terms T. QB QB Q Q (2) Matrixes AQB x , Ay , Az are formed from the left upper blocks of matrixes Ax , Ay , AQz accordingly, while the left lower blocks form vector K. (3) At the second stage, the system of equations in the characteristic form is solved QB QB owing to the diagonal character of matrixes AQB x , Ay , Az , i. e., the equations are represented at once in the most simple form possible. The calculation relations for the first stage of the decomposition are given below.

156

Chapter 3 Computational models of magnetohydrodynamic processes

Eigenvalues of matrix AQ xG x1 D u, x2 D u,

x3 D u,

x4 D u C a,

x5 D u  a.

(3.145)

The following matrix of left eigenvectors (rows of the matrix) corresponds to these eigenvalues there: 1 0 1 0 0 0 0 C B B 0 0 0 1 0C C B  C B DB 0 Sx1 (3.146) 0 1 0 0 C, C B G a C B 1 @ a2 p 0 0 1 A 1 a  p 0 0 1 a2 where the serial number of the matrix row corresponds to the serial number of eigenvalues (3.145). The solution of the set of equations in the form (3.85) gives the following expressions for vector dx components: dx,1 D LQ x1 ,

dx,2

  p Q x 1 Q x L2  L5 , D a 2

dx,4 D LQ x4 ,

dx,5 D

dx,3 D LQ x3 ,

(3.147)

1 Qx L , 2 5

where functions LQ xj are calculated by the Thompson method [125]. Namely, if on the left boundary of the calculation domain (x D x0 / an eigenvalue xj < 0, then LQ xj D Lxj , otherwise LQ xj D 0. And, conversely, if on the right boundary of the calculation domain (x D xL > x0 ) an eigenvalue xj > 0, then LQ xj D Lxj , otherwise LQ xj D 0. These conditions mean that a disturbance going out from the area is to be calculated with use of functions LQ xj , and a disturbance entering into the area is considered equal to zero. Functions Lxj are calculated under the formula: T Lxj D xj lEx,j

@ W1 @x

(3.148)

or in a component-wise notation 0 Lx1

@ D u.1, 0, 0, 0, 0/ @x

B B B B @

 u v w p

1 C C C D u @ , C @x A

0 Lx2

@ D u.0, 0, 0, 1, 0/ @x

B B B B @

 u v w p

1 C C C D u @w , C @x A

157

Section 3.13 A method of splitting for three-dimensional (3D) MHD equations

0 Lx3

@ D u.0, 0, 1, 0, 0/ @x

B B B B @

 u v w p

1 C C C D u @v , C @x A

0

1  B u C     C @ B 1 a x B v C D .u C a/ 1 @ C a @u , , 0, 0, 1 L4 D .u C a/ 2 , C a p @x B a2 @x p @x @w A p 0 1  B u C     C @ B a @u 1 a 1 @ x B C  . ,  , 0, 0, 1 L5 D .u  a/ v C D .u  a/ 2 a2 p @x B a @x p @x @w A p Eigenvalues of matrix AQ yG y

y

y

y

y

1 D v, 2 D v, 3 D v, 4 D v C a, 5 D v  a.

(3.149)

With use of the given eigenvalues the following matrix of the left eigenvectors (rows of the matrix) turns out: 0 .Sy1 /G

B B DB B @

1 0 0 1=a2 1=a2

0 0 1 0 0 0 0 a= p 0 a= p

0 0 1 0 0

0 0 0 1 1

1 C C C, C A

(3.150)

where the serial number of the matrix row corresponds to the serial number of eigenvalues (3.149). The solution of the set of equations in the form of (3.85) gives the following expressions for components of the vector dy : y dy,1 D LQ 1 ,

y dy,2 D LQ 2 ,

dy,3 D

p Q y 1 Q y .L3  L5 /, a 2

Q y, dy,4 D L 4

1 y dy,5 D LQ 5 , 2

where functions LQ yj are calculated with use of the following conditions: y D ymin :

y y y y y if j < 0, then LQ j D Lj ; if j > 0, then LQ j D 0;

y y y y y D ymax : if j > 0, then LQ j D Lj ; if j < 0,

y then LQ j D 0.

158

Chapter 3 Computational models of magnetohydrodynamic processes

Functions L yj are calculated under the formula y T @W 1 Lyj D j lEy,j @y

or in a component-wise notation 0 1  B u C C @ B @ y L1 D v.1, 0, 0, 0, 0/ B v C Dv , B C @y @ @y w A p

0 y L2

@ D v.0, 1, 0, 0, 0/ @y

0 y L3

(3.151)

@ D v.0, 0, 0, 1, 0/ @y

B B B B @

 u v w p

B B B B @

 u v w p

1 C C C D v @u , C @y A

1 C C C D v @w , C @y A

0

1  B u C     C @ B a @v 1 a 1 @ y B C , 0, 1 C , L4 D .v C a/ , 0, v C D .v C a/ a2 p @y B a2 @y p @y @wA p 0 1  B u C     C 1 @ @ B a @v 1 a y C B v C D .v C a/  . L5 D .v C a/ 2 , 0,  , 0, 1 a p @y B a2 @y p @y @wA p Eigenvalues of matrix AQ zG z1 D w,

z2 D w,

z3 D w,

z4 D w C a,

z5 D w  a.

(3.152)

With use of the given eigenvalues the following matrix of the left eigenvectors (the matrix rows) turns out: 1 0 1 0 0 0 0 C B B 0 1 0 0 0C C B C B (3.153) .Sz1 /G D B 0 0 1 0 0 C , C B B 1 0 0 a 1C A @ a2 p 1 a2

0 0

a p

1

159

Section 3.13 A method of splitting for three-dimensional (3D) MHD equations

where the serial number of the matrix row corresponds to the serial number of eigenvalues (3.152). The solution of the set of equations in the form of (3.85) gives the following expressions for components of the vector dz : dz,1 D LQ z1 , dz,4 D

dz,2 D LQ z2 ,

p Q z 1 Q z .L4  L5 /, a 2

dz,3 D LQ z3 ,

(3.154)

1 dz,5 D LQ z5 , 2

where functions LQ zj are calculated with use of the following conditions: z D zmin :

if  zj < 0, then LQ zj D Lzj ;

if  zj > 0, then LQ zj D 0;

z D zmax :

if  zj > 0, then LQ zj D Lzj ; if  zj < 0, then LQ zj D 0.

Functions LQ zj are calculated under the formula T @W1 Lzj D zj lEz,j @z

or in a component-wise notation 0 1  B u C @ B C @ Lz1 D w.1, 0, 0, 0, 0/ B v C Dw , B C @z @ A @y w p

(3.155)

0 Lz2

B @ B D w.0, 1, 0, 0, 0/ B @z B @

 u v w p

1 C C C D w @u , C @z A

1  B u C C @v @ B z Dw , v C L3 D w.0, 0, 1, 0, 0/ B C B @y @ A @z w p 0 1  B u C     C a @w @ B 1 a 1 @ z B C  , , 0, 0,  , 1 L4 D .w C a/ v C D .w C a/ 2 a2 p @z B a @z p @z @w A p 1 0  B u C     C @ B 1 a B v C D .w  a/ 1 @ C a @w . Lz5 D .w  a/ 2 , 0, 0, ,1 C a p @z B a2 @z p @z @w A p 0

160

Chapter 3 Computational models of magnetohydrodynamic processes

Nonstationary boundary conditions at the first stage of the problem solution Multiplying the equation (3.139) from the left side on matrixes .Sx1 /G , .Sy1 /G , .Sz1 /G sequentially and then in the same sequence on reciprocal matrixes (matrixes of the right eigenvectors), we come to the equation: @ W1 D  dx  dy  dz C T. @t

(3.156)

The equation (3.156) represents the boundary condition for vector function W1 on any boundaries of the calculation domain. The structure of vector functions dx , dy , dz is that they contain only derivatives in corresponding directions x, y, z. In other words, if the boundary condition is calculated on a surface x D const, the normal modification of functions in relation with this boundary is considered only in vector function dx , while vector functions dy , dz contain derivatives in tangential directions. Vector function T contains derivatives in all directions, therefore to make the calculation of boundary conditions more obvious, we shall present it in the form of two terms T D Tn C T , (3.157) where 0 B B B B Tn D B B B B @

0 @ .By2 CBz2 / @x @ .Bx2 CBz2 / @y @ .Bx2 CBy2 / @z

0

1 C C C C C, C C C A

0

0

1

B B By @Bx C Bx @By C Bz @Bx C Bx @Bz @y @y @z @z B 1 B B B @Bx C B @By C B @By C B @Bz T D y @x x @x z @z y @z m B B B B @Bx C B @Bz C B @By C B @Bz x @y z @y y @y @ z @x 0

C C C C C. C C C A

Thus, as a result of the first stage calculations the values of functions  pC1, upC1 , v pC1 , w pC1 , p pC1 on the upper time layer are determined in view of the action of magnetic field forces. At the second stage of the calculations, these values will be used for determination of boundary values of components of a magnetic field. Nonstationary boundary conditions at the second stage of the problem solution For determination of values of functions Bx , By , Bz on .p C 1/-th time layer it is necessary to solve the following set of equations: @Bx @Bx @Bx @Bx @u @v @u @w C upC1 C v pC1 C w pC1 D By  Bx C Bz  Bx , @t @x @y @z @y @y @z @z

(3.158)

161

Section 3.14 Splitting method for nonstationary 3D MHD flows

@By @By @By @By @u @v @v @w C upC1 C v pC1 C w pC1 D By C Bx C Bz  By , @t @x @y @z @x @x @z @z

(3.159) @Bz @Bz @Bz @Bz @u @w @v @w C w pC1 C w pC1 C w pC1 D Bz C Bx  Bz C By , @t @x @y @z @x @x @y @y

(3.160) where superscripts .p C 1/ specify that corresponding values of velocity components are taken by results of the first stage. Derivatives of velocities in right parts of the equations are calculated with the use of values from the lower time layer. Thus, in the present section the method, simple enough to realize, is formulated for the calculation of nonstationary boundary conditions. Note that the question of the effectiveness of the given approach in problems of strong MHD interaction is remains open.

3.14 Application of a splitting method for nonstationary 3D MHD flow field, generated by plasma plume in the ionosphere A three-dimensional computational model and the results of the calculation for the dynamics of a plasma plume induced by a pulsed plasma thruster (PPT) in a magnetized flow of the ambient plasma are considered in this section. A nonstationary expansion process of a PPT induced plasma plume is studied in the ionospheric plasma over 100 s after the start of a 6 microsecond plasma pulse with the use of a splitting method described in the previous section. The parameters of the plasma plume in the PPT outlet section simulate the performance characteristics of a pulsed thruster PPT-4 (Pulsed Plasma Thruster 4 [38]) utilizing solid teflon propellant. The magnetohydrodynamic (MHD) model of the process makes it possible to predict the dynamics of a plasma bunch after the pulse termination during its interaction with the ambient medium and the magnetic field as well as the induced electric fields and currents in the vicinity of the expanding plasma. Development of small spacecraft intended for optronic environmental monitoring, communications, and various defense purposes has attracted special attention to modeling and experimental investigation in electric jet propulsion. The high-performance index, simple design, and reliability of pulsed plasma thrusters utilizing solid teflon propellant make these thrusters highly promising for smaller space vehicles. The theory and practice of electrojet engines were intensively developed in the 1960s and 1970s. It was in that period that prototype PPTs were designed that proved their efficiency in the real conditions of operation in space. The theoretical and computational models that were developed provided a comprehensive understanding of the processes in these engines and helped to significantly enhance their efficiency [9, 39].

162

Chapter 3 Computational models of magnetohydrodynamic processes

The physical processes of PPT operation are studied by physical mechanics (plasma dynamics) investigating the laws of the interaction of plasma flows in various conditions. In this case, one is dealing with the interaction of an originally dense plasma plume with a homogeneous flow of ionospheric plasma incident on it at an arbitrary angle. In the course of the plasma spread, its density becomes comparable with that of the rarefied ambient medium so that the reciprocal influence of these two flows becomes inevitable. The authors [86] compared computational data obtained for the problem of the spread of a plasma cloud in the ionosphere by an MHD model and a collisionless plasma model. It was shown that the calculated large scale flow characteristics (shock wave velocity, disturbance range of gas-dynamic parameters and magnetic cavity parameters) are close for both models. This is a very important conclusion because it shows that resource sparing MHD models can be used to describe large scale phenomena accompanying the plasma clouds’ dynamics. The analysis of conditions for the dynamics of a plasma plume induced by a pulsed plasma thruster shows that although the employment of the collisionless plasma model seems more justifiable in this case, at the initial period of plasma plume dynamics until the interaction with the ambient medium has started the MHD model appears to be quite acceptable. The scheme of the problem being solved is shown in Figure 3.1. The symmetry axis of a cylindrical nozzle and a velocity vector of the PPT plasma plume are directed along the z axis. The center of the symmetry of the PPT outlet section has the coordinates x0 , y0 , and z0 . The plasma velocity at the outlet section of the nozzle is assumed constant and equal to V0. For simplicity of representation of the data, two planes (ı and /, possessing the designed symmetry properties, are shown in the calculation domain. In the plane (the plane coordinate being x D x0/, the vectors of the z

PPT

Vinf

zo

β vo

Bo α

z γ

xo δ x

Figure 3.1. MHD Problem scheme.

y

Section 3.14 Splitting method for nonstationary 3D MHD flows

163

unperturbed magnetic field induction B0 and the velocity V1 of the incident flow of the ionospheric plasma are specified. The slopes of these vectors are measured from the axis z: the angle ˛ determines the slope of the unperturbed magnetic field and the angle ˇ determines the vector of the plasma’s incident flow. The plane ı (the plane’s coordinate being y D y0) is perpendicular to the plane . In this plane, the conditions of the symmetry of all three of the functions sought must be satisfied about the straight line with the coordinates (x0, y0). The following input data were used in the computations: 

  





the PPT pulse duration was t D 6 s; the time dependence of the PPT mass flow ratio during this pulse was considered constant; the mass of the teflon used over a single pulse was m D 12  106 g; the radius of the outlet section of the PPT nozzle was R0 D 0.03 m; the averaged density of the plasma plume at the PPT outlet section was 0 D 2.83  105 kg/m3; the averaged plasma pressure at the PPT outlet section p0 D 250 N/m2 . This pressure corresponds to the plasma molecular weight average M† D 19 kg/kmole and the average temperature T0 D 17 400 K; the plasma velocity at the PPT outlet section V0 D 25 km/s. The specified input data correspond to MA D 25.

It is assumed that at the PPT outlet section, the plasma is homogeneous with respect to the radius. Numerical modeling is performed for two orientations of the PPT symmetry axis with respect to the incident flow: for ˇ D 30ı and 60ı (see Figure 3.1). The slope of the magnetic field is ˛ D 60ı . The parameters of the ambient (ionospheric) plasma correspond to the altitude of 150 km:  D 2.0  109 kg/m3, p D 4.5  104 N/m2 , and B0 D 5.0  105 T; the velocity of the incident flow of the ionospheric plasma was assumed equal to V1 D 6 km/s. The calculation domain had the following size Lx D 2.75 m, Ly D 3.25 m, and Lx D 4 m. In the calculations, an inhomogeneous grid, measuring 100  100  100, was used. The calculation results for the dynamics of the plasma cloud for the angle ˇ D 60ı (between the PPT axis and the incident flow of ionospheric plasma) are shown in Figures 3.2–3.4 for one of the time moments. These figures show the density distribution of the plasma bunch and the ambient medium, and the gas-dynamic and magnetic pressure against the background of the vector velocity field in the plane . In order to estimate the numerical concentration of the plasma particles (in cubic meters), the density values shown in the figures must be multiplied by the coefficient 6.34  1016. It is clearly seen from the presented data that after the pulsed thruster stops operation (t D 6 s), the plasma plume moves across the space in the form of a toroid shape plasma bunch. By the maximum density value in the expanding plasma bunch

164

Chapter 3 Computational models of magnetohydrodynamic processes

(a)

(b)

RO

P

20 40 60 Y 80 100

PH

4.0E+02 3.6E+02 3.2E+02 2.8E+02 2.4E+02 2.0E+02 1.6E+02 1.2E+02 8.0E+01 4.0E+01

1.6E+01 1.4E+01 1.3E+01 1.1E+01 9.5E+00 7.9E+00 6.3E+00 4.7E+00 3.2E+00 1.6E+00

0

(c)

40 60 X 80

20 2040

60 Y 80 100

120 100 80 60

1.6E+01 1.5E+01 1.3E+01 1.1E+01 9.7E+00 8.1E+00 6.5E+00 4.9E+00 3.2E+00 1.6E+00

40 60 X 80

20 2040

60 Y 80 100

40 20 40 60 X 80

20

0

Figure 3.2. Total density distribution of the plasma plume and of the ionospheric plasma (a), gas-dynamic (b) and magnetic (c) pressure at t D 100 s. The angle ˇ D 60ı . The tables give the density related to the ambient medium density. Gas-dynamic pressure is related to p D p= Q  V2, while the magnetic pressure is related to those in the ambient ionospheric plasma

(correspondingly  D 1000, 360, 44, 16 for t D 6, 25, 50, 100 s), it is possible to judge the rate of the rarefication of the plasma bunch. At the later time stages of the expansion process (t D 75 and 100 s), rarefication is observed in the wake of the moving bunch as well as the anisotropy in the space distribution of the density, which in the considered configuration points to the influence of the incident ionospheric flow. It is obvious from the presented distributions of gas-dynamic (Figure 3.2 (b)) and magnetic (Figure 3.2 (c)) pressures that as early as time t D 100 s a pressure increase is observed on the side of the incident ionospheric flow due to the collision of the plasma bunch with the ambient medium. The asymmetry of the compression wave is clearly identified from the side of the incident ionospheric flow (Figure 3.3). It should be recalled that the distribution of the magnetic pressure (Figure 3.2 (c)) characterizes the perturbation distribution of the magnetic field module in the vicinity of the plasma spread, and the components of the rotor of the magnetic field induction vector make it possible to obtain the electric currents induced by the expanding plasma. In this connection, the intensity distribution of the induced electric field in the vicinity of the plasma bunch is of great practical interest. Figure 3.4 show the projections of the vector of the intensity of the electric field intensity on the axes y and z, respectively, at the time t D 100 s. Figures 3.5–3.7 show the density, gas-dynamic and magnetic pressure at ˇ D 30ı, as well as flow field and the intensity of the electric field at the time t D 100 s. The plume asymmetry, which is lower than it was before, clearly demonstrates the reduction in the intensity of this interaction (compare Figures 3.3 and 3.6). This points to the fact that in the simulation cases considered, the interaction of plasma flows

165

Section 3.14 Splitting method for nonstationary 3D MHD flows Z 120

100

80

60

40

20

0

0

20

40

60

80

100

Y

Figure 3.3. Distribution of the gas-dynamic pressure and velocity field at t D 100 s. The angle ˇ D 60ı . The pressure in the plasma bunch reaches 401.

(a)

(b)

Z 120

80 Ey = +24

Ey = −18.6 Ey = −24

40 Ey = +18.6

0

0 40

Y

80

80

X

40

40

Y

80

80

X

40

Figure 3.4. Intensity distribution of the electric field Ey (a) and Ez (b) at t D 100 s. The angle ˇ D 60ı .

166

Chapter 3 Computational models of magnetohydrodynamic processes

(a)

(b)

(c)

P

RO 9.3E+00 8.3E+00 7.4E+00 6.5E+00 5.6E+00 4.6E+00 3.7E+00 2.8E+00 1.9E+00 9.4E-01

0

1.4E+01 1.3E+01 1.1E+00 9.8E+00 8.4E+00 7.0E+00 5.6E+00 4.2E+00 2.8E+00 1.4E+00

0 40

Y

80

80

X

40

z 120

PH

2.8E+02 2.6E+02 2.3E+02 2.0E+02 1.7E+02 1.4E+02 1.1E+02 8.5E+01 5.7E+01 2.8E+01

80 40 0

0 40

Y

80

80

X

40

40

Y

80

80

X

40

Figure 3.5. Spatial distribution of the total density of the plasma (a), the gas-dynamic (b) and magnetic (c) pressure at t D 100 s. The angle ˇ D 30ı . Z 120

100

80

60

40

20

0

0

20

40

60

80

Y

Figure 3.6. Distribution of the gas-dynamic pressure and velocity field at t D 100 s. The angle ˇ D 30ı . The maximum pressure is 284.

167

Section 3.14 Splitting method for nonstationary 3D MHD flows

(a)

(b) z 120 80 Ey = −19.8

Ez = −13.5

Ez = +13.5

40

Ey = +19.8 0

0 40 Y

80

80

X

40

40

Y

80

80

X

40

Figure 3.7. The intensity of the electric field Ey (on the left) and Ez in two planes at t D 100 s. The angle ˇ D 30ı.

between each other appears to be stronger than the interaction of the plasma bunch with the ambient magnetic field. If the characteristic scales of the expanding plasma deceleration are calculated by the estimated relations, the radius of the magnetic deceleration to be about 10 m, and the radius of the hydrodynamic braking to be about 1 m, which shows that it is the hydrodynamic braking that will be manifested first. Therefore, both approximate theoretical estimates and the results of the threedimensional numerical calculations of this section show that the motion of the plasma bunch in the conditions considered is primarily influenced by the incident flow of the ionospheric plasma after about 20–40 s after the pulse ends. It is also evident that the intensity of the interaction between the plasma plume with the ionospheric flow depends on their mutual orientation. Therefore, at ˇ D 30ı, it is easier to identify the asymmetry of the plasma plume due to the interaction of the spread plasma with the geophysical magnetic field. Although the physical model employed was approximate, the numerical computations succeeded in predicting practically all important parameters of the plume induced by a pulsed plasma thruster, such as the velocity of plasma expansion, the intensity of the induced electric and magnetic fields, and the induced currents. The fact that these functions are characterized by a spatial distribution enables the researcher to analyze the pattern of interaction of the spread of the plasma with the flow of the ionospheric plasma, and, which is of special significance in providing the safe operation of space vehicles, the interaction of the plasma flows with the spacecraft’s surface. In the considered case, numerical modeling has shown that at a distance of 0.5 m from the outlet section of the PPT4 nozzle, the plasma plume starts its interaction with the incident ionospheric plasma.

Part II

Numerical simulation models of glow discharge

Chapter 4

The physical mechanics of direct current glow discharge This chapter is devoted to computer simulation techniques of glow discharges. Physical models and numerical methods considered in the second part have various applications in modern gas-discharge physics, plasma research and aerophysics. Chapters of the second part of the book give the full necessary physical models and computational physics methods for the creation of computational models of glow discharges. The representations about structure of glow discharge are substantially based on numerous experimental data and on the physical models describing processes in separate areas of glow discharge [12, 83]. Until now a common theoretical-computational model has not been created. Among the existing models it is necessary to note driftdiffusion models of discharge [12, 83, 123], models of quasi-neutral plasma with ambipolar diffusion [97], and also the models based on use of statistical Monte-Carlo methods [5]. Only few of the enumerated models are intended for researching spatial (two- and three-dimensional; 2D and 3D) structures of glow discharge. This is related, first of all, with the high laboriousness of spatial problems solution. However, due to the growth of computer power and the use of parallel computing technology in computational physics, the spatial computational models including all varieties of kinetic processes become more and more realistic. The necessity for this increases is also connected with a new wave of interest in electric discharge phenomena, which is generated by possible applications of these phenomena in hypersonic aircraft. In the book, two classes of models of glow discharge are presented: the driftdiffusion model and the model of quasi-neutral plasma with ambipolar diffusion. Basically, two-dimensional models are considered, and problems of testing of these models are solved with methods of computing physics, or by comparison with the known experimental facts. The majority of the presented results of numerical calculations are concerned with the two-dimensional glow discharge burning in conditions of normal current density. In this case, besides a description of the axial distributions of discharge parameters (from the cathode to the anode), the significant interest is the study of regularities of two-dimensional structures forming. In addition, it has emerged that good test criterion for the two-dimensional models of normal glow discharge are outcomes of calculations under one-dimensional Engel–Steenbeck theory, created at strongly simplified suppositions for a one-dimensional model of a cathode layer. The specified one-dimensional model has appeared simple enough and rather successful so it became the standard for the analysis of experimental data. Numerous verifications of the given model by experimental data allow to consider its relevance for the whole class of spatial computing models.

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Chapter 4 The physical mechanics of direct current glow discharge

There are two well-known types of glow discharges. In normal glow discharge, the current is transferred only through a part of the cathode surface (gas excited by electrons shines out only above a part of the cathode), and in anomalous glow discharge, the current flows through all surface of the cathode, and the whole cathode becomes covered by a luminescence. In normal glow discharge, the law of a normal current density is satisfied: the current through discharge increases with magnification of a voltage drop on a cathode layer due to magnification of a current spot square, thus the current density in the center of a current spot does not vary (varies slightly).

4.1 Fundamentals of the physics of direct current glow discharge. The Engel–Steenbeck theory of a cathode layer The classical glow discharge is characterized by the following parameters:  

total current through discharge I  103–102 A; voltage drop on discharge gap V  200–5 000 V.

Figure 4.1 gives the representation of glow discharge place among other types of discharges between two electrodes. The typical values of voltage and current on the gasdischarge gap of classical glow discharge are shown on the axes of the figure. The majority of works on the experimental and theoretical study of glow discharge has considered the pressure range of p  1–5 Torr. However, this type of discharge is used commonly enough at raised pressure, though in these cases glow discharges are rather unstable V, kV 6 1

2

3

0.5 4

5

1

7 8 10

−4

10

−2

I, A

Figure 4.1. The diagrammatic representation of voltage-current characteristic of electric discharges (boundary values of voltages and currents are approximate). 1–2: area of non-selfmaintained discharge; 2–3: dark, the Townsend discharge; 4–5: normal glow discharge; 5– 6: anomalous glow discharge; 6–7: area of transition from glow to arc discharge; 7–8: arc discharge.

173

Section 4.1 The Engel–Steenbeck theory of a cathode layer

Notwithstanding that the given type of discharge is widely applied in physical devices of various kinds, also in light sources, generators of plasma and in gas-discharge optical quantum generators (lasers), it is necessary to recognize that the theoreticalcomputational description of glow discharges has only been developed in an insufficient degree. First of all it is explained with the complicated structure of glow discharges and a real lack of acceptable computational models allowing to describe all elements of glow discharge structure. The basic elements of structure of classical normal glow discharge in a gasdischarge tube are shown in Figure 4.2. I II III IV V

VI

К

VII A

R0

E

Figure 4.2. The scheme and structure of classical glow discharge: I is the region of the Aston dark space; II is the region of cathode luminescence; III is the region of cathode dark space; IV is the region of negative luminescence; V is the region of the Faraday dark space; VI is the positive column; VII is the anode dark space; K is the cathode; A is the anode; R0 is the Ohmic resistance of the external circuit; E is the value of electromotive force.

The alternation of relatively light and dark areas on the scheme of discharge approximately corresponds to the real alternation of gas luminescence areas in the discharge. However, as a rule, areas I–V adjoin the cathode more close. The typical distribution of electric field strength E and volumetric density of charge  D e.ni  ne / are shown in Figure 4.3. The basic physical processes ensuring the existence of the glow discharge are the secondary ion-electronic emission of electrons from the cathode, acceleration of the electrons in an applied electric field and their collisions with neutral discharge particles whose concentration is always of several orders more than that of the electrons and positive ions in the discharge. Negative ions also play an important role in some gases. The principal source maintaining glow discharge is the electron emission from a surface of the cathode due to the ion-electronic emission. Electrons that abandon the cathode have rather low energy ( 1 eV) which is caused by the ion-electronic emission mechanism. These electrons cannot excite a luminescence of atoms and molecules in the gas-discharge gap, therefore, near to the cathode the dark area is observed. This is the Aston area of glow discharge (see Region I in Figure 4.2). Near to the cathode, the electric field strength is great (E  3–4 kV/cm), there electrons fast gain sufficient energy to cause a luminescence of gas, the so-called cathode luminescence (Region II in Figure 4.2). During the process of increasing energy, the electrons become capable

174

Chapter 4 The physical mechanics of direct current glow discharge V

E I

I II

II III IV V

Cathode

III VI

VII

+ IV V

x − Anode Cathode

−

x

Anode

Figure 4.3. A qualitative picture of the distribution of an electric field strength (a) and a volumetric density of charge (b) along an axis of symmetry from the cathode to the anode. Figures mark areas of glow discharge structure (see Figure 4.2).

of exciting higher and higher energy levels of atoms and molecules, therefore, a gradual modification of luminescence color (from red to violet shadowing) in the field of a cathode luminescence can be observed. Finally, the energy gained by the electrons becomes so high that their inelastic collisions with atoms and molecules result in ionization of the particles. This happens in the field of cathode dark space (the Region III in Figure 4.2), where an avalanche-like process of electron multiplication prevails. Electrons which are born as a result of a neutral particle ionizing event start to gain their energy in an electric field whose strength is essentially lower than near to the cathode. Nevertheless, the large number of electrons accelerated by this small field appears sufficient for excitation of neutral particles, and a zone of a negative luminescence arises in the discharge (the Region IV in Figure 4.2). The distance from the cathode is greater the lower the field strength, and according to this the energy accumulated by the electrons decreases also. This leads to the fact that it is also possible to recognize different spectral regions (from violet to red shadowing) in the luminescence of Area IV. After the negative luminescence area, there is the Faraday dark space area (Region V in Figure 4.2). The energy of electrons in this area is so low that they do not excite a luminescence of neutral particles any more. However, the number of electrons here is already rather great, therefore, it is possible to observe an area of a spatial negative charge, and the electric field strength can change its sign to the opposite shown here (see Figure 4.3). The most extended area of classical glow discharge is the positive column (Region VI in Figure 4.2). It is the area of quasi-neutral plasma with rather low electric field strength and an average energy of thermal motion of electrons of about 1–3 eV. Nevertheless, in the power spectrum, electrons are available with large enough energy, and they cause the excitation and luminescence of neutral particles. In the immediate proximity of the anode, the anode dark space (Region VII in Figure 4.2) is observed. This is a small area of a volumetric negative charge (see Figure 4.3). The anode does not generate ions, but accepts an electron flow.

Section 4.1 The Engel–Steenbeck theory of a cathode layer

175

Further, the classical one-dimensional Engel–Steenbeck model is presented for a cathode layer of normal glow discharge [12, 128]. Let us consider the one-dimensional cathode layer of normal glow discharge. If we suppose that ne,0 is the number of electrons emitted from a surface of the cathode due to ions falling on it, ne,1 is the number of electrons at the end of a cathode layer with thickness d , then through this thickness of the cathode layer (ne,1  ne,0 ) pairs of electrons and ions are generated. We shall assume that exactly this amount of ions “beats out” electrons from the cathode, and the total number of these electrons is ne,0 : ne,0 D .ne,1  ne,0 /,

(4.1)

where is the secondary ion-electronic emission coefficient. It is easy to derive the formula for ne,1 with the use of definition of the first Townsend ionization coefficient ˛.E=p/, being a function of a field strength and pressure: Z d   E ne,1 D ne,0 exp ˛ dx (4.2) p 0 and ne,1 D ne,0

1C .

(4.3)

Whence we receive a condition of maintaining a stationary current in a cathode layer Z

d

exp 0

 1 E dx D 1 C . ˛ p 

(4.4)

Based on numerous measurements of field strength in a cathode layer, Engel and Steenbeck have assumed the following linear dependence: E.x/ D C.d  x/,

(4.5)

where C is the some constant. For the definition of C we shall consider a potential of an electric field inside of a cathode layer Z x

V .x/ D

E.x 0 /dx 0 D C.xd  0.5x 2/.

(4.6)

0

Having set a voltage drop on a cathode layer Vd (it is possible to measure this value; the mode of its calculation will be given later), we shall receive Vd D C

d2 2

(4.7)

C D Vd

2 . d2

(4.8)

or

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Chapter 4 The physical mechanics of direct current glow discharge

So, the distribution of strength and a field potential in a cathode layer is set in the form of 2 , d2 x V .x/ D Vd .2d  x/ 2 . d

E.x/ D Vd .d  x/

(4.9) (4.10)

But from the Poisson equation it follows that 2 d2 V .x/ D 4 D  2 Vd . dx 2 d

(4.11)

What this means is that in the cathode layer the charge density remains constant: D

Vd D e.ni  ne /. 2d 2

(4.12)

Let us rewrite the condition (4.1) concerning the current density je,0 D ji ,0,

(4.13)

where je,0 , ji ,0 are the current densities of electrons and ions on the cathode. Then the total current density on the cathode is equal to j0 D je,0 C ji ,0 D ji ,0.1 C /.

(4.14)

ji ,0 D ni ,0 vi ,dr ,0 D ni ,0i E.x D 0/,

(4.15)

By definition where i is the ion mobility; vi ,dr ,0 is the drift velocity of ions. We shall determine the strength of the electric field on the cathode E.x D 0/ by supposing that ne,0 < ni ,0 , then from formulas (4.9), (4.10) and (4.12) one can derive .x D 0/ D

E.x D 0/ 4d

and

(4.16)

Vd2 .1 C /. (4.17) d 3 Voltage drop on the cathode can be calculated using a condition of self-maintained discharge (4.4). For the definition of the value ˛.E=p/ we shall take an empirical relation which will be often used in the following:   B ˛.E=p/ D A exp  , (4.18) p .E=p/ j 0 D i

where A, B are the empirical factors obtained for various gases [12, 83].

177

Section 4.1 The Engel–Steenbeck theory of a cathode layer

Substituting (4.18) in (4.4), we receive     Z d 1 Bp D Ap dx. ln 1 C exp  E.x/ 0

(4.19)

Now it is necessary to substitute into (4.19) the formula for the calculation of the strength of a field (4.9) " #   Z d 1 Bp ln 1 C exp  2V

(4.20) D Ap  dx. d 0 1 x d

d

Let us choose the value t as an integration variable x 2Vd . d Bpd

tD

Then, integration of (4.20) gives:     AB.pd /2 1 2Vd D , ln 1 C S 2Vd Bpd Z

where

z

S.z/ D 0

  1 dt . exp  t

(4.21)

(4.22)

(4.23)

The last integral was calculated and tabulated. In the book [83] an assumption is used about a constancy of electric field strength in the cathode layer. It has been shown that this allows to calculate the integral (4.19) easily and with a sufficient exactitude for practical needs. The relation (4.22) can be presented in the form of a simple algebraic equation that can be considered a dimensionless characteristic equation for all gases .A, B/ and materials of the cathode . /.  .C1Vd /1=3  S .C1 Vd /1=3 .C2 j0/1=3 D 1, 2=3 .C2 j0/

(4.24)

C1 D

2A ; B ln.1 C 1= /

(4.25)

C2 D

4 ln.1 C 1= / . AB 2 p 2 .pi /.2 C /

(4.26)

where

Engel and Steenbeck [128] investigated the characteristic equation (4.24) and have shown that dependence of C1 Vd from C2j0 looks like a parabola whose minimum

178

Chapter 4 The physical mechanics of direct current glow discharge

meets normal glow discharge and corresponds to values C1Vd D 6 and C2 j0 D 0.67. It allows to obtain the parameters of a cathode layer of normal glow discharge   1 3B ln 1 C , V, (4.27) Vn D A 2 jn 14 AB .i p/.1 C / , D 5.92  10 p2 ln.1 C 1= /

A cm2  Torr

(4.28)

ln.1 C 1= / , cm  Torr, (4.29) A where the index n marks the parameters of normal discharge. As mentioned above, the stated theory of Engel and Steenbeck was excellently confirmed by numerous experimental research studies. Therefore, in the following, the conclusions of this theory (first of all, relations .4.27/–(4.29)) will be repeatedly used for the interpretation of numerical simulation results. dn p D 3.78

4.2 Drift-diffusion model of glow discharge 4.2.1 Governing equations Equations of the drift-diffusion model of stationary glow discharge between two infinite flat electrodes (the scheme of the problem is presented in Figure 4.4) are formulated in the following form [84]: @ne C div  e D ˛ j e j  ˇne nC , @t @nC C div  C D ˛ j e j  ˇne nC , @t div E D 4e.nC  ne /,

(4.30) (4.31) (4.32)

where  e D De grad ne  ne e E,

(4.33)

 C D DC grad nC C nC C E, E D  grad ',

(4.34) (4.35)

ne , nC are the concentrations of electrons and ions in 1 cm3 (further on in the given chapter the functions concerning ions will be designated with the index “C”, as the index “i” will be used in finite-difference schemes); E and ' are the vector and potential of an electric field strength; De , DC are the electron and ion diffusion coefficients; e , C are the electron and ion mobility; ˛ D ˛.E/ is the coefficient of molecule

179

Section 4.2 Drift-diffusion model of glow discharge

x=H

Anode

E x r r=0

Cathode

R0

r=R

Figure 4.4. The scheme of the problem.

ionization by electron impact (the first Townsend coefficient); ˇ is the coefficient of ion-electron recombination. In view of the relation (4.35) the equation (4.32) can be rewritten in the following form: div Œ grad .'/ D 4e.nC  ne /. (4.36) Finite-difference operators will be formulated for two orthogonal geometries, namely for the rectangular and the cylindrical. Typical boundary conditions for equations (4.30)–(4.32) are formulated as following: x D 0: x D H: r D 0: r D R : 1/ 2/

e,x D C,x , ' D 0, nC D 0, ' D V , @nC @' @ne D D D 0, @r @r @r V ne D nC D 0, ' D x, H @nC @' @ne D D D 0, @r @r @r

(4.37) (4.38) (4.39) (4.40) (4.41)

where is the coefficient of ion-electron emission from the cathode surface; V is the voltage drop on the discharge gap. Boundary condition (4.40) sets an undisturbed electric field and lack of charges at a great distance from the center of discharge, and the condition (4.41) states lack of transverse field gradients and particles on the external boundary of the rated area. Really, in calculations there is a recommendation to use a very small value of concentration ( 106 of maximum concentration in the discharge). By methodological calculations it has been found that a change of this “background” concentration by several digits did not affect the parameters of the discharge.

180

Chapter 4 The physical mechanics of direct current glow discharge

The first-mentioned condition on the external side boundary r D R corresponds to a process without radial diffusion of particles. Really, in the no-current area of discharge there is no place for charged particles to appear. Charges move along electric field lines and cannot pass from one line to another. Thus, on each field line, the Townsend condition of stationary self-maintained discharge should be satisfied   Z 1 ˛.l 0 /dl 0 > ln 1 C . (4.42) l If this condition is not satisfied on any field line, charges drifting along this line will abandon the discharge gap, and then this line will become no-current. Diffusion of charged particles and their origin on boundaries of the discharge gap (on the cathode and the anode) can cause current-carrying lines without realization of the condition (4.42). At the same time it is necessary to mention that in the case of diffusion, the force and current lines do not coincide any more, and the condition of stationary self-maintained discharge has non-local character. In this case, there is no basic exclusion for charged particles’ origin on the periphery of the discharge area at r ! R. It supposes the boundary condition (4.41), which assumes the presence of a spatial symmetry in a disposition of current-carrying channels (in a cylindrical case, the discharge looks like coaxial cylinders enclosed in one another, and in a flat case it looks like channels regularly arranged on a plane), or the presence of high enough background concentration of the charged particles. If the solitary discharge is considered, this background level should be small enough to suppress an appreciable influence on the structure of the discharge. By numerical experiments it has been shown that in a wide range of parameters of glow discharge the type of boundary conditions does not essentially influence the results of calculations. At statement of the boundary value problem (4.30)–(4.41), diffusion of the charged particles in an axial direction is neglected. Boundary conditions include a still undetermined value of a voltage drop on discharge gap V . For its determination it is necessary to include conditions in an external circuit (see Figure 4.4). For stationary glow discharge it is possible to write the obvious relation Z R EV D 2 e,x .r , x D H / r m dr , (4.43) eR0 0 which specifies equality of the sum of voltage drops on a resistance R0 and on the discharge gap to the electromotive force "; e,x is the projection of an electron flux density vector on x axis. In equation (4.43) and further, m D 0 corresponds to a flat case, and m D 1 corresponds to a cylindrical case.

181

Section 4.2 Drift-diffusion model of glow discharge

Let us consider a glow discharge in molecular nitrogen: e p D 4.4  105,

C p D 1.45  103,

.Torr  cm2 /=.V  s/,

(4.44)

ˇ D 2  107, cm3 =s, " D 4e D 1.81  106, V  cm, 8   E B ˆ ˆ , > 100; ˆ A  exp  < E=p p ˛ D .cm  Torr/1 ,   ˆ p A B E 1 1 ˆ ˆ exp  , < 100, : E=p E=p p

(4.45)

where A D 12 .cm  Torr/1 ; B D 342 V=.Torr  cm/; A1 D 900 V=.Torr  cm2 /; B1 D 314 V=.Torr  cm/. Note that similar empirical coefficients for other gases can be found in [12, 83]. Diffusion coefficients were defined with the Einstein relations D e D e T e ,

D C D C T C ,

where Te , TC are the temperatures of electrons and ions, eV. With the use of the model it is possible to make calculations of glow discharge structure in various suppositions concerning the temperature of electrons and heavy particles: (a) Te D TC D 0; (b) Te D 11,610 K (Te D 1 eV), TC D 300 K (TC D 0.0258 eV); (c) Te D f .x, r / and TC D f .x, r /. Note that the condition (a) corresponds to lack of physical diffusion.

4.2.2 Reduction of governing equations to a form convenient for numerical solution The charge conservation equations (4.30), (4.31) can be rewritten in the following dimensional form (neglecting diffusion in axial direction): @ne 1 @ @' @ @' C e m ne r m C e ne @t r @r @r @x @x 1 @ @ne D ˛ j e j  ˇne nC ,  De m r m r @r @r 1 @ @' @ @' @nC  C m nC r m  C nC @t r @r @r @x @x 1 @ m @nC D ˛ j e j  ˇne nC ,  DC m r r @r @r

(4.46)

(4.47)

182

Chapter 4 The physical mechanics of direct current glow discharge

To reduce the equations (4.46) and (4.47) to a dimensionless form, multiply them on H H the factor E , where N0 D 109 cm3 is the characteristic concentration of charged e N0 particles in glow discharge. The following definitions will be used below r t H2 x , rQ D , D , t0 D ; H H t0 e,0 E

ne nC ' , NC D ,ˆ D u D Ne D ; N0 N0 E

xQ D

De , DQ e D e,0 E W D

DC DQ C D , e,0 E E , H

Dt

QD

HN0 ; We,0

We,0 . H

For brevity, the sign of a tilde will not be used for nondimensional coordinates r and x, therefore, @ˆ @ˆ 1 @ @Ne 1 @ @ @Ne C m r m Ne C Ne  DQ e m r m @

r @r @r @x @x r @r @r s     @ˆ 2 @ˆ @Ne 2 Q Ne  De C Ne , (4.48) D ˇQNe NC C ˛H @x @r @r

@ˆ C @ @ˆ 1 @ @NC @NC C 1 @ m  r NC  NC  DQ C m r m m @

e r @r @r e @x @x r @r @r s     @ˆ 2 @ˆ @Ne 2 Q  De Ne C Ne . (4.49) D ˇQNe NC C ˛H @x @r @r Using the designations given in Table 4.1, the equations (4.48) and (4.49) can be presented in the following canonical form @u 1 @ 1 @ @u @ C m .r m au/ C .bu/  c m r m Df, @t r @r @x r @r @r

(4.50)

for which a finite-difference scheme will be constructed. The dimensionless Poisson equation for electric field potential can be obtained by analogy " HN0 1 @ m @ˆ @2ˆ D  r C .NC  Ne /. r m @r @r @x 2 W

(4.51)

183

Section 4.2 Drift-diffusion model of glow discharge Table 4.1. Dimensionless parameters for Equation (4.50). a

Electrons

Ions

r

m @ˆ

@r

C @ˆ r m e @r

b @ˆ @x C @ˆ  e @x

c

F r

DQ e

˛H

Ne @ˆ @x

2

 2 Q e @Ne  C Ne @ˆ  D @r @r  ˇQNe NC

r DQ C

˛H

Ne @ˆ @x

2

2  e C Ne @ˆ  DQ e @N  @r @r  ˇQNe NC

Boundary conditions (4.39)–(4.41) also should be transformed to the dimensionless form @NC @ˆ @Ne D D D 0, (4.52) r D 0: @r @r @r R V 1/ Ne D NC D 0, ˆ D x, (4.53) rD : H E @NC @ˆ @Ne D D D 0. (4.54) 2/ @r @r @r Let us consider the derivation and analysis of boundary conditions at the cathode. For this purpose we shall consider expressions for particle flux vectors in view of radial diffusion of charged particles:     @' @' @ne C er e ne  De , (4.55)  e D ex Ce ne @x @r @r     @' @' @nC C er C nC  DC . (4.56)  C D ex C nC @x @r @r The boundary condition (4.37), noted above, establishes a relation between modules of charged particle fluxes to cathode and anode. Note that this condition does not define their angular distributions. For the drift-diffusion model with a constant coefficient of ion-electronic emission, the boundary condition on a cathode is formulated only for axial components of electrons and ions fluxes . e /x D  . C /x . (4.57) If we accept the condition (4.57), then from it directly follows that C . ne D nC e

(4.58)

The condition is also used in the one-dimensional theory of a cathode layer. In view of (4.58) it is possible to define approximate “grid” boundary conditions for ions on the cathode.

184

Chapter 4 The physical mechanics of direct current glow discharge

From equations (4.30) and (4.31) the continuity equation is derived in the following form @Q C div . C   e / D 0. (4.59) @t Considering that at x D 0, ne  nC , and also introducing designation C Deff D De  DC , e one can receive   @ @' 1 @ @nC @nC  .1 C /C . nC  Deff m r m @t @x @x r @r @r

(4.60)

In a dimensionless form this equation looks as follows: @NC @ˆ 1 @ @NC C @ D .1 C / NC  DQ eff m r m , @

e @x @x r @r @r

(4.61)

where Deff DQ eff D . Ee,0 Using continuity equation (4.59) and boundary condition (4.38), we will receive the equation for electrons near to the anode, which will be used for construction of a “grid” boundary condition for electrons @Ne @ˆ 1 @ @Ne @ D  Ne C DQ e m r m . @

@x @x r @r @r

(4.62)

4.2.3 Initial conditions of the boundary value problem for the glow discharge Two kinds of initial conditions are in common use. The first one is the use of numerical simulation results obtained for close boundary value problem. The second method is based on some a priori information on a prospective solution. For example, simulating two dimensional normal glow discharge one can wait for a satisfactory description of the glow discharge in a one-dimensional axial direction whose parameters are predicted by the Engel–Steenbeck theory (see Section 4.1) 2 jn 11 AB C p.1 C / , mA=.cm2  Torr/ D 5.92  10 p2 ln.1 C 1= /

(4.63)

dn p D 3.78A1 ln.1 C 1= /, cm  Torr,

(4.64)

Vn D 3BA1 ln.1 C 1= /, V,

(4.65)

where jn is the normal current density; dn , Vn are the thickness of the cathode layer and the voltage drop on it.

Section 4.2 Drift-diffusion model of glow discharge

185

Let us estimate a level of ion concentration in the cathode layer. For this purpose we shall write the relationship between a current density and the concentration of ions jn D ne e Ex C nC C Ex D .1 C /Ex nC CQ0 , where the coefficient Q0 D 1.6  1016 N0 defines dimension of jn in mA/cm2 . The value Ex can be estimated under the formula: ˇ ˇ ˇ d' ˇ Vn , Ex D ˇˇ ˇˇ  dx dn whence nC 

j n dn D .nC /0 . .1 C /Vn CQ0

Thickness of the cathode layer dn sets a scale of an exponential fall in ion concentration from .nC /0 to value N0 . Initial axial distribution of electron concentration is also built under the exponential law from the value on the cathode .ne /0 D

C .nC /0 e

up to value N0 on distance dn from the cathode. Radial distributions of ion and electron concentrations are built uniformly along axis x under the formula   n.x, r / D n.x, r D 0/ exp .r=rcc/m , where m D 2–4; rcc is the prospective radius of current channel which is estimated by the relation s E  2Vn . rcc Š  R0 jn Iterative calculation of electric field potential is performed with the use of the ne .x, r / and nC .x, r /. These iterations are necessary with the Poisson equation (4.32) for the calculation of voltage drop V on the gas-discharge gap, corresponding to the given distributions of charged particles. The new potential field obtained at each iteration is used for recalculating value V under the formula (4.43). Due to the high nonlinearity of the equation it is necessary to introduce a restriction on a possible change of a voltage drop, for example, Vn  V  E.

186

Chapter 4 The physical mechanics of direct current glow discharge

4.2.4 Glow discharge with heat of gas The self-consistent numerical model of glow discharge is formulated for flat and axially symmetric geometry in which the heat of gas is considered. It is supposed that the discharge exists in a condition of a normal current density mode between two flat electrodes (see Figure 4.4) so the side boundary effects do not influence its structure and parameters. As before, the electrodynamic structure of glow discharge is investigated within the limits of the drift-diffusion model formulated concerning electronic and ionic concentrations together with the Poisson equation, which defines the distribution of an electric potential in the electro-discharge gap. So, in comparison with the problem considered in the previous section, the heat of neutral gas will be considered here. The Joule thermal emission is caused by the heat of neutral gas and is taken into consideration here. It is due to the collision of molecules with electrons which receive energy from an electric field (they are “warmed up” by the electric field inside the glow discharge). Not all the energy transmitted at the collisions of electrons with gas molecules is used on heating them. The significant part of this energy is spent for excitation of molecules’ vibrational degrees of freedom. In the given case, the solution of physical kinetic equations for the determination of the electron energy distribution function, which allows to predict part of the electronmolecular collision energy transformable for heating of gas, is approximately replaced with phenomenological effectiveness ratio of Joule energy emission transfer to heat of gas. It is known that this coefficient can vary over a wide range, and for typical laser mixtures .N2 C CO2 / and conditions it makes 0.15–0.25. The set of equations of the design model is formulated in the following form: @ne @e,x 1 @r m e,r C C m D ˛.E/ j e j  ˇnC ne , @t @x r @r @C,x 1 @r mC,r @nC C C m D ˛.E/ j e j  ˇnC ne , @t @x r @r   1 @ @2 ' m @' C r D 4e.ne  nC /, @x 2 r m @r @r   @T @ @T 1 @ m @T cv D . / C m r  C Q, @t @x @x r @r @r

(4.66) (4.67) (4.68) (4.69)

where  e D De grad ne  ne e E; Q D . jE/;

 C D DC grad nC C nC C E;

j D e. C   e /;

E D  grad ';

T is the temperature of gas; m D 0 is used for flat discharge, m D 1 is used for axially symmetric discharge. Remaining variables and coefficients of the drift-diffusion model are as previously.

187

Section 4.2 Drift-diffusion model of glow discharge

In the case considered, the energy conservation equation for neutral gas (4.69) is formulated in the form of the Fourier–Kirchhoff equation. The important corollary of neutral gas heating is the modification of local values of neutral particles density that renders an essential influence on the value of the collision frequency of electrons with molecules of gas, hence, on such parameters of the driftdiffusion model as frequency of ionization, mobility of electrons and ions. Introducing some effective pressure, the specified variables of the drift-diffusion model can be written in the form of e .p  / D

4.2  105 p

C .p  / D

p D p De D e .p  /Te , cp D 8.314 M† D 28 g=mole,

293 , Torr, T DC D C.p  /T ,

7 1 , J=.g  K/, 2 M† cv D 0.742 J=.g  K/,

 D 1.58  105 8.334  104 D  2 .2.2/

.2.2/ D

1.157 , .T  /0.1472

s

2280 , cm2=.V  s/, p

M† p , g=cm3 , T

  cp M† T 0.115 C 0.354 , W=.cm  K/, M† R0

T , ."=k/ D 71.4 K, ."=k/ R0 D 8.314 J=.K  mole/, T D

 D 3.68 Å,

p is the pressure of gas inside the gas-discharge gap. In this case, the ionization coefficient is calculated under the formula   B , cm1 . ˛.E/ D p  A exp  .jEj=p  /

(4.70)

4.2.5 Estimation of typical time scales of the solved problem According to the statement of the glow discharge problem in molecular nitrogen, the following physical processes are considered: 

ionization of molecules N2 by electron impact;



recombination of the positive ions at collision with electrons; drift of ions and electrons in an external electric field;



188   

Chapter 4 The physical mechanics of direct current glow discharge

diffusion of charged particles; ambipolar diffusion of charged particles; heating of neutral gas.

Let us estimate characteristic time scales for the enumerated processes, and compare them with typical time of space charge relaxation. It will give the possibility to get a representation of the time steps needed for the numerical simulation. Charged particles’ characteristic drift time in glow discharge The drift velocity of electrons and ions in a constant electric field is equal to k vdr D k E,

.k D e, C/.

At pressure p D 5 Torr from .4.44/ one can find e D 8.8  104 cm2 =.s  V /C D 2.9  102 cm2 =.s  V/. There are two typical levels of electric field in glow discharge. These are: Ec D 3 000 V/cm for cathode layer, and Epc D 100 C V=cm for positive column. In these regions of glow discharge we can define correspondingly: C e .vdr /c D 2.64  108 cm=s, .vdr /c D 8.7  105cm=s, e .vdr /pc D 8.8  106 cm=s,

C .vdr /pc D 2.9  104 cm=s.

Each of these regions also has a typical spatial scale: for a cathode layer dc D 0.1 cm, and for a positive column H D 1 cm. Hence, the required typical times are: e .tdr /c D 0.379  109 s,

C .tdr /c D 0.115  107 s,

e C /pc D 0.114  106 s, .tdr /pc D 0.345  104 s. .tdr

Typical time of ionization Let the time of ionization be

i D 1=i, where i .p/ is the frequency of ionization; p is the pressure. Frequency of ionization is connected with electron drift velocity and with the Townsend coefficient of ionization [12, 83] e i D ˛.E, p/vdr . Let us estimate Townsend’s coefficient under formula (4.45) at two values of field strength, Ec D 3 000 V/cm and Epc D 100 V/cm: ˛c  30 cm1 , ˛pc  105 cm1 . Hence, the estimations of ionization frequency and typical time of ionization are: .i /c D 7.92  109 s1 ,

.i /pc D 3.9  102 s1 ;

. i /c D 0.127  109 s, . i/pc D 0.256  102 s.

189

Section 4.2 Drift-diffusion model of glow discharge

Typical time of recombination Frequency of ion-electronic recombination is proportional to the concentration of ions r D ˇnC , s1 , therefore,

r D 1=ˇnC , s. Accepting ˇ D 2  10 1010 cm3 , one can get

7

cm /s and a concentration of ions equal to nC D 3

p  0.5  103 s.

Typical time of charged particles’ diffusion The average squared displacement of particles in a diffusion process is set with the Einstein formula x 2 D 2Dt , where D is the diffusion coefficient. Let us consider three types of diffusion: 

diffusion of electrons with coefficient D e D e T e ;



diffusion of ions with coefficient D C D C T C ;



ambipolar diffusion with coefficient D a D C T e .

In the given formulas the Einstein relation for diffusion coefficients was used, where the temperature of electrons and ions were expressed in eV. Accepting Te D 1 eV, TC D 0.0258 eV, one can get De D 8.8  104 cm2 =s;

DC D 7.5 cm2 =s;

Da D 2.9  102 cm2=s.

For a typical scale of diffusion transfer q ƒ D x 2 D 0.1 cm, which corresponds approximately to thickness of a cathode layer dn and to expansion of a column boundary in a radial direction, we have characteristic times of diffusion:

de D

ƒ2 ƒ2 D 0.569  107 s, dC D D 0.666  103 s, 2De 2DC ƒ2 D 0.345  104 s.

da D 2Da

190

Chapter 4 The physical mechanics of direct current glow discharge

Typical time of relaxation of a space charge Let there be a spatially homogeneous charge  D e.nC  ne / such that div E D 4 D 4e.nC  ne / D " .nC  ne /. From the continuity equation @ C div j D 0, @t where j D E, supposing that due to strong differences in mobilities of electrons and ions, the current is transferred by electrons. Then @ C  div E C E grad  D 0. @t For homogeneous distributions of the charged particles @ @ C  div E D C 4 D 0, @t @t and .t / D .0/ exp.4 t /. By definition, the time of volume charge relaxation, named the Maxwell time, is defined as follows 1 1 1 D D  .

M D 4 4 ne e e " ne e Substituting numerical values, which are characteristic for the considered problem, we define

M Š 0.63  109 s. Typical time of heat conduction by thermal conductivity Typical time of temperature wave propagation is set by the formula

h D

L2cV 

This value is estimated by tens milliseconds for typical glow discharge. Time steps for numerical calculations Dimensionless time , entering into the system of the solved equations, looks like

Dt

t We,0 , D H tH

Section 4.2 Drift-diffusion model of glow discharge

191

where tH D

H We,0

Obviously, this is the time necessary for an electron to fly through the discharge gap under a homogeneous electric field Ex D W D E=H , corresponding to full magnitude of the electromotive power E. For typical calculation cases: (1) E D 2500 V, W  3330 V/cm, and tH  0.256  108 s; (2) E D 500 V, W  667 V/cm, and tH  1.28  108 s. One can see that the times tH are approximately one order of magnitude greater than the times of electron drift in a cathode layer and essentially less than the drift times in a positive column. They surpass the times of ionization in a cathode layer and times of relaxation of a volume charge, but noticeably less than diffusion and recombination times. Relations between time scales Apparently from the estimations given above, the solved problem has essentially different time scales. The fastest processes are ionization of molecules by electron impact and electronic drift in a cathode layer. Their time scale is of  109 s. Volume charge relaxation time has the same scale. The typical time of ion drift in a cathode layer is of  107 s, and it makes the first basic difficulty of the numerical simulation because ions play the same significant role in forming the discharge as electrons do. In the problem under consideration there are some other time scales connected with drift of charged particles in a positive column, with electron and ion diffusion, and also with ambipolar diffusion. Typical times of drift of charged particles in a positive column vary by approximately 102–103 more than corresponding times in a cathode layer, namely for electrons it takes 107 s, and for ions it takes 104 s. The same order holds for times of electron diffusion (107 s) and for ambipolar diffusion  104s. The greatest time scale is for processes of recombination and diffusion of ions, namely 103 s, and also for the process of temperature wave propagation (actually for heating neutral gas). So, the range of time scales is of 109–103s. It is obvious that the direct solution of the considered problem represents excessively greater computing difficulties. Numerical simulation of unsteady or stationary glow discharges with the use of finite different time-dependent methods allows to perform calculations with some certain time steps, which are defined by the stability conditions of the finite-different schemes used. These conditions are defined by types of solved equations, by parameters of the finite-different schemes and by the grids used. It is possible to specify two dimensionless criteria of the solved set of equation (for explicit finite-difference calcu-

192

Chapter 4 The physical mechanics of direct current glow discharge

lation schemes). These are the hyperbolic and parabolic CFL numbers (the Courant– Friedrichs–Lewy stability condition): Q v

2D

 1, CFLP D 2  1, h h where v is the velocity of convective transport; h, are the minimal step in space and the maximal step in time, respectively. The hyperbolic CFL number shows the number of grid points that are passed by any disturbance for one calculation time step. This condition can be derived from the analysis of the stability of the finite-difference scheme used for the simplest hyperbolic equation @f @f Cv D 0, @t @x where v is the transport velocity. p The parabolic CFL number gives the relation of characteristic diffusion length Q to the grid step h, i. e., this is the number of grid points on which a particle 2D

will be transferred by diffusion for one time step. The parabolic CFD number arises from the stability analysis for the simplest equation of the form CFLG D

@f @2 f D DQ , @

@x 2 where DQ is the diffusion coefficient. Using the procedure of normalization of the governing equations one can detect that there is a scale of velocity vQ D e,0 E=H , which corresponds to the velocity of electron motion in an electric field with strength Ex D E=H . For example, the velocity vQ D 2.93108 cm/s is obtained at E D 2 500 V and H D 0.75 cm. It is obvious that characteristic dimensionless velocities for drift of electrons and ions are extremely different e vdr vC C , vdr D dr . vQ vQ For the estimation of the greatest hyperbolic CFL numbers, it is necessary to consider subregions with the greatest velocities and the lowest steps on the grid. In our case, these conditions are satisfied in a cathode layer where drift velocities are maximal and the typical grid step is equal to  0.01 cm. For electrons, it is CFLG e  90 , and for  0.29 . ions is CFLG C Let us consider time step , which is defined by explicit finite-difference schemes from condition CFLG < 1: 1 D 1.11  102 s,

 90 e  vdr

193

Section 4.2 Drift-diffusion model of glow discharge

which corresponds to physical time of tD

1.11  102

H D D 0.434  109 s. We,0 0.256  108

It is clear that the calculation time step obtained corresponds to the least characteristic time scales of elementary physical processes considered above. It allows us to say that the calculation of a nonstationary process with CFLG  1 is possible under the explicit scheme. Numerical modeling of glow discharges by the time-asymptotic method with such time steps is uneconomical, therefore, the implicit methods are in common use. These schemes allow to perform numerical integration at CFLG e  9–90,

CFLG C  0.029–0.29.

It can be seen that the implicitness introduced into the numerical simulation algorithms allows to increase CFLG approximately 100 times in comparison with the explicit ones. Limiting the value CFLG C  1 corresponds to the characteristic physical time of ion C drift in the cathode layer .tdr /c  0.117  107 s. Thus, it is possible to conclude that the process limiting a computing procedure is the ion drift in the cathode layer. From here we can estimate the number of time steps  104 that should be made to take into account ambipolar diffusion and ion diffusion. For preparing the estimations of the parabolic CFL numbers one can obtain: e T e Te De DQ e D D D ; e,0 E e,0 E E then

DQ e  4  104 ;

C T C DC DQ C D D ; e,0 E e,0 E DQ C  3.4  108;

C T e Da DQ a D D ; e,0 E e,0 E

DQ a  1.3  106.

Setting, as before, h  102, we shall obtain characteristic numerical values CFLPe  8 ;

CFLPC  6.8  104 ;

CFLPa  2.6  102 .

Thus, the stable conditions based on the parabolic CFL numbers in the problem under consideration are weaker than ones based on the hyperbolic CFL numbers. In summary, we shall note that the given estimations should be considered approximated, as in conditions of extremely strong heterogeneity of the electric field it is impossible to choose reference features.

194

Chapter 4 The physical mechanics of direct current glow discharge

4.3 Finite-difference methods for the drift-diffusion model 4.3.1 Finite-difference scheme for the Poisson equation Let us introduce an inhomogeneous grid  ˚ ! D .ri , xj /, 1  i  NI , 1  j  NJ in the calculation domain Gf.r , x/, 0  r  R, 0  x  H g, where r1 D 0, x1 D 0, rNI D R, xNJ D H (Figure 4.5). x j+1 j + 1/2 j j − 1/2 j−1 i − 1 i − 1/2 i i + 1/2 j + 1

r

Figure 4.5. Fragment of the finite-difference grid used.

The required finite-difference scheme will be created by the finite-volume method (FVM), therefore, so-called “flux” points are additionally introduced ri C1=2 D

1 .ri C ri C1/, 2

1 ri 1=2 D .ri C ri 1 /; 2

xj C1=2 D

1 1 .xj C xj C1/, xj 1=2 D .xj C xj 1 /. 2 2

Let us consider elementary volume in the neighborhood of a point (i, j ) Z riC1=2 Z xj C1=2 r m dr dx Vi ,j D .2/m ri1=2

D

xj 1=2

.2/m 1 .xj C1  xj 1/.ri C1  ri 1 /.ri C1 C 2ri C ri 1 /m , m C 1 2mC2

where m D 0 is used for a flat geometry, m D 1 for axisymmetric geometry. Let us now integrate the Poisson equation " HN0 H 2 1 @ m @ˆ @2ˆ D  r C .NC  Ne /, R2 r m @r @r @x 2 W

(4.71)

195

Section 4.3 Finite-difference methods for the drift-diffusion model

over the elemental volume Vi ,j , considering, that       Z r 1 iC 2 @ 1 m @ˆ m @ˆ m @ˆ r dr D m .ri C1 C ri / ,  .ri C ri 1/ @r 2 @r i C 1 @r i  1 r 1 @r i 2

Z

2

xj C1=2 xj 1=2

@2 ˆ dx D @x 2



@ˆ @x



  j C1=2

@ˆ @x

2

 , j 1=2

and defining derivatives in “flux” points under the following formulas     @ˆ @ˆ ˆi C1  ˆi ˆi  ˆi 1 D , D , @r i C1=2 ri C1  ri @r i 1=2 ri  ri 1     ˆj C1  ˆj @ˆ ˆj  ˆj 1 @ˆ D , D , @x j C1=2 xj C1  xj @x j 1=2 xj  xj 1 we receive 2.m C 1/ .ri C1  ri 1/.ri 1 C 2ri C ri C1/   H2 m ˆi C1  ˆi m ˆi  ˆi 1  .ri C ri 1 /  2 .ri C1 C ri / R ri C1  ri ri  ri 1   ˆj C1  ˆj 2 ˆj  ˆj 1 C  xj C1  xj 1 xj C1  xj xj  xj 1 D

"HN0 .NC  Ne / . W

(4.72)

After some simple transformations one can receive the canonical five-point finitedifference scheme: Ai ,j ˆi 1,j  Ci ,j ˆi ,j C Bi ,j ˆi C1,j C ANi ,j ˆi ,j 1  CN i ,j ˆi ,j C BN i ,j ˆi ,j C1 C Fi ,j D 0, Ai ,j D Ei Sim1

1 , ri  ri 1

Ci ,j D Ei SimC1 ANi ,j D Ej

1 , ri C1  ri

1 1 C Ei Sim1 , ri C1  ri ri  ri 1

1 , xj  xj 1

CN i ,j D Ej

Bi ,j D Ei SimC1

BN i ,j D Ej

1 , xj C1  xj

1 1 C Ej ; xj C1  xj xj  xj 1

(4.73)

196

Chapter 4 The physical mechanics of direct current glow discharge

Ej D

SimC1 D .m C 1/

2 , xj C1  xj

.ri C1 C ri /m , .ri 1 C 2ri C ri C1/m

Ei D

H2 2 ; 2 R ri C1  ri 1

Sim1 D .m C 1/

.ri C ri 1 /m ; .ri 1 C 2ri C ri C1/m



 "HN0 .NC  Ne / D . W i ,j

Fi ,j

(4.74)

This finite-difference scheme has the second order of approximation in space. We shall consider conservative property of the scheme at m D 1. The desired integral balance equation can be easily obtained from an initial equation for electric field strength div E D " .nC  ne / D A. Having integrated it by all considered volume with the use of the Stokes theorem, one can receive Z

Z

Z

div EdV D

Z

E dS D

V

†

S1

Z  S3

Z

 Z Ex .r , x D H /dS3 D 2  r

0

S2 R

r 0

R

C

Ex .r , x D 0/dS1 C

Z

 @' @x

xDH

dr  R 0

H

Er .r D R, x/dS2  @' @x



 @' @r

xD0

r DR

dr

Z

dx D

AdV . (4.75) V

Let us multiply (4.72) by Vi ,j and sum over all elementary volumes (further m D 1/ NJ 1 NI 1 X X j D2

i D2

C

1 NJ 1 NI X X j D2

D

  H2 ˆi C1  ˆi ˆi  ˆi 1 2 .xj C1  xj 1/ 2 .ri C1 C ri /  .ri C ri 1 / 4 R ri C1  ri ri  ri 1

i D2

NJ 1 NI 1 X X j D2

i D2

  ˆj C1  ˆj 2 1 ˆj  ˆj 1 .ri C1  ri 1/.ri 1 C 2ri C ri C1/  2 4 xj C1  xj xj  xj 1

" H 2 1 N0 .NC  Ne / W 2 23

 .xj C1  xj 1/.ri C1  ri 1 /.ri 1 C 2ri C ri C1/.

197

Section 4.3 Finite-difference methods for the drift-diffusion model

The evaluation of the sum results in 2

NJ 1 X j D2

C 2

  xj C1  xj 1 H 2 rNI C rNI 1 ˆNI ,j  ˆNI 1,j r2 C r1 ˆ2,j  ˆ1,j  2 R2 2 rNI  rNI 1 2 r2  r1 NI 1 X

ri C1  ri 1 ri 1 C 2ri C ri C1 2 4

i D2

D

NJ 1 NI 1 X X j D2

i D2



ˆi ,NJ  ˆi ,NJ 1 ˆi ,2  ˆi ,1  xNJ  xNJ 1 x2  x1



2"HN0 xj C1  xj 1 ri C1  ri 1 ri 1 C 2ri C ri C1 . .NC  Ne /ij W 2 2 4

Under the symmetry condition it is necessary to suppose ˆ1,j D ˆ2,j ,

8j 2 Œ1, NJ , .

Then we shall receive the required finite-difference analogue of an integral balance equation (4.75):  NI 1 X ri C1  ri 1 ri 1 C 2ri C ri C1 ˆi ,2  ˆi ,1   2  2 4 x2  x1 i D2

C

NI 1 X i D2

C 2

ri C1  ri 1 ri 1 C 2ri C ri C1 ˆi ,NJ  ˆi ,NJ 1   2 4 xNJ  xNJ 1

NJ 1 X j D2

D



xj C1  xj 1 H 2 rNI C rNI 1 ˆNI ,j  ˆNI 1,j  2 R2 2 rNI  rNI 1

1 NJ 1 NI X X j D2



i D2

2

" H N0 .NC  Ne /ij W

xj C1  xj 1 ri C1  ri 1 ri 1 C 2ri C ri C1 . 2 2 4

(4.76)

4.3.2 Finite-difference scheme for the equation of charge motion Let us consider the generalized form of the particle motion equation (4.50) @u 1 @ c @ @u @ C m .r m au/ C .bu/  m r m Df. @

r @r @x r @r @r

(4.77)

One can see that two physical processes are included into equation (4.77). These are: the transfer process with velocities a and b, and the dissipation process with diffusion

198

Chapter 4 The physical mechanics of direct current glow discharge

coefficient c. The desired finite-difference scheme will be created in the following two stages. At the first step, only transfer processes of particles will be considered, while at the second step, the diffusion process will be taken into account. We will derive the conservative form for the desired finite-difference scheme. Algorithmically the construction of conservative schemes is realized with the finite-volume method, which was already considered above. Let us integrate the motion equation @u 1 @ @ C m .r m au/ C .bu/ D f @

r @r @x

(4.78)

by the elementary volume Vij introduced in the previous item (see Figure 4.5 and the formula (4.71)) Vi ,j

p upC1 i ,j  ui ,j



C .2/m .xj C1  xj 1/ 

1 2mC1

Z 

1 2

Z

riC1=2 ri1=2

@ m 1 .r au/dr C .2/m @r mC1

.ri C1  ri 1/.ri 1 C 2ri C ri C1/

xj C1=2 xj 1=2

@ .bu/dx D Vij fij . @x

(4.79)

It is well known that the simplest stable finite-difference scheme for the motion equation (4.78) should be built so that derivatives have been approximated “upstream”. However, in the solved problem, coefficients a and b can be sign-changing, therefore in computing algorithms it is necessary to provide the analysis of their signs at each rated point, if necessary to change the approximating formula in the appropriate way. There are a lot of such finite-difference schemes (see, for example, text-books quite popular in the aerospace community [1,65,88]. However, it is practically impossible to offer a uniform recipe for the solution of any specific problem. Therefore, in our case, the elementary finite-difference scheme with “donor cells” is convenient [40, 65, 88]. According to this scheme, values of velocities are defined in “flux” points with a special algorithm, which we will consider by way of example of the integral evaluation Z

riC1=2 ri1=2

@ m .r au/dr D rimC1=2.au/riC1=2  rim1=2 .au/ri1=2 . @r

Values of fluxes in points ri C1=2 and ri 1=2 are defined under formulas C  C aR ui C1,j , .au/i C1=2 .au/R D aR uR D ui ,j aR

(4.80)

C  C aL ui 1,j , .au/i 1=2 .au/L D aL uL D ui ,j aL

(4.81)

Section 4.3 Finite-difference methods for the drift-diffusion model

199

where 1 aL D .ai C ai 1 /; 2

aR D

1 .ai C1 C ai /; 2

1 C D .aR C jaR j/; aR 2

 aR D

1 .aR  jaR j/; 2

1 C D .aL C jaL j/; aL 2

 aL D

1 .aL  jaL j/. 2

Thus,  ui C1,j rimC1=2aR uR  rim1=2 aL uL D rimC1=2aR C  C C ui ,j .rimC1=2 aR  rim1=2 aL /  rim1=2aL ui 1,j . (4.82)

Under similar formulas the second integral in (4.79) is calculated C C   ui ,j C1 C ui ,j .bR  bL /  bL ui ,j 1 . bR uR  bLuL D bR

(4.83)

Substituting relations (4.80)–(4.83) in (4.79), it is simple to receive a standard canonical form of the five-point finite-difference equation similar to equation (4.73) (see below). To take account of particle diffusion in a radial direction there is a necessity for approximation of the integral Z riC1=2 xj C1  xj 1 1 @ @u (4.84) DQ k m r m .2 r/m dr r @r @r 2 ri1=2     xj C1  xj 1 m m @u m @u Q .2/ .r /r  r D Dk 2 @r iC1=2 @r ri1=2 xj C1  xj 1 .2/m D DQ k 2     1 1 ri C1 C ri m ri C1 C ri m  ui C1,j  ui ,j 2 ri C1  ri 2 ri C1  ri      1 1 ri C ri 1 m ri C ri 1 m  ui ,j C ui 1,j . 2 ri  ri 1 2 ri  ri 1 After division of the received approximations for transfer and diffusion processes on Vi ,j , the final form of the finite-difference scheme suitable to a numerical solution can be presented in the following canonical five-point finite-difference equation: pC1   .Ai ,j C Ai ,j /upC1 i 1,j C .Bi ,j C Bi ,j /ui C1,j  .Ci ,j C Ci ,j /ui ,j

C ANi ,j ui ,j 1 C BN i ,j ui ,j C1 C Fi ,j D 0, pC1

pC1

(4.85)

200

Chapter 4 The physical mechanics of direct current glow discharge

Ai ,j D

2Gim aC r m , ri C1  ri 1 L i 1=2

Bij D 

Ci ,j D

2Gim a r m , ri C1  ri 1 R i C1=2

Ai ,j D Di Sim1

1 , ri  ri 1

Bi,j D Di SimC1

1 , ri C1  ri

2Gim 1 2  .r m aC  rim1=2aL /C .b C  bL /, C

ri C1  ri 1 i C1=2 R xj C1  xj 1 R Ci,j D Di SimC1 ANi ,j D

1 1  Di Sim1 , ri C1  ri ri  ri 1

2 b C, xj C1  xj 1 L

BN i ,j D 

p

Fi ,j D

ui ,j

C fi ,j ,

Gim D .m C 1/

2 b, xj C1  xj 1 R

2m , .ri 1 C 2ri C ri C1/m

SimC1 D .m C 1/

.ri C1 C ri /m , .ri 1 C 2ri C ri C1/m

Sim1 D .m C 1/

.ri C ri 1/m .ri 1 C 2ri C ri C1/m

Di D DQ r ,k

H2 2 , 2 R ri C1  ri

.k D e, C/.

In the given formulas, index p marks functions on the previous (the lower) time layer.

4.3.3 Conservative properties of the finite-difference scheme for the motion equation Let us consider the motion equation in an integral form. It is completely similar to the continuity equation introduced, for example, in hydrodynamics [2, 17, 78, 88]  Z  Z @u C div dV D Q dV , (4.86) @t V V where Q is the source term, V is the arbitrary volume for which the finite-difference scheme is created. Let us transform equation (4.86) with the use of the Gauss theorem Z Z Z @u dS D Q dV , (4.87) dV C V @t † V

201

Section 4.3 Finite-difference methods for the drift-diffusion model

where † is the square of the surface of the considered volume, which will be presented as the sum of three parts, as earlier for cylindrical geometry: Z

Z †

 dS D 

Z S1

x dS C

m

D .2/



Z

R



S2

Z

r dS C

S3

x dS

m

r x .r , x D 0/dr C R

0 R

C

Z

 r m x .r , x D H /dr .

m

Z

H

r .r D R, x/dx

0

(4.88)

0

For greater definition, the flux projections on coordinate axes will be presented in the explicit form using formulas (4.33) and (4.34) (separately for ions and electrons): Z

  Z R  @u @' m m dV C .2/  r C u dr C Rm @x xD0 0 V @t  Z H @' @u  DC C u  dr @r @r r DR 0   Z R  @' m r C u dr C @x xDH 0 Z QdV , u D nC , D

(4.89)

 Z R   @u @' dV C .2/m  r m e u dr @x xD0 0 V @t  Z H @u @'  De dx CR e u @x @r 0   Z R  @' C r m e u dr @x xDH 0 Z QdV , u D ne . D

(4.90)

V

Z

V

For example, let us consider the second type of boundary conditions at r D R: @u @' D D 0. @r @r

202

Chapter 4 The physical mechanics of direct current glow discharge

In this case, we have for ions (u D nC ) Z V

Z R @u @' r m .C u/xD0dr  .2/m dV C .2/m @t @x 0  Z R  Z @' m  r u dr D QdV @x xDH 0 V

(4.91)

and for electrons (u D ne ) Z V

  @' r m e u dr @x xD0 0  Z R  Z @' C .2/m r m e u dr D QdV . @x xDH V 0

@u dV  .2/m @t

Z

R

(4.92)

For the undisturbed axial electric field and for a small constant concentration of particles at r D R, one can derive a more bulky expression that contains particle flux components on the specified boundary. To prove the conservative properties of the finite-difference scheme, we will multiply the equation (4.85) by Vi ,j and sum the finite-difference correlations obtained throughout all internal points of the calculation domain: NI 1 NJ 1 X X i D2

j D2

pC1

pC1

Vi ,j f.Ai ,j C Ai ,j /ui 1,j C .Bi ,j C Bi,j /ui C1,j pC1 pC1 N N  .Ci ,j C Ci,j /upC1 i ,j C Ai ,j ui ,j 1 C Bi ,j ui ,j C1 C Fi ,j g D 0.

The received correlation can be simplified if the following conditions are taken into account: (1) The axial symmetry condition follows to pC1 upC1 2,j D u1,j ,

C  .aL /2,j D .aL /2,j D 0;

(2) Boundary conditions follow to pC1

pC1

uNI ,j D uNI 1,j ,

C  .aR /NI 1,j D .aR /NI 1,j D 0.

(4.93)

Section 4.3 Finite-difference methods for the drift-diffusion model

203

Then NI 1 NJ 2 X X i D2

pC1

Vi ,j

j D2 m

 .2/

NI 1 X i D2

pC1

p

ui ,j  ui ,j

ri C1  ri 1 .ri 1 C 2ri C ri C1/m 2 .m C 1/  2m pC1

C   Œui ,1 .bL /i ,2 C ui ,2 .bL /i ,2  C .2/m pC1 C  Œui ,NJ .bR /i ,NJ 1

D

1 NI 1 NJ X X i D2

C

NI 1 X

ri C1  ri 1 .ri 1 C 2ri C ri C1/m 2 .m C 1/  2m

i D2 pC1 pC1   ui ,NJ 1 .bR /i ,NJ 1 C ui ,NJ 1 .bR /i ,NJ 1 

Vi ,j fi ,j .

(4.94)

j D2

For considering the direct current glow discharge there are specific physical boundary  /i ,2 D conditions on the cathode and the anode @'=@x > 0, therefore, for electrons .bL C 0, and for ions .bR /i ,NJ 1 D 0. Normalizing the integral balance equations (4.91) and (4.92) and comparing them with the corresponding relation following from (4.94), we conclude that the constructed finite-difference scheme has the conservative property.

4.3.4 The order of accuracy of the finite-difference approximation used. The mesh diffusion This problem is analyzed in many books on computational physics and computational fluid dynamics (CFD) [1,65,81,88]. It was shown that at constant transport velocity u the donor-sell finite-difference scheme has the second order of accuracy for convective transport if a joint solution of equations of motion and continuity is considered. But, at the same time, it is well known that for the transport equation only the scheme has the first order of accuracy. To introduce the definition of the mesh diffusion, the following boundary value problem will be analyzed: @u @u Ca D 0, @t @x

a > 0,

u.x, t D 0/ D 0 .x/

(4.95) (4.96)

204

Chapter 4 The physical mechanics of direct current glow discharge

with the use of the simplest explicit upstream finite-difference scheme (referred to in the following as “scheme” for short) pC1

ui

p

p

p1

 ui u  ui Ca i

h 0 ui D i .xi /.

D 0,

(4.97)

Suppose that function u.x, t / has the continuous regular limited derivatives, which are not below the second order. Let us expand this function in Taylor’s series in neighborhood points i, i  1 (of a spatial grid), and then p and p C 1 (of a time grid). Then, we will receive that the scheme (4.97) has the following principal part of an approximation error:

@2 u h @2 u ED a . (4.98) 2 2 @t 2 @x 2 But, proceeding from (4.95),       2 @2 u @2 u @ @u @ @u @ @u 2@ u D a D a D a a D a D . @t 2 @t @t @x@t @x @t @x @x @x 2 we have

 ED

a2 ah  2 2



@2 u a @2u ah  1  D  . @x 2 2 h @x 2

(4.99)

Analyzing the form of the principal part of the approximation error, it is simple to determine that using approximation (4.97), we will actually solve the equation of the following form: @2 u @u @u Ca  Dm 2 D 0, (4.100) @t @x @x where a

ah .1  /. (4.101) Dm D 2 h Note that the mesh diffusion coefficient Dm contains the hyperbolic CFL number CFLG D a. = h/, which was discussed above. Thus, the considered equation of convective transport describes some fictitious (“mesh”) dissipative process. For a solution of problems of heat and mass transfer, the mesh diffusion appears useful in some cases. Some effective computing algorithms are based on its use. There are also a lot of cases when the additional term "c .@2 u=@x 2/ is introduced into the equation in an explicit form. The value "c is named “artificial viscosity (diffusion, dissipation)”. Dissipative processes connected with the artificial dissipation also appear in finite-difference equations when using the smoothing (regularization) of a solution, and also in some approximations of time derivatives (time-dependent viscosity, diffusion or dissipation).

205

Section 4.3 Finite-difference methods for the drift-diffusion model

As to the numerical simulation of a drift-diffusion model of glow discharge, the occurrence of mesh diffusion is extremely undesirable. Therefore, the mesh dissipative properties will be studied below in detail. First we will set the form of the mesh diffusion coefficient Dm in the case of the pC1 p steady-state transport equations. It can be made by substituting ui D ui in equation (4.97), and then having expanded the remaining part of the equation in Taylor’s series. It gives the following 1 S D ah. Dm (4.102) 2 Comparing (4.101) and (4.102), we have drawn the conclusion that a nonstationary solution of the motion equation with the numbers CFLG , which strongly differ from unity, leads to a significant increase in the mesh diffusion coefficient. This implies the practically important conclusion about the necessity of integration of the transport equation with different time steps for the estimation of the influence of mesh diffusion. In the numerical method formulated, only CFLG e numbers for electrons sufficiently differ from unity (generally near to the cathode), while the greatest interest is represented with their radial diffusion on the edge of the current channel (at the beginning of the positive column). Numbers CFLG + for ions remain less than unity everywhere. Thus, S in the further estimations we will be guided by the expression for Dm D 12 ah. From the viewpoint of practical needs for performing computational physical experiments, there is always a principal problem: how the mesh dissipation influences the numerical solution obtained? More precisely: how is the value of mesh diffusion related to other significant physical processes which are taken into account in the equation? From the structure of the dissipative term appearing in the equation (4.100) it follows that mesh diffusion will show up substantially in the region of sharp particle concentration differences and high velocity of particle motion, and also at a solution of the problem on a coarse difference grid. If a conditional mesh diffusion flux is assigned for the artificial dissipation process, then comparing this flux with drift and physical diffusion fluxes it is possible to obtain a representation about the influence of the mesh dissipation on the desired numerical solution. Dimensional relations for the determination of the drift and physical diffusion fluxes in axial and radial directions look like .e,dif /r D De

@ne , @r

.e,dif /x D De

@ne ; @x

.e,dr /r D ne e

@' , @r

.e,dr /x D ne e

@' ; @x

@nC , @r

.C,dif /x D DC

@nC ; @x

.C,dif /r D DC

.C,dr /r D nC C

@' , @r

.C,dr /x D nC C

where labels “dif” and “dr” mark the diffusion and drift fluxes.

@' , @x

(4.103)

206

Chapter 4 The physical mechanics of direct current glow discharge

Corresponding mesh diffusion fluxes are defined so: e .e,m /r D Dm,r

@ne , @r

e .e,m /x D Dm,x

@ne ; @x

C .C,m /r D Dm,r

@nC , @r

C .C,m /x D Dm,x

@nC . @x

(4.104)

Using the formula (4.102), one can determine explicit expressions for coefficients of the mesh diffusion e Dc,z D e

@' 1 z, @z 2

C Dc,z D C

@' 1 z, @z 2

z D .r , x/. Estimating

@' @z



' ,

z

one can receive e Dc,z 

e ' , 2

C  Dc,z

C ' . 2

(4.105)

Considering that coefficients of physical diffusion are defined by the Einstein relations, the ratio of physical diffusion fluxes to mesh diffusion fluxes look like e,dif 2Te ,  e,m '

C,dif 2TC ,  C,m '

(4.106)

and the ratio of drift fluxes to mesh diffusion fluxes e,dr ne 2 , e,m ne

C,dr nC 2 . C,m nC

(4.107)

It is obvious that an optimum case is realized when all given ratios are significantly superior to unity. It superimposes rather rigid requirements on the selection of the difference grid. For example, at a potential drop in the cathode layer by a value of 150 V per grid step, it follows from (4.106) that the number of calculation points should be so large as is necessary to ensure the condition ' < 2Te on one spatial grid step. Unfortunately, it has been proved inconvenient for a solution of a two-dimensional problem. Less severe constraints are dictated by the relations (4.107). Numerical simulation results show that ion concentrations in the cathode layer drop approximately ten times, whence it follows that the number of grid points for ions should be within the limits of 10–30. About the same number of points are necessary for describing electron motion in the cathode layer. At realization of these conditions, drift fluxes of particles will surpass mesh diffusion only at some times. It does not guarantee a lack of influence of the mesh dissipation processes on the results of numerical simulation.

Section 4.3 Finite-difference methods for the drift-diffusion model

207

The simplest way to decrease mesh diffusion (within the frame of the finitedifference scheme considered) is improving the finite-difference grid, for example, by the use of nonuniform grids. However, as is known, here there are significant restrictions too, which, on the one hand, require not so sharp a variation of grid steps, and, on the other hand, preservation of a certain proportion of the difference grid point disposition in various directions. Another constructive approach to a solution of this problem is transition to schemes of the second and higher accuracy orders with the subsequent application of regularization methods [1, 65]. But unfortunately, in this case there is the possibility of occurrence of oscillations in weak function variation areas adjoining areas of significant drops. So, this problem should be investigated in each calculation case. Often for the estimation of dissipative properties of finite-difference schemes the concept of Reynolds grid number [88] is used Re D

uh , 

where u,  are the velocity and viscosity (diffusivity). Value Re is the Reynolds number, obtained at a local velocity of the stream, its dynamic viscosity and a characteristic size equal to one step of the calculation grid. If instead of , the coefficient of mesh diffusion is substituted, one can obtain the Reynolds grid number Rem D

uh . Dm

It is obvious that the condition Rem Re is necessary to provide smallness of grid dissipation in comparison with a physical one. Formulas for calculation of different flux kinds and corresponding numerical coefficients allowing to obtain the fluxes in mA/cm2 are given in Table 4.2. These data are useful for the analysis of the results of the calculations.

4.3.5 The finite-difference grids For calculations of glow discharge, the problem of selection of acceptable finitedifference grids is no less important than in problems of computational fluid dynamics [1, 47, 65, 81]. Even in cases of elementary flat or axially symmetric geometry, essentially homogeneous grids are usually used. Examples of some of them are given below. Nonuniform scales for transverse coordinates A typical nonuniform scale on radial coordinate is built (analytically or numerically) with clustering grid points in the neighborhood of a current column with use of any

208

Chapter 4 The physical mechanics of direct current glow discharge

Table 4.2. Definition of particle fluxes. Fluxes along radius, mA/cm2

Fluxes along an axis x, mA/cm2 @Ne S3 @x @ˆ S3 .je,dr /x D Ne @x x @ˆ @Ne .je,m /x D S3 2 @x @x

@Ne .je,dif/r D DQ e S1 @r @ˆ .je,dr /r D Ne S1 @r r @ˆ @Ne .je,m /r D S1 2 @r @r

.je,dif /x D DQ e

.jC,dif/r D DQ C

@NC S1 @r @ˆ .jC,dr /r D NC S2 @r r @ˆ @NC S2 .jC,m/r D 2 @r @r

.jC,dif /x D DQ C

N0 E e , R N0 E S3 D 1.6  1016 e , H

N0 E C , R N0 E S4 D 1.6  1016 C . H

S1 D 1.6  1016

@NC S3 @x @ˆ .jC,dr /x D NC S4 @x x @ˆ @NC S4 .jC,m /x D 2 @x @x S2 D 1.6  1016

function of crowding. If the calculation of glow discharge in an extended area is necessary, it has being reasonable to introduce a multiple block grid. For example, the two-block grid is built as follows. The grid with modification of points under the law of an arithmetical progression near an axis of a symmetry is calculated under the formula ri D dmin.i  1/ C .i  1/.i  2/0.5Hw , i D 2, 3, : : : , Ni w , r1 D 0,

(4.108)

where Ni w is the number of grid points in internal area; dmin is the grid least step near to an axis. The value dmin is selected as the certain part of normal thickness of the cathode layer dn (for example, dmin D 0.1–0.03dn/, which is known before the beginning of calculations (it is defined, for example, under the Engel–Steenbeck theory). The arithmetical ratio Hw is determined under the formula: Hw D 2

rcc  dmin.Ni w  1/ , .Ni w  1/.Ni w  2/

where rcc is the presupposed radius of the current channel; this is the boundary of the first block.

Section 4.3 Finite-difference methods for the drift-diffusion model

209

The calculation grid in region r 2 Œrcc , R is built starting from a first step H1 D .ri w  ri w1/P , where P D 1.5–2 is the scale factor. That is, the first step in the external area is uniquely connected with the last step in the internal area. Grid coordinates in the external area are calculated under the formula rNiw Ci D rNiw C H1 i C i.i  1/ DD2

D , 2

(4.109)

.R  rcc/  .NI  Ni w /H1 . .NI  Ni w /.NI  Ni w  1/

where NI is the full number of points along a radius. One more example of a radius-inhomogeneous grid in the second (external) block is shown below. The tangential law of grid modification by a variable r is set concerning a point of the prospective boundary of the current channel. The new variable is introduced as follows y D arctg ˇ.r  rcc /, in which the step of the calculation grid was supposed constant and equal to Hy D

ymax  ymin , NI2  1

where ymax D arctg ˇ.R  rcc /; ymin D arctg.ˇrcc / D arctg ˇrcc ; ˇ is the grid compression factor; NI2 is the number of points in the external area. Calculating formulas for the determination of ri look like 1 tgyi C rcc , ˇ yi D yi 1 C Hy , i D 2, 3, : : : , NI2 , ri D

(4.110)

y1 D ymin , r1 D 0. Note that for numerical simulation of glow discharge dynamics in flat channels it is reasonable to use a homogeneous or mobile grid in the transversal direction (analog of variable r in two-dimensional axially symmetric geometry). Nonuniform scale on longitudinal coordinates Unlike the previous case, inhomogeneous grids are always used for the axial direction between cathode and anode. In cases where the anode is arranged opposite to the cathode (see Figure 4.4), as a rule, the whole calculation domain between the cathode and the anode is divided in two subregions. The first one is the cathode region

210

Chapter 4 The physical mechanics of direct current glow discharge

between the cathode and a prospective point of the cathode potential drop extremity (see Figure 4.3). The second one is the external region from the point of the cathode layer extremity to the anode. Thickness of the cathode layer dn can be estimated by the Engel–Steenbeck theory (see, for example, (4.29)). In the internal area (the cathode layer) the grid is built, for example, under the formula 1 xj D .j  1/dn C .j  1/.j  2/ Djw , 2 j D 2, 3, : : : , Njw , Djw D 2

(4.111)

dn  .Njw  1/dcc , .Njw  1/.Njw  2/ x1 D 0,

where dcc has the same value as at the construction of the calculation grid along transversal coordinate. In the external area the grid is built so that its greatest crowding was on the anode potential drop xj 1 D xj  Aj ,

j D NJ , NJ  1, : : : , Njw C 1,

Aj D dan C Dje .m  1/, Dje D 2

(4.112)

m D 1, 2, : : : , NJ  Njw ,

.H  dn /  .NJ  Njw /dan , .NJ  Njw /.NJ  Njw  1/

where NJ is the full number of points of the calculation grid in x direction. The value dan is the least step on an axial coordinate near to the anode. As a rule, dan D dn . Function y D arctgˇx can also be recommended for construction of calculation grids between electrodes, because it allows to receive a crowding of grid points near to the cathode and the anode. Sometimes there is a necessity to rebuild using grids and to re-interpolate arrays of calculated functions on new finite-difference grids. It is recommended to use the curve fitting ability. In this case, the laws of step modification of the old and new grids can be independent.

4.3.6 Iterative methods for solving systems of linear algebraic equations in canonical form In the considered two-dimensional problem of direct current glow discharge there is a necessity to integrate the following three finite-difference equations: the continuity

211

Section 4.3 Finite-difference methods for the drift-diffusion model

equations for ions and electrons, and also the Poisson equation for electric field potential. The important feature of the integration is also the algorithmic realization of these equations’ solution sequence (the so-called external iterative process). A separate section of this chapter will be devoted to the consideration of the external iterative process. But here we will concentrate our attention on methods of integration of finitedifference equations in canonical form. As it was discussed above, each of the enumerated equations being written in finitedifference form can be presented in the canonical form, which is determined by using the five-point grid pattern Ai ,j uSi1,j C Bi ,j uSiC1,j  Ci ,j uSi,j C ANi ,j uSi,j 1 C BN i ,j uSi,j C1 C Fi ,j D 0, i D 2, 3, : : : , NI  1,

j D 2, 3, : : : , NJ  1,

(4.113)

where S is the iterative index. The boundary conditions are presented in the following general form: i D 1: i D NI : j D 1: j D NJ :

u1,j D ˛,j u2,j C ˛,j , uNI ,j D ,j uNI 1,j C ,j , ui ,1 D i ,˛N ui ,2 C i ,˛N , ui ,NJ D i ,N ui ,NJ 1 C i ,N .

(4.114) (4.115) (4.116) (4.117)

Numerous publications on classical numerical analysis and computing physics are devoted to methods of solving the boundary value problem (4.113)–(4.117) [1,61,74]. Information on the methods presented in the quoted works, as a rule, appears enough for construction of codes for performing numerical simulation. However, it is necessary to take into account that a solution of the glow discharge problem, as a rule, is performed by iterative methods, so effectiveness of this or that finite-difference method should be estimated finally by analysis of the total effectiveness of the numerical simulation. In other words, the chosen finite-difference method should have high convergence rates both on rough and on almost exact solutions, and besides it should have relatively low laboriousness. These requirements are very often rather inconsistent. So the problem of the choice of a computing method is rather difficult. Three numerical methods are considered below. These methods have shown their effectiveness in solving various problems of glow discharge. These are: (1) nonlinear iterative ˛-ˇ algorithm [18]; (2) successive over-relaxation (SOR) method; (3) explicit method.

212

Chapter 4 The physical mechanics of direct current glow discharge

Nonlinear iterative ˛-ˇ-algorithm The method has been offered in the work [18]. The principal advantage of the ˛-ˇ iteration scheme, which causes its effectiveness for the solution of the glow discharge problem, is the lack of the necessity for knowledge of any a priori information on a spectrum of difference operator or of the necessity to define its boundaries. In other words, from the practical point of view this method is autonomous and does not demand special preliminary operations for a solution. Solution of the finite-difference scheme (4.113) is searched for in the form of ui ,j D ˛i C1,j ui C1,j C ˇi C1,j ,

(4.118)

ui ,j D i 1,j ui 1,j C di 1,j ; ui ,j D ˛N i ,j C1ui ,j C1 C ˇNi ,j C1,

(4.119)

ui ,j D Ni ,j 1ui ,j 1 C dNi ,j 1.

Sequentially substituting (4.118) and (4.119) in (4.113), it is possible to receive formulas for eight run factors which are connected among themselves. The situation is facilitated by the fact that it is possible to select two independent groups of those factors. The computing process begins with a set of zero approximations of a matrix of factors NiS,j . Below, examples of the special statements of the zero approximations NiS,j will be considered, intended to accelerate convergence of an iterative process. For the first time it is possible to use the boundary conditions (4.117) NiS,j D i ,N , i D 1, 2, : : : , NI ; j D 1, 2, : : : , NJ . In a direct cycle of the so-called “˛-process”, calculations for the modification of an index j D 2, 3, : : : , NJ  1 are made, and there factors are determined Bi ,j

SC 1

2 D ˛i C1,j

Ci ,j 

SC 1 ˛i ,j 2 Ai ,j

1

SC  ˛N i ,j 2 ANi ,j  NiS,j BN i ,j

,

i D 2, 3, : : : , NI  1, (4.120)

SC 1 i 1,j2

D

Ai ,j 1

1

SC SC Ci ,j  i ,j 2 Bi ,j  ˛N i ,j 2 ANi ,j  NiS,j BN i ,j

,

i D NI  1, : : : , 2, (4.121)

BN i ,j

SC 1

2 ˛N i ,j C1 D

Ci ,j 

SC 1 ˛i ,j 2 Ai ,j

1

1

SC SC  ˛N i ,j 2 ANi ,j  i ,j 2 Bi ,j

, i D 2, 3, : : : , NI  1. (4.122)

213

Section 4.3 Finite-difference methods for the drift-diffusion model

In an inverse cycle of the ˛-process the index j varies in the opposite direction j D NJ  1, NJ  2, : : : , 2 and the following factors are calculated ˛iSC1 C1,j D

Bi ,j Ci ,j 

˛iSC1 ,j Ai ,j



SC 1 ˛N i ,j 2 ANi ,j



N NiSC1 ,j Bi ,j

, i D 2, : : : , NI  1, (4.123)

iSC1 1,j D

Ai ,j 1

SC N Ci ,j  iSC1 N i ,j 2 ANi ,j  NiSC1 ,j Bi ,j  ˛ ,j Bi ,j

, i D NI  1, : : : , 2, (4.124)

NiSC1 ,j 1 D

ANi ,j SC1 SC1 N Ci ,j  iSC1 ,j Bi ,j  ˛i ,j Ai ,j  Ni ,j Bi ,j

,

i D 2, : : : , NI  1. (4.125)

For realization of the specified runs, it is necessary to use the boundary conditions SC1=2 D ˛,j , ˛2,j

SC1=2 NI 1,j D ,j ,

(4.126)

SC1 SC1 D ˛,j , NI ˛2,j 1,j D ,j , j D 2, : : : , NJ  1 SC1=2

˛N i ,2

D i ,˛N , NiS,NJ 1 D i ,N ,

NiSC1 ,NJ 1 D i ,N ,

i D 2, : : : , NI  1.

After completion of the direct and inverse cycles, the estimation of ˛-process convergence is made.  NiS,j j j NiSC1 ,j (4.127) If "D > "˛ .103 /, j NiSC1 j ,j then appropriation is made .i D 2, : : : , NI  1; j D 2, : : : , NJ  1/, NiS,j D NiSC1 ,j and the ˛-process is repeated. Otherwise the “ˇ-process” begins. Prior to the beginning of the ˇ-iterative process, the following auxiliary functions are calculated ˛,i ,j

D Ci ,j  ˛i ,j Ai ,j  ˛N i ,j ANi ,j  Ni ,j BN i ,j ,

,i ,j

D Ci ,j  i ,j Bi ,j  ˛N i ,j ANi ,j  Ni ,j BN i ,j ,

˛,i N ,j

D Ci ,j  ˛i ,j Ai ,j  ˛N i ,j ANi ,j  i ,j Bi ,j ,

,i N ,j

D Ci ,j  ˛i ,j Ai ,j  i ,j Bi ,j  Ni ,j BN i ,j ,

i D 2, : : : , NI  1;

j D 2, : : : , NJ  1.

(4.128)

214

Chapter 4 The physical mechanics of direct current glow discharge

Zero approximations for coefficients ˛N i ,j are set as follows ˛N iS,j D i ,N ,

i D 2, : : : , NI ;

j D 2, : : : , NJ .

(4.129)

Direct cycle of iterative ˇ-process is made at modification of index j D 2, : : : , NJ 1: 1

SC 12 ˇi C1,j

D

1

SC SC Fi ,j C ˇi ,j 2 Ai ,j C ˇNi ,j 2 ANi ,j C dNiS,j BN i ,j

,

˛,i ,j

i D 2, : : : , NI  1; SC 1 di 1,j2

SC 12

D

Fi ,j C di ,j

(4.130)

1

SC Bi ,j C ˇNi ,j 2 ANi ,j C dNiS,j BN i ,j

,

,i ,j

i D NI  1, : : : , 2; 1

SC 12 ˇNi ,j C1

D

1

(4.131) 1

SC SC SC Fi ,j C ˇi ,j 2 Ai ,j C ˇNi ,j 2 ANi ,j C di ,j 2 Bi ,j

,

˛,i N ,j

i D 2, : : : , NI  1.

(4.132)

In an inverse cycle of the ˇ-process the index j varies in an opposite direction j D NJ  1, : : : , 2: 1

ˇiSC1 C1,j D

N SC 2 N N SC1 N Fi ,j C ˇiSC1 ,j Ai ,j C ˇi ,j Ai ,j C di ,j Bi ,j

,

˛,i ,j

i D 2, : : : , NI  1;

(4.133)

1

diSC1 1,j D

N SC 2 N N SC1 N Fi ,j C diSC1 ,j Bi ,j C ˇi ,j Ai ,j C di ,j Bi ,j

,

,i ,j

i D NI  1, : : : , 2; dNiSC1 ,j 1 D

SC1 N SC1 N Fi ,j C ˇiSC1 ,j Ai ,j C di ,j Bi ,j C di ,j Bi ,j ,i N ,j

i D 2, : : : , NI  1.

(4.134) , (4.135)

Section 4.3 Finite-difference methods for the drift-diffusion model

215

Boundary conditions of the following form are used in the ˇ-process: SC 12

D ˛,j ,

ˇ2,j SC1 D ˛,j , ˇ2,j

SC 1

2 dNI 1,j D ,j ,

SC1 dNI 1,j D ,j ,

ˇNi ,2 D i ,˛N , dNiSC1 ,NJ 1 D i ,N ,

(4.136)

j D 2, : : : , NJ  1;

1

SC dNi ,NJ21 D i ,N ,

i D 2, : : : , NI  1.

Convergence of the ˇ-process is checked for functions dNiSC1 ,j . If  dNiS,j j jdNiSC1 ,j > "ˇ . 103/, "D jdN S j

(4.137)

i ,j

with the subsequent recurring of the ˇthen appropriation is made dNiS,j D dNiSC1 ,j process. Otherwise, the final calculation of required functions is made ui ,1 D

dNisC1 ,1  i ,˛N C i ,˛N

i D 2, : : : , NI  1;

(4.138)

N SC1 ui ,j D NiSC1 ,j 1 ui ,j 1 C di ,j 1 , i D 2, : : : , NI  1;

j D 2, : : : , NJ  1; (4.139)

1  i ,˛N NisC1 ,1

,

ui ,NJ D i ,N ui ,NJ 1 C i ,N , u1,j D ˛,j u2,j C ˛,j , uNI ,j D ,j uNI 1,j C ,j ,

i D 2, : : : , NI  1;

(4.140)

j D 1, 2, : : : , NJ ;

(4.141)

j D 1, 2, : : : , NJ .

(4.142)

Note that there is a possibility to calculate mesh functions ui ,j with the use of other formulas, for example (4.118). Then it is necessary to change the order of use of boundary conditions. Numerous research studies of the ˛-ˇ-iteration algorithm have shown that the ˛process takes essentially less time for calculation than the ˇ-process. Therefore, basic efforts for a diminution of calculation time are usually focused on an optimization of the ˇ-process. For example, there is the possibility to use the over-relaxation method for the ˇ-process. An appreciable reduction in the of number of internal ˛- and ˇ-iterations can be reached due to some rise of computer RAM memory space. It is recommended to N SC1 keep NiSC1 ,j 1 and di ,j 1 for each of the solved equations after convergence of the ˛ˇ-process. These remembered factors are used in the subsequent calculation as initial approximations.

216

Chapter 4 The physical mechanics of direct current glow discharge

Finite-difference boundary conditions for the ˛-ˇ-iterative method For the Poisson equation we have i D 1:

u1,j D u2,j ,

i D NI :

˛,j D 1, ˛,j D 0,

j D 1, : : : , NJ ;

a) uNI ,j D uNI 1,j , ,j D 1, ,j D 0,

j D 1, : : : , NJ ;

b) uNI ,j D j D 1:

ui ,1 D 0,

j D NJ :

ui ,NJ D

xj V , E

,j D 0, ,j D

xj V , j D 1, : : : , NJ ; E

i ,˛N D 0, i ,˛N D 0, V , E

i ,N D 0, i ,N D

V , E

i D 1, : : : , NI ; i D 1, : : : , NI . (4.143)

Boundary conditions for electrons and ions in axial directions have the first order, and in a radial direction they can be of both the first and the second order. If the radial diffusion is considered, and the equations have the second order in the radial direction, the boundary conditions look like (for ions and electrons they are identical) i D 1:

u1,j D u2,j ,

i D NI : a) uNI ,j D uNI 1,j , b) uNI ,j D 0,

˛,j D 1, ˛,j D 0, j D 1, : : : , NJ ; (4.144) ,j D 1, ,j D 0, j D 1, : : : , NJ ; ,j D 0, ,j D 0, j D 1, : : : , NJ .

If particle diffusion is not considered, the equations of the first order are solved. In this case it is necessary to formulate some “effective” boundary conditions which have no physical meaning, and as a rule, these are the direct consequence of the solved equations. In the radial direction at great distance from an axis of symmetry, the boundary conditions of the second kind can be used @nC @ne D D 0, (4.145) @r @r which are the corollary of a stationary continuity equation under the condition of homogeneous background particle concentration. Grid boundary conditions in the axial direction have been formulated in Section 4.2.2 (see formulas (4.61) and (4.62)). In finite-difference notation these conditions will have the following appearance: (a) At the anode

i ,N

SC1 uSC1 i ,NJ D i ,N ui ,NJ 1 C i ,N ,     1 @ˆ 1 Pb , D Zi @x i ,NJ 1 xNJ  xNJ 1

(4.146)

217

Section 4.3 Finite-difference methods for the drift-diffusion model

 p  1 ui ,NJ p p  AOi ,NJ ui 1,NJ  BO i ,NJ ui C1,NJ , Zi

  @ˆ 1 1  Ci,NJ , Zi D C Pb

@x i ,NJ xNJ  xNJ 1   @ˆ 1 1 Zi D C Pb  Ci,NJ ,

@x i ,NJ xNJ  xNJ 1

i ,N D

AOi ,NJ D Di Sim1

1 1 , BO i ,NJ D Di SimC1 , ri  ri 1 ri C1  ri

H2 2 CO i ,NJ D AOi ,NJ C BO i ,NJ , Di D DQ e 2 , Pb D 1, R ri C1  ri 1 where DQ e is the dimensionless factors of diffusion; Sim1 , SimC1 are defined above (see (4.85)). Iterative index S is used in (4.146) instead of index of a time layer p to mark that for the calculation of grid functions at each time layer it is necessary to perform some internal iterations (S). (b) At the cathode SC1 uSC1 i ,1 D i ,˛N ui ,2 C i ,˛ ,     Pb @ˆ 1 i ,˛ D  , Zi x2  x1 @x i ,1  p  1 ui ,1 S S C AOi ,1ui 1,1 C BO i ,1 ui C1,1 , i ,˛ D Zi

  @ˆ 1 1  Ci,1, Zi D  Pb

@x i ,1 x2  x1

AOi ,1 D Di Sim1

(4.147)

1 1 .1 C /, BO i ,1 D Di SimC1 .1 C /, ri  ri 1 ri C1  ri

C H2 2 , Di D DQ C 2 . CO i ,1 D AOi ,1 C BO i ,1, Pb D .1 C / e R ri C1  ri 1 Note that despite neglecting diffusion in the axial direction, grid boundary conditions have included factors of radial diffusion of ions and electrons. This demonstrates the fact that we use a corollary of the solved equation as a grid boundary condition.

218

Chapter 4 The physical mechanics of direct current glow discharge

The Gauss–Seidel and successive relaxation methods The computing process of this kind of relaxation method is built as follows [1, 78]: pC1

ui ,j

p

pC1=2

D .1  !/ui ,j C !ui ,j

,

(4.148)

where Fi ,j C Ai ,j ui 1,j C Bi ,j ui ,j 1 C ANi ,j ui C1,j C BN i ,j ui ,j C1 p

upC1=2 i ,j

D

pC1

pC1

p

p

Ci ,j

,

(4.149)

p is the index of a time step or the iterative index, ! is the relaxation parameter. It was supposed at writing of (4.149) that the passage of a calculation grid is carried out in an order of magnification of indexes i and j (see Figure 4.5). Functions in the points marked by bubbles are taken on the upper time layer, and in the points marked by rectangles on the lower. In our problem, such an order of tracking is convenient for the calculation of the electrons’ properties. If the concentration of ions is to be calculated, for which the boundary condition is set on the anode, then tracking with a diminution of an index j and magnification of an index i is more natural, i. e., pC1=2 ui ,j

D

p pC1 pC1 p p Fi ,j C Ai ,j ui 1,j C BN i ,j ui ,j C1 C Bi ,j ui ,j 1 C ANi ,j ui C1,j

Ci ,j

.

(4.150)

For a solution of the Poisson equation the direction of tracking is indifferent. A choice of relaxation parameter ! has the great importance for the effectiveness of the method considered. The theorem is known [88], according to which the method of a successive relaxation converges at 0 < ! < 2. More often the method of relaxation at ! < 1 is named as the method of successive under-relaxation, at ! > 1 it is named as the method of successive over-relaxation, and at ! D 1 the Gauss–Seidel method. In the majority of practically important cases, it is possible to solve a problem of the determination of the optimum parameter !. The following algorithm is in common use. First a calculation under formulas (4.149) or (4.150) at ! D 1 is made. As a result of evaluations on three successive time steps it is possible to estimate a maximum eigenvalue 1 of transition operator for the Gauss–Seidel method: v u P pC1 p u .u  uij /2 ij u u ij 1  u P p , (4.151) t .u  up1 /2 ij

ij

ij

whence we obtain the estimation of the relaxation parameter !

2 p . 1  1  1

(4.152)

Section 4.3 Finite-difference methods for the drift-diffusion model

219

Note that the correct choice of the optimum relaxation parameter can essentially reduce the number of iterations necessary for full convergence. In the work [88] the convincing example illustrating dependence of calculation errors on the choice of the ! parameter is given. Cited data are especially useful because they show the character of a diminution of an error as approaching the optimum !. The sharp fall in the calculation error is observed only in the immediate proximity of optimum !, so that at its practical definition it is always reasonable to make numerical experiments for reaching the greatest effectiveness. The practice of calculations of glow discharges shows that computing effectiveness of relaxation methods increases with use of the Thomas algorithm along coordinate lines j D const or i D const (the SOR by lines method; the SLOR method) [1]. In particular, outcomes of calculations of the glow discharges presented in the following section testify more rapid convergence of the problem solution as a whole with use of the Thomas algorithm for a solution of flow continuity equations for the charged particles in the direction from the cathode to the anode. Boundary conditions for methods of successive relaxation Calculation of ion concentration. The calculation of ion concentrations is performed from the anode to the cathode. The first line j D NJ  1 is nonstandard only. On the left boundary of the calculation area (i D 1/ the symmetry condition is used u1,NJ 1 D u2,NJ 2 . If we consider that an upper bound of a boundary condition is known: ui ,NJ D i ,N ui ,NJ 1 C i ,N , then at the first calculation point it is necessary to use the formula  pC1=2 p p u2,NJ 1 D F2,NJ 1 C BN 2,NJ 1 2,N C B2,NJ 1 u2,NJ 2 p 1 C A2,NJ 1 u3,NJ 1 CO 2,NJ 1 ,

(4.153)

where CO 2,NJ 1 D C2,NJ 1  A2,NJ 1  BN 2,NJ 1 2,N . Let us use values A2,NJ 1 and BN 2,NJ 1 from (4.85), then one can see that O C2,NJ 1 ¤ 0. Since the point i D 3 is in line j D NJ  1 we can begin calculation  pC1=2 pC1=2 ui ,NJ 1 D Fip,NJ 1 C BN i ,NJ 1 i ,N C B2,NJ 1 up 2,NJ 2 C Ai ,NJ 1 ui 1,NJ 1 p C ANi ,NJ 1 ui C1,NJ 1  CO i1 i D 3, : : : , NI  1, (4.154) ,NJ 1 , where CO i ,NJ 1 D Ci ,NJ 1  BN i ,NJ 1 i ,N .

220

Chapter 4 The physical mechanics of direct current glow discharge

All the next lines i D NJ  2, NJ  3, : : : , 2 begin from the satisfaction of the symmetry boundary conditions

p pC1=2 pC1 p p  1 u2,j D F2,j C BN 2,j u2,j C1 C B2,j u2,j 1 C AN2,j u3,j CO 2,j , (4.155) CO 2,j D C2,j  A2,j and then the subsequent calculation on the basic relation (4.150). After completion of the calculations in all internal points, the boundary values of the desired functions are calculated using conditions (4.114)–(4.117). As has been shown above, grid boundary conditions on the cathode are formulated in a standard form, therefore, their use does not involve any difficulties. Calculation of electron concentrations. These calculations are performed in a reverse order compared to the calculation of ions: from the cathode to the anode. The direction of motion along the radius is the former, i. e., from an axis to a periphery. Calculations along line i D 2 for all j and along line j D 2 for all i are performed in a nonstandard way. At the point i D 2, j D 2 the grid function is calculated under the formula

pC1=2 p  O 1 N u2,2 D F2,2 C B2,2 2,˛N C BN 2,2 up (4.156) 2,3 C A2,2 u3,2 C2,2 , CO 2,2 D C2,2  A2,2  B2,22,˛N . It guarantees that CO 2,2 ¤ 0 due to values A2,2 , B2,2, 2,˛N . The line j D 2 is passed with the formula  1

pC1 O N p D Fip,2 C Bi ,22,˛N C BN i ,2 up upC1=2 i ,2 i ,3 C Ai ,2 ui 1,2 C Ai ,2 ui C1,2 Ci ,2 ,

(4.157)

CO i ,2 D Ci ,2  Bi ,2i ,˛N . Concentrations of ions near to a symmetry axis are calculated as following

pC1=2 p pC1 p  1 D F2,j C BN 2,j u2,j C1 C B2,j u2,j 1 C AN2,j u3,j CO 2,j , u2,j where CO 2,j D C2,j  A2,j . Tracking of all interior points is made with use of the basic formula (4.150). At the end of the calculation loop the boundary values of functions are improved. Boundary conditions (4.114), (4.115), (4.117) are applied for this purpose. Explicit method of calculation of ion concentrations The explicit method in the considered statement is reasonable only for calculating ion concentrations. As has been shown above, calculation of glow discharge parameters is performed with characteristic times corresponding to CFLG C < 1,

CFLPC < 1.

(4.158)

Section 4.3 Finite-difference methods for the drift-diffusion model

221

It allows to use the explicit method for ions. In the following we will assume conditions (4.158) as fulfilled. Let us take the equation (4.85), and introduce the following designation Ci ,j D 1 C Ei ,j , Ei ,j D

2Gim

ri C1  ri 1

C  .rimC1=2aR  rim1=2aL /i ,j

(4.159) 2  C .b C  bL /i ,j , xj C1  xj 1 R

then the explicit form of the finite-difference equation gains the following form

pC1 p p p p p p  ui ,j D Ai ,j ui 1,j C Bi ,j ui C1,j  Ei ,j ui ,j C ANi ,j ui ,j 1 C BN i ,j ui ,j C1 C Fi ,j . (4.160) The calculation for ions begins from the anode and from a symmetry axis where boundary conditions have been fixed. Unlike an implicit method, the computing cycle ends with the use of boundary conditions. That is, at the beginning zero approximations of desired functions are used for the numerical solution. Then with engaging conditions (4.114)–(3.117) the boundary values of functions are defined.

4.3.7 An iterative algorithm for the solution of a self-consistent problem The determination of self-consistent distributions of charged particles and electric field potential is no less important a problem than the construction of a finite-difference scheme and the choice of numerical methods for the solution of finite-difference equations. Basically, it is possible to use various iterative schemes, beginning from completely explicit and finishing with completely implicit. A completely explicit scheme of the iterative process is understood as the simple sequence of solutions of the Poisson equation, and the continuity equations for ions and electrons. As a rule, this scheme appears over-uneconomical, due to stability conditions similar to (4.158). Improving the efficiency of the iterative process can be achieved by the introduction of implicitness in the algorithm. This algorithm is named the “explicit-implicit approximation – simple iteration”. The computing cycle on the (p C 1) time layer begins with the determination of the electric potential field by integrating the Poisson equation. This field will not be recalculated any more, and is assigned to (p C 1)-th time layer. So, in the given computing scheme, the field is defined by an explicit mode. The implicit part of the iterative process consists in a sequential improvement of ions and electrons concentrations at the fixed velocity fields of drift motion (they are defined by the fixed electric field). During these so-called “internal” iterations the weight scheme for the calculation of concentrations of ions and electrons is used 

nSC1 (4.161) D nSe C w neSC1=2  nSe , e 

SC1=2 D nSC C w nC  nSC , nSC1 C

222

Chapter 4 The physical mechanics of direct current glow discharge SC1=2

SC1=2

where concentrations ne and nC are the results of numerical integration of the finite-difference equations. The weight w is defined as follows. First, the maximum relative error is determined for electron concentrations on a symmetry axis during two sequential iterations ˇ ˇˇ S .ne /1,j  .nSC1 /1,j ˇ e e "max D max . (4.162) j .nSC1 /1,j e Then a ratio of maximum admissible error "w , which is introduced as some a priori parameter, to received error is obtained w0 D

"w . "emax

The iterative weight w is selected by the following condition ( 1, w0 1, wD w0, w0  1.

(4.163)

(4.164)

Numerous calculations have shown that the basic shortcoming of this scheme is the explicit mode of representation of the ˛.E) function. Besides, the explicitness of such an algorithm is in the calculation method of the source term ˛eS  ˇnSC nSe . Attempting to use the quasi-linearization method for calculating the term does not bring any tangible effect. The completely implicit iterative scheme appears the most economic. The basic difference of this computing scheme from previous one is the inclusion of an electric potential calculation in a united iterative process. As a result, an improvement of source terms happens also in the internal iterations. At first sight, the solution of the Poisson equation on each internal iteration is very labor-consuming. However, such an iterative process leads to a considerable increase of the integration time step. In this computing scheme, the application of a weight principle for correction of concentration fields of ions and electrons is also convenient. The greatest effectiveness is reached on the first iterations.

4.3.8 Characteristic properties of a solution of a two-dimensional problem about glow discharge in a nonstationary statement A stationary solution of the glow discharge problem is determined, as a rule, by the time-asymptotic method. The unique difference from a truly nonstationary problem consists in the use of the relation (4.43) for an external circuit instead of real nonstationary boundary conditions.

Section 4.3 Finite-difference methods for the drift-diffusion model

223

But at the unsteady forming of glow discharge, the accumulation of charges on electrodes can render an appreciable influence on a current of glow discharge, especially for fast electrodynamic processes. Let us consider the Maxwell equation   1 "0 @' 4 rot j r , (4.165) B D 0 c c @t where B is the magnetic induction vector; "0, 0 are the dielectric and magnetic permeabilities. Taking into account the obvious relation   1 div rot B D 0, 0 and having applied operation div to the equation (4.165), one can receive @' 4 divj D "0 div grad  @t @ div 4j  "0 grad ' D 0. @t

or

Further we will use the Gauss theorem Z Z "0 @' nj ds  n grad ds D 0. 4 @t P

(4.166)

(4.167)

P

The surface integrals close to an infinitely thin electrode (Figure 4.6) can be rewritten Z Z Z Z "0 "0 @' @' nr ds C nr ds D 0. nj ds  n j ds  @t @t S1 S1 4 S3 S3 4 Assuming constant electric potential of the conductor, one can receive a total current Z Z Z @' "0 I D nr ds  njds D n jds. @t S3 S1 4 S1 But near to the electrode @'=@r D 0, therefore, Z Z "0 @ @' ds  n jds. I D S1 4 @t @x S1 Considering the presence of an external circuit, we receive Z E  V .t / @Q D  njds, @t eR0 S1

(4.168)

224

Chapter 4 The physical mechanics of direct current glow discharge

→

S3

n

→

n

S2

S1

→

n

→

j

Figure 4.6. Diagram for illustrating the boundary conditions statement for a glow discharge electrode.

where V .t / is the voltage drop on the gas-discharge gap; R0 is the Ohmic resistance; "0  1; Q is the charge accumulated on an electrode: QD

1 4e

Z S1

@ ds. @x

The equation (4.168) reflects the fact that the accumulation of charges on the electrode happens because there is a difference between the velocities of their outflow in the .t / ) and inflow from the discharge gap. external circuit ( EV R0 If equation (4.168) is normalized as above, we will receive dQQ H D .E  V /  HRmC1 d

e,0 W RQ 0 WRmC1 QQ D N0 " where

QQ D

Q , .2/m N0

Z

1 0

Z

1 0

@ˆ Ne r m dr, @x

@ˆ m r dr, @x

RQ 0 D 1.6  1019.2/m N0 R0 ,

(4.169)

(4.170) " D 4e.

Calculation of QQ under the formula (4.170) with its subsequent use in (4.169) for the determination of potential V is not optimum. Insignificant errors in the definition of @ˆ=@x can lead to an appreciable modification of voltage drop V , that leads, in turn, to increasing numerical instabilities. The following algorithm is recommended for the numerical solution. If we assume that on the p-th time layer V p , QQ p and I p are known, the desired solution on the

225

Section 4.4 Numerical simulation of the one-dimensional glow discharge

(p C 1)-th layer starts with the definition of the concentration fields of ions and electrons, then with the use of (4.169) the new value of QQ is calculated (while the iterative value, but not on the (p C 1)-th layer)   Z 1 @ˆ H Ne r m dr . ."  V S /  HRmC1 (4.171) QQ SC1 D QQ p C

e W RQ 0 0 @x Now it is required to define the modification of electric potential on the electrodes due to the received modification of charge density (QQ SC1  QQ p /. For this purpose the equation (4.170) is used. However, from this equation it is impossible to receive V pC1 in an explicit form, therefore, it is necessary to use one more internal iterative process. Setting some trial value of potential Vt , the corresponding charge QQ t is determined under the formula (4.170). Then the estimation of potential V SC1 can be received with the use of linear interpolation between V p and Vt V SC1 D V p C .Vt  V p /

QQ SC1  QQ p . QQ t  QQ p

(4.172)

Now it is possible to suppose Vt D V SC1 and to repeat the process the necessary number of times up to convergence of value V . However, the desired solution of governing equations on the (p C 1)-th time step does not come to the end because the self-consistent solution of charged particle continuity and Poisson equations has not yet been obtained. The iterative process becomes closed on its beginning, and so it proceeds until convergence of values QSC1 , V SC1 is reached. The iterative algorithm should also contain the block of automatic control of numerical time step. If convergence of the iterative process is not reached with NITER iterations (NITER is the initially defined number of internal iterations; for example, NITER D 10) , the time step decreases twice, and the solution on the p-th step has been chosen again as zero approximation. From the other side, the automatic increasing of calculation time steps can be performed when the number of iterations does not surpass some previously defined number, say 3.

4.4 Numerical simulation of the one-dimensional glow discharge The one-dimensional numerical simulation model of glow discharge in a nonstationary statement will be considered below. The advantage of such an approach is the possibility of deriving both truly nonstationary discharge, and corresponding nonstationary solutions. The choice of the nonstationary variant of the glow discharge electrodynamics equations is also connected with the desire to achieve methodological unity with the technique of the two-dimensional problem solution.

226

Chapter 4 The physical mechanics of direct current glow discharge

4.4.1 Governing equations and boundary conditions The governing equations in the one-dimensional statement follow from the 2D formulation given in Section 4.2.2, and have the following form @u @2 u @ C .Pb bu/ C Db 2 D f , @

@x @x s   @ˆ @Ne 2 f D ˛H  DQ ex Ne  ˇQNe NC , @x @x @2 ˆ N0 .NC  Ne /, D " H 2 @x W where

( Pb D

(

C =e for ions, 1

for electrons; bD

Db D

(4.173)

(4.174) (4.175)

DQ C for ions, DQ e for electrons;

@ˆ . @x

Boundary conditions are formulated as follows: For ions (u D NC ): x D 1, u

DQ Ce @u @ˆ D , @x 2C @x

(4.176)

x D 0,

@u D 0. @x

(4.177)

For electrons (u D Ne ): x D 0, u

C @ˆ 1 @u @ˆ D C NC , @x 2 @x e @x

(4.178)

x D 1,

@u D 0. @x

(4.179)

For electric potential (u D ˆ): x D 0, u D 0, x D 1, u D V =E. An important point in the problem solution is the set of the fixed value of voltage drop for gas-discharge gap V (furthermore this boundary condition will be repeatedly used also in two-dimensional calculations).

227

Section 4.4 Numerical simulation of the one-dimensional glow discharge

Boundary conditions for ions and electrons are obtained from the following considerations. Let us present a summarized particle flux and partial fluxes of electrons and ions in the form of the sum of two flux components in the positive and the negative direction of the x axis Г–

Г+

H

0

x

C  P D P  P ,

P D C  e ,

C  C D C  C ,

(4.180) e D eC  e .

Now let us assume that on the cathode ions are not emitted in gas-discharge gap x D 0,

C D 0, C

(4.181)

and the electron flux is connected with a flux of incident ions by the factor of a secondary electron-ionic emission  eC D C .

(4.182)

Let us write the flux components in the form of eC D e ne

@' 1 @ne  De , @x 2 @x

(4.183)

1 @ne e D De , 2 @x

(4.184)

@nC 1 C , C D  DC 2 @x

(4.185)

 C D C nC

@' @nC 1 C DC . @x 2 @x

(4.186)

It is easy to be convinced that substituting (4.183), (4.184) in the fourth condition (4.180), and relations (4.185), (4.186) in the third relation, we will come to common expressions for axial particle fluxes. Boundary conditions (4.181) and (4.182) can now be formulated in terms of concentration and a potential @nC x D 0, D 0, (4.187) @x @' 1 @ne @' @nC 1 e ne  De D nC C C DC @x 2 @x @x 2 @x

228

Chapter 4 The physical mechanics of direct current glow discharge

or

@ne 2 @' 2 C @' D e ne  nC . @x De @x De @x On the anode we set a condition of nonemission of particles e D 0,

(4.188)

(4.189)

 D 0. C

Substituting here (4.184) and (4.186), we receive @ne D 0, @x @' 2C @nC D nC . @x DC @x

(4.190) (4.191)

Using the usual procedure of normalization, we will come to conditions (4.176)– (4.179).

4.4.2 The elementary implicit finite-difference scheme By analogy with the two-dimensional case, sequential application of the finite-volume method and the donor cells approximation for the equation (4.173) leads to the following finite-difference scheme ANj uj 1 C BN j uj C1  Cj uj C Fj D 0,

(4.192)

where .Pb b/C .Pb b/ R L  ANj ; BN j D   BN j ; hj hj  1 1  .Pb b/C  Cj ; Cj D R  .Pb b/L C hj

Cj D Aj C Bj ; 1 1 ; Bj D Db Ax ; Aj D Db Ax xj  xj 1 xj C1  xj 2 1 1 D ; hj D .xj C1  xj 1 /. Ax D xj C1  xj 1 hj 2 ANj D

With the purpose of preservation of methodological generality with a solution of the two-dimensional problem, three-point finite-difference equations (4.192) are solved by the Thomas algorithm. For this purpose, it is necessary to formulate boundary conditions in the standard form u1 D ˛1u2 C ˇ1 ,

(4.193)

uNJ D !uNJ 1 C ı;

(4.194)

229

Section 4.5 Diffusion of charges along a current line

then, for ions ˛1 D 1, ˇ1 D 0;    @ˆ Db 1 Db C , !D 2Pb hj @x NJ 2Pb hj

x D 0, x D 1,

ı D 0,

(4.195) (4.196)

hj D xNJ  xNJ 1 ; and for electrons    @ˆ Db Db 1  , x D 0, ˛1 D  2h1 @x 1 2h1      @ˆ @ˆ C Db 1 ˇ1 D .NC /1  , h1 D x2  x1 ; e @x 1 @x 1 2h1 x D 1,

! D 1,

ı D 0.

(4.197) (4.198) (4.199)

Similarly, the equation for electric potential is solved. Coefficients of the Thomas algorithm and boundary conditions look like ANj D

Ax , xj  xj 1

Cj D ANj C BN j ,

BN j D Ax D

Ax ; xj C1  xj

(4.200)

2 ; xj C1  xj 1

x D 0, ˛1 D 0, ˇ1 D 0; x D 1, ! D 0, ı D V =E. Note that the given finite-difference scheme allows to solve problems of the structure of capacitive high-frequency of glow discharges. In these cases, the typical frequency of electric field oscillations is equal to f  0.1–100 mHz.

4.5 Diffusion of charges along a current line and effective method of grid diffusion elimination in calculations of glow discharges Physical diffusion of charges in a direction of drift motion can appear in areas of strong longitudinal heterogeneities, in particular near to electrodes in glow and RF capacitive discharges. Its inclusion into governing equations demands the statement of additional boundary conditions. Usually this is done without additional physical substantiation. The account of longitudinal diffusion often appears illusory because of large grid diffusion, which is present in numerical schemes.

230

Chapter 4 The physical mechanics of direct current glow discharge

The sequential derivation of boundary conditions is given below, and the example of one-dimensional glow discharge influence on physical and grid diffusion is shown. Diffusion effects are inspected by means of specially developed and complicated scheme of the fourth accuracy order, where grid diffusion is actually excluded. The effective method of elimination of grid-diffusion errors within the limits of standard Thomas schemes of the first and second accuracy orders is offered.

4.5.1 Governing equations for the one-dimensional case The glow discharge is described by the standard set of equations of the drift-diffusion model for electron and ion densities ne , ni and for electric field E: @ne @ne @e C D ˛e  ˇne nC , e D ne e E  De , @t @x @x @nC @nC @C C D ˛e  ˇne nC , C D CnC C E  DC , @t @x @x

(4.201) (4.202)

@' @E D 4e.nC  ne /, E D  , (4.203) @x @x where  are the fluxes of charges;  is their mobility; D is the coefficient of physical diffusion; ˛.E/ is the Townsend ionization coefficient; ˇ is the coefficient of recombination; ' is the electric potential. On the cathode (x D 0) ' D 0, on the anode (x D L/ ' D V . Without taking into account diffusion (De,C D 0) on the cathode it is supposed that e D  C , where is the coefficient of the secondary ion-electron emission; on the anode it is supposed C D 01.

4.5.2 Boundary conditions Expression (4.201) for e is based on the Lorentz representation of electron velocity v distribution f .v/ D f0 .v/ C f1 .v/ cos # , where # is the angle between v and an axis x [83]. One-sided fluxes on electrodes eC and e in positive and negative directions of the axis x (e eC  e ) submit to physical real conditions. By integrating f .V/ within corresponding hemispheres, we can see: “ ˙ v cos # .f0 C f1 cos # /v 2dv 2 sin # d# e D D 1

  ne vN e @ne e ne vN e 1 ˙ D ˙ e ne E  De , E < 0. 4 2 4 2 @x

(4.204)

In the given model with ˛.E/, the nonlocal effects are ignored. In this model the Faraday region, where electron diffusion is also actually significant, does not appear.

231

Section 4.5 Diffusion of charges along a current line

The average velocity vN e is defined by temperature Te ; De D e Te D l vN e =3, where l is the electron free path. Let the cathode not absorb electrons, but reflect and emit them, i. e., eC D e  i . The condition e D  i holds valid. If the anode does not reflect electrons, then e D 0,

3 El l @ne D  . ne @x 2 Te

(4.205)

Drift velocity at the anode ved D e jEj  vN e , therefore jEjl  Te . The function ne .x/, being linearly continued for absorption surface of the anode, converts in zero on the so-called extrapolated length lex D .2=3/l (in more strict theory lex D 0.71 l/. The anode layer is many times larger than l, and it is assumed that error in calculation ne .x/ will not be large, if it is supposed that lex D 0. Then on the anode ne D 0. Because of smallness of ionic temperature TC we have vCd vN C . Therefore, it makes no sense to consider the longitudinal diffusion of ions generally. However, if we take into account the longitudinal diffusion, the condition C .L/ D 0 is kept, as the anode C D 0. reflects ions; on the cathode C

4.5.3 Numerical methods for the one-dimensional calculation case Below we are going to consider numerical simulation results obtained with two finitedifference methods, ensuring the first and the fourth order of accuracy. Completely implicit iterative schemes will be used. In the scheme of the first order (furthermore referred to as “the scheme 1”), particle concentrations are calculated in integer points of a computational mesh, and streams are calculated in half points. Calculations with the fourth order of spatial accuracy are made with use of Petukhov’s decomposition method (“the scheme 4”). In the frame of the method, the equations (4.201), (4.202) are formed in common form accepted for partial differential equations of the first order on t and of the third on x:  0 3 3 X X m0 C ui mi D k0 C ui ki  p2 uP 2  p3 uP 3 , i D1

(4.206)

i D1

where “prime” means coordinate derivation and “point” means time derivation. With reference to (4.201), (4.202) Z

x

u1 D

n.x 0 /dx 0 ,

0

m1 D D,

u2 D n,

m2 D a' 0 ,

u3 D u02 ;

k0 D F ,

m0, m3 , k1 , k2 , k3 D 0;

p2 D 1,

p3 D 0.

(4.207)

Here, values are reduced to a dimensionless form: densities are measured in terms n0 D 109 cm3 , x in L, t in L2=e V , ' in V . At the same time, a D 1 for electrons

232

Chapter 4 The physical mechanics of direct current glow discharge

and (a D C =e / for ions, and F is the right-hand side of the continuity equations (4.201), (4.202): ˇ ˇ F D ˛L ˇne ' 0 C De n0e ˇ  ˇbne ni , b D L2n0 =e V , (4.208) where ˛ and ˇ are dimensional. At integration of (4.206) the second order of time approximation accuracy is ensured. A peculiarity of the application of this method is the necessity to calculate coefficients for integer and half points of a computational mesh. In integer points function ' 0 , i. e., E, is approximated by central differences, and in half points under the formula of the first order of accuracy. Values n in half points can be calculated as a half-sum of nodal values. The calculation of F in integer and half points is performed by (4.208). The stationary glow discharge condition is obtained by solution of the nonstationary problem by the time-asymptotic method. In addition, it is possible to enter either V and then to calculate a current density j at t ! 1, or to fix j D j0 on one of the electrodes and to search for V .t ! 1/. It is obvious that j.x, 1/ D j0. The volt-ampere characteristic (VAC) of stationary discharge V .j / has a minimum. Both modes give equally growing branches of VAC, but it is possible to receive a falling one if only j is set. If V is set, this branch never grows, as it is unstable.

4.5.4 Results of 1D numerical simulation Numerical simulation results will be considered for N2 at p D 5 Torr, L D 0.4 cm, D 1=3. Coefficients ˛, ˇ, e,i were borrowed from [12]; Te D 0.1–5.0 eV (at the account of ion diffusion with Ti D 0.026 eV). Figure 4.7 shows the discharge structure at current density j D 2.5 mA/sm2 and Te D 1 eV, which is close to normal. Calculations were made under schemes 1 and 4 with average steps of h1 D 0.014 cm, h4 D 0.002 cm accordingly. At approximation of gas-discharge finite-difference equations and boundary conditions in the first scheme, there is greater grid diffusion which results in additional losses of electrons because of their returning to the cathode. As a result, the electric field in the cathode layer, cathode voltage drop Vc and, as a corollary, voltage drop on electrodes V increases. This is required for amplification of ionization which should complete the specified nonphysical losses. It follows from the equation (4.201) and boundary conditions without recombination when charges disappear only on electrodes, the index of electron multiplication is equal to   Z L 1 1 I D C 1 D 1.39 .for D /, (4.209) ˛ dx D ln 3 0 regardless of physical diffusion, i. e., to value Te . For compensating the recombination losses, the amplification should also be more significant. In the considered conditions, recombination losses are low, and parameter I calculated under the scheme 4, which excludes grid diffusion, really only differs a little from 1.39: I D 1.40–1.42

233

Section 4.5 Diffusion of charges along a current line

20

n(4) +

16 E

(1)

12

E, kV/cm

n, 109cm–3

2.8 320

n(1) +

2.1 240 φ(1)

8 E(4)

φ(4)

4 0

φ, V

n(4) e n(1) e 0.08

0.16

0.24

1.4 160 0.7 80 n(1,4) n(1,4) e + 0 0.32 0.4

Figure 4.7. Distributions of charge densities ne , nC , field strength E and potential ', calculated under schemes of first and fourth orders (superscripts 1 and 4) (nitrogen, L D 0.4 sm, V D 215 V, j D 2.5 mA/cm2, Te D 1 eV).

at Te D 0.1–5.0 eV. A small increase of I with growth of T se , is probably connected with redistribution of ne,C .x/, i. e., with the modification of recombination and with errors of calculation. In calculations under the scheme 1, I is essentially larger (I D 1.91). As already mentioned, this phenomenon is connected with increasing electron multiplication degree, which is caused by the necessity of compensating for their additional losses. R The overwhelming part of integral ˛ dx relates to the cathode layer. Its thickness d Rd we will define by coordinate x D d for which 0 ˛ dx makes 99 % of I , and a cathode drop will be defined by the condition of Vc D '.d /. The volt-ampere characteristics of the cathode layer Vc .j / obtained by different schemes and different Te are presented in Figure 4.8. It is evident how at the given j the value Vc steadily grows with increasing grid diffusion, which is more intensive because the calculation grid net is rougher in the scheme 1. At calculation under the scheme 4, values VK and d very poorly depend on Te . Thus, physical diffusion in a longitudinal direction renders an insignificant influence on the structure and parameters of glow discharge, and, apparently, it should not to be taken into consideration in the problem. A much greater and rather appreciable influence is rendered by grid diffusion, when the calculation is conducted under finitedifference schemes with grid diffusion, especially with rough meshes. Use of schemes of the high order of accuracy, such as the scheme 4, is not always justified, as the accuracy is reached by large complexity and increased order of expenditures of computational time. However, the much simpler and more economic scheme 1 can be essentially improved by appreciably excluding most essential grid diffusion effects, namely losses of electrons due to their nonphysical returning to the cathode. Let us consider a numerical model of glow discharge without taking into account longitudinal physical diffusion, as has been done in the overwhelming majority of

234

Chapter 4 The physical mechanics of direct current glow discharge Vc, V 200

4 150

3 2 1

2

4

6

8

j

Figure 4.8. VAC in the cathode layer, derived from calculations under different schemes: (1) scheme 4, Te D 0.1 eV, step h D 0.002 cm; (2) scheme 4, Te D 5 eV, h D 0.002 cm; (3) scheme 1, Te D 1 eV, step h D 0.002 cm; (4) scheme 1, Te D 1 eV, step h D 0.014 cm.

works. The boundary condition on the cathode e D  C with purely drift fluxes immediately connects particle densities in calculation point x D 0 at the cathode: ne0 D .C =e /nC0 . At the presence of diffusion, the boundary stream condition has the same appearance; only the expression for flux e is redefined. It is offered to include grid diffusion electron flux (ionic diffusion flux is small) in a flux e in the same manner as would be made to take account of physical diffusion. In this case, the correct resulting balance of charges on the cathode is kept. As it was shown in the work [84], coefficient of grid electron diffusion De D 0.5e ', where ' is the potential drop at grid step h. Note that with De D 0.5e jEjh we can transfer from the differential boundary condition e ne E  De

@ne D  i ni E, @x

E D  jEj

(4.210)

to corresponding finite difference condition ne0 D

i 1 ni 0 C .ne1  ne0 /, e 2

(4.211)

where ne0 , ni 0 concern the calculation point x D 0 on the cathode, and ne1 concerns the next point x D h. Calculation under the scheme 1 with the former step h D 0.014 cm, but with the advanced boundary condition (4.211), has given distributions of E, ne , ni , which are almost indistinguishable from those received under scheme 4. The integral of multiplication has decreased from I D 1.91 to 1.57 (against 1.42 in scheme 4). The remained discrepancy is partly connected with errors of the evaluation of the integral from the very sharp function ˛.E/.

Section 4.5 Diffusion of charges along a current line

235

4.5.5 Method of fourth order accuracy for the solution of the drift-diffusion model equations In the previous section it was shown that the method of the fourth order of accuracy allows to avoid the grid diffusion effects in numerical simulation. As the specified method has appeared rather effective, below we will derive all equations of the method. The Petukhov decomposition method has been developed for the solution of partial differential equations of the following parabolic type dM D K C ˛P , d

(4.212)

where M D m1 u1 C m2 u2 C m3u3 C m4, K D k1 u1 C k2 u2 C k3 u3 C k4 ; @u2 @u3 C p3 , P D p2 @ @ @u2 @u3 u1 D D u02 , u2 D D u03 , @ @

(4.213)

(4.214)

where factors mi D mi ., , u1 , u2 , u3 /,

(4.215)

ki D ki ., , u1 , u2 , u3 /, pi D pi ., , u1 , u2 , u3 / are the set functions of their arguments. The equation (4.212) is solved in the calculation domain 0   ı and  0. The finite-difference mesh in directions and  can be inhomogeneous. Here we will receive computing relations for a numerical integration of system (4.212) along a variable (i. e., at ˛ D 0/. Boundary conditions for the differential equation (4.212) are formulated in the following form: at D 0

at D ı

11u1 C 12u2 C 13u3 C 14 D 0,

(4.216)

21u1 C 22u2 C 23u3 C 24 D 0;

(4.217)

31u1 C 32u2 C 33u3 C 34 D 0,

(4.218)

where factors ij are known.  Connection between functions u j on the left boundary of the segment and funcC  tions uC j on the right boundary of the segment D C 2 is set in the following form: 3 3 X X C C   aiC4 C aij uj D ai4 C aij uj , i D 1, 2, 3, (4.219) j D1

j D1

236

Chapter 4 The physical mechanics of direct current glow discharge

where factors aiC4 and ai4 are subject to definition,  is half of the range between the left and right boundaries of the segment. To discover these factors, three additional conditions are used: Z C Z M0 d D K d , (4.220)  2  Z  C  2 u3 d D .uC (4.221) 3 C u3 /  .u2  u2 / =3, Z

2 

2 

C   2 u2 d D .uC 2 C u2 /  .u1  u1 / =3.

(4.222)

Formulas (4.221) and (4.222) are Simpson’s formula for the integration of function   C    Z @ @  2=3 C    . d D . C C   /   (4.223) @ @ 2  For deriving the first connection between the required factors, we will take the equation (4.220) in the following form (the right term is calculated under the Simpson formula): C C C C C C        mC 1 u1 C m2 u2 C m3 u3 C m4  m1 u1  m2 u2  m3 u3  m4  C .K C K  C 4K0/, (4.224) D 3

where K0 is calculated by values of functions u0j in the center of the interval K0 D k10 u01 C k20 u02 C k30 u03 C k40 ,

(4.225)

which, in turn, are expressed through values of functions on boundaries of the interval u01 D

3 C 3  1 C 1  u2  u  u1  u1 , 4 4 2 4 4

u02 D

3 C 3  1 C 1  u3  u  u2  u2 , 4 4 3 4 4

1 1   C   u03 D uC u C u . 3 C u3  2 2 4 2 4 2 As a result of reduction (4.224) to the form (4.219) we receive  C C .k  k10 /, D mC a11 1  3 1  C  2 0 C 0 .k k  k10 , D mC   k / C a12 2 2 3 2 3 3  C C .k C 2k30/  k20 , D mC a13 3  3 3  C C D mC a14 .k C 2k40/ 4  3 4

(4.226)

(4.227)

Section 4.5 Diffusion of charges along a current line

237

and   .k  k10 /, 3 1    2 0 0 .k k  k10 , D m C  k / C 2 2 3 2 3 3   .k C 2k30 /  k20 , D m 3 C 3 3   D m .k C 2k40 /. 4 C 3 4

 D m a11 1 C  a12  a13  a14

(4.228)

For deriving the second connection between the factors we will take the relation (4.221) with reference to function M : Z 

M d D .M C C M  /  MC  M  2 =3, (4.229) 2 

but

Z

MC D K C M d D

and 2 

and M D K   .M C C M  C 4M 0/, 3

(4.230) (4.231)

Therefore, substituting .4.230/ and .4.231/ in (4.229), we will receive 2 C D  mC a21 1  3 2 C D  mC a22 2  3 2 C D  mC a23 3  3 2 C D  mC a24 4  3

 2 C  0 k C m, 3 1 3 1  2 C  0  2 0 k2  m01 C m C m3 , 3 3 2 3  2 C 2 k3  m02   m03 , 3 3 2  C 2 k   m04 3 4 3

(4.232)

and 2  D   m a21 1  3 2  D   m a22 2  3 2  D   m a23 3  3 2  D   m a24 4  3

 2   0 k  m , 3 1 3 1  2   0  2 0 k2  m01  m C m3 , 3 3 2 3  2  2 k3  m02 C  m03, 3 3 2   2 k C  m04. 3 4 3

(4.233)

238

Chapter 4 The physical mechanics of direct current glow discharge

The third connection between factors aj˙i is determined directly under the formula (4.222)  2 C , a32 D  , 3  2  , a32 D D  , 3

C D a31  a31

C a33 D 1,

C a30 D 0;

  a33 D 1, a30 D 0.

(4.234) (4.235)

˙ are defined under formulas (4.227), (4.228), Thus, in the ratio (4.219) all factors aij (4.232)–(4.235). Now it is necessary to relate the specified factors with factors ij in formulas for boundary conditions (4.216)–(4.218) and to receive final correlations for calculation. For boundary D 0, instead of .4.216/ and (4.217) we write

14 C

3 X

1j u j D 0,

(4.236)

2j u j D 0,

(4.237)

j D1

24 C

3 X j D1

and for boundary D ı instead of .4.218/ we write 34 C

3 X

3j uC j D 0.

(4.238)

j D1

Let us assume that in each fixed point of the computational mesh, for example , functions u j satisfy two equations  14 C

3 X

  1j uj D 0,

 24 C

j D1

3 X

  2j uj D 0,

(4.239)

j D1

 where factors  1i and 2i (i D 1, 2, 3, 4) coincide with 1i and 2i on boundary D 0 and are subject to definition inside of the calculation area. Together with (4.239) we will consider relations (4.219) and (4.235) C C C C C C C AC 3 D a31 u1 C a32 u2 C a33 u3 C a34 D

 2   u C  u 2 C u3 . 3 1

(4.240)

Whence we receive

 2  u   u 2. 3 1 Having substituted (4.241) in (4.239), we will receive C u 3 D A3 

 C Q  Q  . i 4 C i 3 A3 / C i 1 u1 C i 2 u2 D 0,

(4.241)

i D 1, 2,

(4.242)

Section 4.5 Diffusion of charges along a current line

239

where   Q i 1 D  i 1  i 3 a31 ,

(4.243)

  Q i 2 D  i 2  i 3 a32 .

(4.244)

The set of equations (4.242) can be resolved concerning functions u i in the following form: C u i D 1, 2, (4.245) i D i 4 C i 3 A3 , where   Q Q Q 12 Q 12 24  22 14 23  22 13 , 13 D , Det Det   Q Q Q 21 Q 21  14  11 24 13  11 23 , 23 D ; D Det Det Det D Q 11 Q 22  Q 12 Q 21.

14 D 24

(4.246)

 Thus, the use of formulas (4.241) and (4.245) allows to calculate functions u 1 , u2 and C  u3 , if function A3 is known. Now we will consider the set of equations (4.219) for i D 1, 2. Assuming that in each point of computational mesh ( C 2 ) required functions satisfy the relations

C 14 C

3 X

C C 1j uj D 0,

j D1

C 24 C

3 X

C C 2j uj D 0,

(4.247)

j D1 C where factors C 1i and 2i (i D 1, 2, 3, 4) are subject to definition in view of a boundary condition at D ı . Let us substitute in (4.219) the functions discovered above C u 1 D 14 C 13 A3 ,

(4.248)

C u 2 D 24 C 23 A3 ,

(4.249)

33AC 3 ,

(4.250)

u 3

D 34 C

where   33 D 1  13 a31  23 a32 ,   34 D 14 a31  24 a32 .

240

Chapter 4 The physical mechanics of direct current glow discharge

Let us exclude u i (i D 1, 2, 3) from (4.219) at i D 1, 2, for this purpose we will rewrite (4.219) in the form of AC i

D

aiC4

C

3 X

C C aij uj

D

ai4

j D1

C

3 X

 aij .j 4 C j 3 AC 3 /, i D 1, 2

(4.251)

j D1

or     AC i D ai 4 C .ai 1 14 C ai 2 24 C ai 3 34 /

C .ai1 13 C ai2 23 C ai3 33/AC 3 ,

i D 1, 2.

(4.252)

Expressions in parentheses can be written in the form of    Q  ij D ai 1 1j C ai 2 2j C ai 3 3j ,

(4.253)

where i D 1, 2, j D 3, 4, so (4.252) gains a more compact form  Q Q C AC i D ai 4 C i 4 C i 3 A3 ,

i D 1, 2.

(4.254)

Let us now pay attention to the structure of formula (4.254). Functions AC i (i D C 1, 2, 3), entering in (4.254), contain factors aij and functions uC (see formula (4.251)). j At their substitution in (4.254), the system of two equations of the form (4.247) will C turn out, whence factors C 1i and 2i (i D 1, 2, 3, 4) will be ascertained C Q C C 11 D 13 a31  a11 , C Q C C 12 D 13 a32  a12 , C Q C C 13 D 13 a33  a13 , C  Q C 14 D .a14  a14 / C 14 ;

C 21

D

C Q  23 a31



(4.255)

C a21 ,

C Q C C 22 D 23 a32  a22 , C Q C C 23 D 23 a33  a23 ,  C Q C 24 D .a24  a24 / C 24 ,

(4.256)

C D 0 is taken into account. where a34 Now it is possible to formulate the calculation algorithm.

Step 1. In a cycle j D 2, 3, : : : , NJ auxiliary factors are calculated sequentially. C  under formulas (4.227), (4.228), (4.232), (4.233) and and aij (a) Calculation of aij (4.234), (4.235).

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 241  (b) Calculation of  1j , 2j (j D 1, 2, 3, 4/. At D 0 (on the left boundary of a segment) these factors coincide with factors of boundary conditions (4.216) and (4.217)   1j D 1j , 2j D 2j , j D 1, 2, 3, 4.

On the right boundary of elementary segment ( C 2 / the specified factors are defined after realization of the following two stages (these factors are equated to C C 1j , 2j /. (c) Calculation of 13 , 14, 23 , 24 under formulas (4.246). At the same time it is reasonable to remember values of the specified factors in each point of a computational mesh. C (d) Calculation of C 1j , 2j (j D 1, 2, 3, 4/ under formulas (4.255) and (4.256).

Step 2. At reaching the rated point j D NJ the set of equations is solved C C C C C C C 11 u1 C 12 u2 C 13 u3 D 14 , C C C C C C C 21 u1 C 22 u2 C 23 u3 D 24 , C C C C C C C 31 u1 C 32 u2 C 33 u3 D 34 ,

whence unknown functions are ascertained on an upper bound of the rated area D ı . Step 3. In a cycle j D NJ  1, NJ  2, : : : , 1 the following functions are calculated sequentially. (a) Functions AC 3 under the formula (4.240) (first half of equality); (b) Unknown functions uC i (j D 1, 2, 3/ under formulas (4.248)–(4.250). In conclusion, it should be noted that the given method ensures accuracy of calculation of required functions proportional to   4.

4.6 Two-dimensional structure of glow discharge regarding neutral gas heating A significant role in the development of our understanding of the nature of glow discharges in gas flows is played by numerical modeling of these discharges in the various conditions representing our practical interest. After the first works dedicated to calculations of glow discharges [129, 130], significant progress has been reached in the beginning of 1980s in the field of numerical modeling of their two-dimensional structure [42], and the models developed afterwards [77, 84, 112] have allowed to carry out regular research of glow discharge structure by means of numerical modeling.

242

Chapter 4 The physical mechanics of direct current glow discharge

In order for computer models of glow discharges to be practically useful in aerophysical research, the further development of these models is required, regarding the account of interaction of discharges with gas flows and an external magnetic field, interactions of discharges with moving gas at super- and hypersonic velocities, and also the account of the physical and chemical transformations which occur in the region of the discharge. In the given section a rather special, but current problem is solved about the influence of gas heating on the structure of glow discharges.

4.6.1 Statement of two-dimensional axially symmetric problem The numerical model of glow discharge, in which gas heating is considered, is formulated for axially symmetric geometry. It is supposed that the discharge exists in the normal mode between two flat electrodes (Figure 4.1), so boundary effects in a radial direction do not influence its structure. The structure of glow discharge is described within the limits of the drift-diffusion model, formulated concerning electronic and ionic components, and also the Poisson equation, defining distribution of an electric potential in an electro-discharge gap (see Section 4.2.4). It is supposed that a source of gas heating is the Joule thermal emission. As already mentioned, not the whole energy of an applied electric field transmitted at electron collisions with molecules of gas goes to their heating. The significant part of this energy can be spent for excitation of molecules’ vibrational degrees of freedom. For its definition it is necessary to solve the kinetic equation for the distribution of the electron energy function. At the same time, it is obvious that in various space zones of glow discharge a relation between the Joule thermal emission energy and the energy going into excitation of internal degrees of freedom will be different. In the offered statement, the solution of this equation is replaced with the introduction of phenomenological effectiveness ratio of electric field energy transmission into gas heating, which does not vary by space, and this ratio is a parameter of the problem. In the work [76] the phenomenological model of the account of a spatial modification of electron temperature and kinetics of excitation of vibrational conditions of molecular nitrogen was offered. With use of this model it was shown that in conditions similar to those considered here, this ratio can reach 90 %. A calculation model of glow discharge in view of gas heating is formulated in the following form: @ne C div e D ˛.E=p/ j e j  ˇni ne , @t

(4.257)

@ni C div i D ˛.E=p/ j i j  ˇni ne , @t

(4.258)

div. grad '/ D 4e .ne  ni / ,

(4.259)

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 243

@T D div. grad T / C Q, @t  e D De grad ne  ne e E;  i D Di grad ni C ni i E; cV

where

Q D .j  E/;

j D e. i   e /;

(4.260)

E D  grad ';

T , p are the temperature and pressure of the gas; ne , ni are the concentration of electrons and ions in 1 cm3 ; E and ' are the vector of electric field strength and its potential;  e ,  i are the vectors of fluxes of electrons and ions; De , Di are the diffusivities of electrons and ions; e , i are the mobilities of electrons and ions; ˛ D ˛.E=p/ is the coefficient of collisional ionization of molecules by electrons (the first Townsend coefficient), E D jEj; ˇ is the coefficient of ion-electron recombination. In the solution of equations (4.257)–(4.260) an orthogonal cylindrical frame is used .x D r /. Boundary conditions for these equations look like

or

@ni D 0, @y

y D 0,

e,y D i ,y ,

y D H,

@ne D 0, @y

x D 0,

@ni @' @ne D D D 0; @x @x @x

x D R,

1/ ne D ni D 0, 2/

ni D 0,

' D 0;

' DV;

'D

(4.261) (4.262) (4.263)

V y, H

(4.264)

@ni @' @ne D D D 0. @x @x @x

Here is the coefficient of ion-electron emission from the cathode surface; V is the voltage drop on the discharge gap, e,y , i ,y are the projections of electron and ion fluxes on axis y; R, H are the coordinates of the calculation domain in directions x and y. The boundary conditions on electrodes for the charged particles are approximate. Boundary conditions include the value of a voltage drop on discharge gap V that has not been determined yet. For its determination it is necessary to include conditions in an external circuit (see Figure 4.1). In conditions of the steady-state glow discharge it is possible to write the obvious relation EV D 2 eR0

Z

R

e .x, y D H /x dx,

(4.265)

0

which postulates equality of the sum of resistance drops R0 and drops across discharge gap to electromotive power E. The calculation model is intended for the study of glow discharge in molecular nitrogen at pressure p D 1–20 Torr, therefore, the following values of the factors

244

Chapter 4 The physical mechanics of direct current glow discharge

entering into the mathematical statement of the problem are set: e p D 4.4  105,

i p D 1.45  103 .Torr  cm2 /=.V  s/,

ˇ D 2  107 cm3 =s,   B ˛ D A  exp  .cm  Torr/1 , p E=p

(4.266)

(4.267)

where A D 12.cm  Torr/1 , B D 342 V=.cm  Torr/. The empirical formula (4.267) for the first Townsend coefficient was recommended in the work [12] for the following range of ratios of electric field strength to a pressure: 100 < E=p < 600 V=.cm  Torr/. In the calculations, an appreciable influence of the ionization coefficient approximation used on integral discharge properties (total current through discharge, voltage drop across gas-discharge gap) can be observed. Therefore, having the intention to achieve numerical proximity of calculated and experimental data it is necessary to concern rather ourselves carefully with the choice of this approximation. The same is true for the choice of secondary ion-electron emission coefficient . Diffusivities can be defined by Einstein’s relations D e D e T e , D i D i T i , where Te , Ti are the electron and ion temperatures, eV. The energy conservation equation for neutral gas is formulated in the form of the Fourier–Kirchhoff equation (4.260) without taking convective gas motion into account. The stationary solution for this equation is searched for in the same manner as for the concentration of the charged particles. The important consequence of gas heating is the modification of local values of a density of neutral particles that makes an essential influence on the value of the collision frequency of electrons with gas molecules and, hence, on such parameters of the drift-diffusion model as the frequency of ionization, mobility of electrons and ions. The account of gas heating in functional associations of the specified parameters is given in Section 4.2.4.

4.6.2 Numerical simulation results Calculation of glow discharge is performed in the geometry shown on Figure 4.1. The axially symmetric glow discharge is considered in molecular nitrogen (N2 / between two flat infinite electrodes in the normal current density mode or close to it. Pressure in the electric discharge gap has been varied in a range of 1–20 Torr. Distance between electrodes H D 2 cm. Boundary conditions on an external cylindrical surface are set on distance x D R D 2 cm and 12 cm, that was quite enough to check their influ-

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 245

ence on the structure of glow discharge. Electromotive force of the power supply has been varied in the range of E D 600  4 000 V. The coefficient of a secondary electronic emission, resistance of an external circuit and electron temperature have been supposed constant and equal to D 0.1, R0 D 300 K , Te D 11 610 K, respectively. Computations were performed in the following sequence. First, the structure of glow discharge was calculated at different pressures without taking its heat into account. In addition, a series of methodological calculations with different meshes has been fulfilled. It is especially necessary to underline the necessity of the careful choice of computational meshes for the solution of the glow discharge problem. This is due to the fact that, as discussed above, the mesh diffusion caused by use of the finite-difference schemes of the first order of accuracy or by use of various methods of correction of numerical flows in schemes of the second and higher orders of accuracy, as a rule, appreciably influences calculation results. Therefore, for the first step, meshes were defined that do not cause appreciable disturbances in calculations. At initial data p D 5 Torr, E D 3 kV, the calculations were performed on meshes NI D 30  NJ D 30, 30  59, 30  117. In these calculations, the number of points by a radial variable (NI D 30) did not vary. This number of points has appeared sufficient for a reliable description of glow discharge structure, as the discharge is localized near to a symmetry axis where point crowding is high enough. Besides, it is well known that field strength in a radial direction is noticeably lower then field strength in an axial direction, in particular near to the cathode. It was of interest to study the influence of mesh detail in an axial direction on the discharge structure. The subsequent calculations with more detailed meshes (117  201) confirmed the validity of the specified choice of the computational mesh. Results of calculations of glow discharge on a mesh of 30  117 at p D 5 Torr and E D 3 kV are shown in Figures 4.9 and 4.11. The discharge is localized in a radial direction near to a symmetry axis where the initial approximation has been set. Current densities in the center of the cathode and anode spots achieve, respectively, jc D 6.0 mA/cm2 and ja D 23.3 mA/cm2 , and a radial size of the cathode spot equal to Rc  0.8 cm. The spatial structure of the glow discharge is shown in Figure 4.9 (a,b), where the area of spatial charges near to the cathode and the anode is visible well. Axial distribution of electric field strength is shown in Figure 4.9 (c). The maximum value of the strength is observed on the cathode Ey,max D 4700 V/cm. Then there is the area of a positive column, in which the field strength is constant at a level of Ey  170V/cm. In the immediate proximity of the anode, the small field strength increase is observed up to Ey  330 V/cm. Note that the calculations with use of the drift-diffusion model do not allow to describe the area of the Faraday dark space. The distributions of ion and electron concentrations in the glow discharge at E D 3 kV and p D 20 Torr are shown in Figure 4.11. The comparison of Figures 4.9 and 4.11 illustrates the influence of pressure on the glow discharge structure.

246

Chapter 4 The physical mechanics of direct current glow discharge y, cm

y, cm 2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

6

1.5

5

1

2.50E+01 2.33E+01 2.16E+01 1.99E+01 1.81E+01 1.64E+01 1.47E+01 1.30E+01 1.13E+01 9.57E+00 7.86E+00 6.14E+00 4.43E+00 2.71E+00 1.00E+00

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15

13

11

9.37E+00 8.74E+00 8.12E+00 7.49E+00 6.87E+00 6.24E+00 5.62E+00 5.00E+00 4.37E+00 3.75E+00 3.12E+00 2.50E+00 1.87E+00 1.25E+00 6.24E−01

4 9

(a)

0.5

3

(b) 7 5

0

15

0

3

1

2

12

14

10

0.5 5 000

1

1

0 x, cm

0.5

1

x, cm

E, V/cm (c)

4 000 3 000 2 000 1 000 0 0 0.25 0.5 0.75 1 0.25 0.5 0.75 2 y, cm

Figure 4.9. (a) Concentration of ions (a) and electrons (b) in glow discharge at p D 5 Torr, E D 3 kV; concentration is referred to 109 cm3 ; (c) Distribution of electric field strength along axis of symmetry of glow discharge at p D 5 Torr, E D 3 kV.

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 247 j, mA/cm2 60 3

40 2 1 20

2 1

0

0

0.5

1 x, cm

Figure 4.10. Current density on the cathode and the anode at E D3kV: (1) p D 5 Torr, (2) p D 10 Torr, (3) p D 20 Torr.

At the raised pressure the discharge is strongly contracted, and the sizes of nearelectrode layers decrease. With the use of rough computational meshes, a non-monotone spatial distribution of electric potential is often observed near to the cathode voltage drop. It also results in oscillations of ion concentrations near to the cathode boundary of a positive column and non-monotone (oscillating) behavior of electron concentration distribution in an axial direction. Common regularities in the behavior of the numerical simulation results at a refinement of computational meshes consist in the following. First of all, the increase of current density on the cathode and the anode is observed. So, at E D 3 kV and p D 10 Torr the following results are observed: 

for the mesh 30  30  jc D 6.2 mA/cm2 and ja D 11.3 mA/cm2 ;



for the mesh 30  59  jc D 17.95 mA/cm2 and ja D 37.5 mA/cm2 ;



for the mesh 30  117  jc D 21.1 mA/cm2 and ja D 38.0mA/cm2.

At the same time, the current in the external circuit varies slightly, therefore, the radius of the cathode and anode spots decreases noticeably. The maximum strength of electric field Ey increases at the cathode; in the positive column and near to the anode it varies slightly. Improving the computational mesh leads to the disappearance of the nonmonotony in the electric potential distribution. The important consequence of the computational mesh refinement is the increase in the concentration of charged particles in the positive column that is connected with a diminution of numerical diffusion losses.

248

Chapter 4 The physical mechanics of direct current glow discharge y, cm

y, cm

2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

1 2

1.5 3

1

4

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

3.22E+02 1 2.65E+02 2.19E+02 3 1.80E+02 1.49E+02 1.22E+02 5 1.01E+02 8.31E+01 7 6.85E+01 5.65E+01 4.65E+01 9 3.84E+01 3.16E+01 2.60E+01 11 2.15E+01

5.82E+01 5.43E+01 5.04E+01 4.65E+01 4.26E+01 3.88E+01 3.49E+01 3.10E+01 2.71E+01 2.33E+01 1.94E+01 1.55E+01 1.16E+01 7.75E+00 3.88E+00

5 13

(a)

0.5

(b)

6 15

0 15

0.5

1.0 x, cm

0.5

1.0 x, cm

Figure 4.11. Concentration of ions (a) and electrons (b) in glow discharge at p D 20 Torr, E D 3 kV; concentration is referred to 109 cm3 .

Results of calculations of glow discharge for three pressures p D 5, 10, 20 Torr at E D3 kV are shown in Figures 4.10 and 4.12, where the radial distributions of current densities on electrodes and the axial distribution of the concentration of charged particles along the glow discharge symmetry axis are given. At increasing gas pressure, the following modifications in glow discharge structure are observed: the current density on the cathode and on the anode increases, radial sizes of glow discharge decrease, thickness of the cathode and anode layers decreases, electric field strength in all electric discharge gaps increases, concentration of electrons and ions in the positive column increases (at the same time, the displacement of the charged particle concentration maxima in a positive column from the anode to the cathode is observed), the concentration of ions in the cathode layer increases. Let us note that the specified behavior of the glow discharge is in good correspondence with numerous experimental data and with the classical Engel–Steenbeck theory. It confirms the adequacy of the numerical model developed for the investigated phenomena. At the same time, it is necessary to mention that some elements of the glow discharge structure are described by the given model quite approximately (for areas of a spatial charge immediately adjoining electrodes, area of the Faraday dark space).

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 249

350

ne, n+, 109cm−3

300 250 200 150 100 3 50 2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 y, cm

Figure 4.12. Distribution of ion (continuous curves) and electron (dashed curves) concentrations along a symmetry axis of discharge at E D 3 kV: (1) p D 5 Torr, (2) p D 10 Torr, (3) p D 20 Torr.

As already mentioned, the Joule heating in glow discharges is caused by energy transmission from electrons, heated up by an external electric field, to gas molecules at their collisions. Part of this energy is spent for excitation of molecular oscillations. It is known that molecules of N2 have great vibrational excitation relaxation time that leads to part of the electron energy not having time to be transformed into heat and it is carried away by the vibrational excited molecules from the area of discharge, for example, at convective motion. In the statement considered, the relaxation processes and gas motion are approximately considered by the introduction of the effectiveness ratio of transformation of the electron energy in heat of the gas, which varies in calculations. Despite the obvious boundedness of such a statement, the specified model allows to investigate the configuration of glow discharge and regularities of origin of areas with increased gas temperature in it taking account of self-consistent electrodynamic and thermal processes. In the first series of calculations of glow discharge in view of gas heating, the pressure p D 5 Torr has been chosen. For two electromotive forces E D 2 kV and 4 kV calculations were performed both with and without taking into account the gas heating. Effectiveness of transformation of electric field energy in gas heating was supposed equal to D 0.5. In all cases, the cathode and the anode were supposed cooled, Tw D 300 K. Calculations have shown that at E D 2 kV the thermal state of discharge is stabilized after 1 millisecond, and at E D 4 kV after 5 milliseconds, which confirms the estimations of characteristic times fulfilled earlier.

250

Chapter 4 The physical mechanics of direct current glow discharge y, cm

y, cm

2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

9

1.5 7

5

1 3

2.50E+01 1.99E+01 1.58E+01 1.25E+01 9.97E+00 7.92E+00 6.29E+00 5.00E+00 3.97E+00 3.16E+00 2.51E+00 1.99E+00 1.58E+00 1.26E+00 1.00E+00

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15 13 11 9 7 5

6.73E+00 6.28E+00 5.83E+00 5.38E+00 4.94E+00 4.49E+00 4.04E+00 3.59E+00 3.14E+00 2.69E+00 2.24E+00 1.79E+00 1.35E+00 8.97E−01 4.49E−01

3 1

1

(a)

0.5

0

15

0

(b)

13 11 9

0.5

1

x, cm

0.5

1

x, cm

Figure 4.13. Concentration of ions (a) and electrons (b) in glow discharge at p D 5 Torr, E D 2 kV; concentration is referred to 109 cm3 .

The distributions of ion and electron concentrations in glow discharge in the first series of calculations (E D 2 kV) both without and with taking into account gas heating are shown in Figures 4.13 and 4.14 respectively, and in the second series (E D 4 kV) are shown in Figures 4.15 and 4.16. From Figure 4.17 it is evident that taking account of gas heating leads to an increase of current density on the anode from ja D 15.5 mA/cm2 up to 24.1 mA/cm2 (E D 2 kV) and from ja D 14.5 mA/cm2 up to 32.7 mA/cm2 (E D4 kV). The current density on the cathode at gas heating falls from 5.8 mA/cm2 to 4.0 mA/cm2 .E D 2 kV) and from 7 mA/cm2 to 4.0 mA/cm2 (E D 4 kV). Taking account of gas heating leads to improving conditions for ionization by electron impact of molecules. The thermal expansion of gas leads to a diminution of local values of particle concentration, so the parameter E=N defining effectiveness of ionization increases. At E D 2 kV gas in the discharge gap heats approximately up to 355 K. The temperature field for this case is shown in Figure 4.18. The existence of two local maxima of axial temperature distribution near to the cathode and anode attracts attention. The analysis of distributions of current density and

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 251 y, cm

y, cm

2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

12

1.5 11

. 9

1.75E+01 1.44E+01 1.19E+01 9.78E+00 8.06E+00 6.64E+00 5.47E+00 4.51E+00 3.72E+00 3.06E+00 2.52E+00 2.08E+00 1.71E+00 1.41E+00 1.16E+00

15 13 11 9 7 5

(a)

7

0.5

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

(b)

3 1

5 3

0

6.68E+00 5.13E+00 3.94E+00 3.03E+00 2.33E+00 1.79E+00 1.37E+00 1.06E+00 8.11E−01 6.23E−01 4.79E−01 3.68E−01 2.83E−01 2.17E−01 1.67E−01

1 13 119

15

0

1

x, cm

1

x, cm

Figure 4.14. Concentration of ions (a) and electrons (b) in glow discharge at p D 5 Torr, E D 2 kV, D 0.5; concentration is referred to 109 cm3. y, cm

y, cm

2

8

1.5

7 5

1

3 1

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

2.97E+01 2.35E+01 1.86E+01 1.47E+01 1.17E+01 9.25E+00 7.32E+00 5.80E+00 4.59E+00 3.63E+00 2.88E+00 2.28E+00 1.80E+00 1.43E+00 1.13E+00

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15

13 11 9 7

6.41E+00 5.99E+00 5.56E+00 5.13E+00 4.70E+00 4.28E+00 3.85E+00 3.42E+00 2.99E+00 2.57E+00 2.14E+00 1.71E+00 1.28E+00 8.55E−01 4.28E−01

5

0.5

3

(a)

(b) 1

1

0

15

0

13

11 9 7 5 3

1

x, cm

1

x, cm

Figure 4.15. Concentration of ions (a) and electrons (b) in glow discharge at p D 5 Torr, E D 4 kV; concentration is referred to 109 cm3.

252

Chapter 4 The physical mechanics of direct current glow discharge y, cm

y, cm

2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

13

1.5 11

1

9

1.69E+01 1.39E+01 1.15E+01 9.47E+00 7.80E+00 6.43E+00 5.30E+00 4.37E+00 3.60E+00 2.97E+00 2.44E+00 2.01E+00 1.66E+00 1.37E+00 1.13E+00

7

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15

13 11 9 7 5

(a)

0.5

1.23E+01 1.15E+01 1.06E+01 9.83E+00 9.01E+00 8.19E+00 7.37E+00 6.55E+00 5.73E+00 4.91E+00 4.10E+00 3.28E+00 2.46E+00 1.64E+00 8.19E−01 (b)

3

5 1 3

0

15

14

1

13

11

0

x, cm

1

1

x, cm

Figure 4.16. Concentration of ions (a) and electrons (b) in glow discharge atp D 5 Torr, E D 4 kV, D 0.5; concentration is referred to 109 cm3 .

j, mA/cm2 40 4

2 20 1 3 2

3 1 0

4 0

0.5

1

1.5 x, cm

Figure 4.17. Radial distribution of current density on the anode (continuous curves) and on the cathode (dashed curves) at p D 5 Torr: (1, 2) E D 2 kV; (3, 4) E D 4 kV; (1, 3) without taking into account gas heating; (2, 4) in view of gas heating.

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 253 y, cm

y, cm

2

1.5

15

1 13

1

3

11

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

3.51E+02 3.47E+02 3.43E+02 3.39E+02 3.35E+02 3.31E+02 3.28E+02 3.24E+02 3.20E+02 3.16E+02 3.12E+02 3.08E+02 3.05E+02 3.01E+02 2.97E+02

15

9 5

0.5

7

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

13

4.70E+02 4.58E+02 4.46E+02 4.35E+02 4.23E+02 4.11E+02 3.99E+02 3.87E+02 3.76E+02 3.64E+02 3.52E+02 3.40E+02 3.28E+02 3.17E+02 3.05E+02

11

(a)

(b) 9

9 11 13 15

0

0

7 5

1

x, cm

3

1

1

x, cm

Figure 4.18. Temperature (K) in glow discharge at p D 5 Torr, E D 2 kV (a) and at p D 5 Torr, E D 4 kV (b); D 0.5.

electric field strength allows to explain such a distribution of temperature. Near to the cathode the electric field strength is great, but the density of electric current is small, the reason for that in turn is a low electron concentration. Near to the anode, the situation varies conversely: electron concentration is great and the electric field strength is small. Therefore, origination of non-monotone axial distribution of the Joule thermal emission qJ D .j  E/ is natural. Nevertheless, the numerical research performed has shown the possibility of realization of different configurations of gas heating depending on parameters of the problem p, ", , and the set boundary conditions for taking account of heat exchange. Let us note one more peculiarity of the account of gas heating. The concentration of ions in the cathode layer decreases from 2.8  1010 cm3 up to 1.85  1010 cm3 at E D 2 kV (Figure 4.19) and from 3.2  1010 cm3 up to 1.8  1010 cm3 at E D 4 kV (Figure 4.20). The concentration of the charged particles in the positive column increases. Heating of the gas leads to the local maximum of particle concentrations in the positive column being displaced to the anode. Let us consider the influence of pressure on heating glow discharge. Besides the calculations with E D 4 kV, p D 10 Torr (Figures 4.21 and 4.22) the calculations with E D 4 kV, p D 20 Torr (results are not given here) have been performed.

254

Chapter 4 The physical mechanics of direct current glow discharge

ne, ni, 109cm−3 30

20

10

2 1

0

0

0.25

0.5

0.75

1

1.25 1.5 1.75 2 y, cm

Figure 4.19. Distribution of the concentration of ions (continuous curves) and electrons (dashed curves) along a symmetry axis of discharge at p D 5 Torr, E D 2 kV: (1) without heating; (2) with heating.

ne, ni, 109cm−3 30

20

10 2 1 0

0

0.25

0.5

0.75

1

1.25

1.5

1.75 2 y, cm

Figure 4.20. Distribution of the concentration of ions (continuous curves) and electrons (dashed curves) along a symmetry axis of discharge at p D 5 Torr, E D 4 kV: (1) without heating, (2) with heating.

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 255 y, cm

y, cm 2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

1.5

9.89E+01 8.15E+01 6.72E+01 5.53E+01 4.56E+01 3.76E+01 3.10E+01 2.55E+01 2.10E+01 1.73E+01 1.43E+01 1.18E+01 9.71E+00 8.00E+00 6.59E+00

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15 13

2.00E+01 1.87E+01 1.74E+01 1.60E+01 1.47E+01 1.34E+01 1.20E+01 1.07E+01 9.35E+00 8.02E+00 6.68E+00 5.35E+00 4.01E+00 2.67E+00 1.34E+00

11

1 7

9 7 5

(a)

(b)

5

0.5 3

3

1 1

15

0 0

0.25

13

0.5

0.75 x, cm

0.25

0.5

0.75 x, cm

Figure 4.21. Concentration of ions (a) and electrons (b) in glow discharge at p D 10 Torr, E D 4 kV; concentration is referred to 109 cm3.

To finish the analysis of influence of pressure it is necessary to also add data from Figures 4.24 and 4.25. With pressure growth, the gas heats more distinctly, and more significant modifications of glow discharge structure are observed. Taking into account gas heating at p D 10 Torr and E D 4 kV leads approximately to a twofold drop of the cathode current density and to a threefold increase in the anode current density (Figure 4.23). The electric field strength near to the cathode falls a little: from Ey D 9 000 V/cm to Ey D 7 000 V/cm. At the same time, the radius of the current spot on the cathode increases, and on the anode it decreases.

256

Chapter 4 The physical mechanics of direct current glow discharge y, cm

y, cm

y, cm

2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

1.5

1

5.03E+01 4.69E+01 4.36E+01 4.02E+01 3.69E+01 3.35E+01 3.02E+01 2.68E+01 2.35E+01 2.01E+01 1.68E+01 1.34E+01 1.01E+01 6.70E+00 3.35E+00

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15 13 11

9

4.00E+01 3.74E+01 3.47E+01 3.20E+01 2.94E+01 2.67E+01 2.40E+01 2.14E+01 1.87E+01 1.60E+01 1.33E+01 1.07E+01 8.01E+00 5.34E+00 2.67E+00

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

15

13 11

9 7

7 5

5

(a)

0.5

(b)

3

(c) 3

1

0

5.92E+02 5.72E+02 5.52E+02 5.32E+02 5.12E+02 4.92E+02 4.73E+02 4.53E+02 4.33E+02 4.13E+02 3.93E+02 3.73E+02 3.53E+02 3.33E+02 3.13E+02

2

2.5

2

x, cm

2.5

1

x, cm

2

2.5

x, cm

3

Figure 4.22. Concentration of ions (a) and electrons (b) in glow discharge at p D 10 Torr, E D 4 kV, D 0.5; concentration is referred to 109 cm3 ; (c) Temperature (K) in glow discharge at p D 10 Torr, E D 4 kV, D 0.5.

120

j, mA/cm2

100 80 2

60 1

40

1

20

2 0

0

0.5

1 x, cm

Figure 4.23. Radial distribution of current density on the anode (continuous curves) and on the cathode (dashed curves) at p D 10 Torr, E D 4 kV: (1) without heating; (2) with heating.

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 257

120 110 100 90 80 70 60 50 40 30 20 01 0

ne, ni, 109cm−3

2 1 0

0.25

0.5

0.75

1

1.25

1.5

1.75 0 y, cm

Figure 4.24. Distribution of concentration of ions (continuous curves) and electrons (dashed curves) along a symmetry axis of discharge at p D 10 Torr, E D 4 kV: (1) without heating; (2) with heating. T, K 600

600

600

600 0

0.25

0.5

0.75

1

1.25

1.5

1.75 2 y, cm

Figure 4.25. Axial distribution of temperature in glow discharge: p D 5 Torr, E D 2 kV (continuous curve), p D 5 Torr, E D 4 kV (dashed curve), p D 10 Torr, E D 4 kV (dash-dot curve).

The distributions of particle concentrations for these two cases are shown in Figure 4.24. It is well evident that near to the anode the concentration of charged particles is more than doubled, and the concentration of ions at the cathode is more than halved. Growth of thickness of the cathode layer at heating is remarkable too.

258

Chapter 4 The physical mechanics of direct current glow discharge

Let us note that by taking account of heating, the calculation of discharge parameters becomes more labor-consuming not only owing to the necessity of integrating the equations describing the structure of discharge till thermal relaxation times, but also owing to the necessity of using more accurate computing meshes (the cathode spot extends, and the anode spot is narrowed). Besides, with heating of gas the maximum admissible time step of numerical integration becomes smaller, ensuring stability of the calculations, decreases. The distributions of temperature along a symmetry axis in different conditions are shown in Figure 4.25. At pressure p D 10 Torr is gas heated up to temperatures T  600K while at p D 5 Torr the temperature does not exceed  400 K. Such a heating leads to significant growth of charged particle concentration in the positive column and, in particular, near to the anode, where ne  ni D 4.3  1010 cm3 (Figure 4.24). The further growth of pressure (up to 20 Torr) does not lead to an appreciable heating of glow discharge as its sizes are reduced, so the radial heat-conducting losses increase. The current density on the anode increases up to 200 mA/cm2 (from 80 mA/cm2 without heating), and the current density on the cathode falls to 27 mA/cm2 (from 70 mA/cm2 without heating). The positive column of glow discharge is narrowed to an even greater degree. The maximum temperature in discharge  T D 590 K, however, concentration of the charged particles in the positive column reaches the value of 1011 cm3 , and near to the cathode it is of 1.4  1011 cm3 . In addition, numerical investigations of the relationship of the glow discharge parameters and regularities of its heating from effectiveness ratio of the Joule heat transmission (in the range of 0.1–0.75) and from value of secondary ion-electron emission coefficient (in the range of 0.01–0.3) have shown objective modifications in its structure. Let us now consider the results of calculations of glow discharge for the least investigated pressure p D 1 Torr. Here, gas heating appears insignificant, however, in calculations the important modification of glow discharge structure in a radial direction is observed: the maximum of the charged particle concentrations and flux densities on the anode are displaced from a symmetry axis to the periphery. That is, the discharge gains a torus-like form. The distributions of ion and electron concentration at two values of electromotive force are shown in Figures 4.26 and 4.27, and distributions of current densities on electrodes are shown in Figure 4.28. Attention is attracted to the close fit of the normal current density law: with variation of the power supply electromotive force by more than five times, the current density on the cathode practically does not vary (Figure 4.28). One more important singularity of glow discharge predicted by the classical Engel–Steenbeck theory and confirmed by the calculations, is that practically in all calculation domains there are no areas of quasi-neutral plasma. This means that the use of quasi-neutral plasma models in aerophysical research at low pressure is unreasonable or, at least, demands additional substantiations.

Section 4.6 Two-dimensional structure of glow discharge regarding neutral gas heating 259

1 2 3 4 5 6 7 8 9 1.00E-02 1.68E-02 2.81E-02 4.70E-02 7.88E-02 1.32E-01 2.21E-01 3.70E-01 6.20E-01 y, cm

(a) 2 1.5 1 0.5 0

1 3

5

7 0

1

2

3

9

4

5

6

7

8

9

10

11 12 x, cm

1 2 3 4 5 6 7 8 9 5.16E-03 1.03E-02 1.55E-02 2.07E-02 2.58E-02 3.10E-02 3.61E-02 4.13E-02 4.65E-02 y, cm

(b) 2 1.5 1 0.5 0

9

3 1 0

1

2

3

7

4

5

5

6

7

8

9

10

11

12

x, cm

Figure 4.26. Concentration of ions (a) and electrons (b) in glow discharge at p D 1 Torr, E D 0.6 kV; concentration is referred to 109 cm3 . Axis of symmetry is located at x D 6 cm. 1 2 3 4 5 6 7 8 9 1.00E-02 1.68E-02 2.82E-02 4.75E-02 7.98E-02 1.34E-01 2.25E-01 3.79E-01 6.37E-01 y, cm (a)

2 1.5 1 0.5 0

3

1 5 7 0

1

9 2

3

4

5

6

7

8

9

10

11

12 x, cm

1 2 3 4 5 6 7 8 9 5.16E-03 1.03E-02 1.55E-02 2.07E-02 2.58E-02 3.10E-02 3.61E-02 4.13E-02 4.65E-02 y, cm (b)

2 1.5 1 0.5 0

9

7 7

3

5

1 0

1

2

3

4

5

6

7

8

9

10

11

12 x, cm

Figure 4.27. Concentration of ions (a) and electrons (b) in glow discharge at p D 1 Torr, E D 2 kV; concentration is referred to 109 cm3. Axis of symmetry is located at x D 6 cm.

260

Chapter 4 The physical mechanics of direct current glow discharge j, mA/cm2 0.15

0.1

0.05

0 0

1

2

3

4

5

6 x, cm

Figure 4.28. Distribution of current densities along the anode (continuous curves) and the cathode (dashed curves) at p D 1 Torr: 1 – E D 0.6 kV; 2 – E D 2 kV.

In conclusion, we shall note that the change of glow discharge radial structure discovered by calculations is one of many effects of essentially more common regularity of structure formation in strongly nonequilibrium physical systems, in particular, in one of the classical models of such systems as a glow discharge. The regular research of glow discharge structure performed at conditions of practical interest for aerophysical applications has allowed to determine a series of important singularities which are necessary for considering in the analysis of experimental data and for the prediction of use of such discharges for modification of a flow field in a neighborhood of constructional elements of various aircraft. Namely, in a range of pressure p < 5 Torr gas heating is negligible in a wide range of electric field strength modification, and at low pressure the glow discharge gets a torus-like form at normal current density conditions. With a raise in pressure of more than 5 Torr and in electric field strengths, significant heating of neutral gas is observed. The specified gas heating, in turn, strongly changes the electrodynamic structure of discharges. It should be stressed that the specified regularities are well known to experimental physicists who investigate discharges of the given type. It is important that the calculation models allow to predict the specified regularities. Nevertheless, it is obvious that the further development of the calculation models is required in view of heating of gas and regarding taking account of gas motion (in particular, supersonic) and account of nonequilibrium physical and chemical processes.

Chapter 5

Drift-diffusion model of glow discharge in an external magnetic field The theory and computing model of glow discharge in two-dimensional flat geometry in view of an external magnetic field are presented in the given chapter. It is supposed that the magnetic field is perpendicular to a plane where the glow discharge is considered. The problem considered is of significant interest for basic physics of gas discharge as it allows to investigate the behavior of glow discharges in external magnetic fields in view of positive columns and near-electrode layers of a spatial charge. It was shown in the previous chapter that the drift-diffusion model of glow discharge allows to predict its performances with a sufficient accuracy for practical needs in a wide range of pressures p  0.5–50 Torr and voltages on electric discharge gap V  0.3–10 kV. Considering that the glow discharge represents partially ionized gas with typical concentrations of the charged particles of  1011 cm3 against the background of  1017 cm3 of neutral particles and with small absolute values of currents across the discharge (tens milliamps), it is obvious that an external magnetic field B  0.01– 1 T can strongly influence on structure of the glow discharges. Further, we will use the assumption that the specified magnetic field will not be distorted by the discharge. The given assumption is laid as a basis of the glow discharge numerical model in an external magnetic field.

5.1 Derivation of the equations for calculation model The flat two-dimensional discharge in molecular nitrogen between two infinite flat electrodes is considered. The configuration of the discharge and the external magnetic field are shown in Figure 5.1. It is supposed that the column of glow discharge is not limited in the direction of the z axis. Equations of the drift-diffusion model are formulated for concentrations of electrons ne and positive ions ni , and for electric field potential ', which defines a vector of an electric field strength E D  grad ' @ne @ e,x @ e,y C C D ˛.E/j e j  ˇni ne , @t @x @y @ i ,x @ i ,r @ni C C D ˛.E/j e j  ˇni ne , @t @x @y @2 ' @2 ' C D 4e.ne  ni /, @x 2 @y 2

(5.1) (5.2) (5.3)

262

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field Anode y

E

Positive column B

Cathode

x R0

Figure 5.1. The scheme of glow discharge with external magnetic field.

where  e D De grad ne  ne e E,  i D Di grad ni C ni i E; Q D .jE/,

j D e. i   e /,

˛.E/ and ˇ are the ionization and recombination coefficients;  e ,  i are the densities of fluxes of electrons and positive ions; e , i are the mobilities of electrons and ions; De , Di are the diffusion coefficients of electrons and ions. To introduce a magnetic field into the computing model, the following equations expressing a momentum conservation law for electronic and ionic liquids are used [2, 17]:  1 @ue e C e .ue  r/ue D rpe   e  e ne E C Œue B @t c  me en ne .ue  un /  me ei ne .ue  ui /, (5.4) i

 1 @ui C i .ui  r/ui D rpi   i C e ni E C Œui B @t c  mi i e ni .ui  ue /  mi i n ni .ui  un /,

(5.5)

where ue , ui are the velocities of electronic and ionic liquids; e , i are the densities of electronic and ionic liquids; e D me ne , i D mi ni ; me , mi are the masses of electrons and ions; un is the average velocity of neutral particles; pe , pi are the pressures of electrons and ions; e , i are the viscous stress tensors of electronic and ionic liquids; en , ei , i n are the collision frequencies of electron impacts with neutral particles, with ions, and also ions with neutral particles; B is the magnetic field induction vector; c is the speed of light. If it is taken into account that me  mi (5.6) and using the condition of smallness of inertial motion of electrons e .ue  r/ue  i .ui  r/ui ,

(5.7)

Section 5.1 Derivation of the equations for calculation model

263

then equation (5.4) can be essentially simplified:  1 rpe  e ne E C Œue B  me en ne .ue  un /  me ei ne .ue  ui / D 0. (5.8) c Even for hypersonic gas flow velocities the inequality is true ue un , ui .

(5.9)

Therefore. the equation (5.8) supposes the further simplification: kTe rne C e ne E C

e ne Œue B C .me e /ne ue D 0, c

(5.10)

where it is supposed that pe D ne kTe . Having divided (5.10) on the product in parentheses, we will receive ne ue D De rne  e ne E 

e ne Œue B, c

(5.11)

where e D e=me e is the mobility of electrons; De D .kTe =e/e is the coefficient of electronic diffusion; e D en C ei . In the considered statement the vector of electronic liquid velocity has two components ue D fue,x ; ue,y g, while the vector of magnetic field induction has only one component Bz , therefore, @ne 1    n E D e e e x 1 C be2 @x  @ne be   D , n E  e e e y 1 C be2 @y

e,x D ne ue,x D

@ne 1    D n E e e e y 1 C be2 @y  @ne be   D , C n E e e e x 1 C be2 @x

(5.12)

e,y D ne ue,y D

(5.13)

where

e Bz , (5.14) c Ex , Ey are the components of electric field. It is significant that cross derivatives in the considered statement are cancelled. By analogy, assuming that in the considered range of velocities it is possible to omit the left part of the equation (5.5), we receive be D

 1 rpi C e ni E C Œui B  mi i e ni .ui  ue /  mi i nni .ui  un / D 0. (5.15) c

264

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

Considering that mi i e ni .ui  ue / D me ei ne .ue  ui /,

(5.16)

and also that in the considered case un D 0,

(5.17)

me e  mi i n ,

(5.18)

and it is possible to receive ni ui D Di rni C i ni E C

i ni Œui B, c

(5.19)

where i D e=mi i n is the mobility of ions; Di D .kTi =e/i is the diffusivity of ions. Furthermore, as is the case for electronic liquid, the average velocity of ions has only two nonzero components ui D fui ,x ; ui ,y g, and the induction of the external magnetic field has only one component Bz , therefore, @ni 1  C  D n E i i i x @x 1 C bi2 @ni bi  C  D n E C i i i y , @y 1 C bi2

i ,x D ni ui ,x D

@ni 1  C i ni Ey Di 2 @y 1 C bi  @ni bi   D , n E C i i i x @x 1 C bi2

(5.20)

i ,y D ni ui ,y D

(5.21)

where bi D .i =c/Bz . Let us introduce an effective electric field for electrons and ions with the following components: Ee,x D

be Ey  Ex , 1 C be2

Ee,y D 

Ei ,x D

Ex C bi Ey , 1 C bi2

Ei ,y D

be Ex C Ey , 1 C be2

Ey  bi Ex . 1 C bi2

(5.22)

(5.23)

Substituting values of densities of electron and ion fluxes with effective electric fields in the continuity equations of electronic and ionic liquids (5.1) and (5.2), one

265

Section 5.1 Derivation of the equations for calculation model

can receive @  De @ne @ne C e ne Ee,x  @t @x 1 C be2 @x  @ De @ne e ne Ee,y  D ˛je j  ˇne ni , C @y 1 C be2 @y @  @ni Di @ni C i ni Ei ,x  @t @x 1 C bi2 @x Di @ni @ i ni Ei ,y  D ˛je j  ˇne ni , C @y 1 C bi2 @y

(5.24)

(5.25)

q 2 C 2 . where je j D e,x e,y Introducing the factors be and bi , which take into account the magnetic field in model (5.24) and (5.25) be D

!e e Bz D , c e

bi D

!i i Bz D , c i n

(5.26)

and are well known in plasma physics. These are the Hall parameters of electronic and ionic liquids. Factors !e D eBz =me c and !i D eBz =mi c in (5.26) are the Larmor frequencies of electron and ion spin in a magnetic field. Boundary conditions for ion and electron concentrations, and also for electric potential are formulated in the following form: yD0:

@ni D 0, @y

y D H : ni D 0,

e D i , @ne D 0, @y

' D 0;

(5.27)

V ; E

(5.28)

'D

xD0:

@ni @' @ne D D D 0; @x @x @x

(5.29)

xDL:

@ni @' @ne D D D 0. @x @x @x

(5.30)

Initial conditions for the integration of equations of the drift-diffusion model of glow discharge were discussed in Chapter 4. It is important to keep in mind that raw initial conditions can result not only in essential deceleration of the integrating procedure, but also to its divergence. Therefore, even at the analytical formulation of initial conditions it is desirable to use the correct configuration of the discharge and reasonable values of the concentration of the charged particles. It is obvious that the integrating procedure

266

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

could be accelerated essentially, if the preliminarily received solutions were used for the calculation of new variants. In the considered problem, heating up of neutral gas has not been taken into account, therefore, the mobilities of electrons and ions, and also the diffusivities are supposed constant: e .p/ D

4.2  105 cm2 2280 cm2 , , i .p/ D , , p Vs p Vs

De D e .p/Te ,

Di D i .p/T ,

where p is the pressure in glow discharge. The coefficient of an ion-electronic recombination and the temperature of electrons have been taken as constant: ˇ D 2  107 cm3 /s, Te D 11 610 K. The ionization coefficient (the first Townsend coefficient) is set in the form of h ˛.E/ D pA exp 

B i , cm1 , .jEj=p/

where for molecular nitrogen A D 12 cm 1Torr , B D 342 cm VTorr . The equations are solved together with the equation for external electric circuit which for a direct current looks like E D V C IR0 , where V is the voltage drop on electrodes; I is the total current through the discharge gap; E is the electromotive force of the power supply; R0 is the resistance of an external circuit.

5.2 Numerical simulation results Calculations were performed for the following input data: 

gas N2 ; pressure p D 2–50 Torr; E D 2–9 kV; R0 D 300 kOhm;



D 0.01–0.33;

Hc D 1–2 cm; Rc D 2–4 cm; B D 0–0.1 T.

To show the influence of an external magnetic field on the discharge structure we will firstly consider the glow discharge at p D 5 Torr, E D 2 kV, D 0.1 in the absence of a magnetic field. Numerical simulation results for the glow discharge configuration shown in Figure 5.1 are presented in Figures 5.2 and 5.3. Here concentrations of charged particles are referred to value of N0 D 109 cm3 . For the calculation of total current across the gas-discharge gap it was supposed that the length of discharge along the z-direction is equal to 1 cm. The fields of electron and

267

Section 5.2 Numerical simulation results y, cm 2 1.5 1 0.5 0

0

1

2

3

4 x, cm

Figure 5.2. Examples of computational mesh (141  61).

ion concentrations are shown in Figures 5.3 (a) and 5.3 (b). The distributions of current densities along the cathode and the anode are shown in Figure 5.3 (d), and the distributions of electron and ion concentrations along a y axis are shown in Figure 5.3 (e). Analyzing the data presented it is possible to define three basic regions in the glow discharge: 

The area of positive volume charge near to the cathode, or the cathode layer (Figures 5.3 (a), (b), (c));



The area of negative volume charge near to the anode, or the anode layer (Figure 5.3 (e));



The area of quasi-neutral plasma, or the positive column of glow discharge (Figures 5.3 (a), (b), (e)).

The specified areas of the normal mode glow discharge are clearly visible in Figure 5.3 (c), where the distribution of electric potential is shown. In the cathode layer (y  0.1 cm), sharp growth of the electric potential is observed; in the positive column (0.1  y  0.9 cm), the distribution of the potential is almost linear; in the anode layer, a small increase in the potential is observed again. As it was mentioned above, the important advantage of the drift-diffusion model, which has great importance for various applications of computer models in gasdischarge physics, is the possibility of electric conductivity prediction. Results of calculations of electric conductivity in the absence of a magnetic field e D ee ne , (Ohm  cm)1 are given in Figure 5.3 (f). The results of numerical modeling of glow discharges without a magnetic field, which are represented in Figure 5.3, will be used below for the analysis of the influence of various factors, including parameters of numerical modeling. Similar research performed in Chapter 4 concerned axially symmetric glow discharge. The common view of the influence of various input data on the structure of the normal mode gas discharge can be obtained from Table 5.1, where numerical results

268

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm

2

0.1

(a)

1.5

(b)

1.5

1

1

2.7 2.4 2.0 1.7 1.40.7 1.1 0.4

0.5 0

y, cm

2

0

1

2

0.5 3

y, cm 2 0.186

1

4 x, cm

0

0

1

2

3

4 x, cm

(d) Carrent Density on Anode Carrent Density on Cathode

10

0.213

8

0.080 0.106 0.0270.053

0

1

12

0.133 0.160

0.5

2.2

10.0

0

0.5 0.2 1.3

j, mA/cm2

(c)

0.266 0.239

1.5

2.2 0.1 0.8 0.3

6 2

3

4 x, cm

4

ne, ni, 109cm3

2

(e)

20

0

0 0.5 1 1.5 2 2.5 3 3.5 4 x, cm y, cm

15 2 10

(f)

10 9 8 7 6 5 4 3 2 1

1.5 E−07

1 5 0

0

0.5

1

1.5

2 y, cm

0

1.00

0.5 0

0.5

1

2.51E−05 6.30E−06 1.26E−05

1.5

2

2.5

3

SIGMA 5.00E−05 2.51E−05 1.26E−05 6.30E−06 3.16E−06 1.58E−06 7.94E−07 3.98E−07 1.99E−07 1.00E−07

3.5 4 x, cm

Figure 5.3. Results of numerical modeling of discharge without a magnetic field at p D 5 Torr, E D 2 kV, D 0.1, H D 2 cm, I D 4.85 mA, jc,max D 3.58 mA/cm2, ja,max D 6.03 mA/cm2, V D 533 V: (a) electron concentration; (b) ion concentration; (c) electric potential (numbers at curves correspond to '=E/; (d) current density on the anode (solid line) and on the cathode (dashed line); (e) distribution of ion (continuous line) and electron (dashed line) concentrations along axis of glow discharge; (f) electrical conduction (Ohm1  cm1/. Levels of concentration are referred to value of N0 D 109 cm3. The parameters of one-dimensional normal glow discharge calculated under the Engel–Steenbeck theory are: dn D 0.15cm, Vn D 205 V, jn D 1.37 mA/cm2.

for various pressures, electromotive forces of a power supply, and also coefficients of a secondary electronic emission are presented. In this table, the values of the normal current density on the cathode jn , predicted by the Engel–Steenbeck theory are also given.

269

Section 5.2 Numerical simulation results

As mentioned above, this theory also allows to calculate a voltage drop on the cathode Vn and thickness of the cathode layer .pd /n for the glow discharge in the mode of normal current density: Vn D

2.72B  1 ln C1 , A

.pd /n D

jn D p 2

.i p/Vn2 . 4e.pd/3n

3.78  1 ln C1 , A

Comparing the results of numerical modeling with the data of the Engel–Steenbeck theory, it is necessary to take into account that the basis of the given theory is the one-dimensional model of a cathode layer and a linear increase of electric potential in the cathode layer. Nevertheless, this has been confirmed by numerous experimental data that allows to consider the correspondence of numerical simulation results to the Engel–Steenbeck theory as evidence of the computing model’s adequacy.

Table 5.1. Numerical simulation results for flat glow discharge without an external magnetic field. p, Torr

E, kV



I , mA

jc,max , mA/cm2

ja,max, mA/cm2

jn , mA/cm2

5 10 5 5 5

2 2 2 3 4

0.1 0.1 0.3 0.1 0.1

4.85 3.90 4.85 8.19 11.5

3.58 7.03 4.97 4.34 4.79

6.03 8.89 6.70 6.22 5.01

1.37 5.49 2.65 1.37 1.37

In Table 5.1 are presented: I is the total current; jc,max, ja,max are the maximal current densities on the cathode and the anode. Numerical simulation results for other input data are shown on Figures 5.4–5.8. The configuration of glow discharge at raised pressure p D 10 Torr is shown in Figure 5.4. It is evident that in comparison with the previous calculation variant (p D 5 Torr), the cross sizes of current columns have been noticeably reduced (Figure 5.4 (a)); the thickness of the cathode layer (Figure 5.4 (b)) has decreased. The current across the discharge gap has decreased from I D 4.85 mA (p D 5 Torr) to I D 3.9 mA (p D 10 Torr). However, at the same time, the current density on the cathode and the anode has increased. One-dimensional Engel–Steenbeck theory also specifies the same. An important point is that with the growth of pressure the increasing difference from the Engel–Steenbeck model should be observed owing to the amplification of influence of the two-dimensional effects.

270

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm

2

1.4 2.1 3.2

1.5 1

(a)

0.9

1

5.0

0.2

0

1

0.1

2

3

y, cm

0.5 0

0

(c)

0.414

1

0

1

3

4 x, cm

12 (d) Carrent Density on Anode Carrent Density on Cathode

10 8 6 2

3

ne, ni, 109cm−3

4 x, cm

4 2

(e)

40

0

0 0.5 1 1.5 2 2.5 3 3.5 4 x, cm y, cm

35 30 2

25

(f)

1.5

20 15

0.5

5 0

0.5

1

1.5

2 y, cm

0

10 9 8 7 6 5 4 3 2 1

1.00E− 07 1.26E−05

1

10 0

2

j, mA/cm2

0.372 0.331 0.290 0.248 0.207 0.165 0.124 0.083 0.041

0 45

4 x, cm

2.51E−05

1

0.5 0.3 0.2 0.1 0.8 3.6 2.2 6.0 10.0

0.5

2 1.5

(b)

1.5

0.5 0

y, cm

2

SIGMA 5.00E−05 2.51E−05 1.26E−05 6.30E−06 3.16E−06 1.58E−06 7.94E−07 3.98E−07 1.99E−07 1.00E−07

6.30E−06

0

0.5

1

1.5

2

2.5

3

3.5 4 x, cm

Figure 5.4. Results of numerical modeling of glow discharge without a magnetic field at p D 10 Torr, E D 2 kV, D 0.1, H D 2 cm, I D 3.9 mA, jc,max D 7.03 mA/cm2, ja,max D 8.89 mA/cm2, V D 835 V: (a) concentration of electrons; (b) concentration of ions; (c) electric potential (numbers at curves correspond '=E/; (d) current density on the anode (solid line) and on the cathode (dashed line); (e) distribution of electron (dashed line) and ion (solid line) concentrations along axis of glow discharge; (f) electrical conduction in gas-discharge gap, (Ohm1  cm1/. Levels of concentration are referred to value of N0 D 109 cm3. Parameters of one-dimensional normal glow discharge (under the Engel– Steenbeck theory): dn D 0.0755 cm, Vn D 205 V, jn D 5.49 mA/cm2.

Modifications in the structure of the glow discharge as the secondary electronic emission coefficient increases can be seen in Figure 5.5 (p D 5 Torr, D 0.3/ comparing these data with similar data in Figure 5.3 (p D 5 Torr, D 0.1/.

271

Section 5.2 Numerical simulation results y, cm (a)

2 1.5

1.5

1 2.7

0.5 0

1

3.0

1

2

3

y, cm 12 (c) 10

0.3 0.1 0.2 0.5 0.8 1.3 2.2 3.6

0.5

2.4 1.4 1.7 2.0

0

y, cm (b)

2

0.1 0.7 0.4

4 x, cm

0

0

20

1

10.0

2

3

4 x, cm

1.5

2 y, cm

ne, ni, 109cm−3 (d)

Carrent Density on Anode Carrent Density on Cathode

15

8 6

10

4 5

2 0

0 0.5 1 1.5 2 2.5 3 3.5 4 x, cm

0

0

0.5

1

Figure 5.5. Results of numerical modeling of glow discharge without a magnetic field at p D 5 Torr, E D 2 kV D 0.3, H D 2 cm, I D 4.85 mA, jc,max D 4.97 mA/cm2, ja,max D 6.7 mA/cm2, V D 454 V: (a) electron concentration; (b) ion concentration; (c) current density on the anode (solid line) and on the cathode (dashed line); (d) distribution of electron (dashed line) and ion (solid line) concentrations along an axis of glow discharge. Parameters of onedimensional normal glow discharge (under the Engel–Steenbeck theory): dn D 0.0924 cm, Vn D 125 V, jn D 2.65 mA/cm2.

With increasing the current in the discharge gap will practically not vary, but the current density on the cathode and on the anode has increased. It is obvious that at the same time cross sizes of the current channel have decreased. One more singularity of modifications in the structure of the discharge is some diminution of thickness of the cathode layer that also corresponds to the approximated one-dimensional theory. The results of calculations of flat glow discharge at increased electromotive force E D 3 kV are shown in Figure 5.6, and at E D 4 kV in Figure 5.7. Based on the given calculated data we can specify the main regularities of glow discharge structure formation without a magnetic field: (1) The positive charge area (the cathode layer) has a small expansion along axis y. At p D 5 Torr, E D 2 kV D 0.1, H D 2 cm (Figure 5.3 (b)) the height of the cathode layer achieves approximately dc Š 0.15 cm. It coincides well with the thickness of a cathode layer dn Š 0.151 cm predicted by the Engel–Steenbeck theory.

272

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm

2

(a)

Level 10 9 8 7 6 5 4 3 2 1

1

1.5 2 10

1

6 9

3 5

0.5

7

8

Ne 3.00 2.68 2.36 2.03 1.71 1.39 1.07 0.74 0.42 0.10

4

0

0

1

2

3

4

5

6 x, cm

y, cm 2 (b)

Level 10 9 8 7 6 5 4 3 2 1

1.5 5

7

1 1

0.5 0

4

2

6 7

10

0

1

2

j, mA/cm2 6 (c)

4

4

3

Carrent Density on Anode Carrent Density on Cathode

Ni 17.00 9.61 5.43 3.07 1.73 0.98 0.55 0.31 0.18 0.10

25

6 x, cm

5

ne, ni, 109cm−3 (d)

20 15 10

2

5 0

0

1

2

3

4

5

6 x, cm

0

0

0.5

1

1.5

2 y, cm

Figure 5.6. Results of numerical modeling of discharge without a magnetic field at p D 5 Torr, E D 3 kV, D 0.1, H D 2 cm, I D 8.19 mA, jc,max D 4.34 mA/cm2, ja,max D 6.22 mA/cm2, V D 555 V: (a) concentration of electrons; (b) concentration of ions; (c) current density on the anode (solid line) and on the cathode (dashed line); (d) distribution of concentrations of electrons (dashed line) and ions (solid line) along axis of glow discharge. Levels of concentration are referred to value N0 D 109 cm3. Parameters of one-dimensional normal glow discharge (under the Engel–Steenbeck theory) are: dn D 0.15 cm, Vn D 205 V, jn D 1.37 mA/cm2.

273

Section 5.2 Numerical simulation results y, cm 2

(a)

4 1

1.5

Level 10 9 8 7 6 5 4 3 2 1

3 2 5

1

10 6 7

0.5 0

0

1

9 8

2

3

4

5

Ne 3.00 2.68 2.36 2.03 1.71 1.39 1.07 0.74 0.42 0.10

6 x, cm

y, cm 2 (b) 1.5

Level 10 9 8 7 6 5 4 3 2 1

7 5

1 1 2 3 4

0.5 0

6

Ni 17.00 9.61 5.43 3.07 1.73 0.98 0.55 0.31 0.18 0.10

10

0

1

2

4

3

j, mA/cm2 6 (c)

5

6 x, cm

Carrent Density on Anode Carrent Density on Cathode

4

2

0

0

1

2

3

4

5

6 x, cm

Figure 5.7. Results of numerical modeling of discharge without a magnetic field at p D 5 Torr, E D 4 kV D 0.1, H D 2 cm, I D 11.5 mA, jc,max D 4.79 mA/cm2, ja,max D 5.01 mA/cm2, V D 561 V: (a) concentration of electrons; (b) concentration of ions; (c) current density on the anode (solid line) and on the cathode (dashed line). Levels of concentration are referred to value N0 D 109 cm3 . Parameters of one-dimensional normal glow discharge (under the Engel–Steenbeck theory): dn D 0.15 cm, Vn D 205 V, jn D 1.37 mA/cm2.

274

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

(2) At the parameters specified above, the glow discharge exists in conditions of subnormal glow discharge. In the condition of normal current density with a modification of a discharge current across the gas-discharge gap (for example, other parameters being equal, with increasing the electromotive force E), the current density on the cathode remains practically constant, and cross sizes of the current column vary. In the case considered here, cross sizes of discharge are comparable to the thickness of the cathode layer, therefore, the law of normal current density is not satisfied exactly because charge losses in a cross direction are too great. Nevertheless, this discharge in subnormal mode is close enough to the normal discharge. Increasing the discharge current in a range of I D 4.85–11.5 mA (at growth of electromotive force from E D 2 kV up to 4 kV) the greatest current density on the cathode varies only in the range jc D 3.58–4.79mA/cm2 . The Engel–Steenbeck theory for these conditions predicts jn D 1.37 mA/cm2 . (3) Growth of electromotive force E of an external electric circuit leads to an increase of cross sizes of the glow discharge, both the positive column and the cathode layer. This tendency is observed for all investigated pressures p D 2.5, 10, 50 Torr and for various factors of a secondary electronic emission ( D 0.01–0.3/. (4) The diminution of pressure in the gas-discharge gap leads to an appreciable increase not only in cross sizes of discharge, but also in the height of the cathode layer. Numerical results of the specified research are confirmed in the Engel–Steenbeck theory. (5) Numerical calculations confirm that the secondary electronic emission coefficient plays an important role in glow discharge structure formation and is one of the defining parameters of numerical modeling. From the physical point of view, this parameter, defining the effectiveness of “breaking-away” electrons from the cathode by an ion stream falling on the cathode, allows to model various materials which the cathode is made of. From the calculated data presented by Table 5.1, the strong influence of is obvious. In particular, from the comparison of numerical simulation results at p D 5 Torr and E D 2 kV there follows that although at growth value from 0.1 up to 0.3 the total current practically does not vary, the significant change of a current density on the cathode from 3.58 mA/cm2 up to 4.97 mA/cm2 is observed. In these conditions, the Engel–Steenbeck theory predicts an increase in the current density in the normal glow discharge cathode from 1.37 mA/cm2 to 2.65 mA/cm2 . It is significant that in all listed cases there was a steady-state solution for the structure of glow discharge. However, insignificant oscillations of all parameters of the glow discharge were observed. Most likely, the principal causes of the specified oscillations are singularities of a numerical solution of the problem.

275

Section 5.2 Numerical simulation results

The influence of an external magnetic field on the structure of glow discharge has been studied for the same initial data as without a magnetic field. In calculations, the induction of the magnetic field varied in the following range: Bz D 0.01–0.1 T. The configuration of glow discharge at p D 5 Torr, E D 2 kV, D 0.1 and magnetic induction of an external field Bz D 0.01 T is shown in Figure 5.8. Let us note two singularities of the data presented: (1) The calculations were performed for a negative orientation of the magnetic field induction vector Bz D 0.01 T (see the field configuration in Figure 5.1); (2) Two instant configurations of the glow discharge are shown in Figure 5.8 (an instant in time t D 50 and 150 s after energizing of a magnetic field), moving in a positive direction of axis x. The velocity of the specified motion will be analyzed below. Let us concentrate on the distribution of the charged particle concentration fields in a cross magnetic field at various initial data. For investigating the influence of a magnetic field on the electrodynamic structure of glow discharge, additional calculations at Bz D 0.05 T have been performed. Results of the specified calculations are shown in Figure 5.9. It is evident that with increasing the magnetic field induction, the glow discharge’s configuration deformation increases and a velocity of the discharge transverse motion in the magnetic field increases considerably. If the discharge’s current column displacement velocity was measured using the electron isolines ne D 1.1  109 cm3 , then at Bz D 0.01 T this velocity is equal to ux  5.5  103 cm/s, at Bz D 0.05 T it is ux  3.25  104 cm/s. 2

y, cm

y, cm 0.1

(a)

(c)

0.4

1.5 2.7

1

2.4

0.5 2

0.1 0.2 0.30.8 0.5 1.3 2.2

2.0 1.4 1.1 0.7

1.7

0.1

(b)

0.4

1.5

10.0

(d)

2.4

1

0.1 0.3 2.2 0.2 0.5 0.8 1.3 10.0 2.2

2.0

0.5

1.7 1.4 1.1 0.7

0

2.2

0

1

2

3

0 x, cm

1

2

2.2 6.0

3

4 x, cm

Figure 5.8. Concentration of electrons (a), (b) and ions (c), (d) (referred to 109 cm3/ in glow discharge with a magnetic field at p D 5 Torr, E D 2 kV D 0.1, H D 2 cm, Bz D 0.01 T at sequential time instants: (a), (c) t D 50 s; (b), (d) t D 150 s.

276

2

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm

y, cm 2.0 1.7 1.4 0.7 0.1 0.4 2.4 1.1 2.7

(a)

1.5

(c)

2.2

1

0.1 0.2 0.3

3.0

0.5 2 (b)

2.0 2.4 2.7

1.5

3.0

1

1.7 1.4

0.5 0.8

3.6

10.0

(d)

1.3 2.2

1.1 0.7 0.4 0.1

0.5 0

1.3

0.1

0.2

0.3 0.5 0.8 3.6 10.0

0

1

2

3

0 x, cm

1

2

3

4 x, cm

Figure 5.9. Concentration of electrons (a), (b) and ions (c), (d) (referred to 109 cm3 / in glow discharge with a magnetic field at p D 5 Torr, E D 2 kV D 0.1, H D 2 cm, Bz D 0.05 T at sequential time instants: (a), (c) t D 10 s; (b), (d) t D 30 s.

It was observed that the change of the magnetic field induction vector direction leads to displacement of glow discharge in an opposite direction. Results of the calculation of glow discharge are shown in Figure 5.10 at p D 5 Torr, E D 2 kV, D 0.1 in cross magnetic field Bz D C0.05 T. The dynamics of glow discharge for increased pressure (p D 10 Torr) in cross magnetic field Bz D 0.1 T is shown in Figure 5.11. The average velocity of discharge displacement in this case is estimated by the value of ux  4.5104 cm/s. The discharge is displaced, virtually keeping the configuration. The numerical simulation results on a moving glow discharge in cross magnetic field Bz D 0.05 T, which are shown in Figures 5.12 and 5.13, give a representation of the influence of electromotive force on glow discharge structure. For this purpose, increasing the electromotive force from E D 3 kV (Figure 5.12) up to E D 4 kV (Figure 5.13) was studied. It is expedient to compare the presented data on the electrodynamic structure of glow discharge with the numerical simulation results shown in Figure 5.9 corresponding to electromotive force E D 2 kV. The specified increase in electromotive force is actually equivalent to growth of the total current through the discharge, but it does not mean proportional growth of current densities on electrodes and an appreciable change of concentration of the charged particles in the near-electrode layers and in the positive column. The comparison of the instant configurations of gas discharge at various electromotive forces allows to draw a conclusion on the appreciable deceleration of displacement of discharge at growth of E (at growth of total current across discharge).

277

Section 5.2 Numerical simulation results

2

y, cm

y, cm

0.1 0.4 1.1 2.0 2.4 0.7 2.7 1.7

1.5

1.3 0.1 0.8

3.0

0.5

0.3 0.2 0.5

3.6

10.0

2

1.4 1.7 2.0 4.4 1.1 2.7 3.0 0.7

1.5 1

(b)

(d)

2.2 1.3 0.3

0.5

0.4 0.1

0.5

(c)

2.2

1.4

1

0

(a)

3.6

0.2

0.1

0.8

10.0

0

1

2

3

0 x, cm

1

2

4 x, cm

3

Figure 5.10. Concentration of electrons (a), (b) and ions (c), (d) (referred to 109 cm3 / in glow discharge with a magnetic field at p D 5 Torr, E D 2 kV D 0.1, H D 2 cm, Bz D C0.05 T at sequential time instants: (a) t D 10 s; (b) t D 30 s.

2

y, cm

y, cm 2.8 1.7 3.4 0.6 3.9 1.2

(a)

1.5 1

(c)

0.1

2.2 3.6

0.3

4.5 5.0

0.5

0.2

0.5

0.8

1.0

2

1.2 1.7 6.0 2.3 2.8 3.4 0.1 3.9 4.5 5.0

(b) 1.5 1 0.5 0

6.0

1.3

(d)

6.0 2.2

0.3 0.2 0.1

0.5

0.8

3.6 1.3

6.0 6.0

0

1

2

3

0 x, cm

1

2

3

4 x, cm

Figure 5.11. Concentration of electrons (a), (b) and ions (c), (d) (referred to 109 cm3 / in glow discharge with a magnetic field at p D 10 Torr, E D 2 kV D 0.1, H D 2 cm, Bz D 0.1 T at sequential time instants: (a) t D 10 s; (b) t D 30 s.

The phenomenon of glow discharge motion in a cross magnetic field, which has been investigated in the present section, is well-known in basic electrodynamics and in plasma physics as the Hall effect of current origin at motion of charged particles in crossed electric and magnetic fields [2, 17].

278

2 1.5 1 0.5 2 1.5 1 0.5 0

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm

y, cm (a)

Level 10 9 8 7 6 5 4 3 2 1

Ne 3.00 2.68 2.36 2.03 1.71 1.39 1.07 0.74 0.42 0.10

1 24 35 6 8 7

(c) Level Ni 6 10 17.00 9 9.61 8 5.43 1 2 3 4 7 3.07 5 6 1.73 5 0.98 4 0.55 (d) 3 0.31 2 0.18 1 1 0.10

9

10 1 2 3 4 5 6

(b)

7

8 10 9

7 10 5 4 3

6

7

2

10

0

1

2

3

4

5

0 x, cm

1

2

3

4

5

6 x, cm

Figure 5.12. Concentration of electrons (a), (b) and ions (c), (d) (referred to 109 cm3/ in glow discharge with a magnetic field at p D 5 Torr, E D 3 kV D 0.1, H D 2cm, Bz D 0.05 T in sequential time instants: (a) t D 17 s; (b) t D 51 s.

2 1.5 1 0.5 2 1.5 1 0.5 0

y, cm

y, cm

3 (a) 1 24 7 5 8 9 6 Level Ne 10 3.00 9 2.68 10 8 2.36 7 2.03 6 1.71 5 1.39 123 4 7 8 5 6 4 1.07 (b) 3 0.74 9 2 0.42 1 0.10 10

(c) Level Ni 10 17.00 9 9.61 8 5.43 7 3.07 6 1.73 5 0.98 4 0.55 3 0.31 2 0.18 1 0.10

6 2

1

35

7

4 10

(d)

6

5 1

2

4 3

7 10

0

1

2

3

4

5

0 x, cm

1

2

3

4

5

6 x, cm

Figure 5.13. Concentration of electrons (a), (b) and ions (c), (d) (referred to 109 cm3/ in glow discharge with a magnetic field at p D 5 Torr, E D 4 kV, D 0.1, H D 2 cm, Bz D 0.05 T in sequential time instants: (a) t D 17 s; (b) t D 51 s.

The magnetic field induction, used as input data for the calculated variants, corresponds to the Hall parameter for electrons of the order of unity. It should be stressed that in the general case of three-dimensional geometry and at an arbitrary orientation of a magnetic field induction, a computing model with magnetic field will be essentially more complicated and contain cross derivatives, which have been omitted in the considered case owing to a special choice of geometry and configuration of a magnetic field.

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 279

5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas As mentioned above, the unique property of a glow discharge is the nonequilibrium weakly ionized gas, in which electrons have the temperature Te  10 000–20 000K, while the ions and neutral particles have a low temperature close to that of the room. The principal reason for this is the great mass distinction between electrons me and heavy particles Mn , Mi (neutrals and ions). The basic physical mechanism of the existence of glow discharge plasma is the electronic emission from a surface of the cathode under impacts of incident ions, and ionization of atoms and molecules at their collisions with electrons accelerated in an external electric field. The quantitative characteristic of this process, as is well known, is the frequency of ionization by electron impact. In the drift-diffusion theory of glow discharge, the ionization coefficient ˛ is widely used. Its analytical approximation has been offered by Townsend and named as the first Townsend coefficient h ˛ D Ap exp 

B i . .E=p/

(5.31)

This coefficient is connected with the frequency of ionization i by the following formula: i i , (5.32) ˛D D vdr,e e E where vdr,e is the electron drift velocity. From the physics of the ionization process and from the relation (5.32) it is obvious that the frequency of ionization i depends on the concentration of neutral particles nn which at the given pressure p is immediately connected with the temperature of neutral particles. This means that despite a huge difference between electron and neutral particle temperatures, the temperature of the neutral particles substantially defines the rate of the ionization process. It is well known from laser physics that in gas-discharge lasers the gas inside laser can be heated up to temperatures  600 K. At higher temperatures the ionization process is intensified so much that the glow discharge loses its stability. Instead of homogeneous glow discharge in such a heated weakly ionized gas the contracted discharge channels are observed. So, the study of glow discharge at various levels of heating of neutral gas is of great scientific and applied interest. In the previous section, numerical modeling of axially symmetric glow discharge in view of heating gas was performed. It is obvious that the presence of an external magnetic field complicates a common scheme of description of glow discharges in view of heating of neutral gas. In the given section, the computing model of such a discharge in two-dimensional flat geometry is considered.

280

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

5.3.1 Problem formulation The schematic diagram of the gas-discharge gap with an external magnetic field, which will be investigated here, is shown in Figure 5.1. If one of the cathodic sections is considered, then the scheme of the solved problem can be presented in the form of Figure 5.14. The glow discharge with heating will be considered between infinite and flat electrodes. y

Anode

y=H

y=0 x=0

x = xc x=L Cathode section

x

Figure 5.14. The calculation domain.

The electrodynamic structure of the glow discharge is described with the use of the drift-diffusion model for electron concentration ne and positive ions ni together with the Poisson equation for electric potential ' and electric field strength vector E D  grad '. In addition, the Fourier–Kirchhoff equation is formulated for the temperature of neutral particles: @  De @ne @ne C e ne Ee,x  C @t @x 1 C be2 @x @  De @ne C e ne Ee,y  D ˛j e j  ˇne ni , @y 1 C be2 @y Di @ni @ni @  C i ni Ei ,x  C @t @x 1 C bi2 @x Di @nC @  i ni Ei ,y  D ˛j e j  ˇne ni , C @y 1 C bi2 @y @2 ' @2 ' C D 4e.ne  ni /, @x 2 @y 2 @T @  @T @  @T D  C  C QJ , cV @t @x @x @y @y

(5.33)

(5.34)

(5.35) (5.36)

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 281

Ee,x D

be Ey  Ex , 1 C be2

Ei ,x D

Ee,y D 

Ex C bi Ey , 1 C bi2

Ei ,y D

be Ex C Ey , 1 C be2

Ey  bi Ex . 1 C bi2

(5.37) (5.38)

where  e ,  i are the densities of electron and ion fluxes: q 2 C 2 ; je j D e,x e,y B is the vector of external magnetic field induction (this vector is shown in Figure 5.1); QJ D .jE/, j D e. i   e /; ˛.E/ and ˇ are the first Townsend ionization coefficient and coefficient of ion-electron recombination; e , i are the mobilities of electrons and ions; De , Di are the diffusivities of electrons and ions; is the part of electron energy gained in the electric field and intended for heating up of gas; the Hall parameters for electrons and ions are: !e e Bz D , c e !i i Bz D . bi D c i n

be D

(5.39)

The Larmor frequency for electrons is !e D

eBz ; me c

(5.40)

eBz . mi c

(5.41)

and the Larmor frequency for ions is !i D

5.3.2 Constitutive thermophysic and electrophysic parameters The discharge in molecular nitrogen is considered, therefore, e .p  / D p D p

4.2  105 cm2 , , p Vs

293 , Torr, T

i .p  / D

De D e .p  /Te ,

2280 cm2 , , p V  s

Di D i .p  /T ,

M† p g , ,  D 1.58  105 T cm3 s 8.334  104 cp M† T  D , 0.115 C 0.354 M†  2 .2.2/ RQ

W , cm  K

(5.42)

282

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

.2.2/

7 1 , 2 M†

J g , M† D 28 , gK mole 1.157 T , ."=k/ D 71.4 K,  D 3.68 Å, D , T D  0.1472 .T / ."=k/ J J cV D 0.742 , RQ D 8.314 ; gK K  mole cp D 8.314

p is the pressure; nn D 0.954  1019.p=T / is the concentration of neutral particles. The coefficient of ion-electron recombination ˇ and electron temperature Te are chosen as constants: ˇ D 2  107 cm3 =s, Te D 11 610 K. The first Townsend ionization coefficient in view of gas heating is formulated as follows: i h B , (cm  Torr)1 (5.43) ˛.E/ D p  A exp  .jEj=p  / where A D 12 .cm  Torr/1 , B D 342 V/(cm  Torr). The set of equations (5.33)(5.35) is solved together with the equation for an external electric circuit: E D V C IR0 , (5.44) where V is the voltage drop on electrodes; I is the total current; E is the electromotive force of power supply; R0 is the Ohmic resistance of an external circuit.

5.3.3 The method of numerical integration Continuity equations for the charged particles and the Poisson equation are formulated in the following canonical form: @  @u @  @u @u @au @bu C C D D C D Cf, (5.45) @

@xQ @yQ @xQ @xQ @yQ @yQ 2

y x where xQ D H , yQ D H , D tt0 , t0 D He,0 E ; u D fne , ni , 'g. Coefficients a, b, D and functions u, f are set by the following formulas: 

For electrons uD

ne , N0

f D fe D ˛H ˇ where Q e D

e e,0 ,

ˆD

aD Qe s 

uQ e

@ˆ , @xQ

b D Q e

@ˆ , @yQ

D D DQ e ,

@ˆ @u 2  @ˆ @u 2  DQ e  DQ e C uQ e @xQ @xQ @yQ @yQ

H 2 N0 ue ui , e,0 E ' E,

(5.46)

DQ e D

De e,0 E ,

(5.47) ue D

ne N0 ,

ui D

ni N0 ;

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 283

e,0 is the characteristic value of electron mobility (e,0 D 8.8  104cm=.s  V/; N0 is the typical concentration of electrons in positive column (N0 D 109 cm3 /; 

For ions uD where Q i D



@ˆ @ˆ ni , b D Q i , D D DQ i , f D fe , , a D Q i N0 @xQ @yQ DQ i D

i e,0 ,

(5.48)

Di e,0 E ;

For electric potential a D b D 0,

D D 1,

f D "Q .ui  ue /,

(5.49)

2

where "Q D 4e H EN0 , 4e D 1.86  106 V  cm Boundary and initial conditions for the solution of the set of equations (5.45) were discussed in Section 5.1.

5.3.4 The finite-difference scheme For the solution of the boundary value problem (5.45)(5.49) the following five-point finite-difference scheme was used: mC1 mC1 mC1 mC1 N N Ai ,j umC1 i 1,j C Bi ,j ui C1,j C Ai ,j ui ,j 1 C Bi ,j ui ,j C1  Ci ,j ui ,j C Fi ,j

mC1=2

D 0, (5.50)

Ai ,j D

C aL Di 1=2,j C , pi pi pi

Bi ,j D

 aR Di C1=2,j C , pi piCpi

(5.51)

Ai ,j D

C bL Di ,j 1=2 C , qj qj qj

Bi ,j D

 bR Di ,j C1=2 C , qj qjC qj

(5.52)

  aC  aL b C  bL 1 C R C R

pi qj D D Di 1=2,j 1 Di ,j 1=2 1 i C1=2,j i ,j C1=2 C C C C , pi pi qj qi piC qjC

Ci ,j D

Fi ,j D

um i ,j

mC1=2

C fi ,j

,

(5.53)

(5.54)

284

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

where 1 pi D .xi C1  xi 1 /, pi D xi  xi 1, piC D xi C1  xi ; 2 1 qj D .yj C1  yj 1 /, qj D yj  yj 1 , qjC D yj C1  yj ; 2 1 1 Di ˙1=2,j D .Di ,j C Di ˙1,j /, Di ,j ˙1=2 D .Di ,j C Di ,j ˙1/; 2 2 1 1 aR D .ai ,j C ai C1,j /, aL D .ai ,j C ai 1,j /; 2 2 1 1 ˙ ˙ D .aR ˙ jaR j/, aL D .aL ˙ jaL j/; aR 2 2 1 1 bR D .bi ,j C bi ,j C1/, bL D .bi ,j C bi ,j 1/; 2 2 1 1 ˙ ˙ D .bR ˙ jbR j/, bL D .bL ˙ jbL j/; bR 2 2 m is the index of a time layer. Boundary conditions were formulated in the following canonical form: ui ,1 D ˛i ui ,2 C ˇi ,

i D 1, 2, : : : , NI ;

(5.55)

ui ,NJ D ˛Q i ui ,NJ 1 C ˇQi , i D 1, 2, : : : , NI ; u1,j D j u2,j C ıj , uNI ,j D Qj uNI 1,j

j D 1, 2, : : : , NJ ; Q C ıj , j D 1, 2, : : : , NJ .

(5.56) mC1=2

A singularity of the solved problem is the strong nonlinearity of function Fi ,j which contains a function of sources of the charged particles (see (5.54)) mC1=2

fi ,j

D Œ˛.Ei ,j / j e ji ,j  ˇnei,j nCi,j mC1=2 .

,

(5.57)

The stability of the numerical algorithm used is provided by the method of approximation of summand ˛.E/ j e j, because ˛.E/ is the exponential function of electric field strength and concentration of charged particles. The module of electron flux density is approximated in the form of q j e ji ,j D .e,x /2i ,j C .e,y /2i ,j , (5.58) where mC1=2

.e,x /i ,j D

mC1=2 C ue,i 1,j Ex,i ,j

.e,y /i ,j D

C umC1=2 e,i ,j 1 Ey,i ,j

C

mC1=2  ue,i C1,j Ex,i ,j

C

 umC1=2 e,i ,j C1 Ey,i ,j

C DQ e,i ,j

xQ i C1  xQ i 1 mC1=2

C DQ e,i ,j

mC1=2

ue,i C1,j  ue,i 1,j

;

mC1=2

ue,i ,j C1  ue,i ,j 1 yQj C1  yQj 1

;

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 285 ˙ Ex,i ,j D

1 1 ˙ .Ex,i ,j ˙ jEx,i ,j j/, Ey,i ,j D .Ey,i ,j ˙ jEy,i ,j j/; 2 2 ˆi C1,j  ˆi 1,j ˆi ,j C1  ˆi ,j 1 ; Ey,i ,j D  ; xi C1  xi 1 yj C1  yj 1 q 2 2 ue D ne =N0 . Ei ,j D Ex,i ,j C Ey,i ,j ;

Ex,i ,j D 

The superscript “m C 1=2” specifies the necessity of the organization of additional iterations between equations for ne , ni and ' on each time layer. The finite-difference equation (5.50) is solved on an inhomogeneous mesh !hD D fxi , i D 1, 2, : : : , NI ; x1 D 0, xNI D L; yj , j D 1, 2, : : : , NJ ; yj D 0, yNJ D H ; t mC1 D t m C , m D 0, 1, : : :g

(5.59)

with use of the successive under-relaxation method (! D 0.5–0.75/ by runs along the x axis. Inside of each time layer some iterations (3–5) should be performed for the coordination of functions ne , ni , '. The specified iterations also led to local convergence of a voltage drop on electrodes V when satisfying the equation for external circuit

where I D

RL 0

V D E  IR0 ,

(5.60)

e ne e E.y D H , x/dx.

5.3.5 The method of numerical integration of the heat conductive equation Firstly, we will consider the solution of the thermal conduction equation for a neutral gas in the absence of its motion @T @  @T @  @T cV D  C  C QJ (5.61) @t @x @x @y @y under the following boundary conditions: x Dxc :

@T D 0; @y

(5.62)

xDL:

@T D 0; @y

(5.63)

yD 0:

T D Tw ;

(5.64)

y DH :

T D Tw .

(5.65)

Here,  is the density of the gas; cV is the specific thermal capacity at constant volume; T is the temperature;  is the coefficient of thermal conduction; y, x are the axial and

286

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

transversal variables; QJ is the volumetric power of the Joule thermal emission; L is the transversal size of the calculated domain. We shall note that the equation (5.61) is written for flat geometry. In the given section, the finite-difference scheme which was used in Chapter 4 for the solution of the axially symmetric problem will be used for the calculation of glow discharge in flat geometry. For the scaling of (5.61) we shall take the following parameters: 0, cV ,0 , T0 , t0 D

H2 , e,0 E

where H is the characteristic dimension of the problem (for example, height of interelectrode gap); E is the electromotive force of the power supply; e,0 is the mobility of electrons in cold gas. With the use of the dimensionless variables the heat conduction equation will take the form @  Q @T @  Q @T @T D  C  C QQ J cV (5.66) @

@x @x @r @r 2 where Q D e,0 QQ J D QJ He,0 E . The remaining functions and arguments are E, received by the division of dimensional values on corresponding dimensionless values. Boundary conditions are reduced to the form

x D xc : xD

L : H

yD0: yD1:

@T D 0; @y

(5.67)

@T D 0; @y Tw ; T D T0 Tw T D . T0

(5.68) (5.69) (5.70)

The finite-volume method will be used, as before, for deriving the five-point finitedifference scheme. We shall integrate (5.66) by volume of an elementary computational mesh: Z Wi ,j f g D

Z

 mC1

d

m

Z

xiC1=2

yj C1=2

dx xi1=2

f gdy,

(5.71)

yj 1=2

where f g means all terms in (5.66). If f g D 1, then Wi ,j D pq , where 1 q D .xi C1  xi 1 /, 2

D mC1  m ,

1 p D .yj C1  yj 1 /. 2

(5.72)

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 287

The finite-difference mesh that was used has the following parameters !hC D fp C D yj C1  yj , p  D yj  yj 1 , j D 1, 2, 3, : : : , NJ ; q C D xi C1  xi , q  D xi  xi 1, i D 1, 2, : : : , NI ;

D mC1  m , m D 0, 1, 2, : : :g.

(5.73)

It is schematically shown in Figure 5.15. y yi+1 yi+½ yi yi−½ yi−1 xj−1 xj−½

xj

xj+½ xj+1

x

Figure 5.15. Fragment of the finite-difference mesh.

Applying the integral operator (5.71) sequentially to all items in the equation (5.66), we will receive n @T o D im,j cVm,i ,j .TimC1  Tim (5.74) (1) Wi ,j cV ,j /Wi ,j ; ,j @

n @  @T o

p Q D C Q i C1=2,j TimC1 (2) Wi ,j C1,j @x @x q  Q Q i 1=2,j mC1 p Q i C1=2,j  p C (5.75) Ti ,j C  i 1=2,j TimC1 1,j , qC q q where Q i ˙1=2,j D 1 .Q i ,j C Q i ˙1,j /; 2

n @  @T o

q D C Q i ,j C1=2TimC1 Q ,j C1 @y @y p  Q Q i ,j 1=2 mC1

q i ,j C1=2 Ti ,j C  Q i ,j 1=2TimC1  q C ,j 1 , pC p p where Q i ,j ˙1=2 D 1 .Q i ,j C Q i ,j ˙1/; (3)

Wi ,j

(5.76)

2

(4)

Wi ,j fQQ J g D qp QQ J .

(5.77)

288

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

Uniting (5.74)–(5.77) in the uniform equation, we will receive the five-point finitedifference scheme of the following form: mC1 mC1 mC1 N Ai ,j TimC1 C ANi ,j TimC1 ,j 1 C Bi ,j Ti ,j C1  C i ,j Ti ,j 1,j C Bi ,j Ti C1,j C Fi ,j D 0, (5.78)

where Ai ,j D BN i ,j

Q i ,j 1=2 , pp 

Bi ,j D

Q i ,j C1=2 , pp C

ANi ,j D

Q i 1=2,j , qq 

im,j cVm,i ,j m Q i C1=2,j Ti ,j C QQ J , D , Fi ,j D qq C

Ci ,j D Ai ,j C Bi ,j

C ANi ,j C BN i ,j C

(5.79)

im,j cVm,i ,j

. (5.80)

The finite-difference scheme received has the second order of approximation on space and the first order of approximation on time.

5.3.6 Numerical simulation results for glow discharge in a magnetic field in view of heating of gas Calculations of glow discharge were performed in the calculation domain shown in Figure 5.14. For all calculations the following initial data were common: H D 2 cm is the height of electro-discharge gap, x D xc D 2 cm is the coordinate of the center of the plasma column at an initial instant, x D L D 4 cm is the breadth of the electric discharge gap. For the estimation of initial charged particle concentrations, the theory of normal current density was used. It should be stressed that the fixed location of discharge can be ensured with the initial conditions and the use of a computing scheme which does not break a labile equilibrium position of glow discharge. Without an external magnetic field, the glow discharge is localized in the region where it has been initiated. Let us start by analyzing the numerical simulation results obtained for gas pressure p D 10 Torr, electromotive force E D 2 kV, secondary electronic emission coefficient D 0.1, and effectiveness ratio of transformation of electric field energy in heating of gas D 0–0.3. Each of the listed parameters of glow discharge strongly influence its structure, therefore, first of all the qualitative analysis of the basic tendencies of this influence will be presented. The data shown in Figure 5.16 correspond to the absence of a magnetic field (B D 0). One-dimensional Engel–Steenbeck theory predicts increasing thickness of cathode layer and voltage drop on it, and also a diminution of normal current density with decreasing . We should remember that the secondary emission coefficient is defined by the ratio of number of secondary electrons to the total number of ions falling on the cathode surface. All this is due to a diminution of actually means the decrease of

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 289

2

y, cm (a) 10

1.5

1

9

1

2

8

0.5

Level 10 9 8 7 6 5 4 3 2 1

Ne 4.00E+00 2.06E+00 1.06E+00 5.43E−01 2.79E−01 1.43E−01 7.37E−02 3.79E−02 1.95E−02 1.00E−02

Level 10 9 8 7 6 5 4 3 2 1

Ni 3.00E+01 1.23E+01 5.06E+00 2.08E+00 8.55E−01 3.51E−01 1.44E−01 5.93E−02 2.43E−02 1.00E−02

6 4 3

2 (b) 1.5

1

7 1

8

2

0.5

6

3 4

0

5

8

0 ne, ni, 109cm−3

50 (c) 45 40 35 30 25 20 15 10 5 0 0

9

1

10

2

Ne Ni

3 4 x, cm j, mA/cm2 Carrent Density on Anode 10 (d) Carrent Density on Cathode 8 6 4 2

0.5

1

1.5

2 y, cm

0 0 0.5 1 1.5 2 2.5 3 3.5 4 x, cm

Figure 5.16. Structure of two-dimensional flat glow discharge at p D 10 Torr, E D 2.0 kV D 0.1; the computational mesh is 141  61: (a) concentration of electrons; (b) concentration of ions; (c) distribution of ions and electrons along axis of symmetry of glow discharge; (d) current density on the cathode and the anode.

290

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

flux of electrons abandoning the cathode that reduces the effectiveness of ionization processes and impedes the process of glow discharge existing. It is the reason for the increasing voltage drop across the discharge gap (V D 451 V at D 0.33 and V D 555 V at D 0.1/ and the diminution of the total current (I D 3.5 mA at D 0.33 and I D 3.2 mA at D 0.1; E D 1.5 kV, p D 5 Torr). Increasing the voltage drop on the gap leads to increasing cross sizes of the cathode spot. Let us note also the twofold drop of concentration of the charged particles in the positive column (ne D ni D 2.8  109 cm3 at D 0.33 and ne D ni D 1.4  109 cm3 at D 0.1/. A growth of gas pressure leads to the proportional diminution of the thickness of the cathode layer and quite an abrupt increase in the current density on the cathode. According to the Engel–Steenbeck theory, the cathode drop does not depend on pressure, therefore, the current density should increase by pressure quadratically. Two-dimensional calculations give good agreement with this theory, in spite of the fact that at pressure increasing the voltage drop varies on the whole electric discharge gap also. The results of the calculations of glow discharge at pressure p D 10 Torr are shown in Figure 5.16. An appreciable diminution of cross sizes of the cathode layer and the positive column should be noticed at increasing pressure. With an increase in pressure the voltage drop on the gas-discharge gap has increased from 544 V to 835 V, and the current density on the cathode has increased from 3.58 mA/cm2 to 6.94 mA/cm2 , the current density on the anode has increased from 6.03 mA/cm2 to 8.89 mA/cm2 . The total current in the circuit has decreased from 4.86 mA to 3.91 mA. It is important to note an abrupt increase in the concentration of ions in the cathode layer from 1.9  1010 cm3 to 4.6  1010 cm3 . By comparing Figures 5.16 and 5.17 it is possible to fix the modification of the glow discharge structure under consideration of gas heating. Isolines of electron and ion concentrations in glow discharge at p D 10 Torr and E D 2.0 kV are shown in Figure 5.17. Because the gas heating leads to a drop in effective local pressure (see (5.42)), the discovered tendency of the discharge integral parameters diminution answers to some decreasing of pressure. The voltage drop on the gap decreases to 756 V, the current in the circuit increases to 4.13 mA, the current density on the cathode drops to 5.36 mA/cm2. The concentration of ions in the cathode layer also decreases (to 3.5  1010 cm3 /. However, the concentration of the charged particles in the positive column in this case increases up to a value of 6.0  109 cm3 . The results of the calculations of glow discharge in an external magnetic field Bz D C0.1 T in view of gas heating ( D 0.3/ are shown in Figures 5.18 and 5.19 for successive time instants. The principal peculiarity of the calculation case considered is the discharge motion perpendicular to the directions of the magnetic field and the current. At comparatively high inductions of the magnetic field (in the considered case, at Bz D C0.1 T) the current channel of glow discharge is displaced so fast that neutral gas does not have time to get warm. As the initial condition for calculations with presence of the magnetic field, the results of calculations of glow discharge in view of

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 291

2

y, cm (a) 10

1.5 9

1 8

0.5

Level 10 9 8 7 6 5 4 3 2 1

Ne 4.00E+00 2.06E+00 1.06E+00 5.43E−01 2.79E−01 1.43E−01 7.37E−02 3.79E−02 1.95E−02 1.00E−02

Level 10 9 8 7 6 5 4 3 2 1

Ni 3.00E+01 1.23E+01 5.06E+00 2.08E+00 8.55E−01 3.51E−01 1.44E−01 5.93E−02 2.43E−02 1.00E−02

1 7

3 6

2 (b) 1.5 8

1 1 2

0.5

3

7 4 5

0

6 8

0

9

1

ne, ni, 109cm−3 50 (c) 45 Ne Ni 40 35 30 25 20 15 10 5 0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 y, cm

10

2

3 j, mA/cm2 14 (d) 12 10 8

4 x, cm Carrent Density on Anode Carrent Density on Cathode

6 4 2 0

0 0.5 1 1.5 2 2.5 3 3.5 4 x, cm

Figure 5.17. Structure of two-dimensional flat glow discharge at p D 10 Torr, E D 2.0 kV D 0.1, D 0.3; the computational mesh is 141  61: (a) concentration of electrons; (b) concentration of ions; (c) distribution of ions and electrons along glow discharge symmetry axis; (d) current density on the cathode and the anode; (e) distribution of temperature along a symmetry axis of discharge.

292

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field T, K 500

(e)

450 400 350 300 0

0.5

1

1.5

2 y, cm

Figure 5.17. (continued.)

gas heating were used at the same values ofp, E, , therefore, in the process of glow discharge motion in the magnetic field the temperature of neutral gas inside the zone of the current column has decreased. At Bz D C0.1 T the velocity of the current column motion in the direction of an axis x was vx  0.3  105 cm/s. Figure 5.20 gives the representation of the glow discharge structure with a relatively weak magnetic field (Bz D C0.01 T). These calculations were performed from initially heated gas. At B D 0.01 T the current column motion in x direction is observed with velocity vx  0.2  104 cm/s. Thus, in this section we have considered the possibility of the two-dimensional driftdiffusion model of glow discharge in flat geometry with a transverse magnetic field and neutral gas heating. A distinctive peculiarity of the research performed is the self-consistency with electrodynamic structure account of neutral gas heating in regions occupied by the discharges. It has been established that molecular nitrogen in glow discharge at p  5–10 Torr and E  2–3 kV is heated up to temperatures of  500 K, and its current column in a magnetic field Bz D C.0.01–0.1/ T moves with a velocity ofvx  0.2  104– 0.3  105 cm/s, with no time to heat up neutral gas to the specified temperatures, but leaving a trail of warmed gas. Note that the numerical simulation research allows to improve the understanding of the processes in glow discharges. In particular, the results of the calculations allow to suggest the possibility to use an external magnetic field as the control parameter near to streamline surfaces.

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 293 y, cm

2

(a) 7

1.5

10 9

1 8

Level 10 9 8 7 6 5 4 3 2 1

Ne 3.00E+00 1.59E+00 8.45E−01 4.48E−01 2.38E−01 1.26E−01 6.69E−02 3.55E−02 1.88E−02 1.00E−02

Level 10 9 8 7 6 5 4 3 2 1

Ni 3.00E+01 1.23E+01 5.06E+00 2.08E+00 8.55E−01 3.51E−01 1.44E−01 5.93E−02 2.43E−02 1.00E−02

0.5 6 5

1

2 (b) 1.5

8

7

1 1 6

2

0.5

3

5

4

2

Level 10 9 1.5 8 7 6 5 1 4 3 2 1 0.5

T (c) 4.68E+02 4.50E+02 4.33E+02 4.15E+02 3.98E+02 3.80E+02 3.63E+02 3.45E+02 3.28E+02 3.10E+02

8

9

10

1 2 3 7 4 8 5 6

9 10

0

0

1

2

3

4 x, cm

Figure 5.18. Structure of two-dimensional flat glow discharge at t D 0, p D 10 Torr, E D 2.0 kV, Bz D C0.1 T, D 0.1, D 0.3; the computational mesh is 14161: (a) concentration of electrons; (b) concentration of ions; (c) temperature.

294

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm

2

Level 10 9 8 7 6 5 4 3 2 1

1.5

1

Ne 3.00E+00 1.59E+00 8.45E+01 4.48E−01 2.38E−01 1.26E−01 6.69E−02 3.55E−02 1.88E−02 1.00E−02

10 9

5 4

8

3 7

2 6 1

0.5 (a) 2 Level 10 9 8 7 6 5 4 3 2 1

1.5

1

Ni 3.00E+01 1.23E+01 5.06E+00 2.08E+00 8.55E−01 3.51E−01 1.44E−01 5.93E−02 2.43E−02 1.00E−02

1

7

2 6

8

3

0.5

4

5

(b) 9

2 Level 10 9 1.5 8 7 6 5 1 4 3 2 1 0.5

T 4.68E+02 4.50E+02 4.33E+02 4.15E+02 3.98E+02 3.80E+02 3.63E+02 3.45E+02 3.28E+02 3.10E+02

1 2 3 4

5 6 7

(c) 0

0

10

1

10 9

8

2

3

4 x, cm

Figure 5.19. Structure of two-dimensional flat glow discharge at t D 20 s, p D 10 Torr, E D 2.0 kV, Bz D C0.1 T, D 0.1, D 0.3; the computational mesh is 141  61: (a) concentration of electrons; (b) concentration of ions; (c) temperature.

Section 5.3 Glow discharge in a cross magnetic field in view of heating of neutral gas 295 y, cm

2

Level 10 9 8 7 6 5 4 3 2 1

1.5

1

Ne 2.00E+00 1.11E+00 6.16E−01 3.42E−01 1.90E−01 1.05E−01 5.85E−02 3.25E−02 1.80E−02 1.00E−02

1 10 2

3 9

0.5

4 7

(a)

5

2 Level 10 9 8 7 6 5 4 3 2 1

1.5

1

Ni 3.00E+01 1.23E+01 5.06E+00 2.08E+00 8.55E−01 3.51E−01 1.44E−01 5.93E−02 2.43E−02 1.00E−02

8

1

7

2 3

0.5

6 5

4

(b)

8

9

2 Level 10 9 1.5 8 7 6 5 1 4 3 2 1 0.5

T 4.62E+02 4.45E+02 4.28E+02 4.11E+02 3.95E+02 3.78E+02 3.61E+02 3.44E+02 3.27E+02 3.10E+02

1

2

3

4

5

6

7

10

8 9 10

(c) 0

0

1

2

3

4 x, cm

Figure 5.20. Structure of two-dimensional flat glow discharge at t D 450 s, p D 10 Torr, E D 2.0 kV, Bz D C0.01 T, D 0.1, D 0.3; initially warmed gas; the computational mesh is 141  61: (a) concentration of electrons; (b) concentration of ions; (c) temperature.

296

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field The analysis of characteristic time scales of elementary physical processes in the glow discharges that have been considered in the previous chapters allow to draw a conclusion that approximately in p  104s after electric breakdown the basic relaxation electrodynamic processes are completed. If it is taken into account that the characteristic spatial scale of the discharge channel is of L  1 cm, then we can omit consideration of the influence of a gas flow with velocities V  0.1.L= p /  103 cm/s on the electrodynamic structure of gas discharges. The given estimations allow to confirm that the influence of gas motion on the electrodynamic structure of glow discharge cannot be neglected at supersonic and, in particular, at hypersonic velocities of a gas flow (V  105 cm/s). It is obvious that the fact of high-speed motion of gas should be considered in computational models of glow discharge. In the given chapter the computing model of two-dimensional flat glow discharge in a cross gas flow and in an exterior magnetic field is given.

5.4.1 Computational model of glow discharge with cross gas flow The schematic of the problem is shown in Figure 5.21.

B 9

y 2 Y

H

6 4 8

5 3 z

1

XL

x

7

Figure 5.21. The scheme of flat glow discharge in a cross gas flow: (1) cathode; (2) anode; (3) cathode layer; (4) anode layer; (5) positive column of gas discharge; (6) gas flow at entry in electro-discharge gap; (7) active resistance of external electric circuit; (8) power supply; (9) external magnetic field.

The physical model is based on the equations of motion of multi-fluid partially ionized gas mixture, which are derived from the Boltzmann equation with use of instant procedure [2, 17, 64]  1 @ue C ne me .ue r /ue D rpe   e C ne Fe  e ne E C Œue H ne me @t c  me en ne .ue  un /  me ei ne .ue  ui /, (5.81)

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

ni mi

 @ui 1 C ni mi .ui r /ui D rpi  i C ni Fi C e ni E C Œui H @t c  mi en ni .ui  ue /  mi i n ni .ui  un /,

297

(5.82)

where ne , ni are the volumetric concentration of electrons and ions; me , mi are the masses of an electron and an ion; ue , ui , un are the average velocities of electronic and ionic liquids, and also of neutral gas; pe , pi are the pressures of electronic and ionic liquids;  e ,  i are the components of viscous stress tensor of electronic and ionic liquids; Fe , Fi are the volumetric forces effecting on particles of electron and ion liquids; e is the electron charge; c is the speed of light; E, H are the strengths of electric and magnetic fields; en , ei are the frequencies of collisions of electrons with neutral particles and with ions; i n , i e are the frequencies of collisions of ions with neutral particles and with electrons. Let us fix some inequalities and introduce some assumptions for the further simplification of the initial equations (5.81) and (5.82): (1) me  mi ; (2) A degree of ionization of gas ˛ < 104, i. e., the concentration of the charged particles is more than four orders of magnitude lower than the concentration of neutral particles ne  ni  nn ; (3) The characteristic velocities of a neutral gas flow are of V0 D 104–3  105 cm/s; (4) There is no influence of the glow discharge plasma on the gas dynamics of neutral particle flow. A glow discharges in molecular nitrogen or in air will be considered below, therefore, for numerical estimations it is possible to take the following approximations of particle collision cross sections:  p (5.83) eN2 D 2.5  1011nN2 Te 1 C 9.3  103 Te , c1 ,  p eO2 D 1.5  1010nO2 Te 1 C 4.2  102 Te , c1 , p eO D 2.8  1010nO Te , c1 ,

(5.84) (5.85)

where Te is the temperature of electrons, K. Let us consider molecular nitrogen at p D 5 Torr Š 6.58  103erg/cm3 , and T D 300 K, then p nN2  D 1.59  1017 cm3 . kT Supposing Te D 11 610 K, one can estimate eN2 Š 9.2  1010 s1 .

298

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

For comparison we shall adduce values of collision frequencies of electrons with molecules O2 and atoms O (all other parameters being equal) eO2 Š 1.42  1010 c1 ,

eO Š 4.79  109 c1 .

The collision frequency of ions with neutral particles is estimated by the formula [83] s p 2 2 2 8kT a nn , (5.86) i n D 3 me  where a is the effective radius of a neutral particle in relation to the interaction of ions with molecules and atoms. For our estimations, it is quite enough to use the Bohr radius under a D 0.529  108 cm, then

i n D 1.42  108 s1 .

To estimate the frequency of collision of electrons with ions it is possible to apply the approximated relation, which considers shielding of a field of the given charged particle by other charged particles ei D 5.5Te3=2 ne ln.220Te ne1=3 /, s1 .

(5.87)

For the initial data discussed ei  2.27  106 s1 . Further we will consider the basic scales of the most significant processes rates: (1) A characteristic velocity of a gas flow is V0 D 104–3  105 cm/s; (2) A thermal velocity of electrons (under the supposition of the maxwellization of their motion) s p 8kTe ve,T D  6.21  105 Te  6.69  107 cm/s; (5.88)  me (3) Drift velocities of electrons and ions in an electric field strength Epc D 300 V/cm (in a positive column) ve dr D e Epc D 2.52  107 cm/s,

(5.89)

vi dr D i Epc D 8.64  104 cm/s,

(5.90)

299

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

where for the mobilities of electrons and ions in molecular nitrogen the following values are chosen 4.2  105 cm2 , (5.91) e D p Vs i D

1.44  103 cm2 p Vs

(5.92)

where the pressure is measured in Torr. The estimations presented allow to draw the important conclusion about the presence of the two scales of velocities in the problem under consideration: (1) The velocities of electronic motion (a thermal velocity, a drift velocity) have the scale of ve  2  107 cm/s; (2) The velocities of ionic motion and motion of neutral particles of vi  V0  104 – 105 cm/s. Let us consider the left part of the equation (5.81), introduce a characteristic time p and a spatial scale L D 1 cm, and estimate orders of its summands: e (a) ne me @u  ne me upe  2  105 ; @t e  ne me (b) ne me ue @u @x

u2e L

 3.6  102.

The summands in the right-hand part of the equation (5.81) can be estimated as: (a) rpe 

ne kTe L

 0.16;

(b) e ne E  1.44  104; (c) e ne c 1 ue H  3.2  102, where strength of a magnetic field is accepted as equal to H D 104 Oe; (d) me en ne .ue  un /  me en ne ue  167.6; (e) me ei ne .ue  ui /  me ei ne ue  5.05  103. We shall fulfill similar estimations with the left and right parts of the equation (5.82). The left part of the equation (5.82) i  ni mi upi  4.65  103; (a) ni mi @u @t i  ni mi (b) ni mi ui @u @x

u2i L

 4.65  102.

300

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

The right part of the equation (5.82) (a) rpi 

pi L

D

ni kTi L

 4.14  103 ;

(b) e ni E  1.44  104; (c) e ni c 1 ui H  1.6, (d) mi i e ni .ui  ue /  mi i e ni ue  1.29; (e) mi i n ni .ui  un /  mi i nni ui  66. In the estimations considered, the analysis of the possible influence of viscous stress tensor components of electron and ion liquids has been omitted. To estimate the values of the items corresponding to the viscous stress tensor components of electron and ion gases, we shall present them in the form of 1  e D e ue C e grad . div ue /, 3 1  i D i ui C i grad . div ui /, 3

(5.93) (5.94)

where e , i are the coefficients of dynamic viscosity of electron and ion liquids. The phenomenological set of the specified coefficients of viscosity (for the absence of a strong magnetic field) gives kTe , me ee kTi , i D ni mi mi i i

e D ne me

(5.95) (5.96)

or for the considered conditions e  5.78  108

g , cm  s

i  0.149  108

g . cm  s

Considering the scales of electronic and ionic velocities (according to 2107 cm/s and 104–105 cm/s) it is possible to conclude that in regions of large gradients of velocities of ion and electron liquids it is impossible to neglect viscous stress tensor components. But further we will assume the insignificance of the given terms in the considered problem about the dynamics of glow discharge. Such an assumption is based on the fact that in the considered cases the possible change of discharge parameters across a streamline is insignificant. The comparison of the order of values in the equations (5.81) and (5.82) allows to draw a conclusion that under the considered conditions the equation for the determination of an average velocity of electrons and ions can essentially be simplified.

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

Having taken the specified estimations, we can write ( me en ne ue D rpe  e ne .E C c 1 Œue H/, mi i n ni ui D rpi  mi ni i nun C e ni .E C c 1 Œui H/.

301

(5.97)

Let us suppose that

Then

rpe D kTe rne ,

(5.98)

rpi D kTi rni .

(5.99)

kTe e  1 r ne  E C Œue H , me en ne me en c 1 kTi e  ui D  E C Œui H . r ni C un C mi ni i n mi i n c

ue D 

By definition e D then

e But D e D therefore,

e , me en

i D

e , mi i n

(5.100)

kTe e kTe kTe D e . D  me en me en e e k e Te

! De D e ke Te , where

k e Te

is the electron temperature in eV,

kTe D e Te ŒeV D De , me en

(5.101)

and by analogy, kTi e D i Ti ŒeV D Di . (5.102) mi i n e In view of (5.101), (5.102) we will receive expressions for the averaged velocities of electrons and ions  1 1  e E C Œue H , (5.103) ue D De rne ne c  1 1 C un C i E C Œui H . (5.104) ui D Di rni ni c Let us rewrite the equations (5.103) and (5.104) in x and y components (see Figure 5.21) 8 1 @ne 1 ˆ ˆ < ue,x D De n @x  e Ex  e c ue,y Hz , e (5.105) ˆ 1 @ne 1 ˆ : ue,y D De  e Ey C e ue,x Hz ; ne @y c

302

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

8 1 @ni 1 ˆ ˆ < ui ,x D un,x  Di n @x C i Ex C i c ui ,y Hz , i ˆ 1 @ni 1 ˆ : ui ,y D un,y  Di C i Ey  i ui ,x Hz . ni @y c

(5.106)

From (5.105) it is possible to derive x and y components of a velocity of electrons. For a velocity ue,x  De @ne e e 2 De @ne C Hz C Hz e Ey  e Ex , ue,x 1 C 2e Hz2 D  c ne @x c ne @y c where be D is the Hall parameter for electrons. Then ue,x D e Ee,x 

e H z , c

be De 1 @ne De 1 @ne C , 2 1 C be ne @x 1 C be2 ne @y

where Ee,x D

Ex  be Ey . 1 C be2

(5.107)

(5.108)

(5.109)

By analogy for a velocity ue,y we shall write the expression: ue,y D e Ee,y 

be De 1 @ne De 1 @ne  , 1 C be2 ne @y 1 C be2 ne @x

where Ee,y D

Ey C be Ex . 1 C be2

(5.110)

(5.111)

We shall adduce similar calculations for ions. The component of a velocity ui ,x ui ,x D

bi Di 1 @ni un,x C bi un,y Di 1 @ni  , C i Ei ,x  2 2 n @x 1 C bi 1 C bi i 1 C bi2 ni @y

where Ei ,x D

Ex C bi Ey . 1 C bi2

(5.112)

(5.113)

The component of a velocity ui ,y : ui ,y D

bi Di 1 @ni un,y  bi un,x Di 1 @ni C , C i Ei ,y  2 2 n @y 1 C bi 1 C bi i 1 C bi2 ni @x

where Ei ,y D

Ey  bi Ex . 1 C bi2

(5.114)

(5.115)

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

303

Let us introduce densities of particle flux vectors  e D ie,x C je,y ,

(5.116)

 i D ii ,x C ji ,y ,

(5.117)

where: e,x D ne ue,x , e,xy D ne ue,y , i ,x D ni ui ,x , i ,y D ni ui ,y Continuity equations for electron and ion liquids are formulated, as before, in view of the ionization of neutral gas and an ion-electron recombination @ne C div  e D ˛e  ˇne ni , @t @ni C div  i D ˛e  ˇne ni . @t Let us consider the equation of continuity of an electronic liquid @e,x @e,y @ne C C D ˛e  ˇne ni . @t @x @y Having substituted in it a relation (5.108), we shall receive @  De @ne D be De @ne E @ne C ne e Ee,x  C @t @x 1 C be2 @x 1 C be2 @y De @ne D be De @ne E @  ne e Ee,y  C D ˛e  ˇne ni . C @y 1 C be2 @y 1 C be2 @x Let us consider a combination of terms noted by the angular brackets. We shall suppose that De varies slightly, then @  be De @ne @  be De @ne  @x 1 C be2 @y @y 1 C be2 @x be De @2ne @  be De @ne be De @2 ne @  be De @ne C   D @x 1 C be2 @y 1 C be2 @x@y @y 1 C be2 @x 1 C be2 @x@y @  be De @ne @  be De @ne   0. D @x 1 C be2 @y @y 1 C be2 @x Therefore, the continuity equation for electrons will take the following aspect: De @ne @  @ne C ne e Ee,x  @t @x 1 C be2 @x De @ne @  ne e Ee,y  D ˛e  ˇne ni . C @y 1 C be2 @y

(5.118)

304

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

By analogy we will define the continuity equation for ions @  un,x C bi un,y Di @ni @ni C n C n  E  i i i i ,x @t @x 1 C bi2 1 C bi2 @x Di @ni @  un,y  bi un,x D ˛e  ˇne ni , (5.119) n C n  E  C i i i i ,y @y 1 C bi2 1 C bi2 @y Let us introduce the effective diffusivities into reviewing DQ e D

De , 1 C be2

DQ i D

Di . 1 C bi2

Then in the final form the equations for the definition of the concentration of electrons and ions are formulated as follows: @ne @ne @ne @  @  C ne e Ee,x  DQ e C ne e Ee,y  DQ e @t @x @x @y @y D ˛e  ˇne ni , @ni @  un,x C bi un,y @ni Q C n C n  E  D i i i i ,x i @t @x @x 1 C bi2  @ un,y  bi un,x Q i @ni D ˛e  ˇne ni . n C n  E  D C i i i i ,y @y @y 1 C bi2

(5.120)

(5.121)

These equations testify that in the conditions considered the motion of neutral particles immediately influences the distribution of the concentration of ions. There is no direct influence of the motion of neutrals on electron behavior. Nevertheless, such influence is available through the electric field connecting the dynamics of electron and ion liquids, and also through the process of a recombination. It is evident from the equation (5.121) that in the absence of a magnetic field a motion of neutral particles adds an additive component to a drift velocity of ion motion in an electric field. So, for example, if neutral gas has moved in x direction with a constant velocity, the glow discharge would simply be blown off by the stream of gas in the same direction. However, in reality everything is more complicated. Due to the boundary layer there is a distribution of a longitudinal velocity by the electric discharge channel height. So, at a different distance from the electrodes the gas flow will influence the motion of ions differently. The influence of neutral gas motion on a configuration of glow discharge will become considerably more complicated in the presence of a magnetic field.

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

305

5.4.2 Simplified hydrodynamic part of the problem under consideration. The Couette flow The Couette flow (gas flow in a laminar condition between two flat surfaces) is characterized by the distribution of a longitudinal velocity inside of the flat channel [1,78,88] vD

1 @p 2 .h  y 2 /. 2 @x

If this equation is integrated by height of the channel, it will turn out Zh GD

vdy, h

@p 3 2h3 b @p ! D  3 G, GD 3 @x @x 2h b 1  3G 2 3G v D   3 .h  y 2/ D 3 .h2  y 2 / D A.h2  y 2/, 2 2h b 4h b where G is the value of the gas flow in unit time through a rectangular area b2h; h D 0.5YH . If it is supposed that the velocity along a symmetry axis equal to V0, V0 D Ah2

!

AD

V0 , h2

then

 V0 2 y2 2 1  . .h  y / D V 0 h2 h2 For the qualitative analysis of gas flow dynamics in the rectangular channel a distribution of a longitudinal velocity was set in the following form: vD

v D V0Œ1  .y= h/m ,

(5.122)

where m D 2 or 6. We shall emphasize that the basic purpose of the given chapter is demonstrating the neutral particle stream influence on the structure of glow discharge. Therefore, the distribution of an axial velocity by height of the channel is not significant in principle. In a more complete problem statement the distributions of gasdynamic functions depend on the parameters of discharge.

5.4.3 Glow discharge in neutral gas flow. Numerical simulation results Calculations were performed with the following initial data: gas pressure p D 5 Torr, electromotive force of the power supply E D 2 kV, resistance of an external electric circuit R0 D 300 kOhm; the distance between flat surfaces of the cathode and

306

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

the anode is YH D 2 cm, and length of the flat channel XL D 6 cm (see Figure 5.21). The induction of the magnetic field directed along an axis z varied in the range of B D 0–0.05 T. All numerical simulation results are related to 1 cm along an axis z. In an initial instant the plasma cloud was placed near to the cathode at x0 D 3 cm with concentration of the charged particles n 0 D 1011 cm3 . In calculations with the absence of a magnetic field and zero velocity of gas, the relaxation regime of formation of the glow discharge plasma in a condition of a normal current density was observed. The typical relaxation time is about  10–20 s after the beginning of the process. It should be stressed that from the computational point of view this part of the problem is the most labor-consuming. It is explained by the nonlinearity of the solved problem and by strong disturbances introduced in the distributions of the required functions by the setting of arbitrary initial conditions. The stationary solution of the problem about glow discharge in two-dimensional flat geometry was used afterwards as initial conditions for the solution of the problem about the dynamics of glow discharge in a magnetic field and in a gas flow. The specified solution is presented in Figures 5.22 and 5.23. In this case, the electron temperature was supposed constant and equal to Te D 1 eV, D 0.1. ne, ni, 109cm−3

20

5

(a)

Ey(Y/E) (b)

4

15

3 10 2 5

0

1

0

0.5

1

1.5

2 y, cm

0

0

0.5

1

1.5

2 y, cm

Figure 5.22. Axial distribution of electron (dashed curve) and ion (continuous curve) concentration (a), and axial distribution of electric field strength Ey (b) at p D 5 Torr, E D 2 kV.

Concentrations of electrons and ions in the given glow discharge are shown in Figure 5.24. In the figures, near-electrode areas of glow discharge (the cathode and anode layers), and also the area of the positive column are well visible. These areas are especially evident in Figure 5.22 (b), where the axial distribution of the electric field strength is shown. The greatest strength is reached near to the cathode Ey D 4350 V/cm.

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

307

j, mA/cm2 Carrent Density on Anode Carrent Density on Cathode

4

2

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 x, cm

Figure 5.23. Current density on the anode (continuous curve) and the cathode (dashed curve); p D 5 Torr, E D 2 kV. y, cm 2 1.5 1 0.5 0

0

1

2

3

4

5

(a)

6 x, cm

y, cm 2 1.5 1 0.5 0 (b)

0

1

2

3

4

5

6 x, cm

Ne 2.10 1.96 1.81 1.67 1.53 1.39 1.24 1.10 0.96 0.81 0.67 0.53 0.39 0.24 0.10

Ni 16.00 14.94 13.88 12.82 11.76 10.70 9.64 8.58 7.52 6.46 5.40 4.34 3.28 2.22 1.16 0.10

Figure 5.24. Volumetric concentration of electrons (a) and ions (b), 109 cm3; p D 5 Torr, E D 2 kV.

308

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

According to the distribution of the electric field strength, the thickness of the cathode layer is estimated as  0.15–0.18cm, that is, in good correspondence with the Engel–Steenbeck theory. The strength of the electric field in the positive column is practically constant and equal to Ey D 180 V/cm. We shall emphasize that the strength of the electric field in the positive column is pre-breakdown, but sufficient for compensating for electron losses from the positive column due to diffusion and drift. We shall pay attention also to some growth of the field strength near to the anode where it reaches Et D 290 V/cm. The normal mode of glow discharge proves to be true by Figure 5.23, where the distribution of the current density to the cathode and the anode is given. In the calculations an additional acknowledgement of the normal mode of glow discharge is received: with an increase of electromotive force of the power supply the current density on the anode increases, and the current density on the cathode varies slightly. At the same time, of course, the breadth of the cathode spot increases, so the full current through the discharge gap varies slightly. The account of dependence of the spectrum-averaged electron energy Te on the electric field strength [84] has not led to appreciable modifications of the electrodynamic structure of the glow discharge. To be convinced of this, it is enough to compare the distributions of the concentration of the charged particles and current densities to the electrodes, which are shown in Figures 5.25 and 5.26, with corresponding distributions received at constant temperature of electrons (see Figures 5.24 and 5.23). The calculated field of electron temperature Te is shown in Figure 5.25 (c), whence it is evident that, as follows from approximation (5.123), the electron temperature is high enough only in the cathode layer where the concentration of electrons is still small. The approximation of electron temperature was used in the form of E Te D 29.96 ln C 24.64, (5.123) T p where E=p is measured in V/(cm  Torr). This approximation has been offered in the work [76] on the basis of the experimental data discussed in the work [12]. As mentioned above, the discovered distributions of the concentrations and electric potential in glow discharge were used as initial conditions for the calculations in view of gas motion and the presence of external magnetic field. We shall notice that the use of the empirical formula (5.123) is reasonable in the drift-diffusion model for the estimation of the role of electronic diffusion only. For the solution of problems of physical and chemical kinetics, the use of Te means acceptance of assumptions about the thermalization of electrons, which mismatches reality. At presence of a gas flow in a positive x direction, the discharge comes to motion in the same direction that is caused by the conditions of the computing experiments. These conditions are those that the glow discharge is in a condition of a labile equilibrium near to the area of starting initialization. However, according to the calculation model, the electrons do not sense motion of neutral particles explicitly, and first of

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field y, cm 2 1.5 1 0.5 0

0

1

2

3

4

5

(a)

6 x, cm

y, cm 2 1.5 1 0.5 0

0

1

2

3

4

5

(b)

6 x, cm

y, cm 2 1.5 1 0.5 0 (c)

0

1

2

3

4

5

6 x, cm

309

Ne 2.10 1.96 1.81 1.67 1.53 1.39 1.24 1.10 0.96 0.81 0.67 0.53 0.39 0.24 0.10

Ni 16.00 14.94 13.88 12.82 11.76 10.70 9.64 8.58 7.52 6.46 5.40 4.34 3.28 2.22 1.16 0.10

Te 2.70 2.64 2.57 2.51 2.44 2.38 2.31 2.25 2.19 2.12 2.06 1.99 1.93 1.86 1.80

Figure 5.25. Volumetric concentration of electrons (a) and ions (b), 109 cm3, and electronic temperature (in eV) in the gas-discharge gap. The Te depends on jEj; p D 5 Torr, E D 2 kV.

all motion of the glow discharge is caused by motion of ions. However, a velocity of their displacement is defined not only by a velocity of neutral particles, but also by drift and diffusion velocities. Besides, distribution of a velocity of neutral particles is inhomogeneous. From here it follows that the velocity of motion of glow discharge can essentially differ from an average velocity of a gas flow. In the case under consideration, the analysis becomes more complicated due to the inhomogeneous distribution of longitudinal velocity (5.122). The cross velocity was supposed equal to zero.

310

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field j, mA/cm2 Carrent Density on Anode Carrent Density on Cathode

4

2

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 x, cm

Figure 5.26. Current density on the anode (continuous curve) and the cathode (dashed curve); p D 5 Torr, E D 2 kV.

The important peculiarity of the calculations is the oscillation of total current and voltage drop on the gas-discharge gap. For the case under consideration, the oscillation amplitude of current and voltage makes  17 %. The specified singularity is inherent in practically all calculated data. The dynamics of the glow discharge motion along x direction is shown in Figures 5.27 (distributions of electron concentrations), 5.28 (ions) and 5.29 (electron temperature). In the work [123] it has been established that the glow discharge in an external cross magnetic field (Bz / moves either in a positive, or in a negative x direction depending on the direction of the magnetic field. With the acting of two factors on glow discharge (the magnetic field and the gas flow) it is possible to obtain various cases of both acceleration and deceleration of motion of gas discharge, also up to inverse motion (against to the gas flow). This is also realized in the following three series of calculations. The dynamics of gas discharge is shown in Figures 5.30–5.32 in the gas-discharge gap with gas flow and with the cross magnetic field Bz D C0.05 T. Calculations were also performed for Bz D C0.01 T and Bz D C0.02 T. If in the case Bz D C0.01 T an appreciable deceleration of the gas-discharge motion along the gas flow is observed, then in the case Bz D C0.02 T the discharge is slowly displaced towards the gas flow. Finally, in the case Bz D C0.05 T the gas discharge is displaced to the left with a velocity that many times exceeds the velocity of the gas flow.

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

311

y, cm 2 1.5 1 0.5 0 (a)

0

1

2

3

4

5

6 x, cm

1

2

3

4

5

6 x, cm

y, cm

2 1.5 1

Ne 2.10 1.96 1.81 1.67 1.53 1.39 1.24 1.10 0.96 0.81 0.67 0.53 0.39 0.24 0.10

0.5 0 (b)

0

Figure 5.27. Distribution of electron concentrations, 109 cm3 ; p D 5 Torr, E D 2 kV: (a) t D 0 s; (b) t D 160 s. y, cm 2 1.5 1 0.5 0 (a)

0

1

2

3

4

5

6 x, cm

1

2

3

4

5

6 x, cm

y, cm

2 1.5 1

Ni 16.00 14.94 13.88 12.82 11.76 10.70 9.64 8.58 7.52 6.46 5.40 4.34 3.28 2.22 1.16 0.10

0.5 0 (b)

0

Figure 5.28. Distribution of ion concentrations, 109 cm3 ; p D 5 Torr, E D 2 kV: (a) t D 0 s; (b) t D 160 s.

312

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm 2 1.5 1 0.5 0 (a)

0

1

2

3

4

5

6 x, cm

1

2

3

4

5

6 x, cm

y, cm

2 1.5 1

Te 2.70 2.64 2.57 2.51 2.44 2.38 2.31 2.25 2.19 2.12 2.06 1.99 1.93 1.86 1.80

0.5 0 (b)

0

Figure 5.29. Distribution of electron temperature, eV; p D 5 Torr, E D 2 kV: (a) t D 0 s; (b) t D 160 s. y, cm 2 1.5 1 0.5 0 (a)

0

1

2

3

4

5

6 x, cm

1

2

3

4

5

6 x, cm

y, cm

2 1.5 1

Te 2.10 1.96 1.81 1.67 1.53 1.39 1.24 1.10 0.96 0.81 0.67 0.53 0.39 0.24 0.10

0.5 0 (b)

0

Figure 5.30. Concentration of electrons, 109 cm3; p D 5 Torr, E D 2 kV, Bz D 0.05 T: (a) t D 20 s; (b) t D 60 s.

Section 5.4 Glow discharge in the cross flow of neutral gas and in the magnetic field

313

y, cm 2 1.5 1 0.5 0 (a)

0

1

2

3

4

5

6 x, cm

1

2

3

4

5

6 x, cm

y, cm

2 1.5 1

Ni 16.00 14.94 13.88 12.82 11.76 10.70 9.64 8.58 7.52 6.46 5.40 4.34 3.28 2.22 1.16 0.10

0.5 0 (b)

0

Figure 5.31. Concentration of ions, 109 cm3; p D 5 Torr, E D 2 kV, Bz D 0.05 T: (a) t D 20 s; (b) t D 60 s. y, cm 2 1.5 1 0.5 0 (a)

0

1

2

3

4

5

6 x, cm

1

2

3

4

5

6 x, cm

y, cm

2 1.5 1

Te 2.70 2.64 2.57 2.51 2.44 2.38 2.31 2.25 2.19 2.12 2.06 1.99 1.93 1.86 1.80

0.5 0 (b)

0

Figure 5.32. Electron temperature, eV; p D 5 Torr, E D 2 kV, Bz D 0.05 T: (a) t D 20 s; (b) t D 60 s.

314

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

With increasing magnetic field induction, ever stronger distortions of the electrodynamic structure of the glow discharge are observed. Numerical experiments confirm that the Hall currents generated in the plasma are rather inhomogeneous by thickness of the electro-discharge gap. A similar conclusion can be made concerning the discharge plasma in a gas flow with accelerating magnetic field in x direction. Thus, the two-dimensional computer model of glow discharge in gas flows and cross magnetic fields has been considered in the given chapter. But, let us recall that the calculation model is limited by conditions jue j jui j, jun j, that are usually fulfilled with greater margins. Nevertheless, at greater hypersonic velocities this inequality ceases to be true. With the use of the given computer model the investigation of glow discharge dynamics in cross magnetic field and in cross gas flow can be performed. Nevertheless, the braking conditions by the cross magnetic field for glow discharges in gas flows and the motion of the discharges towards gas flows are estimated by the calculations in the section.

5.5 Computing model of glow discharge in electronegative gas Until now we have considered computer models of glow discharges consisting only of positive ions and electrons. The specified models describe well glow discharges, for example, in molecular nitrogen (N2 /. However, the creation of a theory and computing models of glow discharges in air (dry and humid) and in other gases containing negative ions is of great practical interest. In the gases containing atoms and molecules, which possess electron affinity, steady negative ions form. Such gases are referred to as electronegative. Halogens, dry and, in particular, humid air are rated as electroneg      ative gases. Ions O 2 , H2 O , F , Br , I , Cl2 , OH , etc. are formed in them. Taking into account the formation of such negative ions leads to appreciable electron losses (to an electron attachment). The following mechanisms of electron attachment are commonly considered: 





Dissociative attachment

e C M , A 1 C A2 ;

(5.124)

e C M C N , M C N;

(5.125)

e C A , A C h,

(5.126)

Attachment in triple collisions

Photo capture

Section 5.5 Computing model of glow discharge in electronegative gas

315

where e is the electron; M is the molecule consisting of atoms A1 and A2; N is any third particle which takes part in the collision; h is the emitted quantum of electromagnetic radiation. In the schemes (5.124)–(5.126) reactions of the destruction of negative ions are also specified. The recombination of negative ions happens at their collisions with various particles (more effectively with excited particles), at a photodetachment, and as a result of ion-molecular reactions. Mechanisms of formation and loss of negative ions are considered, for example, in book [83]. In the same book, methods for calculating elementary process probabilities of the formation and destruction of negative ions are given.

5.5.1 Computational model The drift-diffusion model of glow discharge in electronegative gas is formulated in the same assumptions, as well as in the previous chapters. The basic difference to the previous formulation is the necessity to add the equations of motion and continuity for the negative ions. The equations of motion for three kinds of particles (electrons, positive and negative) are formulated with the use of the momentum procedure applied to the Boltzmann equations: me ne

@ue me ne Fe C me ne .ue  r/ue D rpe C  @t me   e  e ne .E C c 1 Œue B/  me ne en .ue  un /  me ne eC .ue  uC /  me ne e .ue  u /,

mCnC

(5.127)

mC nC FC @uC C mCnC .uC  r/uC D rpC C @t mC 1   C C e nC .E C c ŒuCB/  mC nC Ce .uC  ue /  mCnC Cn .uC  un /  mC nC C .uC  u /,

m n

(5.128)

m n F @u C m n .u  r/u D rp C @t m 1     e n .E C c Œu B/  m n e .u  ue /  m n n .u  un /  m n C .u  uC /,

where

(5.129)

@ p˛,i ,m @ p˛,j ,m @ p˛,k,m j k ; @ xm @ xm @ xm p˛,.i ,j ,k/,m are the components of a viscous stress tensor (summation by repeating indexes is supposed); ˛ D .e, C, , n/; F˛ is the mass volumetric force acting on  ˛ D i

316

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

particles ˛; me , m˛ are the masses of electrons and ions; n˛ is the volumetric concentration of particles of a class ˛; u˛ is the averaged velocity of particles of a class ˛; p˛ is the pressure of particles of a class ˛; e is the electron charge; c is the velocity of light; E D iEx C jEy is the vector of electric field strength; B D kBz is the vector of magnetic field induction; ˛ˇ is the collision frequency for particles of ˛ and ˇ classes. For partially ionized gas, as before, we suppose nn .nC , ne , n /.

(5.130)

Let us consider glow discharge in air at p D 5 Torr and electromotive force of external power supply E D 1 kV. In this case nn  1017 cm3 , ne  nC D 1011 cm3 , n  108 cm3 . Volumetric mass forces and viscosity are supposed negligible, i. e., F˛ D 0,  ˛ D 0. For the estimation of collision frequencies for electrons with ions, and ions and electrons with neutral particles we will use the following formulas: eC D 5.5Te3=2 n ln.220Te =n /, s1 ; s p 2 2 2 8kT Cn D a nn , s1 ; 3 me  1=3

p eN2 D 2.5  1011 nN2 Te .1 C 9.3  103 Te /, s1 , p p eO2 D 1.5  1010 nO2 Te .1 C 4.2  102 Te /, s1 , p eO D 2.8  1010 nO Te , s1 ,

(5.131) (5.132)

(5.133)

where n D ne D nC ; a is the effective radius of neutral particles; Te is the temperature of electrons. Further, we will consider the physical processes averaged on a period t maxf en , e+ , e , +n , + g, where ˛ˇ D 1=˛ˇ . In this case, it is possible to neglect the left parts of the equations (5.127)–(5.129). Let pe D ne kTe ,

p+ D n+ kT ,

p D n kT ,

(5.134)

where Te , T are the temperatures of electrons and all heavy particles. We shall suppose these temperatures constant. Then it is possible to rewrite (5.127)–(5.129) in the form of  kTe rne  e ne E  e ne c 1 Œue B  me ne en ue  me ne eC .ue  un /  me ne e .ue  uC / D 0, (5.135)

Section 5.5 Computing model of glow discharge in electronegative gas

317

 kT rnC C e nC E C e nC c 1 ŒuC B  mCnC Ce .uC  ue /  mC nC Cn uC  mC nC C .uC  u / D 0, (5.136)  kT rn  e n E  e n c 1 Œu B  m n e .u  ue /  m n n u  m n C .u  uC / D 0. (5.137) Let us suppose that the velocity of neutral particles un is smaller then other velocities in (5.135)–(5.137), and consider estimations of values of some terms in (5.135). From .5.131/–.5.133/ it follows that en .eC , e /, therefore, instead of (5.135) it is possible to write kTe rne  e ne E  e ne c 1 Œue B D me ne ue en or

ne ue D De rne  e ne E  e ne c 1 Œue B,

(5.138)

where e , me en  kT  e e D D Te ŒeV e . e me en e D

De D

kTe me en

(5.139) (5.140)

Instead of (5.136) for positive ions it is possible to write  kT rnC C e nC E C e nC c 1 ŒuC B  mCnC uC C C mCnC .Ce ue C C u / D 0, (5.141) where C D Ce C Cn C C  . For juC j  ju j and Cn C  the last two terms in (5.141) are equal to .mC nC uC C C mC nC ue Ce /. But for C C e (even for ue uC / it is possible to write mCnC uC C D kT rnC C e nC E C e nC c 1 ŒuCB or

nC uC D DC rnC C CnC E C C nC c 1 ŒuC B,

(5.142)

where e , mC C  kT  e D TŒeV C . D e mCC C D

DC D

kT mCC

(5.143) (5.144)

318

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

By analogy the equations for negative ions have been obtained  kT rn  e n E  e n c 1 Œu B  m n u  C m n .e ue C C uC / D 0, (5.145) where  D e C n C  C . As C juC j   ju j and e jue j   ju j, it is possible to write m n u  D kT rn  e n E  e n c 1 Œu B or

n u D D rn   n E   n c 1 Œu B,

(5.146)

where e , m   kT  e

 D D D

kT D m 

e

m 

(5.147) D TŒeV  .

(5.148)

Equations .5.138/, .5.142/ and .5.146/ will be used for the further transformations. In the two-dimensional statement considered (the configuration of an external magnetic field is shown in Figure 5.33) Œu˛ B D i u˛y B  j u˛x B,

B D Bz ,

(5.149)

therefore, projecting equation .5.138/ on axes x and y, one can receive @ne ue,y Bz  e ne Ex  e ne , @x c @ne ue,x Bz D De  e ne Ey C e ne . @y c

ne ue,x D De

(5.150)

ne ue,y

(5.151)

The given set of equations can be solved relative to ue,x and ue,y @ne 1    D ne ue,x D n E e e e x 1 C be2 @x @ne be    D ,  n E e e e y 1 C be2 @y @ne 1    D n E ne ue,y D e e e y 1 C be2 @y @ne be    D  n E e e e x , 1 C be2 @x

(5.152)

(5.153)

Section 5.5 Computing model of glow discharge in electronegative gas

319

The gap between electrodes is filled by an electric-negative gas E y=H Anode layer B

Positive column y x=0 x = L/2 z Cathode layer

x=L R0

x

Figure 5.33. Glow discharge in an external magnetic field.

where be D e B=c.

(5.154)

Now we can introduce the effective electric field into relations (5.152) and (5.153) in the following way ne ue,x D e ne Ee,x 

@ne @ne be 1 C D e,x , De De 2 2 1 C be @x 1 C be @y

(5.155)

ne ue,y D e ne Ee,y 

@ne @ne be 1  D e,y , De De 2 2 1 C be @y 1 C be @x

(5.156)

where Ee,x D

Ex  be Ey , 1 C be2

(5.157)

Ee,y D

Ey  be Ex . 1 C be2

(5.158)

By analogy one can transform equation .5.142/ for the positive ions @nC uC,y Bz C CnC Ex C C nC , @x c @nC uC,x Bz C CnC Ey  C nC , D DC @y c

nC uC,x D DC

(5.159)

nC uC,y

(5.160)

or nC uC,x D CC nC EC,x 

@nC bC DC @nC  D C,x , DC 2 @x 2 @y 1 C bC 1 C bC

(5.161)

nC uC,y D CC nC EC,y 

@nC bC DC @nC C D C,y , DC 2 @y 2 @x 1 C bC 1 C bC

(5.162)

320

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

where bC D

C Bz , c

(5.163)

EC,x D

Ex C bC Ey , 2 1 C bC

(5.164)

EC,y D

Ey  bCEx . 2 1 C bC

(5.165)

Equations for the negative ions will be received from .5.146/ @n u,y Bz   n Ex   n , @x c @n u,x Bz D D   n Ey C  n , @y c

n u,x D D

(5.166)

n u,y

(5.167)

or n u,x D  n E,x 

D @n @n b D C D ,x , 2 2 1 C b @x @y 1 C bC

(5.168)

n u,y D  n E,y 

@n b D @n  D ,y , D 2 @y 2 1 C b 1 C b @x

(5.169)

where

  Bz ; c

(5.170)

E,x D

Ex  b Ey ; 2 1 C b

(5.171)

E,y D

Ey C b Ex . 2 1 C b

(5.172)

b D

Now we will consider continuity equations for all kinds of the charged particles. Here we will consider the expressions obtained for densities of fluxes of particles ˛,x and ˛,y , and also kinetic processes for the charged particles @ne @e,x @e,y C C D ˛.E/j e j  ˇe nC ne  a ne C kd nn n , @t @x @y

(5.173)

@C,x @C,y @nC C C D ˛.E/j e j  ˇe nC ne  ˇ n nC , @t @x @y

(5.174)

@,x @,y @n C C D a ne  kd nn n  ˇ n nC , @t @x @y

(5.175)

321

Section 5.5 Computing model of glow discharge in electronegative gas

where ˛.E/ is the coefficient of ionization; ˇe is the coefficient of electron-ion recombination; ˇ is the coefficient of recombination at collision of positive and negative ions; a is the frequency of attachment; kd is the detachment coefficient. Using formulas for densities of electron and ion fluxes (5.155), (5.156), (5.161), (5.162), (5.168) and .5.169/ we will receive a system of equations of the drift-diffusion model of glow discharge in electronegative gas @ne @  @  De @ne De @ne C e ne Ee,x  C  n E  e e e,y @t @x 1 C be2 @x @y 1 C be2 @y D ˛.E/j e j  ˇe nC ne  a ne C kd nn n , (5.176) @  DC @nC DC @nC @nC @ C n E  CC nC EC,x  C C C C C,y 2 @x 2 @y @t @x @y 1 C bC 1 C bC D ˛.E/j e j  ˇe nC ne  ˇ n nC , (5.177) @  @  D @n D @n @n C  n E,x  C  n E    ,y 2 @x 2 @y @t @x 1 C b @y 1 C b D a ne  kd nn n  ˇ n nC , (5.178) @2 ' @2 ' C D 4e.ne C n  nC /, @x 2 @y 2

(5.179)

E D  grad '.

(5.180)

Here the components of the effective electric field are defined under formulas .5.157/, (5.158), (5.164), (5.165), (5.171), (5.172). To integrate the equations (5.176)–(5.180) it is necessary to define kinetic coefficients in the right-hand side of equations (5.176)–(5.178). Frequency of attachment is defined under the formula a D .˛a =p/vdr  p, where ˛a =p is the coefficient of attachment; vdr is the drift velocity of electrons. The following approximate relations for air are used: ˛a 1 E B D 0.005 at < 40 ; p cm  Torr p cm  Torr hE i ˛a 1 E B D 0.005 C  40  0.35  103 , at > 40 . p p cm  Torr p cm  Torr

322

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

The coefficient of ion-ion recombination is estimated for the following reaction  NC 2 C O2 ! N2 C O2

and is supposed equal to ˇ D 1.6  107 cm3 /s. The detachment coefficient is estimated by value of kd D 1014 cm3 /s. The coefficients of ionization ˛.E/ and recombination ˇe are taken, as before, in the form of h ˛ B i 1 D A exp  , , p .E=p/ cm  Torr ˇe D 2  107 cm3 /s, where

1 V , B D 342 . cm  Torr cm  Torr Also other approximations [12, 83] are used A D 12

˛ p Aair D 15 or

air

1 , cm  Torr

Bair D 365

˛ p

h B i 1 air , , D Aair exp  .E=p/ cm  Torr

air

for

V cm  Torr

D 1.17  104

E p

for

 32.2 ,

E V 2 Œ100–800 p cm  Torr 1 cm  Torr

V E 2 Œ44–176 . p cm  Torr

Let us notice that the selection of the electro-physical constants describing the kinetics of physical and chemical processes in partially ionized electronegative gases is the extremely important stage of construction of a computing model. The detailed analysis is necessary for the adequate account of all processes in each concrete case. It is possible to recommend the works [12, 83]. Boundary conditions for the problem are formulated in the form: yD0:

@nC @n D D 0, @y @y

y D H : nC D 0,

e D C,

@ne @n D D 0, @y @y

' D 0, ' DV,

Section 5.5 Computing model of glow discharge in electronegative gas

xD0:

@nC @n @' @ne D D D D 0, @y @y @y @y

xDL:

@nC @n @' @ne D D D D 0, @y @y @y @y

323

where is the coefficient of the secondary ion-electron emission.

5.5.2 Numerical simulation results Numerical modeling of glow discharge in electronegative gases was performed in the calculation domain shown in Figure 5.33. The following initial data were used: the height of the gas-discharge gap y D H D 2 cm, the breadth of the calculation area x D L D 4 cm, initial position of glow discharge x D L=2 D 2 cm. The computational mesh shown in Figure 5.2 was used for the calculations. Calculations have been performed for glow discharge at pressure p D 5–10 Torr, emf of the power supply is E D 2 kV, coefficient of the secondary electronic emission D 0.05, the induction of cross external magnetic field B D ˙0.05 T. Notice that in the previous section, numerical simulation results were considered at D 0.1. First of all, let us compare calculation results for electropositive and electronegative gases. Current densities on electrodes of glow discharge in electropositive gas (molecular nitrogen, N2 / at two pressures are shown in Figure 5.34 (a) and 5.34 (b). Figures 5.35 and 5.36 show concentrations of electrons and ions for pressures p D 5 and 10 Torr. A common tendency of the modification of the glow discharge structure with the growth of pressure is the diminution of the thickness of near-electrode layers and substantial growth of the current density on the cathode. According to Engel– Steenbeck theory, the voltage drop on the cathode layer does not depend on pressure, therefore, the current density on the cathode should increase quadratically by pressure. There is one more important singularity of the glow discharge, namely the cross sizes of the cathode spot and the positive column decrease with the growth of pressure. For example, at double increase in pressure the voltage drop between electrodes increases from 544 V to 827 V, the maximum current density on the cathode increases from 3.95 mA/cm2 to 7.3 mA/cm2 . At the same time, the total current through the discharge decreases from 4.86 mA to 3.91 mA. Also at D 0.1 appreciable growth in the concentration of ions in the cathode layer is observed. These numerical simulation results for glow discharge in an electropositive gas were used as initial conditions for calculating of the glow discharge in an electronegative gas. These calculations were performed both without an external magnetic field, and with an external cross magnetic field. The structure of such a glow discharge in air at p D 5 Torr, E D 2 kV and D 0.05 is shown in Figure 5.37. Comparing the data presented with the results of calculations of glow discharge in molecular nitrogen, one can note that modifications in the structure of the discharge are not so essential. In the given calculation case, the concentration of negative ions is

324

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field j, mA/cm2

10 9 8 7 6 5 4 3 2 1 0

Carrent Density on Anode Carrent Density on Cathode

(a)

0

0.5

1

1.5

2

2.5

3

3.5 4 x, cm

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

j, mA/cm2 Carrent Density on Anode Carrent Density on Cathode

(b)

0

0.5

1

1.5

2

2.5

3

3.5 4 x, cm

Figure 5.34. The current density on the anode (solid line) and on the cathode (dashed line) in N2 at E D 2 kV D 0.05: (a) p D 5 Torr; (b) p D 10 Torr.

rather low (see Figure 5.37 (c)). The current densities on the cathode and anode, cross sizes of the positive column, and concentrations of positive ions and electrons have decreased a little. Most likely, the principal reason for this is the use of some other approximations of the Townsend ionization coefficient. Numerical simulation results for glow discharge in a cross magnetic field are shown in Figures 5.38–5.40. As before, the cross magnetic field leads to motion of glow discharge plasma along an axis x. The magnetic field induction gets higher, the higher the velocity of this motion is. Figures 5.38 and 5.39 show glow discharges in electronegative gas for two values of the magnetic field inductions, B D  0.01 T and B D  0.05 T. In the first case (B D  0.01 T) the instant configuration of charged particles is shown at t D 99 s after energizing the magnetic field, and in the second case (B D  0.05 T) the structure of glow discharge is shown at t D 19 s. It is obvious that in the second case the glow discharge moves essentially faster, and its structure is distorted to a greater degree by this motion. Calculations of glow discharge at B D  0.05 T up to t D 19 s show that it is displaced practically without changing its configuration. As already mentioned with respect to research on glow discharge in molecular nitrogen, the direction of discharge motion depends on the direction of the cross magnetic field. It is natural that this regularity is also kept for discharges in electronegative gases. For an example, the glow discharge for conditions similar to the previous calculation, but at B D C 0.05 T is shown in Figure 5.40. Results of these calculations correspond to the instant t D 19 s. Notice that the full symmetry of the calculation results in relation to the variant B D 0.05 T not only confirms physical regularities, but also supports the adequacy of the developed numerical model.

Section 5.5 Computing model of glow discharge in electronegative gas

325

y, cm

2

(a)

1.5

1

0.5

2.00 1.79

0

0

1

1.58

1.37

0.94 1.16 0.73 0.10

2

3

y, cm

4 x, cm

2 (b)

1.5

3.42

1 0.10 0.58 0.18

0.32

0.5

1.90

0

20.00

0

1

2

6.16

1.05 3.42

3

4 x, cm

Figure 5.35. Fields of concentrations of electrons and ions (related to 109 cm3) in electropositive gas at p D 5 Torr, E D 2 kV, D 0.05: (a) electrons; (b) ions.

Thus, in the given chapter, calculations of the electrodynamic structure of glow discharge have been performed with the use of two-dimensional model glow discharge in electronegative gases. Nevertheless, we shall emphasize that the question of numerical research of glow discharges dynamics in electronegative gases is left open, since in the conditions considered the presence and behavior of negative ions practically have not affected the structure of the discharge. Such calculations should be made in the future.

326

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field y, cm

2

3.24

(a)

5.00

1.5

1

0.5

0

2.10 1.36

0.10

0

1

2

3

4 x, cm

3

4 x, cm

y, cm 2 (b)

1.5

1 0.37 0.57 0.88

0.5

1.36 5.00 5.00

0

0

1

2

2.10 3.24

0.24 0.15 0.10

Figure 5.36. Fields of concentrations of electrons and ions (related to 109 cm3) in electropositive gas at p D 10 Torr, E D 2 kV, D 0.05: (a) electrons; (b) ions.

327

Section 5.5 Computing model of glow discharge in electronegative gas

2

y, cm 0.27

(a)

0.10

1.5

2.00

1.43

1

0.5 1.03 0.74

0.53

2 (b) 1.5

1 1.90

0.5 1.90

3.42 20.00

2

6.16

11.10

3.11E−04

(c) 1.5

1 2.13E−04

0.5 3.11E−04

0 0

2.06E−03

1

1.41E−03

2

3

4 x, cm

Figure 5.37. Structure of glow discharge in electronegative gas (air) in the absence of a magnetic field at p D 5 Torr, E D 2 kV: (a) concentration of electrons (109 cm3 /; (b) concentration of positive ions (109 cm3/; (c) concentration of negative ions (109 cm3/.

328

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

2

y, cm (a)

1.79 2.00

1.5 1.58

1 0.52

1.37

0.5

0.31 1.16 0.94 0.10

2 (b) 1.5

1.62

0.53

1 0.17 0.10

0.5

0.30

0.93 1.62

2.82 4.93

15.00

2 (c)

1.00E−04

1.5

1

2.39E−04

0.5 1.54E−04

2.39E−04

0 0

3.68E−04

2.10E−03

1

2

3

4 x, cm

Figure 5.38. Concentration of the charged particles (109 cm3/ at p D 5 Torr, E D 2 kV, B D  0.01 T, at the instant t D 99 s: (a) for electrons; (b) for positive ions; (c) for negative ions.

329

Section 5.5 Computing model of glow discharge in electronegative gas

2

y, cm 0.10 0.42

(a)

0.74

1.07

1.39

1.71 2.03

1.5

2.36

1 2.68 3.00

0.5

2 (b)

1.00

1.39 1.95

1.5

1 2.71

0.5 3.79

2 (c)

1.39 3.79 20.00 14.34 7.37

1.00E−04 1.54E−04

1.5

2.39E−04

1 3.68E−04

0.5 5.69E−04

0 0

5.00E−03

1

2

3

4 x, cm

Figure 5.39. Concentration of the charged particles (109 cm3 / at p D 5 Torr, E D 2 kV, B D  0.05 T, at the instant t D 19 s: (a) for electrons; (b) for positive ions; (c) for negative ions.

330

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

2

y, cm 0.74

0.42

1.5

1.39

(a)

0.10

1.71 2.03

1.07

0.10 2.36

1 2.68 3.00

0.5

2 1.00

1.39

(b) 1.95

1.5

1 2.71

0.5

20.00

2

3.79

(c)

1.00E−04 1.54E−04

1.5

2.39E−04

2.39E−04

1

3.68E−04

0.5

5.69E−04

0 0

5.00E−03

1

2

3

4 x, cm

Figure 5.40. Concentration of the charged particles (109 cm3/ at p D 5 Torr, E D 2 kV, B D C 0.05 T, at the instant t D 19 s: (a) for electrons; (b) for positive ions; (c) for negative ions.

Section 5.6 Glow discharge between electrodes arranged on a surface

331

5.6 Numerical modeling of glow discharge between electrodes arranged on the same surface The drift-diffusion model has been applied above to numerical modeling of glow discharge between two flat infinite electrodes (the cross scheme of discharge). Such a scheme of a discharge is the most convenient for researching the normal mode of glow discharges. However, for applications the longitudinal scheme of discharges, when two electrodes are arranged on the same plane, is of the greatest practical interest (shown in Figure 5.41). It is obvious that such a scheme of a discharge is most acceptable for assumed schemes of discharge disposition on various streamline surfaces. B y

z

xc1 xc2

xa1 xa2 R0

x

E

Figure 5.41. The scheme of surface glow discharge.

The glow discharge of this kind has the following important peculiarities: (1) The discharge cannot exist in a condition of normal current density because it is limited by the boundaries of the cathode and anode sections; (2) It is necessary to expect abrupt growth in electric field strength near to the boundaries of the electrode sections; (3) The glow discharge region becomes non-symmetric: one of its border is located on a dielectric surface, and other one is a non-disturbed gas of external gas flow. The specified properties of the surface direct current glow discharges are manifested in the formulation of the calculation model and the boundary conditions.

5.6.1 The equations of the drift-diffusion model for surface glow discharge Surface glow discharge is considered in molecular nitrogen between two flat electrodes (see Figure 5.41). The glow discharge is described by the drift-diffusion model of motion of electrons and ions together with the Poisson equation for the definition

332

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

of electric potential ' and electric field strength E D  grad ', and the energy conservation equation describing neutral gas heating-up: @ne @  De @ne C e ne Ee,x  @t @x 1 C be2 @x @ De @ne e ne Ee,y  D ˛je j  ˇne ni , (5.181) C @y 1 C be2 @y Di @ni @ni @  C i ni Ei ,x  @t @x 1 C bi2 @x Di @ni @  i ni Ei ,y  D ˛je j  ˇne ni , (5.182) C @y 1 C bi2 @y @2 ' @2 ' C D 4e.ne  ni /, @x 2 @y 2 @T @  @T @  @T D  C  C qJ , cV @t @x @x @y @y

(5.183) (5.184)

 e D ue ne D ne V  De grad ne  ne e .E C ue  B/,

(5.185)

 i D ui ni D ni V  Di grad ni C ni i .E C ui  B/,

(5.186)

and ions;  e ,  i are the vecwhere ne , ni are the volumetric concentration of electrons q 2 2 ; B is the external magtors of densities of electron and ion flows je j D e,x C e,y netic field (its configuration is shown in Figure 5.41); qJ D .j  E/; j D e. i   e /; ˛.E/ and ˇ are the coefficients of ionization and recombination; e , i are the mobilities of electrons and ions; ue , ui are the averaged velocities of electrons and ions; De , Di are the diffusivities of electrons and ions; is the part of energy of the electric field used for heating up the neutral gas; V is the velocity of gas motion (for example, along a surface); !e !i e Bz i Bz be D D D , bi D (5.187) c e c i 

are the Hall parameters for electrons and ions; !e D



(5.188)

eBz eHz D mi c mi c

(5.189)

is the Larmor frequency for electrons; !i D



eHz eBz D me c me c

is the Larmor frequency for ions;

333

Section 5.6 Glow discharge between electrodes arranged on a surface

T , , cV are the temperature, density and specific thermal capacity at constant volume;  is the coefficient of thermal conduction. Components of the external effective electric field are expressed, as before, by the following relations: Ee,x D

be Ey  Ex , 1 C be2

Ee,y D 

Ei ,x D

Ex C bi Ey , 1 C bi2

Ei ,y D

be Ex C Ey ; 1 C be2

Ey  bi Ex . 1 C bi2

(5.190) (5.191)

It is supposed that the glow discharge does not distort an external magnetic field. The setting of constitutive thermodynamic relationships was discussed in previous sections, therefore, here they are presented without arguing: e .p  / D

4.2  105 cm2 , , p Vs

i .p  / D

2280 cm2 , , p V  s

(5.192)

293 , Torr, De D e .p  /Te , Di D i .p  /T , T M† p g 7 1 J g , M† D 28 ,  D 1.58  105 , cp D 8.314 , , 2 M† g  K mole T cm3 s T W cp M† 8.334  104 /, , .0.115 C 0.354 D 2 .2.2/ Q M† cm  K  R p D p

" 1.157 T , D 71.4 K,  D 3.68 Å, , T D  0.1472 .T / ."=k/ k cV D 0.742 J/(g  K), RQ D 8.314 J=.K  mole/,

.2.2/ D

p is the pressure in non-disturbed gas. Coefficients of recombination ˇ and electronic temperature Te are supposed constant: ˇ D 2  107cm3 =s, Te D 11610 K. The coefficient of ionization is defined as follows (the first Townsend coefficient): h i B 1 ˛.E/ D p  A exp  , , (5.193) .jEj=p  / cm  Torr where

V 1 , B D 342 . cm  Torr cm  Torr The equations (5.181)–(5.183) are solved together with the equation for an external electric circuit, which in the stationary case looks like A D 12

E D V C IR0 ,

(5.194)

where V is the voltage drop on the electrodes; I is the discharge current; E is the emf of the power supply; R0 is the external ballast resistance.

334

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

The total current in the discharge is calculated under the formula Z L Z L .jn/c dx D .jn/a dx, I D 0

(5.195)

0

where n is the unit cathode (c) and anode (a) surface normal vector; j is the current density. Heat emission in partially ionized gas due to the discharge current is calculated under the formula qJ D .j  E/ D 1.6  1019 ŒnE 2 .i C e / C .De  Di /E grad n, where is the effectiveness of transformation of the electric field energy in heating up neutral gas (  0.1–0.9/. For the calculation of the parameters of the surface glow discharge in flat geometry, all integral parameters (for example, the total current) should be related to length unit in z direction (see Figure 5.41). In the given chapter, the simplified gas-dynamic model is used. Validity of the equation u D const along each streamline along a surface is supposed.

5.6.2 Boundary conditions for surface discharge Boundary conditions are formulated in the following form: At y D 0, x 2 Œxc1 , xc2  (surface of the cathode) ne D ni i ,

@ni D 0, @y

' D 0;

(5.196)

at y D 0, x 2 Œxa1, xa2 , (surface of the anode) ni D 0, at y D 0, x … Œxc1, xc2 ,

@ne D 0, @y

' DV;

(5.197)

x … Œxa1 , xa2  (a surface of a dielectric) ne D ni D n0 ,

@' D 0; @y

(5.198)

@ni @' @T @ne D D D D 0; (5.199) @y @y @y @y @ne @ni @' x D 0, L : D D D 0; (5.200) @x @x @x x D 0 : T D T1 ; (5.201) @T xD L: D 0. (5.202) @x Here n0 is the typical concentration of electrons on a surface of dielectric (n0 is by several digits less than in electro-discharge plasma above electrodes, for example, y!1:

Section 5.6 Glow discharge between electrodes arranged on a surface

335

n0  103 –107 cm3 /; V is the potential of the anode relative to the cathode. Notice that electro-physical boundary conditions on the dielectric surface are substantially defined by its catalytic ability. The following coordinates of cathode and anode sections were used in calculations: xc1 D 1.476 cm, xc2 D 2.111 cm, xa1 D 5.861 cm, xa2 D 6.496 cm.

5.6.3 Initial conditions of numerical modeling Clouds of quasi-neutral plasma above the cathode and the anode were set as the initial conditions. Initial distribution of a potential was obtained from the solution of the Laplace equation under given boundary conditions (5.196)–(5.200). The temperature has been supposed constant in all the calculation domain, T D 300 K.

5.6.4 Numerical simulation results of surface glow discharge The typical configuration of surface glow discharge is shown in Figure 5.42. This discharge corresponds to the following initial data: p D 5 Torr, E D 500 V, R0 D 12 kOhm, D 0.1, D 0.9. Concentrations of charged particles are related to value of N0 D 109 cm3 . The calculations performed have confirmed some singularities of surface glow discharges which are well known from experiments. The positively charged area of the discharge (the cathode layer) is well visible in Figure 5.42 (a). The increased concentration of positive ions is also observed near to the boundaries of the anode with a dielectric, though on the anode surface (according to the boundary conditions) their concentration is equal to zero. The concentration of electrons in the cathode layer is very low (Figure 5.42 (a)). The greatest concentration of electrons is observed near to the boundaries of the anode. The numerical simulation results show that the area of quasi-neutral plasma is pushed aside from the dielectric surface at a distance of  0.5–1 cm. Figure 5.42 (c) shows the distribution of the electroconductivity of discharge plasma and vectorial electric field. Let us consider the basic peculiarities of the presented distributions: (1) The maximum electric field strength ( 3 kV/cm) is observed in the cathode layer near to the cathode. This result is in full correspondence with the theory of glow discharges. (2) The electric field strength in the anode layer is commensurable with the field strength in the plasma layer near to the dielectric surface between electrodes. (3) The local maxima of electric field strengths ( 4 kV/cm) are observed near to the boundaries of the cathode with the dielectric surface. It corresponds well to the known experimental fact of a strength step on boundaries of electrodes. For this reason, in real designs the electrodes with smoothed boundaries are used, as the

336

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

2

y, cm

1.5 1 0.5 0 0

1 1.4E−01

Ni:

1.0E−01

2

2.8E−01

2.0E−01

y, cm

3

2

5.7E−01

4.0E−01

5

4 (a) 1.1E+00

8.1E−01

2.3E+00

1.6E+00

7

6 4.6E+00

3.3E+00

x, cm

8

9.3E+00

6.5E+00

1.3E+01

1.5 1 0.5 0

1

SIGMA:

3.4E−06

2.8E−06

2

2 5.1E−06

4.2E−06

y, cm

3

4 (b)

7.5E−06

6.2E−06

5

1.1E−05

9.1E−06

6

1.6E−05

1.3E−05

7

4.4E−05

2.0E−05

x, cm

8

3.4E−05

2.9E−05

4.3E−05

1.5 1 0.5 0 0

1

T:

3.0E+02 3.0E+02

2

2

3

3.2E+02

3.1E+02

3.3E+02

3.2E+02

4 (c)

5

6

3.4E+02

3.5E+02

3.6E+02

3.3E+02

3.5E+02

3.6E+02

7

x, cm

8

3.7E+02

3.7E+02

3.8E+02

y, cm

1.5 1 0.5 0 0

1

2

3

4 (d)

5

6

7

x, cm

8

Figure 5.42. Configuration of surface glow discharge at p D 5 Torr, E D 500 V, R0 D 12 kOhm, D 0.1, D 0.9: (a) concentration of electrons; (b) concentration of ions; (c) electroconductivity (Ohm  cm)1 and electric field strength E (the arrow shows electrostatic intensity 3 000 V/cm); (d) temperature, K.

Section 5.6 Glow discharge between electrodes arranged on a surface

337

specified strength steps lead to the development of breakdown and instabilities of discharge plasma. The temperature of the neutral gas in the surface glow discharge is shown in Figure 5.42 (d). The assumption of a cooled surface was used, therefore, the maximum of the temperature field is reached at some distance from the surface (approximately at a distance of 0.2 cm). Neutral gas near to the anode of the discharge is heated up to a lesser degree. In the subsequent figures (Figures 5.43 (a)–(d)), similar function distributions of surface glow discharge in a cross gas flow with the velocity of 365 m/s are given (the gas flows from left to right). It is obvious that the gas flow appreciably distorts the distributions of electron and ion concentrations (Figure 5.43 (a) and 5.43 (b)). The temperature field (compare Figures 5.42 (d) and 5.43 (d)) is also greatly deformed. As it follows from the theory of glow discharges, the cross magnetic field strongly influences the electro-discharge structure at be  1 (and, especially, at bi  1/. Calculations performed on surface glow discharge with a cross magnetic field of jBz j  0.03–0.5 T have confirmed this supposition. In the specified cases jbe j D 0.264–0.44 and jbi j D 0.000864–0.00144. A significant influence of the magnetic field induction vector direction on the discharge structure has been observed. Figures 5.44 and 5.45 show the electrodynamic structure of the surface glow discharge at Bz D 0.03 and C0.03 T. First of all, we shall heed significant modifications in fields of electric conductivity in immediate proximity to the surface. As it follows from the theory, the superposition of a magnetic field changes the configuration of an effective electric field. It is well visible in Figure 5.44 and Figure 5.45. Figure 5.46 illustrates the amplification of this influence by growth of the magnetic field induction (compare Figures 5.45 (a) and 5.46 (a), and also Figures 5.45 (b) and 5.46 (b)). The distributions of the charged particle concentration presented here correspond to the values of emf E D 500 V. An increase of the voltage drop on electrodes (in our case, emf) leads to an abrupt increase of all parameters of the surface glow discharge, and then to its breakdown. As an example, Figure 5.47 shows fields of electron concentration at E D 900 V. The increased level of concentration is well visible. It should be emphasized that these are only preliminary numerical simulation results for surface glow discharge between the electrodes arranged on the same surface. The problem has appeared rather complicated and demanding carrying out additional research on numerical methods for the solved equations (in particular near to electrode–dielectric boundaries), formulation of a charged particle deposition model for dielectric surfaces, approximation of ionization processes in locally strong fields. Nevertheless, it is possible to state that the given model reflects the basic regularities known from experiments.

338

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field Ne:

1.3E−01

(a) 1.0E−01 y, cm 2

2.1E−01

1.6E−01

3.4E−01

1.1E+00

7.0E−01

2.6E−01

5.5E−01

3

4

8.9E−01

1.8E+00 3.0E+00

1.4E+00

2.4E+00

1.5 1 0.5 0 0 Ni:

1

2

5

6

7

8 x, cm

1.5E−01 3.1E−01 6.6E−01 1.4E+00 3.0E+00 6.4E+00 1.4E+01

(b) 1.0E−01 y, cm 2

2.1E−01 4.5E−01 9.7E−01 2.1E+00 4.4E+00 9.4E+00 2.0E+01

1.5 1 0.5 0 0

1

2

3

4

5

6

7

8 x, cm

SIGMA: 4.9E−06 7.2E−06 1.1E−05 1.6E−05 2.3E−05 3.4E−05 5.0E−05

(c) 4.0E−06 y, cm 2

5.9E−06 8.7E−06 1.3E−05 1.9E−05 2.8E−05 4.1E−05 6.1E−05

1.5 1 0.5 0 0 T:

1

2

3

4

5

6

7

8 x, cm

3.0E+02 3.0E+02 3.0E+02 3.1E+02 3.1E+02 3.2E+02 3.2E+02

(d) 3.0E+02 y, cm 2

3.0E+02 3.0E+02 3.1E+02 3.1E+02 3.2E+02 3.2E+02 3.2E+02

1.5 1 0.5 0 0

1

2

3

4

5

6

7

8 x, cm

Figure 5.43. Configuration of surface glow discharge at p D 5 Torr, E D 500 V, R0 D 12 kOhm, D 0.1, D 0.9, V1 D 365 m/s: (a) concentration of electrons; (b) concentration of ions; (c) electroconductivity (Ohm  cm)1 and electric field strength E (the arrow shows electrostatic intensity 3 000 V/cm); (d) temperature, K.

339

Section 5.6 Glow discharge between electrodes arranged on a surface

4.0E−02

Ne:

(a)

3.0E−02

7.2E−02

5.4E−02

1.3E−01

9.6E−02

2.3E−01

1.7E−01

4.1E−01

3.1E−01

7.4E−01

5.5E−01

1.3E+00

9.9E−01

1.8E+00

y, cm

2 1.5 1 0.5 0

0

1

SIGMA:

(b)

2

7.2E−07

3.6E−07

3

1.4E−06

1.1E−06

2.2E−06

1.8E−06

4

5

6

2.9E−06

3.6E−06

4.3E−06

2.5E−06

3.2E−06

4.0E−06

7

8 x, cm

5.0E−06

4.7E−06

5.4E−06

y, cm

2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8 x, cm

Figure 5.44. Configuration of surface glow discharge at p D 5 Torr, E D 500 V, D 0.1, B D 0.03 T: (a) concentration of electrons; (b) electroconductivity (Ohm  cm)1 and an effective electric field Ee (the arrow shows electrostatic intensity 3 000 V/cm).

340

Chapter 5 Drift-diffusion model of glow discharge in an external magnetic field

5.8E−02

Ne:

(a)

7.7E−02

6.7E−02

5.0E−02

1.0E−01

9.0E−02

1.4E−01

1.2E−01

1.9E−01

1.6E−01

2.5E−01

2.1E−01

3.3E−01

2.9E−01

3.8E−01

y, cm

2 1.5 1 0.5 0

0

1

SIGMA:

(b)

2

1.8E−06

1.5E−06

3

2.6E−06

2.2E−06

3.8E−06

3.2E−06

4

5

6

5.7E−06

8.3E−06

1.2E−05

4.7E−06

6.9E−06

1.0E−05

7

8 x, cm

1.8E−05

1.5E−05

2.2E−05

y, cm

2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8 x, cm

Figure 5.45. Configuration of surface glow discharge at p D 5 Torr, E D 500 V, D 0.1, B D C0.03 T: (a) concentration of electrons; (b) electroconductivity (Ohm  cm)1 and an effective electric field Ee (the arrow shows electrostatic intensity 3 000 V/cm).

341

Section 5.6 Glow discharge between electrodes arranged on a surface

1.7E−01

Ne:

(a)

1.4E−01

2.5E−01

2.1E−01

3.7E−01

3.1E−01

5.5E−01

4.5E−01

8.1E−01

6.7E−01

1.2E+00

9.8E−01

1.8E+00

1.4E+00

y, cm

2 1.5 1 0.5 0

0

1

SIGMA:

(b)

2

1.8E−06

1.5E−06

3

2.6E−06

2.2E−06

3.8E−06

3.2E−06

4

5

6

5.7E−06

8.3E−06

1.2E−05

4.7E−06

6.9E−06

1.0E−05

7

8 x, cm

1.8E−05

1.5E−05

2.2E−05

y, cm

2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8 x, cm

Figure 5.46. Configuration of surface glow discharge at p D 5 Torr, E D 500 V, D 0.1, B D C0.05 T: (a) concentration of electrons; (b) electroconductivity (Ohm  cm)1 and an effective electric field Ee (the arrow shows electrostatic intensity 3 000 V/cm).

1.7E−01

Ne:

1.4E−01

2.5E−01

2.1E−01

3.7E−01

3.1E−01

5.5E−01

4.5E−01

8.1E−01

6.7E−01

1.2E+00

9.8E−01

1.8E+00 2.1E+00

1.4E+00

y, cm

2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8 x, cm

Figure 5.47. Distribution of the concentration of electrons in the surface glow discharge at p D 5 Torr, E D 900 V, D 0.1, D 0.9.

Part III

Ambipolar models of direct current discharges

Chapter 6

Quasi-neutral model of gas discharge in an external magnetic field and in gas flow In the given part the quasi-neutral model of partially ionized gas in an external magnetic field is considered. This model will be used for the creation of self-consistent computing models for the numerical investigation of glow discharge parameters in hypersonic flows. The constituent of the quasi-neutral plasma model is the model of the ambipolar diffusion of charged particles in glow discharge, which will be described in Section 6.2 after the definition of some basic notations of plasma physics (Section 6.1).

6.1 The spatial scale of electric field shielding in plasma. The Debye radius The characteristic spatial scale of an electric field shielding in plasma can be defined from the one-dimensional model of non-collisional plasma. Let as assume that the distribution functions of electrons fe and ions fi can be derived from the stationary kinetic equations of the following form [2, 17]: @fe e @fe E D 0, C @x me @ve @fi e @fi  E D 0, vi @x mi @vi

ve

(6.1) (6.2)

where x is the physical coordinate; ve , vi are the velocities of electrons and ions; E is the electric field strength which is defined with a potential ' from the equation of Poisson Z d2 ' e dE D D (6.3) .fe  fi /dv,  dx dx 2 "0 where "0 is the dielectric permeability of the plasma, which is supposed constant. For determination offe we will use the variable separation method fe D V .ve /X.x/.

(6.4)

Substituting (6.4) in (6.1) and changing the electric field strength on the electric potential, we will receive dX 1 dV 1 D D ˇ 2 . e.d'=dx/ dx me ve V dve

(6.5)

346

Chapter 6 Quasi-neutral model of gas discharge in magnetic field

Integrating (6.5) separately by coordinates and velocity, one can find X.x/ D e ˇ or

2 e'

,

V .v/ D Ae ˇ

fe D Ae ˇ

2

2 .m v 2 =2/ e e

Œ.me ve2 =2/Ce'

,

(6.6) (6.7)

where the constant of integration A is defined from the condition of normalization of the distribution function Z (6.8) ne,0 D fe dv. If we suppose ' D 0, then from .6.7/ it follows fe D Ae ˇ

2

.me ve2 =2/

.

(6.9)

Let us admit that motion of electrons is thermalized due to distant collisions, then function fe should look like the Maxwellian distribution function, so, the separation constant ˇ 2 can be supposed equal to ˇe2 D .kTe /1 ,

(6.10)

where Te is the electron temperature. Now, having used the normalization condition (6.8), we can define the constant of integration  m 3=2 e A D ne,0 . (6.11) 2kTe Then instead of .6.7/ we have  m 3=2 2 e fe D ne,0 e .me ve =2kTe /.e' =kTe / . (6.12) 2kTe By analogy for ions we have  m 3=2 2 i fi D ni ,0 e .mi vi =2kTi /.e'=kTi / . 2kTi Having substituted (6.12) and (6.13) in (6.3), one can receive Z h  d2 ' mi 3=2 .mi v2 =2kTi /.e'=kTi / e i D  e n i ,0 dx 2 "0 2kTi  m 3=2 i 2 e  ne,0 e .me ve =2kTe /.e' =kTe / dv 2kTe e D  .ni ,0 e e'=kTi  ne,0 e e'=kTe /. "0

(6.13)

347

Section 6.2 The ambipolar diffusion

Let us suppose ni ,0  ne,0 , Ti  Te D T , then e d2 ' D  ne,0 .e e'=kTi e'=kTe /. 2 dx "0 If we assume that the thermal energy of the charged particles’ motion surpasses the energy of electric field (kT e'/, we can expand the two exponential functions into a series  d2 ' e' e 2ne,0 e' e C     1 C      2 '. 1 C Š  n e,0 dx 2 "0 kT kT "0 kT

(6.14)

Solution of the differential equation (6.14) at x ! 1 has the following appearance: p

' D Ae 

2.x=rD /

,

p where rD D "0kT =e 2ne,0 has dimension of length and is referred to as the Debye shielding radius. This value defines the characteristic spatial scale at which the Coulomb force action becomes small at distances x > rD . It means that at distances x > rD it is possible to consider the plasma as quasi-neutral plasma.

6.2 The ambipolar diffusion The classical model of quasi-neutral plasma of glow discharge is based on the following two equations [12, 83]: @n C div .Vn/ D div .Da grad n/ C !P e , @t

(6.15)

div j D 0,

(6.16)

where Da is the coefficient of ambipolar diffusion; V is the velocity of gas flow; n is the volumetric concentration of the charged particles (ne  ni D n/, j is the vector of current density; !P e D ˛.E/j e j  ˇe n2 is the velocity of charged particle formation, which includes the ionization (the first term) and the recombination. The process of ambipolar diffusion arises in inhomogeneous quasi-neutral plasma because of large difference in the masses of electrons and ions. Essentially easier and consequently more mobile electrons quickly abandon any area of heterogeneity arising in plasma. The electric field of polarization, which arises at this moment between electrons and essentially more slowly moving ions, restrains motion of electrons and accelerates motion of ions. It was shown that the Debye radius can be considered as the characteristic spatial scale of such separation of the charged particles. The specified type of diffusion of particles refers to the ambipolar.

348

Chapter 6 Quasi-neutral model of gas discharge in magnetic field

Let us derive coefficients of the ambipolar diffusion. For this purpose we will consider fluxes of charged particles in the form used in the drift-diffusion model:  e D e ne E  De grad ne ,

(6.17)

 i D Ci ni E  De grad ni .

(6.18)

The Poisson equation will be used for the determination of the polarization electric field strength (for a homogeneous external one) div E D 4e.ni  ne /.

(6.19)

Let us estimate a typical spatial scale of quasi-neutrality in a positive column of glow discharge, having used the relation (see Section 6.1) s s Te ŒK Te ŒeV D 525 . rD D 4.6 3 ne Œcm  ne Œcm3  Let us suppose Te D 11 610K, ne D 1011 cm3 , then rD Š 1.7  103 cm. This means that the ambipolar approximation is true on scales of L rD , for example, L  102 cm. Let us now consider a small heterogeneity of charges on the scale L jni  ne j  ne  ni D n.

(6.20)

In order that the separation of charges does not increase up to a significant value, the fluxes of electrons and ions should be approximately equal (further we shall consider such separation in x direction) x e,x  i ,x ,

(6.21)

but @n , @x @n  Ci Ex n  Di . @x

e,x  e Ex n  De

(6.22)

i ,x

(6.23)

To exclude the field of polarization Ex from .6.22/ and (6.23), we will divide .6.22/ on e , and (6.23) on i , and then sum up the outcomes e,x i ,x De @n Di @n  , C D e i e @x i @x or x

D e C i Di @n e , D C e i e i @x

349

Section 6.2 The ambipolar diffusion

or

i De C e Di @n . (6.24) e C i @x Thus, we have established that the flux of particles (electrons and ions) can be written in the form of the diffusion flux i D e C e D i @n (6.25) x D Da , Da D @x e C i with the diffusivity Da which is called the coefficient of ambipolar diffusion. Since e C and De DC , we can estimate the value of the ambipolar diffusion coefficient for typical conditions in a positive column of glow discharge (Ti  300 K, Te  10 000 K, ne  ni D nn ). Mobility of electrons is calculated under the formula x D 

e D

e 1.76  1015 cm2 , D me e,n e,n Œs 1  V  s

(6.26)

where the typical value of the collision frequency of electrons with neutral particles is 1 1 . (6.27) D 4.2  109 e,n D

e,n s  Torr Therefore, at pressure of p D 1 Torr in nitrogenous plasma with mobility of electrons e .N2 / D 4.5  105

1 cm2 D 4.5  105 , pŒTorr Vs

(6.28)

the mobility of ions is equal to i  1.45  103

1 cm2 , pŒTorr V  s

(6.29)

therefore, cm2 . Vs From here the relation between the mobilities is the following i D 1.45  103

4.5  105 e D  300. i 1.45  103 This enables to simplify the formula for the ambipolar diffusion coefficient: i e T e C e i T i Da D  i T e , e C i where Te is expressed in eV. For Te D 1 eV it turns out as Da  2.28  103 These estimations will be used below.

cm2 . s

(6.30)

(6.31)

(6.32)

350

Chapter 6 Quasi-neutral model of gas discharge in magnetic field

6.3 Ambipolar diffusion in an external magnetic field Now we will consider ambipolar diffusion at the presence of an external magnetic field. Earlier it was shown that in the two-dimensional flat case, densities of particle fluxes are defined under formulas (see Chapter 5) e,x D e nEe,x 

De @n , 1 C be2 @x

(6.33)

i ,x D Ci nEi ,x 

Di @n , 1 C bi2 @x

(6.34)

be D !e en , eB , me c

!e D Ee,x D

bi D !i i n,

Ex  be Ey , 1 C be2

!i D

eB , mi c

Ei ,x D

Ex C bi Ey . 1 C bi2

(6.35) (6.36)

(6.37)

Let us estimate values of the Hall parameters be and bi in an external magnetic field induction of B D 0.5 T. Then !e D 17.5  1010BŒT s 1 ,

!e D 8.7  1010s 1 ,

(6.38)

!i D !NC D 3.3  106BŒT s 1 ,

!i D 1.56  106 s 1 ,

(6.39)

2

1 D 41.6B D 20, 4.2  109 1 D 1.875  104 B  104 , bi D 6  106B 3.2  1010

be D 17.5  1010B

(6.40) (6.41)

so bi  be . Let Ex > be Ey , then Ee,x D

Ex , 1 C be2

Ex , 1 C bi2

(6.42)

e De @n nEx  , 2 1 C be 1 C be2 @x

(6.43)

C Di @n . nEx  2 1 C bi 1 C bi2 @x

(6.44)

Ei ,x D

then relations (6.33), (6.34) can be rewritten e,x D 

i ,x D C

Section 6.3 Ambipolar diffusion in an external magnetic field

351

As before, for elimination of the polarization field from (6.43) and (6.44), we will e i divide the equation (6.43) on 1Cb and sum up their results: 2 , and (6.44) on 1Cb2 e

x

 1 C b2 e

e

or x D  or x  

C

i

1C i

bi2

D Di @n e , D C e i @x

i De C e Di @n e i , e i @x .1 C be2 /i C .1 C bi2/e

@n @n i De C e Di @n D e Da D DQ a , 2 .1 C be /i C e @x @x @x

where e D

e C i , .1 C be2 /i C e

(6.45)

(6.46)

Da is the classical coefficient of ambipolar diffusion; DQ a is the effective coefficient of ambipolar diffusion in an external magnetic field. Note that the effective coefficient of ambipolar diffusion has been used in plasma physics for a long time [2, 17]. Let us estimate coefficient e , at first having made an obvious simplification: e D

e C i e Š . .1 C be2 /i C e .1 C be2 /i C e

In the considered conditions (see (6.30)) i Š 0.3  102e , therefore e D

e 1 . D .1 C be2/  0.3  102 e C e 1 C 0.3  102.1 C be2/

This means that for B D 0.5 T, be D 20 : e D 0.45, for B D 0.2 T, be D 8 : e D 0.84, for B D 0.1 T, be D 4 : e D 0.952. Thus, it is possible to draw the following conclusions. (1) The fluxes of charged particles in the ambipolar diffusion approximation in an external magnetic field can be estimated under the formula  D e Da grad n,

(6.47)

where e is the effective mobility of electrons .6.46/. (2) At magnetic fields B < 0.2 T the influence of an external magnetic field on ambipolar diffusion flux is small (certainly for the conditions considered).

352

Chapter 6 Quasi-neutral model of gas discharge in magnetic field

6.4 Two-dimensional model of ambipolar diffusion in an external magnetic field To finish the description of the quasi-neutral model of glow discharge (6.15)–(6.16) we will consider the transformation of an expression for a vector of the current density j D e. i   e / D eŒex . i ,x   e,x / C ey . i ,y   e,y /,

(6.48)

where ex , ey are the unit vectors along axes x and y; the projections of particle flux densities should be taken from the following relations: e,x D e ne Ee,x 

De @ne @ne be De C , 1 C be2 @x 1 C be2 @y

(6.49)

i ,x D Ci ni Ei ,x 

@ni bi Di @ni  , D 2 @x 2 i @y 1 C bi 1 C bi

(6.50)

e,y D e ne Ee,y 

De @ne @ne be De  , 1 C be2 @y 1 C be2 @x

(6.51)

i ,y D Ci ni Ei ,y 

@ni bi Di @ni C , D 2 @y 2 i @x 1 C bi 1 C bi

(6.52)

where effective components of the electric field are Ee,x D

Ex  be Ey , 1 C be2

Ee,y D

Ey C be Ex , 1 C be2

(6.53)

Ei ,x D

Ex C bi Ey , 1 C bi2

Ei ,y D

Ey  bi Ex . 1 C bi2

(6.54)

First of all, we take into account that in conditions considered bi  1, therefore, components of the effective electric field acting on ions are equal to Ei ,x D Ex ,

Ei ,y D Ey ,

@ni , @x @ni . D Ci ni Ey  Di @y

(6.55)

i ,x D Ci ni Ex  Di

(6.56)

i ,y

(6.57)

The projection of the full density of the flux to coordinate axis x in view of ne  ni  n is expressed in the form of i ,x  e,x D i nEx  Di C

@n e nEx C @x 1 C be2

@n e De @n De  n.be Ey /  be . 2 2 2 1 C be @x 1 C be 1 C be @y

Section 6.4 2D model of ambipolar diffusion in magnetic field

353

Let us introduce the effective mobility of electrons and the effective diffusivity of electrons: e De Q e D , DQ e D , 1 C be2 1 C be2 then @n @n  Q e n.be Ey /  DQ e be . (6.58) Q e /nEx C .DQ e  Di / i ,x  e,x D .i C  @x @y By analogy, for the y direction @n @n i ,y  e,y D .i C Q e /nEy C .DQ e  Di / C Q e n.be Ex / C DQ e be . (6.59) @y @x Let us now neglect the last two terms in (6.58) and (6.59). This means that vectors of particle fluxes are approximated in the form of

Then

 e D Q e nE  DQ e grad n,

(6.60)

 i D Ci nE  Di grad n.

(6.61)

j D e. i   e / D eŒ.i C Q e /nE C .DQ e  Di / grad n.

(6.62)

From here it is obvious that the vector of the electric field strength can be derived from the relation ED

DQ e  Di j  grad n. e n.i C  Q e / n.i C Q e /

(6.63)

It allows to receive densities of fluxes in the quasi-neutral approximation. We shall substitute (6.63) in (6.60) and (6.61) Q e j e D    DQ a r n, e Q e C i j i i D C   DQ a r n, e Q e C i where

Q e Di C i DQ e . DQ a D i C Q e

(6.64) (6.65)

(6.66)

Note that the equations (6.64) and (6.65) can be rewritten using e j  e D  e  e Da r n, e j i  e Da r n. i D C  e Q e C i

(6.67) (6.68)

354

Chapter 6 Quasi-neutral model of gas discharge in magnetic field

As a result, the quasi-neutral model of glow discharge in an external magnetic field can be formulated in the form of @n C div .Vn/ D div .e Da grad n/ C !P e , @t div Œ.i C Q e /nE C .DQ e  Di / grad n D 0,

(6.69) (6.70)

where  Qe D e D

e ; 1 C be2

e C i e Š ; 2 2 .1 C be /i C e .1 C be /i C e De . DQ e D 1 C be2

(6.71) (6.72) (6.73)

The equation (6.70) can be used for the calculation of electric field E, and the equation (6.69) is used for the evaluation of the concentration of the charged particles. The source term in (6.69) can also be modified in view of the magnetic field: !P e D

 ˛   e .p / p E  ˇn2 , p 1 C be2

where the definition of frequency of ionization is used in the form of  ˛ ˛ e,d D e E, i D p p

(6.74)

(6.75)

where ˛=p  is the coefficient of ionization; p  is the effective pressure.

6.5 Illustrative results of numerical simulation The presented theory of the quasi-neutral plasma of glow discharge in an external magnetic field was used for an investigation of the surface glow discharge in a supersonic boundary layer with an external magnetic field. The schematic of the problem is shown in Figure 6.1. Parameters of undisturbed incident gas flow have corresponded to experiments [67]: statistical pressure p1 D 78.4 Pa (0.59 Torr), temperature T1 D 43 K, velocity V1 D 675.5 m/s. It corresponds to the Reynolds number of Re D 1.615  105 (by meter), therefore, the boundary layer was considered as laminar. It is supposed that the surface glow discharge burns between two electrode sections arranged on a plane of a streamline plate. Such a configuration of electrodes can be used as an electromagnetic aerodynamic actuator [92,94]. The temperature of the electrodes is equal TW D 600 K, and the dielectric surface is supposed heat insulated. In

355

Section 6.5 Illustrative results of numerical simulation y

W B A

B

C

D

E

F G x

z R0

E

Figure 6.1. The scheme of an electromagnetic aerodynamic actuator.

the considered case, the discharge exists in a condition of fixed emf of a power supply of the electric circuit, E D 1 200 V. Ohmic resistance of the external electric circuit is equal to R0 D 12 kOhm. The distribution of pressure along a streamline surface with two heated up electrodes with the current flow of I  50 mA is shown in Figure 6.2. The continuous curve in this figure displays the results of the numerical integration of Navier–Stokes equations and the equations of glow discharge electrodynamics. The dashed curve illustrates the distribution of pressure predicted by the asymptotic theory of weak interaction [46]. From the data presented it is well visible that in the conditions considered the pressure increases approximately 1.7 times in the neighborhood of the electrodes. In the cases considered further, the induction of the magnetic field varied within the limits of 0.2– 0.5 T. The pressure distribution above a streamline plate at B D 0.2 T is shown in Figure 6.3. The assumed direction of the magnetic field corresponds to the negative z direction, so that the induced volumetric force acts in the direction orthogonal to the surface. Comparing Figures 6.2 and 6.3, it is possible to draw the conclusion that the external magnetic field appreciably influences the distribution of the surface pressure, in particular above the anode (x > 8 cm). It has been shown in research of induced surface pressure with the use of the elementary ambipolar diffusion model [94] that with growth of the magnetic field induction, there is a monotone increasing of the induced pressure. However, later it was found that a more correct calculation of the magnetic field within the limits of ambipolar diffusion model leads to different results. Figure 6.4 shows the distribution of pressure along the streamline surface at B D 0.5 T. Comparing this result with the previous one it is possible to draw a conclusion on a drop of the induced pressure at growth of a magnetic field induction. Thus, for deriving the greatest influence of a magnetic field on the distribution of pressure along a surface it is necessary to define optimum conditions, and the magnetic field should not be excessively great at all.

356

Chapter 6 Quasi-neutral model of gas discharge in magnetic field p/pinf

3

Present result Interaction theory 2.5

2

1.5

1

0

2

4

6

8

10

12

14 x, cm

Figure 6.2. The pressure induced by surface glow discharge (continuous curve), and predicted by the theory of weak interaction (dashed curve).

3

p/pinf Navier – Stokes The weak interaction

2.5

2

1.5

1

0

2

4

6

8

10

12

14 x, cm

Figure 6.3. The pressure induced by surface glow discharge with magnetic field B D 0.2 T (continuous curve), and predicted by the theory of weak interaction (dashed curve).

Note that the modified model of ambipolar diffusion at small inductions of a magnetic field (see Figure 6.3, B D 0.2 T) gives the numerical simulation results conterminous with the elementary theory of ambipolar diffusion, i. e., numerical modeling proves the estimations made above to be true. It is obvious that the external magnetic field influences not only the distribution of pressure, but also all the other fields

357

Section 6.5 Illustrative results of numerical simulation

3

p/pinf Navier – Stokes The weak interaction

2.5

2

1.5

1

0

2

4

6

8

10

14 x, cm

12

Figure 6.4. The pressure induced by surface glow discharge (continuous curve), and predicted by the theory of weak interaction (dashed curve), B D 0.5 T.

y, cm 3 2 1 0

0 1 (a) y, cm

2

3

4

5

6

7

8

9

10 x, cm

3 2 1 0

0 (b)

1

2

3

4

5

6

7

8

9

10 x, cm

Fi 5.45E−01 5.08E−01 4.72E−01 4.36E−01 3.99E−01 3.63E−01 3.27E−01 2.91E−01 2.54E−01 2.18E−01 1.82E−01 1.45E−01 1.09E−01 7.26E−02 3.63E−02

Fi 7.50E−01 7.00E−01 6.50E−01 6.00E−01 5.50E−01 5.00E−01 4.50E−01 4.00E−01 3.50E−01 3.00E−01 2.50E−01 2.00E−01 1.50E−01 1.00E−01 5.00E−02

Figure 6.5. The electric potential in surface glow discharge with a magnetic field: (a) B D 0.1 T; (b) B D 0.5 T.

358

Chapter 6 Quasi-neutral model of gas discharge in magnetic field

of gas-dynamic and electro-physical functions. To illustrate this in Figures 6.5–6.7, a comparison is given for fields of electric potential, current densities and volumetric concentration in two cases, which differ by magnetic induction: B D 0.1 T and B D 0.5 T. From the results presented it is visible that increasing a magnetic field leads to a drop of concentration of the charged particles in the surface glow discharge. Thus, the fulfilled numerical research on the modified model of ambipolar diffusion has shown that the model can be used for searching for the optimum parameters of an electromagnetic aerodynamic actuator and allows to predict qualitative tendencies of the influence of a magnetic field on the glow discharge, which are known from experimental research. y, cm 5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

9

(a)

10 11 12 13 14 15 x, cm

y, cm 5 4 3 2 1 0

0

(b)

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 x, cm

J 2.38E−01 1.61E−01 1.09E−01 7.38E−02 4.99E−02 3.37E−02 2.28E−02 1.54E−02 1.04E−02 7.06E−03 4.78E−03 3.23E−03 2.19E−03 1.48E−03 1.00E−03

J 7.60E−03 5.58E−03 4.09E−03 3.00E−03 2.20E−03 1.62E−03 1.19E−03 8.72E−04 6.40E−04 4.70E−04 3.45E−04 2.53E−04 1.86E−04 1.36E−04 5.00E−04

Figure 6.6. The module of a current density (A=cm2 / in surface glow discharge with a magnetic field: (a) B D 0.1 T; (b) B D 0.5 T.

Section 6.5 Illustrative results of numerical simulation y, cm 5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

9

(a)

10 11 12 13 14 15 x, cm

y, cm 5 4 3 2 1 0

0 (b)

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 x, cm

359 Ui 6.79E+01 4.26E+01 2.68E+01 1.68E+01 1.05E+01 6.61E+00 4.15E+00 2.61E+00 1.64E+00 1.03E+00 6.44E−01 4.04E−01 2.54E−01 1.59E−01 1.00E−01

Ui 1.94E+01 1.80E+01 1.66E+01 1.53E+01 1.39E+01 1.25E+01 1.11E+01 9.71E+00 8.33E+00 6.94E+00 5.55E+00 4.17E+00 2.78E+00 1.40E+00 1.00E−02

Figure 6.7. Volumetric concentration of particles (in 1010cm3 / in surface glow discharge with a magnetic field: (a) B D 0.1 T; (b) B D 0.5 T.

Chapter 7

Viscous interaction on a flat plate with surface discharge in a magnetic field In the given chapter, in essentially more detail than in the previous one, the problem considered is about viscous interaction of a gas flow with a flat plate, at which direct current surface discharge between two electrodes arranged across a gas flow initiates and steadily burns (see Figure 6.1). For the solution of this problem, selfconsistent thermo-gasdynamic and electrodynamic models are used based on the twodimensional Navier–Stokes equations and the diffusion-drift model of electric discharge in an approximation of quasi-neutral plasma. The electrodynamic equations are closed by the equations of an external electric circuit, therefore, the received solutions meet the fixed values of electromotive power of a direct current source and active resistance of an external circuit. Initial data are also a velocity of an incident gas flow and its pressure, sizes and configuration of electrodes of the electric discharge system. Numerical solutions were received with reference to the conditions of experimental research conducted at Wright State University, Dayton, Ohio, USA [94, 97]. Methods of electrodynamic and magnetogasdynamic control of a gas flow are not new. During more than the last fifty years, in different areas of fundamental and applied physics and mechanics including practical aerodynamics, these methods have been discussed and investigated in detail. In recent years, heightened interest in the given problem is observed [91] because the gas-discharge and magnetogasdynamic technologies promise to be effective for hypersonic vehicles control. Note the problem has been called one of the perspective problems of the next decade in the world space industry. The classical theory of viscous interaction near to the flat plate which is flowed round by a hypersonic gas flow, is stated in the work [46]. Within the limits of the specified theory, the distribution of pressure along a surface of a streamline plate is connected with the thickness of the growing boundary layer and is calculated with the use of the parameter of viscous interaction N D M3 .C =Rex /1=2 ,

(7.1)

1 ; C D 1b b1 ; Rex D 1V11 x ; ,  are the density and coefficient where M D Va1 of dynamic viscosity; V1 , a1 are the velocity of incident flow and the velocity of a sound in it; x is distance along a plate surface from its leading edge; indexes ı and 1 designate properties on the external boundary of a boundary layer and in an incident nondisturbed flow of gas. In the asymptotic theory of weak interaction of Leese–Probstein [46] it is shown  c that for a flat plate at D1.4 and Pr D 1 1p1 D 0.725 where cp1 , 1 are the spe-

Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field 361

cific thermal capacity at constant pressure and a thermal conduction of incident gas, the distribution of pressure induced by viscous interaction between incident flow and a boundary layer is defined by the following law (accurate within the second infinitesimal order of the specified theory) p D 1 C 0.31N C 0.05N 2. p1

(7.2)

Good concurrence of the asymptotic theory (7.2) at N < 3 with Bertram experimental data is shown in the study [46]. We shall notice that the theory of weak viscous interaction allows to predict the distribution of pressure along thin wedges and a plate at small angles of attack (with large Reynolds and Mach numbers or with moderate supersonic Mach numbers and small Reynolds numbers), and also on thick wedges with large Mach numbers. The asymptotic theory of strong interaction yields good outcomes at N > 3. It has been proved accurate within the second infinitesimal order on N 1 in the work [46] p D 0.514N C 0.753. p1

(7.3)

The comparison of Bertram experimental data presented by the above-stated work for a heat-insulated plate with the dependence (7.3) also testifies to their good concurrence. The purpose of the given chapter is to research a possibility of viscous interaction parameters modification by means of the electromagnetic forces arising near to a streamline surface in the case of surface direct current discharge. At the first stage, calculations of a classical problem about viscous interaction near to a flowed round heat-insulated plate were performed, and then outcomes of numerical modeling were compared to the asymptotic theory of weak interaction. At the second stage, the influence of locally heating up two sites on the plate surface, which imitates the heating up of electrodes that have been built in a streamline surface, on the aerodynamics of flow round a plate and on performances of viscous interaction is investigated with use of the computational technique that was developed and tested at the first stage. The temperature of the locally heated surface varied in a range of 300– 800 K, which corresponds to the measured values of electrodes’ surface temperature in the experiments [67]. At the third stage, research on flow round a plate with heated up electrodes and surface direct current discharge burning between them was fulfilled, and at the fourth stage, the cross magnetic field is considered in calculations. Parameters of surface discharge and induction of a cross magnetic field correspond to experimental data [67].

362 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field

7.1 Statement of a problem about viscous interaction The two-dimensional computing model is based on a set of Navier–Stokes equations and a set of equations of the direct current discharge drift-diffusion theory, closed by the equation of an external electric circuit. The investigated model considers the action of both a gas flow on discharge, and gas discharge on a flow, and hence, action on the distribution of pressure along a streamline surface that allows to study regularities of strong thermo-gasdynamic and electrodynamic interaction under the conditions when it cannot be considered small. The physics of such mutual influence consists in the following. The viscous gas flow round a plate leads to the formation of a boundary layer near to its surface. At the boundary layer the velocity and temperature vary from the values on a surface up to values in a nondisturbed stream. Any actions on a gas flow, for example, locally heating up a plate surface, lead to a modification of thermo-gasdynamic parameters in a boundary layer and, consequently, parameters of viscous interaction between a boundary layer and a nondisturbed flow, which is clearly exhibited in a modification of pressure distribution along a streamline surface. Ignition of a direct current surface discharge is one of the methods of action on a flow. However, such a physical entity as a direct current discharge is rather sensitive to conditions in a gas flow, in particular, to its temperature and pressure, as these two parameters define a velocity of ionization of neutral particles in an electric field of discharge. Hence, they essentially affect a current flow and a Joule thermal emission, which in turn in many respects defines heating up of neutral gas in the region of an electric current flow. Thus, the line-up of mutual influences becomes closed. The solution of the discussed self-consistent problem demands construction of an adequate theory and computational methods, allowing to consider such important physical phenomena as changes in a pressure distribution along a streamline plate depending on force of an electric current, “blowing-off” of surface discharge by a gas flow and the development of ionization-overheating instability in gas discharge. In the latter case, there are local overheat areas in discharge where the process of avalanche ionization cannot be stopped or decelerated by convective cooling of gas and, as the corollary, very strong electric current starts to flow through discharge, which leads to loss of its homogeneous configuration and a transfer to the form of arc discharge. Additional action on a flow is made by a magnetic field coupled with an electric current flowing through discharge. It follows to origination of the additional force of volumetric character, which allows to change the aerodynamic performances of a streamline surface. The computational model developed imitates all the listed phenomena up to the derangement of a direct current surface discharge to an electric arc. Navier–Stokes and continuity equations are written in the rectangular Cartesian frame, connected with a streamline plate surface. In the form convenient for numerical

Section 7.1 Statement of a problem about viscous interaction

363

realization they have the following appearance: @f @E @F @Ev @Fv C C D C C ˆ, @t @x @y @x @y 2 3 2 3 2 3  u v f D 4 u 5 , E D 4 uu C p 5 , F D 4 vu 5 , v uv vv C p 2 3 3 3 2 2 0 0 0 1 1 4 xx 5 , Ev D 4 xy 5 , ˆ D 4 FM ,x 5 , Fv D Re Re

xy

yy FM ,y where

(7.4)

n @u 2 h @u  @u @v io @v

xx D  2  C , xy D  C , @x 3 @x @y @y @x n @v 2 h @u @v io

yy D  2  C . @y 3 @x @y

Here x, y are the Cartesian coordinates along a plate surface and normal to it; u, v are the components of velocity V vector along coordinate lines x, y; , p are the density and pressure;  is the coefficient of dynamic viscosity; xx , xy D yx , yy are the components of a viscous stress symmetric tensor; FM ,x and FM ,y are the components of the magnetic force acting on a gas flow; Re D 1V11 L ; L D 1 cm is the length scale. It is assumed that gas is heated up slightly and its molecular weight varies weakly, and the electric discharge which is being initiated near to a surface does not lead to an essential modification of a chemical compound of gas in a boundary layer, i. e., gas is considered weakly ionized. In this case, the energy conservation equation is convenient to present in the form of the Fourier–Kirchhoff equation, i. e., concerning temperature of gas T @T @T @T C cp u C cp v @t @x @y 1 @  @T L 1 @  @T C Qp C Q , D  C  C QJ Re Pr @x @x Re Pr @y @y 1 V1 cp1 T1 (7.5)

cp



c

where Pr D 1 1p1 . Functions Qi in the right member (7.5) describe the effects connected with heating up of gas: QJ is the Joule thermal emission; Qp , Q are the thermal emissions connected with compressibility of gas and viscous dissipation in gas. Last two functions are represented in the following aspect: @p @p Cv , @x @y n h @u 2  @v 2 i h @u @v i2 2  @u @v 2 o C . C  Q D  2 C C @x @y @y @x 3 @x @y Qp D u

(7.6)

364 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field In the given model, Joule thermal emission QJ is calculated under the formula QJ D .j  E/, in which is the effectiveness ratio of transformation of electron-molecular collision energy in the heating up of gas (in the given work it is set as a parameter of the problem, D 0.5); j is the vector of current density; E is the vector of electric field strength. The electrodynamic equations follow from the drift-diffusion theory of direct current discharge, which was used earlier in the analysis of electro-gasdynamics interactions in a wide range of velocities of a gas flow [118–120, 123]. In these works, the development of the specified theory with the purpose of accounting for external cross magnetic field (provided that the electric discharge does not distort a magnetic field, and the parameter of Hall is small) has been fulfilled. The system of the electrodynamic equations has the following appearance: @fM @EM @GM @ED,M @GD,M (7.7) C C D C C QM , @t @x @y @x @y       n nu nv , EM D , GM D , fM D ' 0 0 3 2 2 2 3 @n @n 3 D ni  ˇn2 Da a 7 6 6 @y 7 @x 7 6 7 7 , FD,M D 6 ED,M D 6 7 , QM D 4 6 @n @n 5 . 4 5 5 4 @' Da  Da @' ne @x @y ne @y @x Here n, ' are the concentration of the charged particles and electric potential; e is the mobility of electrons; Da is the coefficient of ambipolar diffusion; i , ˇ are the coefficients of collisional ionization (the first Townsend ionization coefficient) and electron-ion recombination. For the calculation of components of the magnetic force FM acting on a flow, the following relations were used: 2 3 ex ey ez

 (7.8) FM D Œj  B D 4 jx jy jz 5 D ex jy B  ey .jx B/, 0 0 B @' @' , FM ,y D jx B D Bne , @y @x where ex , ey , ez are the unit basis vectors of a rectangular Cartesian frame. In the calculations, the following dimensions of coefficients and functions are used: FM ,x D jy B D Bne

e D 1.6  1019 K, ŒE D V/cm,

Œn D cm3 ,

ŒB D T,

Œj   Œe ne E D W/cm2 ,

Œe  D cm2 /(s  V),

Œj   Œe ne E D A/cm2 , ŒQJ   Œ e ne E 2  D W/cm3 ,

ŒFM   Œ105e ne EB D g/(cms2 ).

(7.9)

365

Section 7.2 Boundary conditions of the problem

7.2 Boundary conditions of the problem The scheme of the solved problem is shown in Figure 6.1. It is supposed that a nondisturbed gas flow attacks a plate with a velocity V1 and the set thermodynamic and transfer properties p1 , 1 , T1 , cp1 , 1 , 1 . The dielectric surface of the plate is considered heat insulated, and the . temperature of electrodes is set as one of the input parameters of the problem. The plate is supposed impenetrable for gas with realization of conditions of an attachment on its surface. The longitudinal coordinates of points, designated by A  G and W on Figure 6.1, were set differently for two calculated variants corresponding to experimental conditions [67]. When the stationary solution is searched for, parameters of nondisturbed incident gas flow were set as initial conditions only for the construction of the numerical method of solution. Boundary conditions were formulated in the following aspect: @' xD0: u D V1 , v D 0, T D T1 ,  D 1 , n D n1 , D 0; @x (7.10) @v @T @n @ @' @u D D D D D D 0; (7.11) x D xG : @x @x @x @x @x @x yD0:

u D v D 0,

@p D 0; @y

(7.12)

xC  x  xD (cathode) :

T D Tw ,

xE  x  xF (anode) :

T D Tw ,

xB  x < xC ,

xD < x < xE ,

@n D 0, @y @n D 0, @y

' D 0; ' D VA ;

xF < x  xG (dielectric surface) : @' @T D D 0; n D 105 n0 , @y @y

where n0 D 1010 cm3 is the typical concentration of the charged particles in the positive column; VA is the potential of the anode section in relation to the cathode. The electric potential of the anode was calculated with use of the Kirchhoff equation for an electric circuit IR0 C VA D E. (7.13) Here E is the emf of a power supply; R0 is the resistance of an external electric circuit; I is the total current in a circuit, defined by the results of the solution of the electrodynamic part of the problem Z Z jdx D jdx. I D x2ŒxC ,xD

x2ŒxE ,xF

366 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field

7.3 Transfer and electro-physical properties of gas In the calculations, thermodynamic and transfer properties of molecular nitrogen were used: p 1 (7.14)  D 2.67105 MA T 2 .2,2/ ,  D 3.68 Å,  T

.2,2/ D 1.157.T  /0.1472, T  D , ."=k/ D 71.4, ."=k/ s T 0.115 C 0.354.cp =R0/MA  D 8.334  104 , MA  2 .2,2/ where cp D 8.317 MA D 28 g/mole,

7 1 , 2 MA

D

p MA , T R0

R0 D 8.314  107 erg/(mole  K).

Frequency of ionization and the Townsend ionization coefficient were defined under empirical formulas in which local heating up of gas is considered that also ensured self-coherence of the solution for the electrodynamic and gas-dynamic parts of the problem: vi D

 ˛ p  Ee .p  /, p

p D p

293 , T

h  ˛ B i , D A exp  p .E=p  /

De D e .p  /Te ,

e .p  / D 4.2  105 ˇ D 2.0  107 –8.0  106 cm3 /s

1 , p

(7.15)

DC D C .p  /TC ,

C.p  / D 1450

Te D 11610 K,

1 , p

A D 15 (cm  Torr) 1 ,

B D 365 V/(cm Torr), where e , i are the nobilities of electrons and ions; De , Di are the diffusivities of electrons and ions. The given coefficients were also used for the calculation of Joule thermal emission: QJ D .j  E/ D 1.6  1019 ŒnE 2 .C C e / C .De  DC/E grad n

(7.16)

Let us notice that the variation of electron-ion recombination coefficient allowed to account for the possible influence of electronegative gases (for example, of oxygen in air) on the results of numerical modeling. The admissibility of the realization of such a method is discussed in the book [83].

367

Section 7.5 Numerical simulation results

7.4 The numerical method of solution The formulated set of equations was solved by a time relaxation method. At each time step, three groups of presented equations were solved sequentially. The Navier– Stokes equations and continuity equations made up the first group. This set of equations was integrated by an explicit method with use of the AUSM finite-difference algorithm [32]. At the second stage, the energy conservation equation, and at the third the electrodynamic equations were integrated. The last two groups of the equations were integrated by an implicit method of sequential under-relaxation and with a direction-alternating run along coordinate lines. Between these two groups of equations additional internal iterations has been fulfilled, their necessity had usually arisen at intensification of the interaction between surface discharge and gas flow.

7.5 Numerical simulation results The following parameters in an incident stream of molecular nitrogen were used in these calculations: 1 D 0.636  105 g/cm3 , p1 D 783.7 erg/cm3 , V1 D 0.676  105 cm/s, which corresponds to Mach number of incident stream M1 D 5.15. The disposition of electrodes on a streamline surface is shown in Figure 6.1. Two variants of their mutual position according to experiments in [67] are considered. Coordinates are represented in Table 7.1, where W is the length of electrodes along an axis z. The breadth of each electrode section (along an axis x) was identical in all calculations, and in the first variant the distance between electrodes along an axis x is longer than  1 cm. A stream cross size of electrodes in the second variant of the experimental research is smaller than  2 cm. In calculations of electrodynamic parameters, cross sizes have been assumed to be 1 cm. Table 7.1. Position of electrodes in two variants of the calculation. Coordinates of electrodes on a plate surface, cm

Variant 1

Variant 2

xA xB xC xD xE xF xG W

0 1.0 3.2 3.835 8.7 9.335 15.0 5.1

0 1.0 1.476 2.111 5.861 6.496 10.0 3.175

The leading edge of a streamline plate was placed at the distance xB D 1 cm from the beginning of the rated area, where the nondisturbed attack flow had been set. The

368 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field mesh along a streamline surface was chosen to be inhomogeneous. It was divided into 5 sites, a special selection of a mesh was made on each of them. The first segment BC (see Figure 6.1) corresponded to an initial site of a current from a leading edge of a plate. In an immediate neighborhood of an edge, where the Navier–Stokes current model used is not completely adequate, there are no expectations of true current description. However, with well-chosen computational meshes at a small distance from a leading edge, good concurrence to asymptotic Lees–Probstein theory is observed, which, as is well known, is also incorrect at small values of x  xB . Considering that the theory of weak interaction has many times been confirmed in experimental research, it is possible to consider the coincidence with it as reliable fidelity criterion for the calculation, and certainly as an adequate choice of viscous interaction parameter (7.1). Parameters of a computational mesh above segments CD and EF (the cathode and the anode) are defined basically by the needs of the electrodynamic part of the problem. There is no direct analogy to principles of construction of a mesh above an initial site of a plate, however, it is necessary to remember that the drift-diffusion model demands a detailed computational mesh both in the immediate proximity of a surface of electrodes, and at an electrode–dielectric boundary. Given the criterion of reasonableness of computational mesh construction, a good coincidence of values of a current through the cathode and the anode can serve as ascertainment of a solution. The fourth DE and the fifth FG sites of the computational mesh meet to form the inter-electrode gap and the current downwards on a stream from the anode to output boundary of the rated area. Besides a good description of a current near a plate, here it is necessary to ensure not so sharp a transition from the mesh above the electrodes to the mesh above the dielectric, and also an accurate description of current at the output section. The sequence of calculations was identical to the two variants of the electrodes’ configuration. As previously mentioned, firstly calculations of flow round a heatinsulating plate were made with the purpose of comparison with the asymptotic theory of weak interaction [46]. In addition, methodological calculations have been fulfilled with the purpose of researching the influence of computational meshes on the characteristics of viscous interaction. At the second stage of evaluations, the influence of the temperature of the electrodes on flow development at a surface was studied. The dielectric surface of the plates was supposed heat insulated, and the temperature of the electrodes (the cathode and the anode) was set constant and equal to Tw D 400, 600, 800 K. Let us notice that the temperature of electrodes Tw D 600 K is close to that measured in experiments on transiting of a current across an electric discharge gap. At the given stage of evaluations, the electric circuit was supposed broken. Preliminary research on the influence of heating up electrodes has allowed to estimate their thermal action on gas dynamics. At the third stage of evaluations, the surface discharge of a direct current was involved, and at the fourth the external magnetic field was imposed.

369

Section 7.5 Numerical simulation results

7.5.1 The heat-insulated plate First we shall consider outcomes of calculations of a flow round a heat-insulated plate. The calculated distribution of pressure along a plate is shown in Figure 7.1, and the comparison with the Lees–Probstein theory of weak (N  3) and strong (N > 3) interaction is given. 3

p/pinf 1 2 3 4 5

2

1

0

0

5

10

15 x, cm

Figure 7.1. Distribution of pressure along a surface of a heat-insulated plate with heated up cross bands. The first configuration of electrodes: (1) numerical solution for a heat-insulated plate; (2) asymptotic theory of weak interaction; (3) asymptotic theory of strong interaction; (4, 5) numerical solution for a heat-insulated plate with heated up electrodes; (4) Tw D 400 K; (5) Tw D 800 K.

Calculations of the viscous interaction N parameter along a plate have shown that the conditions considered correspond to a condition of weak interaction everywhere, except for the nearest neighborhood of a leading edge of a plate, where, however, the asymptotic theories used are untrue. The pressure induced on a surface and caused by viscous interaction between an incident flow and a boundary layer, which is being developed above a plate surface, after an abrupt increase in a neighborhood of its leading edge gradually decreases downstream. As the surface is heat insulated, the temperature of gas immediately at the surface gradually increases downstream. At the same time, the vertical component of a velocity reaches maximum positive values in the field of the greatest pressure, i. e., at the front of the induced shock wave.

370 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field

7.5.2 Heating electrodes Let us consider how the heating up of electrodes arranged on a plate influences a flow field. Dielectric surfaces are still considered heat insulated. Let us note two peculiarities of the calculated data presented on Figure 7.1. Firstly, with growth of temperature of electrodes a natural near-electrode increase of pressure is observed which leads to formation of shock waves departing from areas of local surface heating. Secondly, in the process of growth of temperature, a disturbed pressure profile widens noticeably that indicates disturbance transfer not only downstream, but also against it, so that it is possible to refer to an effect of free interaction. Let us note one more important effect. At high temperature of electrodes (Tw D 800 K) the flow separation is observed behind them, which is indicated by a negative value of the surface friction coefficient Cf < 0 (Figure 7.2) and by the origination of areas with a negative velocity, which are evidently visible in computer visualizations of a flow field. The aerodynamic coefficients shown on Figure 7.2 are calculated under the formulas ˇ ˇ  @u @y ˇyD0 p  p1

w Cp D 1 , Cf D 1 D 1 . (7.17) 2 2 2 2 1 V1 2 1 V1 2 1 V1 Thus, local heating up of a streamline plate leads to an appreciable modification of gas-dynamic structure. At a temperature of Tw D 800 K, pressure near to the first area of heating grows twice, and near to the second area 1.5 times. However, it is necessary to mention that data on relative increasing pressure is tightly connected with a configuration of heated up bands (electrodes). Below it will be shown that for the second variant of disposition of electrodes (the cathode is arranged closer to a leading edge of a plate) pressure at the cathode quintuples.

7.5.3 The surface discharge For researching the influence of surface discharge on gas dynamics, rated variants with a temperature of electrodes Tw D 600 K have been chosen that correspond to experimental data [67]. Before discussing the results of numerical modeling, let us note that deriving the stationary solution of an electrodynamic part of the problem, which describes the spatial distribution of charged particles and an electric potential, appears rather laborconsuming and demands the setting of good initial conditions. A principal cause of this is the exponential dependence of an ionization velocity (the right member of the equation (7.7)) on an electric field strength and gas-dynamic parameters (pressure, temperature). A significant number of pre-computations was spent with the purpose of determining the least parameters of an external circuit at which steady existence of surface discharge has been reached. At the conditions considered, such parameters

371

Section 7.5 Numerical simulation results Cf, Cf 0.07 1 – Cf 2 – Cp 3 – Cf 4 – Cp

0.06 0.05 0.04 0.03 0.02 0.01 0 −0.01 0

5

10

15 x, cm

Figure 7.2. Distribution of aerodynamic coefficients along a surface of a heat-insulated plate with heated up electrodes without discharge: (1, 3) Cf ; (2, 4) Cp ; (1, 2) Tw D 400 K; (3, 4) Tw D 800 K.

have appeared E D 800 V, R0 D 800 Ohm. Parameters of surface discharge in a supersonic stream of gas at great values of emf are calculated by means of gradual increasing the last parameter. Results of numerical simulation E D 1.2 kV and R0 D 12 kOhm are presented in Figure 7.3. A field of current at a plate with discharge indicates the insignificant modifications in the distribution of pressure along a surface in comparison with a case without discharge. Small growth of gas pressure on a surface between electrodes (approximately 1.13 times) is caused, probably, by a small heating of gas in the most surface discharge. It is necessary to emphasize that in the conditions considered, Joule heating up of gas near to electrodes appears smaller than its convective heating up from electrodes. The electrodynamic structure of surface discharge is shown in Figure 7.4. It is visible that the distribution of concentration of charges (a) and the module of a current q 2 2 density jjj D jx C jy (b) above a surface of electrodes is very inhomogeneous, and their absolute values reach 1011 cm3 and 0.5 mA/cm2 , respectively. Let us also note natural asymmetry in the distribution of concentration and current density, connected with a small drift of charges in a supersonic gas flow (from the left to the right). The electric field strength is shown in Figure 7.4 (c). Length of each arrow of a vector field corresponds to an absolute value of ratio jEj=E in the given point of space. The absolute value E=L is given as a scale on the figure margin. A representation of the electrodynamic structure of discharge is given also by Figure 7.4 (d), where the electric potential is shown. Let us note that according to the set boundary conditions

372 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field p/p∞

2

1 2 3 4

1.8

1.6

1.4

1.2

1

0

2

4

6

8

10

12

14

x, cm

Figure 7.3. Distribution of pressure along a surface of a heat-insulated plate with heated up electrodes (Tw D 600 K). The numerical solution for the first configuration of heated up electrodes: (1) without discharge; (2)–(4) with discharge, E D 1.2 kV, R0 D 12 kOhm; (2) B D 0, (3) B D 0.1 T, (4) B D C0.1 T.

y, cm

y, cm 5

5

(a)

3 7

1 0 1 15 14 13 12

9

3 8.11 5.93 4.33 3.16

8

5 11 10 9 8

5 1

3

89

7

9

2.31 1.69 1.23 0.90

7 6 5 4

11 13 15 x, cm 0.66 0.48 0.35 0.26

3 0.19 2 0.14 1 0.10

3

15 14 13 12

5

3 0.46 0.35 0.27 0.20

4

5 11 10 9 8

3 2

7 0.15 0.12 0.09 0.07

1

9 7 6 5 4

3

5

7

9

11 13 15 x, cm

y, cm 5 10 8 9 11

3 2

1 0 1

(b)

1 0 1

1 0 1

3

y, cm 5

(c)

3

(d)

12 13 14

7 4 5 6

0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.05 0.04 0.03 0.02 0.01

15

1

3

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

5

7

9

11 13 15 x, cm

11 13 15 x, cm 0.05 0.04 0.03 0.02

3 0.02 2 0.01 1 0.01

Figure 7.4. Electrodynamic structure of surface discharge at Tw D 600 K, E D 1.2 kV, R0 D 12 kOhm: (a) concentration of the charged particles, 1010 cm3; (b) module of current density, mA/cm2; (c) electric field; scale of a vector field – E/L; (d) electric potential '=E.

Section 7.5 Numerical simulation results

373

on the whole boundary of the rated area, except for electrodes, for an electric potential the Neumann boundary conditions are realized @' @' D D 0. @x @y Calculations of flow round a plate with a direct current surface discharge, but at the increased temperature of electrodes Tw D 800 K at constant parameters of an electric circuit E D 1.2 kV and R0 D 12 kOhm have shown minor changes in the gas-dynamic structure, namely the local growth of pressure near to the first electrode (cathode). Comparing the data obtained with outcomes of calculation without surface discharge, it is possible to conclude that at the specified parameters of an electric circuit and such a high temperature of electrodes, the influence of surface discharge on a flow field is insignificant in comparison with the influence of the heated up electrodes. Let us consider the influence of an external magnetic field on a flow round a plate. Outcomes of calculations of a flow round a plate with surface discharge in an external magnetic field with induction B D 0.1 T at E D 1.2 kV and R0 D 12 kOhm are shown in Figure 7.3. The results obtained are comparable to the case of similar surface discharge without a magnetic field. Presence of an external cross-stream magnetic field leads to the origination of additional volumetric forces acting on a gas flow in the region of surface discharge. Therefore, first of all, changes in the structure of a flow are caused by these forces. In this case, the direction of the magnetic field is such (in a negative z direction, see Figure 6.1) that near to the cathode along an axis x the positive component of volumetric magnetic force has appeared, and near to the anode – the negative component. The vector of a current density j  e ne E coincides in a direction with force lines of the electric field strength, shown in Figure 7.4 (c). It allows to judge a direction of a vector j in each point in space. Then, in accordance with the left-hand rule, the direction of magnetic force is also easily determined. As the current flows towards to a gas flow, basically, in the case considered, the component of magnetic force FM ,y is directed to the streamlined plate. This explains the appreciable raise of pressure upon the plate between electrodes, and the interaction of incident flow with disturbed flow field above a surface of the anode leads to an additional increase of pressure above the anode and to the occurrence of a detached flow zone directly ahead of the anode. More evidently, the given effect is exhibited at growth in the absolute value of magnetic field induction, which leads to a natural increase in magnetic force influence on flow round a plate with discharge. Also the reverse flow area in front of the anode increases noticeably. However, it is necessary to mention that the calculation with the model presented at jBj > 0.1 T is not justified because of a sharp increase in the Hall parameter that is not considered in the model (see Chapter 6). The influence of the raised temperature of electrodes (Tw D 800 K) on flow round a plate with surface discharge in a magnetic field was also studied. The remaining initial

374 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field data were the same as in the previous variant. In this case, increasing the temperature of the electrodes had practically no influence on the gas-dynamic structure of surface discharge. With use of the same initial data an attempt to calculate a structure of discharge was undertaken at the opposite orientation of the magnetic field induction vector, B D C0.5 T. However, such a calculation was not successful due to divergence of numerical results. The greatest value of magnetic field induction at which it was possible to receive the steady-state self-consistent solution of gas-dynamic and electrodynamic parts of the problem was B D C0.2 T. The comparison of the calculation data at B D 0.1 T and B D C0.1 T (Figures 7.3 and 7.5) shows that a change of direction of magnetic force components to their opposite rather noticeably changes parameters of flow round even at not so great inductions of a magnetic field. In front of the anode there is an area of under pressure, though flow separation is not observed. At B < 0 components of a magnetic field change their sign. Now the longitudinal component of magnetic force is negative (decelerates the incident flow) directly behind the cathode, and it is positive (accelerates the flow) in front of the anode. The cross flow component of a magnetic field is positive. 0.06

Cf , Cp 1 2 3 4

0.04

1.02

0 0

5

10

15 x, cm

Figure 7.5. Distribution of aerodynamic coefficients along a surface of a heat-insulated plate with heated electrodes (K) Tw D 600 at, E D 1.2 kV, R0 D 12 kOhm; (1, 3) Cf ; (2, 4) Cp ; (1, 2) B D 0.1 T; (3, 4) B D 0.1 T.

The second series of evaluations was made for another configuration of electrodes on a streamline plate: the cathode was placed closer to a leading edge on distance xC  xB D 0.476 cm, and the distance between electrodes was approximately 1 cm shorter.

375

Section 7.5 Numerical simulation results

In Figure 7.6 the calculation data for a heat-insulated plate with heated up electrodes (Tw D 600 K) are presented. The sharp growth of pressure (five times) near to the first electrode on a stream attracts attention. It is rather an important result, which suggests one possible method of flow control even without use of electric discharge. p/p∞ 5 1 2 3 4 5 6

4 3 2 1 0 0

2

4

6

8

10 x, cm

Figure 7.6. Distribution of pressure along a surface of a heat-insulated plate with heated up electrodes (Tw D 600 K). The second configuration of electrodes: (1) numerical solution for a heat-insulated plate with heated electrodes without discharge; (2) asymptotic theory of weak interaction; (3) asymptotic theory of strong interaction; (4)–(6) numerical solution for a heatinsulated plate with heated up electrodes and with discharge, E D 800 V, R0 D 12 kOhm; (4) B D 0; (5) B D 0.2 T; (6) B D C0.2 T.

Flow round plates with surface discharge, whose parameters of external electric circuit are E D 800 V, R0 D 12 kOhm, Tw D 600 K, is illustrated by the data given also in Figure 7.6. Presence of surface discharge leads to an appreciable diminution of the pressure peak near to the cathode and to a small growth in pressure in the field between electrodes, which was also observed in the first rated case. Such growth of pressure is connected, most likely, with small volumetric heating of gas in surface discharge. Near to electrodes, the Joule thermal emission power reaches 40 W/cm3. The analysis of distributions of the charged particles’ concentration, current density and electric field shows that appreciable increases in all functions near to boundaries of electrodes is common for them. At the analysis of the electrodynamic structure of surface discharge, the relation of volumetric velocities of ionization ni and recombination ˇn2 is important. In the case considered, the velocity of ionization essentially surpasses the velocity of recom-

376 Chapter 7 Viscous interaction on a flat plate with surface discharge in a magnetic field bination even in view of the effective recombination coefficient, which exceeds the usually used value approximately 100 times. Inclusion of an external cross magnetic field B D 0.2 T in the calculation causes appreciable growth of pressure in a surface downstream from the cathode (see Figure 7.6), but practically does not influence the pressure before and above the cathode. As well as in the first rated variant, presence of a magnetic field leads to origination of detached flow in front of the anode. We shall recall that at negative values of a magnetic induction (the field is z directed) the component of magnetic force along an axis x is positive at the cathode and negative at the anode. The component FM ,y is positive above a surface between electrodes. At the magnetic field induction of B D 0.2 T, maximum pressure in front of the anode increases 2.6 times in comparison with the pressure of incident flow, but nevertheless remains smaller than in front of the cathode, where its step is caused by heating up the electrodes, as has been mentioned above. Any basic modifications in the electrodynamic structure of surface discharge at inclusion of a magnetic field has not been observed, however, it is possible nevertheless to note physically natural drop of charged particle concentration and current density, including a total current through electrodes, that leads to an increase in a voltage drop on electrodes (the fact established in experiments on the influence of a magnetic field on integral parameters of an electric circuit). Just like for the first variant of calculations, at change of direction of magnetic field induction vector (B D C0.5), instability of calculations is discovered. The steady self-consistent solution of the problem has been received only at B D C0.2 T. If a calculation variant is chosen without a magnetic field for the comparative analysis, it is possible to state that near to the cathode practically nothing has varied, which confirms the supposition about a leading role of thermal processes in this case. Near to the anode pressure drops below values in incident flow (see Figure 7.6). It meets the fact that a positive component of magnetic force repels gas from a surface. Increasing emf up to several kilovolts in all the rated cases leads to an appreciable raise in the pressure at a surface of the anode and to a natural increase of concentration of the charged particles and a current in the electric discharge gap, that is the reason for growth of magnetic force at the same induction of a magnetic field. In summary, we shall note once again the restrictions of the developed model which are caused by the initial suppositions that led in its basis, namely by the assumption about quasi-neutrality of gas-discharge plasma, and also by neglecting magnetic field influence on a drift of the charged particles and on velocity of ionization (actually by neglecting the Hall parameter). Estimations show that at pressures greater then several Torrs, and at inductions of magnetic field B < 0.1 T the offered model is quite substantiated. With growth of pressure, the upper admissible boundary of magnetic field induction increases also. The calculations fulfilled in the given chapter meet the use admissibility boundary of the specified model.

Section 7.5 Numerical simulation results

377

So, numerical calculations have shown that local heating up of surfaces of electrodes up to Tw D 800 K essentially influences aerodynamic performances of flow round a plate and surpasses the influence of surface discharge burning along it. The cross external field can lead to an increase in pressure upon a surface and exceed the growth of pressure caused by heat of electrodes. The direction of an external magnetic field is the important parameter of the problem. At B > 0 (in a positive z direction) the conditions occurs for origination of instabilities in a flow field and in the electrodynamic structure of discharge. The effect of heating up of the electrode (cathode), first along the flow, increasingly influences the structure of the gas flow in the process of distance diminution between a leading edge and an electrode. In the cases considered, heating up of the cathode was a major factor defining gas dynamics in the region from a leading edge of a plate up to a trailing edge of the cathode. The presence of surface discharge without a magnetic field at the investigated parameters modifies a flow above a surface of a streamline plate to an insignificant degree, reducing pressure jump near to the cathode and increasing pressure in the gap between electrodes and above the anode.

Chapter 8

Hypersonic flow of rarefied gas in a channel with glow discharge in an external magnetic field A two-dimensional computational model is used in this chapter to investigate the gasdynamic structure of a rarefied (pressure of several Torr) hypersonic flow of molecular nitrogen in a plane channel, with a glow discharge maintained between two surfaces of this channel. An asymmetric configuration of electrodes is considered, namely, the cathode is a narrow strip located on the lower surface of the channel, and the upper surface of the channel is a continuous anode. The fundamental possibility is studied of using an external magnetic field transverse to the flow for modifying the shock wave structure of flow in the channel. A two-dimensional conjugate electro-gasdynamic model is considered, which includes equations of continuity, Navier–Stokes equations, equations of conservation of energy, and equations of continuity of charged particles in ambipolar approximation. The real thermophysical and transport properties of molecular nitrogen are included. It is demonstrated that the use of a surface glow discharge in a rarefied hypersonic flow enables one to effectively modify the shock wave structure of flow and, thereby, consider discharges of this type as additional means of control over rarefied gas flows. The present study involves the formulation and investigation of a model of hypersonic flow in a plane channel which simulates a prototype of an air intake unit of a hypersonic flight vehicle (HFV), on the walls of which a glow discharge is initiated between the cathode section and the continuous anode. A hypersonic flow of rarefied nitrogen with the following parameters is preassigned at the inlet to the plane channel: p0 D 2 956 erg/cm3 , 0 D 4.0  106 g/cm3 , T D 257 K, and V0 D 128 500 cm/s. These conditions of incident flow are partly realized in experimental investigations [67] and approximately correspond to the flight of HFV at velocity M D 4 at an altitude of 40 km. The principal objectives of the present investigation include: 

formulating a numerical model of motion of a viscous heat-conducting gas in view of special features of the formation of gas-dynamic structure upon ignition of a glow discharge in a hypersonic rarefied flow and modification of this discharge by an external magnetic field. The special feature of this model consists in the matching of gas-dynamic and electric discharge processes in view of strong nonlinearity of avalanche ionization and of the heating of gas;



studying the possibility of modification of hypersonic rarefied gas flow in channels which simulate the operation of air intake units of HFV, by way of applying glow discharges with magnetic field; and

379

Section 8.1 Model of gas dynamics 

performing numerical investigations of the thus developed model for the purpose of studying its properties relative to the variation of predicted parameters.

8.1 Model of gas dynamics We consider the flow of a viscous heat-conducting partly ionized gas in a square channel, on the surfaces of which a glow discharge is burning and a magnetic field is applied across the flow. The glow discharge is initiated between the electrode section and the opposite plane of the channel. The problem to be solved is shown schematically in Figure 8.1. Anode E y=H U0

B Cathode

y

z

xc1 xc2

x=L

x R0

Figure 8.1. Schematic of the problem.

The set of equations of dynamics for viscous heat-conducting partly ionized gas has the following form: @ C div .V/ D 0, (8.1) @t @u @p 2 @ C div .uV/ D   . div V/ @t @x 3 @x @ h  @u @v i @  @u C  C C2  C fM ,x , @y @y @x @x @x @p 2 @ @v C div .vV/ D   . div V/ @t @y 3 @y @v i @  @v @ h  @u  C C 2  C fM ,y , C @x @y @x @y @y  cp

@p @T C  cp V grad T D +div. grad T / C V grad .p/ C Q C QJ , @t @t

(8.2)

(8.3)

(8.4)

380

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

where x and y are the longitudinal and transverse Cartesian coordinates; V D .u, v/ is the flow velocity vector and its projections on the x and y axes;  and p denote the density and pressure, respectively;  is the dynamic viscosity; cp is the heat capacity at constant pressure; T is the gas temperature;  is the thermal conductivity coefficient; fM x and fMy are components of ponderomotive force; QJ is the volumetric power of heat release, caused by the flow of electric current; and Q is the dissipative function, h  @u 2  @v 2  @v @u 2 2  @u @v 2 i C C . C2 C  Q D  2 @x @y @x @y 3 @x @y

8.2 Model of electrodynamics of glow discharge in a magnetic field A model of quasi-neutral plasma is used for describing electric discharge effects. The classical model of quasi-neutral plasma of glow discharge is based on the following two equations (see Chapter 6): @n C div .Vn/ D div .Da grad n/ C !P e , @t

(8.5)

div j D 0,

(8.6)

where !P e D ˛.E/j e j  ˇe n2 is the volume velocity of formation of charged particles, which is defined by collisional ionization of neutral particles (the first term) and by ion-electron recombination; ˛.E/ is the collisional ionization coefficient (the first Townsend coefficient); ˇe is the coefficient of ion-electron recombination; j e j is the magnitude of the density vector of electron flow; E is the modulus of electric field intensity; Da is the ambipolar diffusion coefficient; j D e. i   e / is the vector of current density;  i and  e are the density vectors of ion and electron flows, respectively. The approximation of quasi-neutral plasma will be employed, therefore n D ni D ne , where ni , and ne denote the concentration of ions and electrons, respectively. Equation (8.6) is one of the system of Maxwell equations and expresses the law of conservation of electric current under steady-state conditions. Equation (8.5) expresses the law of continuity of charged components of partly ionized gas. The following assumptions were made in deriving this equation: 

the velocities of charged and uncharged particles in partly ionized gas are equal, V D Ve D Vi (V is the average velocity of neutral particles; Ve and Vi are the average velocities of electrons and ions, respectively);



the concentrations of ions and electrons are equal (n D ne D ni );



charged particles diffuse with the ambipolar diffusion coefficient Da .

381

Section 8.3 Boundary conditions of the problem

The quasi-neutral model of glow discharge (8.5) and (8.6) was modified in Chapter 6 to include the magnetic field, @n C div .Vn/ D div .e Da grad n/ C !P e , @t

(8.7)

div Œ.i C Q e /nE C .DQ e  Di / grad n D 0,

(8.8)

where  Qe D

e , 1 C be2

e D

e C i e Š , 2 2 .1 C be /i C e .1 C be /i C e

DQ e D

De , 1 C be2

be is the Hall parameter for electrons, and e is the electron mobility. The volume rate of formation of charged particles on the right-hand side of equation (8.5) may likewise be modified upon inclusion of a magnetic field, !P e D ˛.E/Q e E  ˇn2 . Equation (8.8) formulated above is used for finding the electric field which arises under conditions of ambipolar motion of ions and electrons in partly ionized gas. In view of the fact that the electric field is related to electric potential ' by the relation E D  grad ' we use the equation for potential @ h @h @' i @' i .i C Q e /n C .i C Q e /n @x @x @y @y h @ h Q @n i @n i @ .DQ e  Di /  .De  Di / D 0.  @x @x @y @y

8.3 Boundary conditions of the problem The boundary conditions for integration of the set of equations (8.1)–(8.8) are formulated as follows (see Figure 8.1): u D U0, xD0: xDL: y D 0,

v D 0,

T D T0 ,

@' D 0; @x @v @T @n @ @' @u D D D D D D0 @x @x @x @x @x @x p D p0 ,

 D 0 ,

x > xC1 ,

n D n0 ,

u D v D 0,

@p D 0; @y

382

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

y D 0, xC1  x  xC2

y D H; 0  x  L

(cathode section): @n D 0, @y

' D 0;

@n D 0, @y

' D Vc ;

(anode):

y D 0; x < xC1 , x > xC2

(dielectric surface): n D 105n0 ,

@' D 0; @y

y D H, x > 0 : @p D 0. @y The surface temperature Tw D 300 K in all calculations was taken to be constant; here, n0 is the typical concentration of charged particles in glow discharge (n0  1010 cm3 ), and Vc is the voltage drop between the cathode and anode sections. The equation for external electric circuit u D v D 0,

IR0 C Vc D E gives the voltage drop between the electrodes. Here, R0 is the resistance of external circuit; and I is the total current which is determined by integration over the electrode surfaces, Z Z L

I D 0

L

.jn/c dx D

.jn/a dx 0

where n is the unit normal vector to the surface of cathode (c) and anode (a), and j is the current density on the electrodes. All quantities are related to unit length in the direction of the z axis. The unperturbed flow in the channel and the plasma cloud with the concentration of charged particles n0 above the cathode section were preassigned at the initial instant in time. The problem was solved by the time relaxation method; therefore, the form of initial conditions was of no fundamental importance. Nevertheless, poor initial conditions for glow discharge plasma resulted in the divergence of iteration procedure of the solution of the problem.

8.4 Closing relations We consider the motion of molecular nitrogen (N2 /, for which the following relations of molecular kinetic theory are employed: p 1 m D 2.67  105 MA T 2 .2,2/ , g/(cm  s) 

383

Section 8.4 Closing relations

is the dynamic viscosity coefficient,

.2,2/ D 1.157.T  /0.1472,

 D 3.68 Å,

s  D 8.334  10

4

T D

T , ."=k/

."=k/ D 71.4,

cp

T .0.115 C 0.354 R0 MA / , W/(cm  s) MA  2 .2,2/

is the heat conductive coefficient; cp D 8.317  107

7 1 erg/(g K), 2 MA

D

p MA g/cm3 , T R0

MA D 28 g/mole,

R0 D 8.314  107 erg/(mole  K). The ionization rate is calculated using the semi-empirical formulas, which were used in previous chapters i D

 ˛ p  Ee .p  /, p

h ˛ B i , (cm  Torr) D A exp  p .E=p  /

where: p D p

293 , T

De D e .p  /Te ,

DC D C .p  /TC , cm2 /s,

1 1 , C.p  / D 1450  , cm2 /(s  V)  p p 7 3 ˇ D 2  10 cm /s, Te D 11610 K,

e .p  / D 4.2  105

A D 12 (cm  Torr)1 ,

B D 342 V/(cm  Torr),

C is the ion mobility. The heat release in gas owing to the flow of electric current (the Joule heat release) is calculated by the obtained current density and electric field intensity, QJ D .j  E/ D 1.6  1019 ŒnE 2 .C C Q e / C .DQ e  DC/E grad n

(8.9)

where is the coefficient of effectiveness of heating the gas. The pre-assignment of this coefficient in the 0.1–0.3 range enables one to partly take into account the imperfection of the employed model in regard to the description of the kinetic physicochemical processes. In the case where the transverse magnetic field is included, the bulk force acting on partly ionized gas FM D c 1 ŒjB is calculated as well; the projections of this force on the x and y axes are included in equations (8.2) and (8.3).

384

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

8.5 Algorithm of solution of complete set of equations The set of equations (8.1)–(8.4), (8.7), and (8.8) of motion of a viscous heat-conducting partly ionized gas in a plane channel with glow discharge and external magnetic field were solved with the time relaxation method until reaching numerical convergence with a relative error of 105, calculated over the entire flow field for the functions u, v, p, T , n, '. The following groups of equations were successively integrated at every step of fictitious time: 

the equation of continuity and Navier–Stokes equation,



the energy equation, and



the set of equations of electrodynamics of glow discharge in a magnetic field.

The gas-dynamic equations were integrated using explicit method based on the AUSM finite-difference scheme. The energy equation was integrated by implicit method using a five-point finite-difference scheme. The set of electrodynamic equations were integrated by an implicit five-point finite-difference scheme using internal iterations for attaining mutual convergence of the concentration of charged particles and electric potential. This part of the computational algorithm was most laborious by virtue of significant (exponential) nonlinearity of the right-hand part. The second internal iteration process (at each time step) consisted in attaining mutual convergence of the functions in solving the Fourier–Kirchhoff equations (8.4) and the set of electrodynamic equations. This iteration process is essential because the gas temperature has a very strong effect on the rate of ionization of gas, which likewise introduces strong perturbations in the right-hand part of equation (8.7). The significant nonlinearity of equation (8.7) turns out to be very critical to the entire computational algorithm. This also true of the need for pre-assigning reasonable initial approximations for solving the problem of determining the concentrations of charged particles and the electric potential. In our study, such initial conditions were provided by the positive column of glow discharge with parameters determined from the Engel– Steenbeck theory. Then the solution of electrodynamic and energy equations was found by iterations. And only after that was the iteration procedure of searching for the solution of the problem as a whole started.

8.6 Numerical simulation results The parameters of flow of molecular nitrogen (N2 / at the inlet to the channel under investigation are given in Table 8.1; this corresponds to the following criteria: M D 0 U0 L0 U0 D 6 290 for L0 D 20 cm, and Re D 12 580 for L0 D 4.0 cm. a0 D 4, Re D 0

385

Section 8.6 Numerical simulation results

Table 8.1. Conditions in the inlet cross section. p0, erg/cm3 0, g/cm T0 , K V0 , cm/s

3

2 956. 4.0  106 257. 128 500.

Table 8.2. Configuration of electrodes. Lower surface Configuration No. 1 Configuration No. 2

xC 1, cm

xC 2 , cm

L0 , cm

H , cm

2.0 4.0

3.0 5.0

20 40

2.0 4.0

The electrode configurations given in Table 8.2 were considered. The general plan of numerical investigation was as follows. (1) First, the calculation was performed of the flow in the channel of the first configuration (L0 D 20 cm and H D 2 cm) without glow discharge. (2) Then the structure of flow was calculated in the channel of the first configuration with glow discharge and with the gradually increasing emf of the power supply E D 500, 1 000, 1 500 V; the resistance of the external circuit was taken to be invariable, R0 D 12 000 Ohm. (3) The gas-dynamic structure of channel with discharge was investigated for different directions of magnetic field with induction Bz D 0.5 T. (4) A similar computational series was repeated for the channel of the second configuration with discharge. The temperature of the surfaces, including the cathode and anode surfaces, was taken to be constant (Tw D 300 K), thereby eliminating the effect of the heating of electrodes on the gas-dynamic structure of flow. The gas-dynamic structure of flow in the channel of the first configuration without discharge is given in Figure 8.2. The pressure of unperturbed gas flow at the channel inlet is p0 D 2.25 Torr. At a distance of  3.5 cm from the inlet, at the point of intersection of two shock waves arising at the channel inlet from the upper and lower surfaces of the channel, i. e., the equalizing and rise of pressure (a), density (b), and temperature (c). The transition region of appreciable fluctuations of all functions is observed up to a distance of  20 cm, i. e., at approximately ten diameters of the inlet cross section.

386

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

(a)

y, cm

Pres: 2.2 2 1 0

0

y, cm y, cm

(c)

1 0

2.4

2.5

2.7

10

2.9

0.9

0.9

1.0

2

4

6

8

10

12

14

16

18

T: 260

278

295

313

331

348

366

384

401

4

6

8

10

12

14

16

18

1.2

16

3.2

0.8

1.1

14

3.1

6

1.0

12

3.0

4

0

8

2.8

2

Ro: 0.8 2

(b)

2.3

18 1.2

20 1.3

20

2 1 0

0

2

20

(d)

y, cm

QEJ: 1.0E+06 3.4E+06 1.1E+07 3.8E+07 1.3E+08 4.4E+08 2 1 0

0

2

4

6

8

10

12

14

16

18

20 x, cm

Figure 8.2. The pressure in Torr (a), the density =0 (b), and temperature in Kelvins (c) in a plane channel without discharge and the volumetric power of heat release ((d); in erg/cm3) in a glow discharge at E D 1 000 V.

The ignition of glow discharge in the configuration given in Figure 8.1 introduces the following special features into the effect under consideration. The asymmetry of the electric discharge scheme in the channel (segmented cathode and solid anode) results in the asymmetry in the distribution of the fields of gas-dynamic functions. The ignition of glow discharge in the channel implies the emergence of a localized region of heat release in the vicinity of the cathode section. The configuration of this region at E D 1 000 V is given in Figure 8.2 (d). One can see that the maximal power of heat release (QJ ,max  4.4108 erg/(s  cm3)) is attained in direct vicinity of the upstream boundary of the cathode section. In accordance with equation (8.9), the configuration of the region of heat release is defined by two factors, namely, by the electron concentration and by the electric field intensity. The electric field intensity experiences the maximal changes in the neighborhood of the boundaries

387

Section 8.6 Numerical simulation results

of the cathode section. However, because of the gas flow, the concentration of charged particles increases towards the rear boundary of the cathode. One can see in Figure 8.2 (d) that the heat is released in the entire region of electric discharge gap where the current flows. However, the power of this heat release is two orders of magnitude lower than that in the vicinity of the cathode section. The emergence of slight asymmetry in the distribution of pressure is observed (Figure 8.3) as a result of the emergence of a localized region of heat release in the vicinity of the lower boundary of the channel at x  3 cm. One can see that a low intensity shock wave departs from the end of the cathode section. As a result, asymmetry of the pressure field is observed in the entire computational domain and shows up ever more clearly downstream. By and large, one can observe a not too significant effect of local heating of glow discharge on gas dynamics. However, one must take into account the small expenditure of energy for maintaining the glow discharge and for localizing the region of heat release at which the mode of weak energy input to discharge is realized. The total current through the discharge in this case is I D 0.0463 A/cm. Therefore, the electrical power spent for maintaining the discharge is just  25 W/cm (recall that all quantities are related to unit length of electrodes in the direction of the z axis).

y, cm

Pres: 2.2 (a)

2.3

2.4

2.5

2.7

2.8

2.9

3.0

3.1

3.2

2 1 0

0

2

4

6

8

10

12

14

16

18

20 x, cm

2

y, cm

1.5 (b)

1 2.7

0.5 0

2.6 2.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5 x, cm

Figure 8.3. The pressure ((a); in Torr) in a channel with discharge and distribution of pressure (b) in the vicinity of the cathode section; E D 1 000 V, B D 0.

If an external magnetic field is applied to the glow discharge (see schematic in Figure 8.1), bulk forces arise in weakly ionized gas-discharge plasma, which are caused by the effect of this magnetic field on charged particles (the ponderomotive forces).

388

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

A fundamentally important feature of magnetic effect on partly ionized gas is the direction of induction vector Bz . Figure 8.1 gives two configurations of magnetic field investigated in the present study (in the positive and negative directions of the z axis). The results of calculations of the bulk forces for two configurations of Bz are given in Figure 8.4. FMAGX: 2.0E+03

2

1.0E+03

y, cm

1.5

8.1E+03

4.0E+03

3.2E+04

1.6E+04

6.5E+04

(a)

FMAGX: –4.1E+04 –5.0E+04

–2.3E+04

–5.5E+03

–1.4E+04

–3.2E+04

(c)

1 0.5 0

FMAGY: –4.2E+04 –5.0E+04

y, cm

1.5

–2.5E+04

–3.4E+04

–9.1E+03

FMAGY:

2.7E+03

1.0E+03

–1.7E+04

(b)

1.9E+04

1.3E+03

5.1E+04

7.1E+03

(d)

1 0.5 0 2

2.5

3

3.5

4

4.5

5

5.5 2 x, cm

2.5

3

3.5

4

4.5

5

5.5 x, cm

Figure 8.4. The projection of ponderomotive force onto the axes (a,c) x and (b,d) y; E D 1 000 V; (a, b) Bz D 0.5 T, (c, d) Bz D C0.5 T.

The bulk forces acting on partly ionized gas have two components each, with one of these components decelerating (or accelerating) the flow and the other one forcing the gas flow away from (or pressing it against) the surface. The sign of projection of ponderomotive force enables one to predict the impact it makes on moving gas. In the case of magnetic induction Bz D 0.5 T, the ponderomotive force is positive in the direction of the x axis (Figure 8.4 (a)) and negative in the direction of y axis (Figure 8.4 (b)). This means that the ponderomotive force accelerates the gas in the direction of the x axis and presses the gas flow against the lower base. We will compare this calculation result with qualitative reasoning based on the theory of electromagnetism. The ponderomotive force component fM x is positive because the positive direction of current is ascribed to the direction of ion motion (in this case in the downward direction, from the anode to cathode), and the direction of magnetic field coincides with the negative direction of the z axis. By the left-hand rule (corollary of formula FM D c 1 ŒjB/, the force acting on electric current is directed rightwards, i. e., in the positive direction of the x axis.

389

Section 8.6 Numerical simulation results

The ponderomotive force component fM ,y is negative (directed downwards) because a tube of current in which the positive charges move from right to left (toward the main flow of neutral particles) will always be found in the gas discharge region due to the drift of ions by the gas flow in the positive direction of the x axis. By the left-hand rule, the ponderomotive force is directed downwards, i. e., is negative. The results of detailed analysis of the components of ponderomotive force enable one to formulate recommendations for developing electromagnetic actuators: the change of sign of ponderomotive force components may be accomplished by reversing the polarity of the electrode sections. Figures 8.4 (c), (d) give an example of change of sign of ponderomotive force components with changing direction of magnetic field. As previously, the current flows downwards (from the anode to cathode) and leftwards (due to the drift of charges downstream of the gas flow). As a result, we have the force component fM ,x < 0, i. e., the ponderomotive force decelerates the overall flow, and the force component fM ,y > 0, i. e., the ponderomotive force forces the gas flow in the boundary layer away from the surface. However, it must be emphasized that the calculation results given in Figure 8.4 are indicative of significant nonuniformity in the distribution of bulk ponderomotive force. A strong force impact is localized in the vicinity of the downstream boundary of the cathode electrode section. In spite of this localization of the force impact on the flow, appreciable transformation of the shock wave structure is observed in the channel in the entire flow field. Figure 8.5 give the pressure field for the cases of Bz D C0.5 T and Bz D 0.5 T.

(a)

y, cm

Pres: 2.2 2 1 0

0

(b)

y, cm

Pres: 2.2 2 1 0

0

2.3

2.4

2.5

2.7

2.8

2.9

2

4

6

8

10

12

2.3

2.4

2.5

2.7

2

4

6

8

2.8

10

2.9

12

3.0

14

3.1

16 3.0

14

3.2

18 3.1

16

20 3.2

18

20 x, cm

Figure 8.5. The pressure (in Torr) in a channel with discharge at E D 1 000 V: (a) Bz D 0.5 T, (b) Bz D C0.5 T.

In accordance with the results of the analysis given above, a significant increase in pressure on the surface is observed at Bz D 0.5 T. The shock wave departing from

390

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

the place of action of ponderomotive force exhibits a gentler slope to the surface than that of the primary shock wave from the inlet cross section. At Bz D C0.5 T, on the contrary, the shock wave departs from the place of action of localized ponderomotive force, and a zone of rarefaction forms in the vicinity of the surface. The observed special features of the action of ponderomotive force on the flow structure affect both the structure of shock wave interaction in the channel and a number of important aerodynamic characteristics such as the distribution of pressure and convective heat fluxes along the channel surfaces and the variation of the pressure level downstream. Figure 8.6 gives the configurations of regions of heat release for the cases of Bz D C0.5 T and Bz D 0.5 T. One can see that in these two cases the polarization of magnetic fields causes an appreciable variation of the configuration of these regions.

QEJ:

y, cm

2

1.0E+06

3.4E+06

1.1E+07

3.8E+07

1.3E+08

4.4E+08

(a)

1 0 (b) 1 0

0

2

4

6

8

10

12

14

16

18

20 x, cm

Figure 8.6. The volumetric power of heat release (erg/cm3) in a glow discharge at E D 1 000 V; (a) Bz D 0.5 T, (b) Bz D C0.5 T.

Therefore, the use of an electromagnetic actuator involving a glow discharge and external magnetic field enables one to modify a rarefied hypersonic flow owing to the following two principal factors, namely, local heating of gas and locally induced ponderomotive force. In both cases, the stimulation of gas flow exhibits a volumetric localized pattern. Naturally, the foregoing factors are characterized by different mechanisms influencing the flow. In the case under consideration, the thermal action on the flow is lower in intensity than the ponderomotive action. Nevertheless, the thermal action is quite appreciable, and studying the regularities of such action is of practical interest. Figure 8.7 gives the distribution of pressure along the upper and lower surfaces of the channel with a gradually increasing emf of the power supply. The case of E D 0 corresponds to the flow in a channel without glow discharge. One can see that, as E increases, a pressure rise above the cathode section is observed. In so doing, one can see the tendency of the region of abrupt pressure increase to shift towards the flow with increasing E. The monotonic pressure rise in incident shock waves and in rarefaction zones is observed further downstream as well.

391

Section 8.6 Numerical simulation results P/Pinf (a)

P/Pinf (b)

1.4

1.4

E=0 E = 500 B E = 1000 B E = 1500 B

1.2

E=0 E = 500 B E = 1000 B E = 1500 B

1.2

B=0

1

1 0

5

10 x, cm

15

20

0

5

10 x, cm

15

20

Figure 8.7. The distributions of pressure on the lower (a) and the upper (b) surfaces at B D 0 in the channel without discharge (1) and with glow discharge for different values of emf: (2) E D 500 V, (3) 1 000 V, (4) 1 500 V.

2.2 2 B=0 E=0 E = 500 B E = 1000 B E = 1500 B

Qw, W/cm**2

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4

0

5

10 x, cm

15

20

Figure 8.8. The distributions of density of convective heat flux along the lower plane at B D 0 in the channel without discharge (1) and with glow discharge for different values of emf: (2) E D 500 V, (3) 1 000 V, (4) 1 500 V.

The distribution of convective heat fluxes on the lower surface is given in Figure 8.8. The increase in convective heat flux to the cathode surface with increasing emf of the power supply E. On the upper surface, the convective heat flux varies insignificantly. Significantly more pronounced changes of pressure and convective heat flux are observed in the presence of a magnetic field. Figure 8.9 gives the distributions of pres-

392

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

2

P/Pinf

2

(a)

(b)

1.5 1.5 1 1

E = 1000 B E=0 E = 1000 B, B = −0.5 T E = 1000 B, B = +0.5 T

0.5

E = 1000 B E=0 E = 1000 B, B = −0.5 T E = 1000 B, B = +0.5 T

0

0.5 0

5

10 x, cm

15

20

0

5

10 x, cm

15

20

Figure 8.9. The distributions of pressure along (a) the lower and (b) the upper planes in a channel with glow discharge and with a magnetic field.

sure along the lower (cathode) and upper (anode) surfaces without glow discharge, with glow discharge, and with a magnetic field of two polarizations Bz D C0.5 T and Bz D 0.5 T. It was already noted that a significant increase in pressure on the surface is observed for the case of Bz D 0.5 T. It is remarkable that a small rarefaction region is observed directly above the surface of the cathode section. Here, the flow is accelerated in the positive direction x. Conversely, at Bz D C0.5 T, a higher pressure zone of small extent is observed directly before the extensive rarefaction region (where the ponderomotive force forces the flow away from the surface). Here, the flow is decelerated by the ponderomotive force. The situation is different on the upper surface: rarefaction is observed at Bz D 0.5 T, and a significant increase in pressure at B D C0.5 T. The distributions of convective heat fluxes along the lower and upper surfaces are given in Figure 8.10. One can see that, in the presence of a magnetic field, convective fluxes to the surface increase or decrease due to pressing the flow against, or forcing it away from, the surface. The distributions of ponderomotive force and power of heat release, which define their impact on the flow field, in turn depend on the distribution of the concentration of charged particles and on the distribution of current density in the discharge gap. These functions are found when solving the electrodynamic part of the problem. The fields of concentrations of charged particles and of current density are given in Figure 8.11 in the absence of a magnetic field. One can well see the drift of charged particles downstream. The leading front of the ionized region is located above the rear boundary of the cathode section. The concentration of charged particles on the lower dielectric surface of the channel after the cathode section is assumed to be low because of the recombination of particles on the dielectric surface.

393

Section 8.6 Numerical simulation results Qw, W/cm**2 3

Qw, W/cm**2 3

(a)

2.5

(b)

2.5 E = 1000 B E=0 E = 1000 B, B = −0.5 T E = 1000 B, B = +0.5 T

2

E = 1000 B E=0 E = 1000 B, B = −0.5 T E = 1000 B, B = +0.5 T

2

1.5

1.5

1

1

0.5

0.5

0

0 0

5

10 x, cm

15

20

0

5

10 x, cm

15

20

Figure 8.10. The distributions of density of convective heat flux along (a) the lower and (b) the upper planes in a channel with glow discharge and with a magnetic field.

y, cm

Ui: 0.01

(a)

0.02

y, cm

0.13

0.30

0.70

6

8

10

1.65

3.86

9.03

21.13

2 1 0

0

J: 1.0E-04 2 (b)

0.05

1 0

0

2

4 5.4E-04

2

4

2.9E-03

6

8

12

1.5E-02

10

14 8.3E-02

12

14

16

18

20 x, cm

4.4E-01

16

18

20 x, cm

Figure 8.11. The distributions of (a) concentration of charged particles n/n0 (n0 D 1010 cm3/ and (b) current density in A/cm2 in a channel with glow discharge at E D 1 000 V without a magnetic field.

The distribution of charged particles on the upper anode surface, which pre-assigns the actual size of the current spot, is defined by the processes of recombination and convective carryover of electrons and ions from the main part of discharge located somewhat downstream of the cathode along the x axis. The regions of flow of electric current in the gas-discharge gap are well identified by the distributions of current density. The difference between the distributions of current density in the cases with and without magnetic field consists in that, in the case without magnetic field, the current channel is formed immediately behind the cathode before the oppositely

394

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

located anode. In the presence of magnetic field, the current channel is pressed against the dielectric surface behind the cathode. In the case of Bz D 0.5 T, the current flows along the front of departing shock wave. Of considerable practical interest is the distribution of gas-dynamic parameters along the central line of the channel and in its outlet cross section. The distributions of gas-dynamic parameters along the central line help estimate the level of global changes in the flow under the local effect of glow discharge and magnetic field. Figure 8.12 gives the distributions of pressure and Mach numbers in the flow with and without discharge at E D 1 000 V. Figures 8.12 (a)–8.12 (c) actually illustrate the effect of volume heat release generated by glow discharge in the vicinity of the cathode, where the discharge burns without an external magnetic field. Note the obvious tendency for an increase in pressure and a decrease in Mach number with increasing

4

p, Torr

4.4

M

(a)

(b) 4.2

3.5

4

3

3.8 2.5 3.6 2 E=0B E = 1000 B, Bz = 0 E = 1000 B, Bz = −0.5 T E = 1000 B, Bz = +0.5 T

1.5 1

0

5

10

15

3.4

M E = 1000 B, Bz = 0 E = 1000 B, Bz = −0.5 T E = 1000 B, Bz = +0.5 T

3.2 3

20 x, cm

0

5

10

15

20 x, cm

T, K

310

(c) 300 290 280 270 260

T E = 1000 B, Bz = 0 E = 1000 B, Bz = −0.5 T E = 1000 B, Bz = +0.5 T

250 240

0

5

10

15

20 x, cm

Figure 8.12. The distributions of (a) pressure, (b) Mach number, and (c) temperature along the central line of the channel, H D 2 cm, L D 20 cm.

395

Section 8.6 Numerical simulation results

emf. In the case under consideration, the above identified global changes in the flow are generated by low energy local stimulation. Much more pronounced effects are observed in the presence of a magnetic field with which, as was already discussed, the emergence of bulk forces in the gas flow is associated. One can well see in Figure 8.12 that the magnetic field, which also generates a local effect on the flow, causes a very significant variation of the global gas-dynamic structure. The temperature distribution given in Figure 8.12 also illustrates considerable possibilities of electromagnetic stimulation of flow: for different directions of magnetic field induction, it is possible to attain differences in temperature (in separate regions of the flow field) up to 50 K. We will recall that such variations are produced by collisional processes in partly ionized gas with the degree of ionization of  105.

4

p, Torr

450

E=0B E = 1000 B, Bz = 0 E = 1000 B, Bz = −0.5 T E = 1000 B, Bz = +0.5 T

3.8

(a)

(b) 400

3.6

350

3.4

300

3.2

250

3

0

0.5

1

1.5

1.5

T, K

2 y, cm

200

E=0B E = 1000 B, Bz = 0 E = 1000 B, Bz = −0.5 T E = 1000 B, Bz = +0.5 T

0

0.5

1

1.5

2 y, cm

Ro/Ro0 (c)

1.4 1.3 1.2 1.1 1

E=0B E = 1000 B, Bz = 0 E = 1000 B, Bz = −0.5 T E = 1000 B, Bz = +0.5 T

0.9 0.8

0

0.5

1

1.5

2 y, cm

Figure 8.13. The distributions of (a) pressure, (b) temperature, and (c) density in the outlet cross section of the plane channel, H D 2 cm, L D 20 cm.

396

Chapter 8 Gas flow in a plane channel with gas discharge and magnetic field

Analysis of flow parameters in the outlet cross section of a hypersonic channel enables one to gain an impression of global changes throughout the channel height at a distance of 17 cm from the place of local stimulation of flow. Figure 8.13 gives the distributions of density, pressure, and temperature throughout the height of the channel inlet cross section. Least of all, the density varies in the boundary layer in the vicinity of the lower (cathode) surface. On approaching the upper surface, the variations of density in the presence of glow discharge and an external magnetic field become ever more pronounced. The pressure turns out to be most sensitive. However, one must take into account the fact that both zones of appreciable pressure increase and rarefaction zones may be observed in the different cross sections of a hypersonic channel because of the special features of the shock wave structure of flow field in this channel. One can see from the temperature distribution that the effect of glow discharge (with or without a magnetic field) may be observed only in the vicinity of the cathode surface where the source of heat release is located. The foregoing calculation results corresponded to a channel of height H D 2 cm and length L D 20 cm. The effect of glow discharge on the cross sectional structure was also studied for a channel of larger size of H D 4 cm and L D 40 cm. The cathode section 1 cm long was located on the lower plane at a distance of 4 cm from the channel inlet. No fundamental changes are observed in the flow structure. However, the general tendency for the pressure to increase downstream is observed. The effect of a magnetic field, as before, is very appreciable for the entire flow field. The direction of magnetic field induction turns out to be of fundamental importance for the entire flow field, although the ponderomotive force is significant only in a small zone in the vicinity of the cathode boundary. The distribution of pressure along the lower and upper surfaces of the channel of larger size (Figure 8.14) give an impression of the level of this effect.

1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5

P/Pinf (a)

No DCD E = 1000 V, Bz = −0.5 T E = 1000 V, Bz = +0.5 T

0

5

10 15

20 25 30

35 40 x, cm

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9

P/Pinf (b) No DCD E = 1000 V, Bz = −0.5 T E = 1000 V, Bz = +0.5 T

0

5

10 15

20 25 30

35 40 x, cm

Figure 8.14. The distributions of pressure along the lower (a) and the upper (b) planes in the channel of height H D 4 cm.

Section 8.6 Numerical simulation results

397

Numerical analysis of the properties of an electromagnetic actuator involving the use of glow discharge and an external localized magnetic field opens up considerable prospects for control over partly ionized hypersonic rarefied flow. It is demonstrated that local stimulation with low energy expenditure for gas flow in the boundary layer in the form of localized heat release or localized ponderomotive force may result in appreciable restructuring in the entire gas-dynamic field.

Chapter 9

Hypersonic flow of rarefied gas in a curvilinear channel with glow discharge In the given chapter, a two-dimensional model of a glow discharge in a hypersonic channel with a local bulge at the bottom surface, which is normally used for generating a shock wave structure in channels, is considered. The configuration of electrodes of the gas-discharge device is displayed in Figure 9.1. In all the calculations performed, the entire upper surface of the channel served as the anode. The sectioned cathode was positioned at the lower wall of the channel. In the first set of calculations, the perturbations of the flow caused by the glow discharge are generated by a shock wave outgoing from the leading edge of the bulge. In the second set of simulations, perturbations were generated before the arrival of the shock wave. In the present chapter, the following problems are solved: (1) A self-consistent simulation model of gas-dynamic and electric discharge processes in two-dimensional curvilinear channels with a transverse glow discharge was developed; E

Ro

Anode Uo y x

L

Cathode E

Ro

Anode Uo y

Cathode

x

Figure 9.1. Schematics of the positioning of the electrodes in the hypersonic flow channel.

399

Section 9.1 Governing equations

(2) The parameters of a glow discharge in a hypersonic rarefied flow were numerically simulated; (3) The role of the coefficient of transformation of the energy of electrodynamic dissipative processes into heat was examined. In addition, the results of some methodological numerical experiments aimed at enhancing the reliability of calculated quantities are presented. A glow discharge in a hypersonic flow is characterized by the following parameters: the concentration of charged species, n0  1012 cm3 ; the voltage across the electrodes, Vc  500–1 000 V; the current density at the electrodes, j  10 mA/cm2 ; and the overall current flowing through the gas-discharge gap, I  10–100 mA. A steadystate glow discharge is considered, and, therefore, electric current passes through an external circuit (Figure 9.1). The pressure in the oncoming flow of p  10 Torr corresponds to a concentration of neutral species of N  3  1017 cm3 . This means that the degree of ionization of the gas in the glow discharge region is D n0 /N  105; i. e., a partially ionized gas with small  1 is considered.

9.1 Governing equations The motion of a viscous heat-conducting partially ionized gas in a curvilinear channel is described by the system of mass continuity, Navier–Stokes, energy conservation, charged-species continuity, and current continuity equations: @ C div .V/ D 0, @t

(9.1)

@u C div .uV/ @t @ h  @u @v i @  @u @p 2 @  . div V/ C  C C2  , (9.2) D @x 3 @x @y @y @x @x @x @v C div .vV/ @t @ h  @u @v i @ @v @p 2 @  . div V/ C  C C 2 . /, (9.3) D @y 3 @y @x @y @x @y @y  cV

@T C  cV V grad T C p div V D div . grad T / C Q† , @t @n C div .nV/ D div .Da grad n/ C nvi  ˇn2 , @t div .ne grad '  De grad n/ D 0,

(9.4) (9.5) (9.6)

400

Chapter 9 Hypersonic flow of rarefied gas in a curvilinear channel

where x and y are the longitudinal and transversal coordinates of the Cartesian coordinate system, respectively; V D .u, v/ is the gas flow velocity (neutral species) and its projections onto the x and y axes;  and p are the density and pressure;  is the dynamic viscosity; cV is the specific heat at constant volume; T is the temperature;  is the thermal conductivity; n is the number density of charged species (since the quasi-neutral plasma approximation is used, n D ni D ne /; Da is the ambipolar diffusion coefficient; i D i .E=N / is the impact ionization coefficient; ˇ is the electron-ion recombination coefficient; ' is the electric potential (related to the electric field strength as E D  grad '; E D jEj; N is the number concentration of neutral species; Q† is the volumetric rate of heat release by dissipative processes (Q ), pressure force work (Qp ), and passage of electric current through a partially ionized gas (QJ Joule heat). The latter quantity is given by QJ D .j  E/ D 1.6  1019 ŒnE 2 .C C e / C .De  DC/E grad n where is the efficiency of transformation of the energy of dissipative electrodynamic processes into heat (according to laser physics, for molecular nitrogen,  0.1–0.3); j is the current density vector; and e , C are the mobilities of electrons and ions, respectively.

9.2 Boundary conditions and closing relations The boundary conditions for equations (9.1)–(9.6) read as (Figure 9.1): xD0:

xDL:

u D u0 ,

v D 0,

p D p0 ,

 D 0 , n D 1010n0 ,

T D T0 , @' D 0; @x

@v @T @n @ @' @u D D D D D D0 @x @x @x @x @x @x

y D 0 : u D v D 0,

T D Tw ,

@p D0 @y

(9.7)

(9.8) (9.9)

xc,1  x  xc,2 (cathode section): @n D 0, @y

' D Vc ;

(9.10)

x < xc,1 , x > xc,2 (dielectric surface): n D 105n0 ,

@' D 0; @y

(9.11)

401

Section 9.3 Numerical simulation results

y D H (anode): u D v D 0,

T D Tw ,

@n D 0, @y

'DE

@p D0 @y

(9.12) (9.13)

Here, n0 is the characteristic concentration of charged species in the glow gap (n0  1012 cm3 /, Vc is the cathode potential with respect to the zero level, E is the emf of the voltage source, and H is the anode-cathode gap distance. To calculate Vc , it is necessary to write the equation for the external circuit of the glow discharge, which, in the simplest case, takes the form IR0 C Vc D E where R0 is the ballast resistance and I is the total current flowing through the circuit, which is determined by integrating the current density over the current-carrying areas of the surfaces: ZL ZL I D .jn/c dx D .jn/a dx. 0

0

Here, n is the unit vector normal to the cathode (subscripts c) and anode (a) surfaces, and j is the current density vector. Since a flow of weakly ionized molecular nitrogen N2 (with a degree of ionization of  105 ) was considered, the transport coefficients are determined in the first approximation of the Chapman–Enskog theory. All necessary relations are presented in the previous chapter.

9.3 Numerical simulation results Before examining the effect of a glow discharge on the flow field pattern, let us consider the shock wave structure of a hypersonic flow in a channel with a 10% bulge of segmental shape at the lower surface. The longitudinal coordinates of the curvilinear segment were xb,l D 1 cm, xb,r D 5 cm, with the segment height being 0.2 cm. Note that this geometry of the channel is used in the computational aerodynamics for testing simulation models. The input data for numerical simulations corresponded to the experimental conditions in studies of a molecular nitrogen flow [95]: p0 D 2.956  103 erg/cm3 , 0 D 4.000  106 g/cm3 , T0 D 249 K, V0 D 128 500 cm/s, M D 4.0, Re D 62 900. Figure 9.2 shows the pressure field in the channel calculated using the Euler and Navier–Stokes equations. The calculations were performed on rather fine computational meshes, with Ni D 201 and Nj D 1 301 (Ni and Nj are the number of computational mesh points in the directions across and along the channel, respectively).

402

Chapter 9 Hypersonic flow of rarefied gas in a curvilinear channel

(a) y, cm

P

1.00 1.36 1.71 2.07 2.43 2.79 3.14 3.50 3.86 4.21 4.57 4.99 5.29 5.64 6.00

2 1 0

0

2

4

6

8

10

12

14

16

18

20

(b)

y, cm

P

1.00 1.36 1.71 2.07 2.43 2.79 3.14 3.50 3.86 4.21 4.57 4.99 5.29 5.64 6.00

2 1 0

0

2

4

6

8

10 x, cm

12

14

16

18

20

Figure 9.2. Pressure field in the channel (in Torr) in the absence of the discharge. The calculations were performed using the Euler (a) and Navier–Stokes (b) equations on a 201  1301 computational mesh.

Generally, the pressure fields for a viscous and non-viscous gas are similar. Note, however, that in a viscous gas, the shock wave structure is more complex (Figure 9.2 (b)). One peculiarity of a viscous gas flow is the existence of shock waves that arise starting from the inlet cross section of the channel. At a distance of x  3.2 cm from the inlet cross section the shock waves propagating from the upper and lower edges intersect. The shock wave from the leading edge of the buckle arrives approximately at the same point. The intersection of the three waves approximately at the same point is an accidental event, associated with the chosen geometry of the channel and the conditions in the incoming flow. In the region behind the intersection of the wave, the pressure increases by about a factor of 3. At the intersection of the shock waves, the slope of the most intense shock wave produced by the segmental bulge somewhat changes its angle with respect to the x axis: it decreases by several degrees. Downstream from this point, the shock wave front interacts with the boundary layer developing from the upper edge. Two waves generated by this interaction are clearly seen: the first one is generated by the interaction of the incident shock wave with the boundary layer, whereas the second one is reflected from the surface. Note that such a configuration was observed in experiments. Thus, two shock waves are reflected from the upper surface. In the case of a nonviscous gas (Figure 9.2 (a)), only one wave is reflected, since no interaction with the boundary layer occurs – it is absent. Note that the reflection of two shock waves from the upper surface for a viscous gas occurs at a shorter distance from the inlet cross section (xrefl,v  5.6 cm) compared to a non-viscous gas (xref,nv  6.2 cm). Note that a high-pressure region is formed behind the shock waves. The two shock waves outgoing from the upper surface intersects at x  9 cm with the shock wave generated at the right edge of the segmental bulge. For a non-viscous gas, the encounter of the reflected

403

Section 9.3 Numerical simulation results

shock wave with the shock wave outgoing from the right edge of the segmental bulge occurs at a distance of x  10 cm. Shock waves generated at the right boundary of the segmental bulge have a markedly smaller slope with respect to the x axis compared to the wave outgoing from the left surface. This can be explained by the fact that the former are generated formed against the background of a more rarefied flow that is formed behind the bulge. Downstream from this region, the above regularities of the shock wave process repeat themselves against the background of the regular decrease of the intensity of the shock waves. The calculation results discussed above were obtained using fine computational meshes. Such calculations are time-consuming. The computer modeling of the structure of the glow discharge is an even more time-consuming problem. The time it takes to solve the electrodynamic part of the problem exceeds that required to solve the gas-dynamic part by two to three orders of magnitude. Therefore, to solve the combined electrodynamic-gasdynamic problem, we used coarser computational meshes (Ni D 51 and Nj D 301), although this may entail the loss of some details of the flow structure. This is illustrated in Figure 9.3 (a), which shows the pressure field calculated using a 51  301 mesh. When comparing Figures 9.2 and 9.3, one should takes into account a small difference in the pressure scale. It is worthwhile to single out two aspects: first, the peak pressures diminished, and, second, the double shock wave structure arising during the reflection from the external surface is hardly noticeable. As will be demonstrated, this loss of detail of the flow field calculated on a relatively coarse mesh is of minor importance for the problem under consideration. Figure 9.4 shows the pressure profile along the lower and upper surfaces calculated for a viscous and non-viscous gas on various computational meshes. For the viscous gas, a sharp rise of the pressure is observed at the leading edge. When calculations were performed with the fine mesh, the pressure rise was 1.44-fold higher. A further (a)

y, cm

P

2.50 2.87 3.24 3.61 3.97 4.34 4.71 5.08 5.45 5.82 6.18 6.55 6.92 7.29 7.66 8.03 8.39 8.76 9.13 9.50

2 1 0

0

2

4

6

8

10

12

14

16

18

20

(b) y, cm

P

2.50 2.87 3.24 3.61 3.97 4.34 4.71 5.08 5.45 5.82 6.18 6.55 6.92 7.29 7.66 8.03 8.39 8.76 9.13 9.50

2 1 0

0

2

4

6

8

10 x, cm

12

14

16

18

20

Figure 9.3. Pressure field (in Torr) in the channel in the (a) absence and (b) presence of the glow discharge calculated for the first configuration of electrodes on a 51  301 computational mesh at E D 700 V and D 0.25.

404

Chapter 9 Hypersonic flow of rarefied gas in a curvilinear channel

mesh refinement near the leading edge will yield still greater values of the pressure. The numerical solution of the Navier–Stokes equation for an infinitely thin leading edge is incorrect. Therefore, solutions on different computational meshes can be compared only at some distance from the leading edge. In our case, such a comparison is sensible only for the upper surface (Figure 9.4 (b)), where the solutions obtained on different meshes are identical at x D 1 cm. As can be seen from Figure 9.4, the distinctions between the results of calculations on different computational meshes are quite expected: with coarser meshes, the peak pressures within the area of incidence of the shock wave are smaller. In this case, the fronts of perturbations are spread over a larger number computational mesh cells. Overall, the changeover to a coarser mesh influences the results for a non-viscous gas (curves (1) and (2) in Figure 9.4). 3

(a)

2.5

P/Pinf

2 2 4 1.5 3

1

1

0.5 0

(b)

2.5

P/Pinf

2

3

4 2

1.5 1 1 0.5 0 0

5

10

15

20 x, cm

Figure 9.4. Pressure profiles along the lower ((a); cathode-carrying) and upper ((b); anode) surfaces calculated in the absence of the discharge for (1) non-viscous gas flow (1301  201 mesh), (2) viscous gas flow (1301  201), (3) non-viscous gas flow (51  301), (4) viscous gas flow (51  301).

405

Section 9.3 Numerical simulation results

Calculations of the structure of a hypersonic flow with a glow discharge at D 0.25 were performed on a 51  301 computational mesh. The pressure profile upon ignition of the glow discharge with E D 700 V for the first configuration of electrodes (the cathode is positioned on the segmental bulge) is displayed in Figure 9.3 (b). The maximum pressure, p  9.5 Torr, is attained at the upper surface within the area of incidence of the shock wave outgoing from the segmental bulge. The glow discharge burning in the gap between the cathode section on the segmental surface and the upper surface (anode) heats the gas in a volume near the cathode, with the maximum volumetric heat release rate being QJ  1.5  109 erg/(cm3 s). This local volume of heat release coincides with the region of maximum current density. The current density field (Figure 9.5 (a)) features three characteristic areas. The highest current density (j  1 A/cm2 / is observed near the rear boundary of the cathode. It is a very small area located above the boundary between the cathode and dielectric surface. Another characteristic area is associated with the current channel between the cathode and anode. The third area spreads downstream over the segmental surface. This area of the current density field reflects the nature of the hypersonic flow under study, more specifically, the downstream drift of charged species. The fields of the concentration of charged species and of the normalized electric potential are shown in Figures 9.5 (b) and 9.5 (c), respectively.

(a)

y, cm

J: 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.06 0.07 0.09 0.12 0.15 0.19 0.25 0.32 0.41 0.52 0.67 0.86 1.10

2 1 0

0

2

4

6

8

10

12

14

16

18

20

(b) y, cm

UI:

0.01

0.72

1.44

2.15

2.86

3.58

4.29

5.00

5.72

6.43

7.15

7.86

8.57

9.29

10.00

2 1 0

0

2

4

6

8

10

12

14

16

18

20

(c) y, cm

FI: 0.65 0.67 0.69 0.71 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.89 0.91 0.93 0.95

2 1 0

0

2

4

6

8

10 x, cm

12

14

16

18

20

Figure 9.5. Fields of (a) electric current density (mA/cm2), (b) concentration of charged species n/n0 , and (c) normalized electric potential '/E calculated for the first configuration of electrodes on a 51  301 computational mesh at E D 700 V and D 0.25.

406

Chapter 9 Hypersonic flow of rarefied gas in a curvilinear channel

A comparison of the pressure fields in the presence and absence of the glow discharge (Figure 9.3) shows that it produces a marked effect. The pressure rises appreciably not only near the upper surface, but in the central part (over the segmental bulge). How the shock wave structure evolves with changing emf is clearly seen from Figures 9.6 (a) and 9.6 (b), the pressure profiles along the lower and upper surfaces, respectively. A general tendency for the pressure to rise with increasing E is obvious. Note that the pressure profile is very sensitive to the emf, a behavior that can be explained by a strong dependence of the current density (and the degree of ionization) on the voltage applied to the discharge gap. In the second configuration of electrodes (see Figure 9.1 (a)), the cathode was positioned ahead of the segmental bulge, i. e., outside the range of action of the shock

3.5

P/Pinf (a)

3 2.5 3

2

4

1.5

2 1

1 0.5 5

(b) 4

4 3 3 2 2 1 1 0

0

5

10 x, cm

15

20

Figure 9.6. Pressure profiles along the (a) lower (cathode-carrying) and (b) upper (anode) surfaces calculated for a viscous gas flow on a 51  301 computational mesh at D 0.25 and various values of emf (in V): E D .1/ 0 (no discharge), (2) 500, (3) 700, and (4) 900.

407

Section 9.3 Numerical simulation results

wave generated by the bulge. The effect of positioning of the cathode is clearly seen from a comparison of the results of calculations with and without the discharge (Figure 9.7). Note that, in this series of calculations, the bulge was shifted downstream by 3 cm. That the cathode is located upstream from the bulge gives rise to a shock wave propagating from the region of bulk heat release in the discharge area over the cathode. As a result, the pressure in the upper part of the flow increases substantially, which, in turn, causes an increase in the total pressure in the channel due to rereflections of more intense waves. (a)

y, cm

P

1.01

1.64

2.29

2.99

3.57

4.21

4.86

5.50

6.14

6.79

7.49

8.07

8.71

9.36

10.00

2 1 0

0

2

4

6

8

10

12

14

16

18

20

(b)

y, cm

P

1.01

1.64

2.29

2.99

3.57

4.21

4.86

5.50

6.14

6.79

7.49

8.07

8.71

9.36

10.00

2 1 0

0

2

4

6

8

10 x, cm

12

14

16

18

20

Figure 9.7. Pressure field in the channel (in Torr) in the (a) absence and (b) presence of the glow discharge calculated for the second configuration of electrodes on a 51  301 computational mesh at E D 700 V and D 0.25.

The fields of the current density (Figure 9.8 (a)), concentrations of charged species (Figure 9.8 (b)), and electric potential (Figure 9.8 (c)) are qualitatively similar to those observed for the first configuration of electrodes. As discussed above, in analyzing the prospects of using the glow discharge as an electric actuator in the aerophysics of hypersonic flows, the problem of the transformation of the energy of the dissipative processes associated with electric current flow into heat is of paramount importance. To solve this problem, it is necessary to examine the kinetics of excitation of vibrational states of molecular nitrogen (in our case) and to calculate the distribution of electrons over energy throughout the entire region of electric current flow. In what follows, we will illustrate the importance of solving this problem. Figure 9.9 shows the pressure profiles along the lower and upper surfaces at fixed characteristics of the external circuit (E D 700 V, R0 D 12 kOhm) and various values of , a parameter that determines the heat release rate (at D 0.25 and 0.50, QJ  1.5  109 and 3.0  109 erg/(cm3  s), respectively). As mentioned above, the volumetric heat release

408

Chapter 9 Hypersonic flow of rarefied gas in a curvilinear channel

(a) y, cm

J: 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.06 0.07 0.09 0.12 0.15 0.19 0.25 0.32 0.41 0.52 0.67 0.86 1.10

2 1 0

0

2

4

6

8

10

12

14

16

18

20

(b) y, cm

UI:

0.01

0.72

1.44

2.15

2.86

3.58

4.29

5.00

5.72

6.43

7.15

7.86

8.57

9.29

10.00

2 1 0

0

2

4

6

8

10

12

14

16

18

20

(c) y, cm

FI: 0.65 0.67 0.69 0.71 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.89 0.91 0.93 0.95

2 1 0

0

2

4

6

8

10 x, cm

12

14

16

18

20

Figure 9.8. Fields of (a) electric current density (mA/cm2/, (b) concentration of charged species n/n0 , and (c) normalized electric potential '/E calculated for the second configuration of electrodes on a 51  301 computational mesh at E D 700 V and D 0.25.

rate is distributed over the discharge gap nonuniformly. Near the rear boundary of the cathode, this quantity is an order of magnitude higher than in the rest of the discharge gap. The pressure profiles shown in these figures and in Figure 9.10 (compare with Figure 9.3 (b)) confirm that the parameter produces a strong effect on the shock wave structure of the flow, which is quite understandable, since the local zone of intense heat release causes an intense perturbation of the gas flow. On the whole, the simulation results are in close qualitative agreement with the experimental data reported in [95]. The simulation results are suggestive of the possibility, in principle, of controlling the structure of supersonic flows by varying the governing parameters, such as the configuration of electrodes, emf of the voltage source, and ballast resistance of the external circuit. An electric-discharge actuator of this type is virtually inertia less. Among its drawbacks are the loss of stability of the gas-discharge structures at elevated pressures and the possibility of emergence of ionization-overheating instability. The upmost efficiency of glow discharges of this type (low energy expenditures) is observed for rarefied flows, which are of considerable interest for aerophysical applications.

409

Section 9.3 Numerical simulation results

4

P/Pinf (a)

3.5

3

3

2

2.5 2

1

1.5 1 0.5 6 (b) 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

3

2 1

5

10 x, cm

15

20

Figure 9.9. Pressure profiles along the (a) lower (cathode-carrying) and (b) upper (anode) surfaces calculated for a viscous gas flow on a 51  301 computational mesh at E D 700 K and various values of : (1) 0 (no discharge), (2) 0.25, and (3) 0.5.

y, cm

Pressure: 2.5

3.24 3.97 4.71 5.45 6.18 6.92 7.66 8.33 9.19 2 1 0

0

2

4

6

8

10

12

14

16

18 20 x, cm

Figure 9.10. Pressure field in the channel in the presence of the glow discharge calculated for the first configuration of electrodes on a 31  301 computational mesh at E D 700 V and D 0.5.

Appendix

A.1

Fundamental constants

Speed of light c D 2.998  1010 cm/s Stefan–Boltzmann’s constant Q D 5.67  108

W erg D 5.67  105 4 2 mK s  cm2 K4

Universal gas constant R0 D 8.3145

J erg D 8.3145  107 mole  K mole  K

Electron charge e D 4.802  1010 g1=2cm3=2 /s

(CGSE electrostatic system of units)

Boltzmann’s constant k D 1.38  1023 Electron mass Proton mass Atomic mass unit Bohr radius

J erg D 1.38  1016 K K

me D 9.109  1028 g mp D 1.673  1024 g m0 D 1.66  1024 g a0 D 0.529  108 cm

Avogadro number 1 mole Loschmidt’s number (the number of molecules in 1 cm3 at T D 273.15 K, p D 1 atm D 101325 Pa/ N0 D 2.687  1019 cm3 NA D 6.022  1023

Planck’s constant

h D 6.625  1027 erg  s

412

A.2

Appendix

Ratios between units of electricity and magnetism

Charge 1 C D 3  109 SGSE units D 9  1011 V  cm D 6.25  1018 electron charges Current V  cm C D 3  109 SGSE units D 9  1011 s s electron charges D 6.25  1018 s

1A D 1

Voltage 1 SGSE units 300 E ŒV=cm D 300E ŒSGSE units 1V D

Resistance 1 Ohm D Conductivity

h 

1 1 SGSE units D s 11 9  10 30

i 1 1 D  Œs 1  Ohm  cm 9  1011

Capacitance 1 Farad D 9  1011 SGSE units Strength of magnetic field 1 Oersted D 1 SGSE units Inductance 1 Henry D 109 SGSE units (cm) Absolute magnetic permeability of vacuum 0 D 4  107

Henry Henry D 12.566  107 m m

Absolute dielectric permeability of vacuum "0 D

1 Farad D 8.854  1012 2 0 c m

i Eh E V E D 3.3  1016 ŒV  cm2 D 0.33 ŒTd p cm  Torr N N 1 Td D 1017 V  cm2

Section A.2 Ratios between units of electricity and magnetism

The temperature E D 11 610 K k corresponds to 

energy E D 1 eV.

The energy

" D 1 eV D 1.602  102 erg

corresponds to T D "=K D 11 600 K,



temperature



frequency f D "= h D 2.418  1014s1



wavelength  D hc=" D 1.24  104 cm D 12.4 Å



wave number ! D "= hc D 8 067 cm1

413

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Index

˛-ˇ algorithm 211 absorbed power of electromagnetic radiation 24 acceleration of gravity 6, 95, 98, 99 adiabatic index 121 Alfvén Mach number 122 Alfvén velocity 122, 137 ambipolar diffusion 4, 188, 189, 191, 193, 346–352, 355, 358, 364, 380, 400 Ampere law 107 anode dark space 173 anomalous glow discharge 172 arc discharge 21, 172, 362 area of transition from glow to arc discharge 172 artificial viscosity 204 association 3, 54, 141, 244 associative ionization 54 ASTEROID software 30 Aston dark spac 173 atomic mass unit 51, 411 AUSM finite-difference algorithm 367 average thermal velocity of electron 63 Avogadro number 52, 411 axial distribution of temperature in discharge 257 bifurcation 27, 85, 88, 89 Bohr radius 298, 411 Boltzmann equation 296, 315 Boussinesq hypothesis 29 caloric equation of state 11 canonical form of the five-point finitedifference equation 42, 199 Cartesian rectangular coordinates 7 cathode section 280, 378, 386, 387, 390, 392, 396, 400, 405

CFL number 192, 193, 204 Chapman–Enskog method 30 characteristic times of diffusion 189 characteristic velocity of a gas flow 298 charge density 6, 176, 225 chemical symbol of reagent 52 coefficient of ambipolar diffusion 347, 349, 351, 364 coefficient of attachment 321 coefficient of ion-electron emission 179, 243 coefficient of ion-electron recombination 179, 243, 281, 282, 380 coefficient of molecule ionization by electron impact 179 coefficient of physical diffusion 230 coefficient of recombination at collision of positive and negative ions 230, 321 coefficient of viscosity 6 coefficients of binary diffusion 53 collision frequencies of electrons 63, 298 collision frequency of ions 298 collisional ionization 54, 243, 364, 380 complete set of the MHD equations 114 completely conservative form of the MHD equation 117 computational fluid dynamics (CFD) 34, 85, 104, 203, 207 concentration of electron in glow discharge 246 concentration of ion in glow discharge 246 concentrations of electrons and ions 178, 306, 323, 325 condition of detailed equilibrium 58 conservative properties 200, 202 conservative variable 104, 106, 129, 132, 133

424 continuity equation 5 continuum absorption mechanism 30 correction of the Debye–Hueckel 62 Couette flow 305 Coulomb constant 62 Coulomb logarithm 21 Courant–Friedrichs–Lewy stability condition 192 cross section of electron-particle collision 63 cross section of inter-particle interaction 60 current density 6, 105, 111, 113, 115, 171, 172 current density on the cathode and the anode 247, 289, 291 curvilinear channel 398, 289, 291, 399 Dalton law 53, 59 Debye radius 345, 347 detachment coefficient 321 dielectric surface 331, 335, 354, 365, 370, 382, 400 diffusion approximation 45, 351 diffusion coefficient 43, 55, 178, 181, 189, 204, 262, 349 diffusion equation 50 diffusion force 53 diffusion of electron 189 diffusion of ion 189, 191, 217, 231 diffusion velocity 53 direction cosines 66 dissociation 54, 56, 63 dissociative recombination 54 distribution of the concentration of electrons along a symmetry axis of discharge 254 distribution of the concentration of ions along a symmetry axis of discharge 254 donor cells 198, 228 drift time 69, 188, 191 drift velocities of electrons and ion 298 drift velocity 176, 188, 231, 279, 299, 321 drift-diffusion model 171, 178 dynamic variable 34, 41, 46, 47, 49, 64, 65

Index

effect of free interaction 370 effective electric field 319, 321, 333, 337, 239–341, 352 effective radius of a neutral particle 298 eigenvalue 129 eigenvector 129 Einstein relations 181, 206 electric field strength 105, 108, 109, 112, 113 electric induction 6 electrical conduction 6, 268, 270 electro-arc (EA) plasma generator 19 electro-physical properties 366 electromagnetic force 15, 51, 112, 361 electromotive force 173, 180, 282 electron charge 6, 21, 411 electron drift velocity 188, 279 electron mobility 178, 283, 381 electronegative gas 314 electronic gas 21, 55, 58, 60–62 electronic thermal conduction 59 elementary implicit finite-difference scheme 228 energy conservation equation 15, 20, 23, 24, 28, 34, 50, 55, 56, 58, 61, 62, 105, 127, 332 energy conservation equation for atomicionic gas 58 energy conservation equation for atoms and ion 56 energy conservation equation for electronic gas 58 energy conservation equation for the molecular gas 56 energy conservation equation of electronic gas 55 energy losses by heat radiation 58 Engel–Steenbeck theory 172 entering waves 132 enthalpy 41, 44 enthalpy of electronic gas 55 enthalpy of molecular gas 56 equation of ionization equilibrium 58 equation of motion 6, 14, 19, 55 equation of motion of electron 55 equations in the conservative flux form 7 equilibrium chemical reaction 53

425

Index

equilibrium constant 52 exchange reaction 54 Faraday law 107 finite-difference equation for a stream function 42 finite-difference equation for enthalpy 44 finite-difference equation for group radiative energy density 44 finite-difference equation for vortex function 43 finite-difference grids 37, 207 finite-difference scheme for the equation of charge motion 197 finite-difference scheme for the Poisson equation 194 finite-volume method 36, 194, 286 first order of accuracy 203 first Townsend coefficient 179, 243, 266, 333 five-point finite-difference scheme 40, 283 flux form 7, 114, 115, 122, 126 flux form of MHD equations 120 flux of charge 230 flux point 194, 198 force acting on ionized gas 111 forces of viscous friction 6, 20 Fourier law 53 Fourier–Kirchhoff equation 187, 244, 280, 363 frequency of attachment 321 Frequency of ion-electronic recombination 189 frequency of ionization 188 frozen chemical reaction 53 gas-dynamic pressure tensor 11 Gauss–Seidel method 218 Gaussian distribution 93 generalized Ohm law 6, 51, 108, 113 glow discharge 171 gravity force 11, 51 grid boundary condition 183 group absorption coefficients of a hydrogen plasma 75

group absorption coefficients of high temperature air 32 group characteristic 29, 35 group spectral model 31 Hall parameter 6, 21, 265, 281, 350, 381 Hall parameter for electron and ion 281 heat capacity of heavy particles gas 60 heat emission caused by action of electromagnetic forces 112 heat emission due to the discharge current 334 heat flux vector 11, 13, 45 heat-conduction regime 26, 68 heat-insulated plate 369 homogeneous chemically equilibrium plasma 5 hyperbolic CFL number 192, 204 hypersonic flight vehicle (HFV) 378 impulse conservation equation 41, 50 inductive plasma generators (IPG) 14 intensity of the electric field 164, 167 ion mobility 176, 178, 383 ion-electronic recombination 58, 189, 266 ionization energy of atom 55, 58, 61 Jacobian matrix 127, 129, 133 Joule thermal emission 15, 25, 114, 186, 253, 363, 366 k  " model 29 kinetic equation 54, 56 Larmor frequency 265, 281, 332 Larmor frequency for electrons 281, 332 Larmor frequency for ions 281 332 laser power 29 laser radiation absorption coefficient 29, 30 laser supported plasma generators (LSPG) 25, 64, 74 laser supported rocket engine (LSRE) 81 laser supported wave 25, 47, 64, 81, 91 laser supported waves in the field of gravity 91 law of a normal current density 172

426 Lees–Probstein theory 368 left eigenvectors 131, 138, 145, 149, 156– 158 line-by-line absorption coefficients of high temperature air 32 line-by-line calculation 31 local thermodynamic equilibrium (LTE) 12, 19, 28, 30, 42, 49, 84, 92, 95 Lorentz force 51 Lorentz representation of electron velocity 230 LSPG channel 74 LSW 94 Mach number 122 magnetic force 15, 363, 373 magnetic induction 11, 104, 107, 125, 136, 141, 223 magnetic pressure 11, 114, 125 magnetic Reynolds number 121 magnetohydrodynamic model 104 mass force 11, 316 Maxwell equations 6, 20, 50 Maxwell time 190 Maxwellian distribution 59, 346 mesh diffusion 203, 245 method of fourth order accuracy 235 method of unsteady dynamic variable 47 MHD equation 104, 107, 114 MHD equation in projection 115 MHD equations without singular transfer matrixes 140 MHD interaction 161 micro-wave (MW) plasma generator 22 mobility of electron and ion 243 model of quasi-neutral plasma 347, 380 mole formation rate 52 molecular weight 6, 21, 51, 163, 363 momentum conservation law 262 momentum equation 50, 58, 61 Monte-Carlo method 171 natural form of the flow equation 34 natural variable 104 Navier–Stokes equation 36, 38, 61, 83, 106

Index

near infrared spectrum range 31 Neumann boundary condition 30, 373 non-self-maintained discharge 172 nonstationary boundary condition 129, 140, 153, 160 nonuniform chemically equilibrium plasma 49 normal current density 172, 184, 244, 269 normal glow discharge 172, 184, 268 Ohm law 6, 20, 51, 108, 113 One-sided flux 230 order of accuracy 203, 231, 235 outgoing waves 132 P1 -approximation 45 parabolic CFL number 192 parameter of viscous interaction 360 parameters of the Earth ionosphere 121 particle flux vector 183, 303 partition function 63 perfect gas equation of state 107 permittivity 6 Petukhovs decomposition 231 photo-dissociation 31 photo-ionization 31 Pl -approximation of the spherical harmonics method 30 Planck function 12 plasma magnetic inductivity 6 Poisson equation for electric field potential 182 Poisson equation relative to pressure 39 positive column 173 potential of an electric field strength 178 power source 11 Prandtl number 29, 121 precursor’ radiation 28 pulsed plasma thruster (PPT) 161 quasi-neutral model 345, 354 quasi-neutral plasma 171 quasi-neutrality 53, 56, 59, 348 quasi-random sampling 67 quasilinear system of the MHD equation 144

427

Index

radial boundary of the laser beam 29 radiation pressure 6, 11, 13 radiation pressure parameter 13 radiation pressure tensor 11 radiation regime 26 radiative heat flux vector 11 radio-frequency (RF) plasma generator 14 radius of the caustic 68 rate constant of collision atom-electronic ionization 58 rate constants the forward and reverse reaction 52 ratio of specific thermal capacities at constant pressure and volume 107 ray-tracing method 65 recombination 54, 179 region of cathode dark space 173 region of cathode luminescence 173 region of negative luminescence 173 region of the Faraday dark space 173 relative atomic weight 51 resistance 173, 180, 243, 266, 333, 365 Reynolds grid number 207 Reynolds number 121, 207 Saha equation 59, 95 schemes of Roe’s type 131 second order of accuracy 203 self-oscillation 27, 69 similarity parameter 35 singularity of Jacobian matrixe 133 specific internal energy 6, 11, 57, 104, 126 specific thermal capacity at constant volume 6, 285, 333 spectral emissivity of unit volume 12 spectral group absorption coefficients of a hydrogen plasma 75 spectral intensity of heat radiation 11 spectral intensity radiation of the black body 12 spectral scattering indicatrix 12 spectral volume absorption coefficient 12 spectral volume scattering coefficient 12 splitting for three-dimensional (3D) MHD equation 153

splitting method 36, 161 Stefan–Boltzmann constant 13 stoichiometric coefficient 52 strong interaction 361 subsonic flow model 64 successive over-relaxation (SOR) method 211, 218 successive under-relaxation 218, 285 surface discharge 334, 360 temperatures of electrons and ions 21, 181 thermal conductivity coefficient 6, 29, 93, 107, 380 thermal diffusion coefficient 53 thermal emissions connected with compressibility gas and viscous dissipation 363 thermal equation of state 6, 15, 20, 39, 51 thermal velocity of electrons 63, 298 thermal-convective instability 103 thermo-conductive wave propagation 35 thickness of the cathode layer 175, 184, 208, 210, 269 Thompson method 156 time of charged particles’ diffusion 189 time of heat conduction 190 time of ionization 188 time of recombination 189 time of relaxation of a space charge 190 time step for numerical calculation 190 torus-like form 258, 260 total specific energy 11, 107 Townsend condition of stationary selfmaintained discharge 180 Townsend discharge 172 turbulent Prandtl number 29 turbulent viscosity 29 ultraviolet spectrum range 31, 73 universal gas constant 6, 21, 39, 411 vector form of Navier–Stokes equations 106 vector of integral radiation flux 6 viscosity of the plasma 60 viscous interaction 360

428 viscous stress tensor 11, 106, 262, 297, 300, 315 visible spectral region 31 volt-ampere characteristic (VAC) 232 voltage drop 184 voltage drop on the discharge gap 19, 172, 175, 180, 184, 224, 266

Index

voltage-current characteristic 172 volume absorptivity 12 volume energy release power 29 volume formation rate of species 52 volume radiation density 6, 29 weak interaction 355, 360 work of the Lorentz force 51