The Pennsylvania State University. The Graduate School. Department of Engineering Science and Mechanics

Transcription

1 The Pennsylvania State University The Graduate School Department of Engineering Science and Mechanics STUDY OF PLASMA PHENOMENA AT HIGH ELECTRIC FIELDS IN APPLICATIONS FOR ACTIVE FLOW CONTROL AND ULTRA-SHORT PULSE LASER DRILLING A Dissertation in Engineering Science and Mechanics by Alexandre Likhanskii 2009 Alexandre Likhanskii Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2009

2 The dissertation of Alexandre Likhanskii was reviewed and approved* by the following: Vladimir V. Semak Associate Professor in Engineering Science and Mechanics Dissertation Advisor Co-Chair of Committee Akhlesh Lakhtakia Charles Godfrey Binder Professor in Engineering Science and Mechanics Co-Chair of Committee Albert E. Segall Professor in Engineering Science and Mechanics Victor Pasko Associate Professor in Electrical Engineering Judith A. Todd P.B. Breneman Department Head Chair *Signatures are on file in the Graduate School

3 iii ABSTRACT Plasma engineering is one of the most actively growing research areas in modern science. Over the past decade, plasma engineering became a significant part of aerospace engineering, material processing, medicine, geosciences, etc. One of the main goals of plasma research is to discover new perspectives in a wide range of research areas. It makes plasma engineering a truly interdisciplinary subject. Recently, a significant interest in the aerospace community was caused by the possibility of an active flow control using dielectric barrier discharge (DBD) plasma actuators. A number of groups tried to explain the physics of the experimentally observed phenomena. However, the developed models could hardly explain the DBD phenomena even qualitatively. This thesis presents the first complete, comprehensive, physicallybased model, which tracks all essential physics of the DBD plasma actuators and utilizes modern numerical capabilities for efficient simulations. By using the developed model, the physics of the plasma actuators was explained. Based on the understanding of the operation of the conventional DBD, driven by a sinusoidal voltage, a novel configuration was proposed. The sinusoidal driving voltage was substituted by the repetitive nanosecond pulses superimposed on the bias voltage. The advantages of the proposed concept over the conventional one were experimentally validated. The developed model demonstrated flexibility for different plasma engineering areas. The model can be used not only for a description of the DBD plasma actuators, but also for a number of problems involving the gas discharges. By using the developed

4 iv model, plasma generation by the ferroelectric plasma source, which is used in fusion technology, was explained. In the area of material processing, a significant interest was caused by an apparent possibility of precise high intensity ultra-short laser pulse drilling with negligible melt production. However, the experiments did not validate the theories proposed in literature. In order to explain the experimental data and analyze the possibility of reduction of melt generation, a new model for laser pulse drilling was developed in this thesis. The model comprehensively describes laser-material interaction and explains the significant amount of melt production in the case of ultra-short laser pulses. The results of the simulations are in good agreement with the experimental data.

5 v TABLE OF CONTENTS LIST OF FIGURES...viii LIST OF TABLES...xvi ACKNOWLEDGEMENTS...xvii Chapter 1 Introduction...1 Dielectric Barrier Discharge (DBD) Plasma Actuators for Flow Separation Control...3 An Overview of General Concepts of Plasma Aerodynamics...3 Description of the DBD Plasma Actuator and its Application for Flow Separation Control...4 Experimental Study of the DBD Plasma Actuators...7 Numerical Study of the DBD Plasma Actuators and Motivation for the Present Work...10 Study of the Ultra-Short Pulse Laser Drilling of Metals and Motivation for the Present Work...13 Scientific Contributions...15 Chapter 2 Physical Model of the DBD Plasma Actuator...19 Plasma Model...20 Flow Model...27 Summary...28 Chapter 3 Low-Voltage Modeling of the DBD Plasma Actuator...30 Numerical Model...30 Sinusoidal Voltage Modeling...33 Negative half-cycle...34 Positive Half-Cycle...40 Summary of the Force Exerted on the Gas in the Case of Applied Sinusoidal Voltage...44 Optimal Wave Form for Low Voltages: Repetitive Negative Short Pulses with DC Bias...46 Physical Principles...46 Numerical Results...47 Summary...55 Chapter 4 High-Voltage Modeling of the DBD Plasma Actuator...57 Numerical Methods...57 High-Voltage Results...59 Repetitive Negative Nanosecond Pulses plus Positive DC Bias...59 Repetitive Positive Nanosecond Pulses plus Positive DC Bias...63 The role of the Background Electron Number Density...64

6 vi Comparison between High- and Low-Voltage Results...68 Summary...70 Chapter 5 Optimization of the Numerical and Physical Models for the DBD Plasma Actuator Simulations...71 Necessity for the DBD model optimization...71 Numerical improvement parallel programming...72 Numerical Model...72 Numerical Results...74 Physical Improvement Modified Time Step...75 Summary...80 Chapter 6 The Role of Photoionization in the Numerical Modeling of the DBD Plasma Actuator...81 Physical Model of the Photoionization...81 Numerical Results...82 Summary...98 Chapter 7 Application of the developed model for the description of the experiments Numerical Modeling of the DBD Induced Flow Numerical Modeling of the Surface Charge Accumulation on the Dielectric Based on the Experimental Measurements of the Dielectric Surface Potential Summary Chapter 8 Limitations of the DBD Induced Flow Velocity Induction of the Inviscid Gas Flow by the Gas Discharge Study of the Viscosity Effects on the DBD Induced Flow On the Possibility of Increase of the Induced Flow Velocity by a Series of DBD Plasma Actuators Summary Chapter 9 Application of the Developed Model for the Description of Plasma Generation by the Ferroelectric Plasma Sources Introduction Description of the Physical Model Physics of Plasma Generation Summary Chapter 10 Ultra-short pulse laser drilling of metals Physical Model Numerical Results Description of the Drilling and Melting Mechanisms Picosecond Pulses...152

7 Nanosecond Pulses Study of the laser drilling at different laser pulse durations and intensities Summary Chapter 11 Conclusions Recommendations for the Future Work Bibliography Appendix Nontechnical Abstract vii

8 viii LIST OF FIGURES Figure 1-1: Scheme of the DBD plasma actuator....5 Figure 1-2: Demonstration of flow reattachment by the DBD plasma actuator (Roth, 2003)....6 Figure 1-3: Induced gas velocity pattern restored from PIV measurements (Hall, Jumper, Corke, & McLaughlin, 2005)...9 Figure 3-1: Scheme of the DBD plasma actuator Figure 3-2: Applied voltage and calculated current for 1.5 kv, 1000 khz sinusoidal voltage. Solid red line shows the voltage shape, the dashed blue line shows the calculated total current, and the dotted blue line shows the displacement current...34 Figure 3-3: Negative half-cycle: A sine voltage waveform with short active phase; B illustration of processes during the active phase (avalanche breakdown of the gas, charging of the dielectric surface, and propagation of the plasma); C illustration of processes during the decaying phase (positive space charge, ions drift towards the surface)...36 Figure 3-4: Instantaneous distributions of electron number density (a), positive ion number density (b), negative ion number density (c), electric potential (d) and force, acting on neutral gas (e) during the ionization avalanche. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz)...37 Figure 3-5: Instantaneous distributions of electron number density (a), positive ion number density (b), negative ion number density (c), electric potential (d) and force, acting on neutral gas (e) after the ionization avalanche in the negative half-cycle. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz) Figure 3-6: Average force during the negative half-cycle for sinusoidal applied voltage (amplitude 3.5 kv, frequency 100 khz)...40 Figure 3-7: Positive half-period: A sine voltage waveform with short active phase; B illustration of positive corona regime at low frequency (no charge on dielectric surface, electrons are drawn to the exposed electrodes, ion clouds expands upward and away from the exposed electrode, imparting momentum to the gas); C illustration of processes at high frequency (negative charge remaining on dielectric surface from the negative half-period, ions drift towards the surface and away from the exposed electrode, imparting momentum to the gas and recombining with electrons deposited on the dielectric surface) Figure 3-8: Instantaneous distributions of electron number density (a), positive ion number density (b), negative ion number density (c), electric potential (d) and force,

9 ix acting on neutral gas (e) in the positive half-cycle. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz)...43 Figure 3-9: Average force acting on neutral gas during the cycle. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz)...44 Figure 3-10: Current and voltage in a representative repetitive-pulse case in quasi-steadystate regime. The red line shows the voltage and the dashed line shows the current. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V) Figure 3-11: Instantaneous distributions of electron number density (a) (The electron avalanche charges the dielectric surface during the pulse), positive ion number density (b) (The positive ion cloud is formed near the exposed electrode during the pulse), electric potential (c) and force, acting on neutral gas (d) at the peak pulse voltage. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V) Figure 3-12: Instantaneous distributions of electron number density (a) (electron avalanche charged the dielectric by the end of the pulse), positive ion number density (b) (positive ions start to move to the dielectric surface), electric potential (c) and force, acting on neutral gas (d) immediately after the pulse. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V)...51 Figure 3-13: Instantaneous distributions of electron number density (a) (electrons are neutralized on the surface by positive ions), positive ion number density (b) (positive ions drift to the negatively charged dielectric surface), electric potential (c) and force, acting on neutral gas (d) 0.5 μs after the pulse. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V)...52 Figure 3-14: Average force acting on neutral gas during one cycle. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V)...53 Figure 3-15: Simplified physical model (multiple sweeps by a porous piston ) describing the momentum transfer to the neutral gas in repetitive-pulse cases Figure 4-1: Time averaged force on the neutral gas. The left plot corresponds to computations with MacCormack method without FCT, and the right one to FCT method. (Pulse voltage: 1.5 kv amplitude, 4 ns FWHM, 500 khz repetition rate, and 500 V bias) Figure 4-2: Current and voltage in a representative repetitive-pulse case in quasi-steadystate regime. The red line shows the voltage, and the dashed line shows the current. The left plot corresponds to MacCormack method without FCT, and the right one to FCT method. (Pulse voltage: -1.5 kv amplitude, 4 ns FWHM, 500 khz repetition rate, and 500 V bias) Figure 4-3: Reverse breakdown. Applied voltage: negative pulses with positive bias. Peak voltage is -4.5 kv, the bias is 0.5 kv, and the pulse FWHM is 4ns. The Figure shows

10 x the electrons and positive ions number densities, electric potential, and the instantaneous force on the gas at the moment corresponding to the peak voltage...61 Figure 4-4: Reverse breakdown. Applied voltage is negative pulse with positive bias. Peak voltage is -4.5 kv, the bias is 0.5 kv, and the pulse FWHM is 4 ns. The Figure shows the electrons and positive ions number densities, electric potential and the instantaneous force on the gas after the pulse...62 Figure 4-5: Average force acting on neutral gas during one cycle. (Pulse voltage: amplitude is -4.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 0.5 kv) Figure 4-6: Streamer propagation speed in 3 different cases. Green line corresponds to the initial and background electron number density of /m 3, the purple line to /m 3, and the blue line to the initial electron number density of /m 3 and no background charge. Applied voltage waveform: positive pulses with positive bias. Peak voltage is 3 kv, the bias is 1kV, and the pulse FWHM is 4 ns Figure 4-7: Streamer-like ionization wave. Applied voltage waveform: positive pulses with positive bias. Peak voltage is 3 kv, the bias is 1kV, and the pulse FWHM is 4 ns. The Figure shows the electrons and positive ions number densities, electric potential and the instantaneous force on the gas...67 Figure 4-8: Average force acting on neutral gas during one cycle. (Pulse voltage: amplitude is 3 kv, FWHM is 4 ns, the repetition rate is 500 khz, and the bias is 1 kv) Figure 4-9: Momentum transferred to the gas. Blue and green lines correspond to the negative pulses with amplitudes -4.5 and -1.5 kv with positive bias of 0.5 kv, and the red line corresponds to the positive pulses with 3 kv amplitude and positive bias of 1 kv. FWHM for all pulses is 4 ns Figure 5-1: 2D decomposed numerical domain, used for DBD computations Figure 5-2: Comparison of computational time between the original single processor and 4-processor parallel codes...75 Figure 5-3: Comparison of the momentum, transferred to the gas, computed using the regular and modified time steps Figure 5-4: The computational time consumption for different stages of plasma modeling using the modified time step Figure 5-5: Momentum, transferred to the gas (Pulse voltage: amplitude 4 kv, FWHM is 4 ns, and the bias is 1 kv) Figure 6-1: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (3 kv amplitude, 4 ns FWHM) without photoionization or artificial plasma....84

11 xi Figure 6-2: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (3 kv amplitude, 4 ns FWHM) with 10 7 m -3 minimum electron number density Figure 6-3: Electron and positive ion number densities and electric potential distribution 4ns after the start of the pulse (3kV amplitude, 4ns FWHM) with m -3 minimum electron number density Figure 6-4: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (3 kv amplitude, 4 ns FWHM) with m -3 minimum electron number density Figure 6-5: Electron and positive ion number densities and electric potential distribution 4ns after the start of the pulse (4 kv amplitude, 4 ns FWHM) with 10 7 m -3 minimum electron number density Figure 6-6: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (4 kv amplitude, 4 ns FWHM) with 10 7 m -3 minimum electron number density Figure 6-7: Electron and positive ion number densities, difference between number densities of positively and negatively charged species, rate of photoionization production, electric potential distribution and electric field distribution 2.5 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters...90 Figure 6-8: Electron and positive ion number densities, difference between number densities of positively and negatively charged species, rate of photoionization production, electric potential distribution and electric field distribution 2.6 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters...91 Figure 6-9: Electron and positive ion number densities, difference between number densities of positively and negatively charged species, rate of photoionization production, electric potential distribution and electric field distribution 2.7 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters...92 Figure 6-10: Electron and positive ion number densities, difference between number densities of positively and negatively charged species, rate of photoionization production, electric potential distribution and electric field distribution 2.9 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters...93 Figure 6-11: Electron and positive ion number densities, rate of photoionization production and electric potential distribution 3ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters...94

12 xii Figure 6-12: Electron and positive ion number densities, rate of photoionization production and electric potential distribution 5ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters...95 Figure 6-13: Electron and positive ion number densities, rate of photoionization production and electric potential distribution 7ns after the start of the pulse (3kV amplitude, 4ns FWHM). Distances along both horizontal and vertical axes are measured in meters...96 Figure 6-14: Computed streamer propagation velocities for the DBD driven by 4ns positive pulses for voltage amplitudes 3 kv,4 kv and 7 kv based on the concept of minimum electron number density and photoionization...98 Figure 7-1: Air density and velocity distribution 14 milliseconds after the burst start Figure 7-2: Comparison between experimentally observed and simulated Schlieren images for DBD induced flow. The images are 20 mm by 10 mm each and correspond to 7, 14, and 21 milliseconds after the burst start Figure 7-3: Vortex propagation velocity as a function of the distance from DBD at different forces in the interaction region. The maximum velocities of the induced flow at the vicinity of interaction region are designated as max v Figure 7-4: Calculated 2D electric potential distribution based on the experimental data for the electric potential on the dielectric surface right after the voltage applied to DBD was turned off (Negative 4 kv pulses, positive 2 kv dc voltage) Figure 7-5: Experimentally measured voltage on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, negative 2kV dc voltage) Figure 7-6: Calculated charge density on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, negative 2kV dc voltage) Figure 7-7: Experimentally measured voltage on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, positive 2kV dc voltage) Figure 7-8: Calculated charge density on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, positive 2kV dc voltage) Figure 7-9: Experimentally measured voltage on the dielectric surface after 1, 10 and 100 applied pulses (Negative 4kV pulses, positive 2kV dc voltage) Figure 7-10: Calculated charge density on the dielectric surface after 1, 10 and 100 applied pulses (Negative 4kV pulses, positive 2kV dc voltage)

13 Figure 8-1: Single DBD induced flow 1, 4 and 8 ms after the DBD is turned on. Color plots represent gas density. Arrows represent the distribution of the induced gas velocity Figure 8-2: Distribution of horizontal component of the flow velocity in the vicinity of DBD flow interaction region 1, 4 and 8 ms after the DBD is turned on. Black rectangular shows the interaction region Figure 8-3: Distribution of the horizontal component of gas velocity along the vertical axis right after the DBD flow interaction region (1.1 cm from the left boundary of the physical domain) Figure 8-4: Time evolution of the maximum DBD induced gas velocity for a single DBD Figure 8-5: Time evolution of the momentum, transferred to the gas by the DBD, (purple) and momentum of the flow (blue) Figure 8-6: Flow, induce by a series of 2 DBDs, 1, 4 and 8 ms after the DBDs were turned on. Color plots represent gas density. Arrows represent the distribution of the induced gas velocity Figure 8-7: Flow, induce by a series of 3 DBDs, 1, 4 and 8 ms after the DBDs were turned on. Color plots represent gas density. Arrows represent the distribution of the induced gas velocity Figure 8-8: Time evolution of the maximum DBD induced gas velocity for a single DBD (green) and series of 2 (red) and 3 (blue) DBDs Figure 9-1: Scheme of ferroelectric plasma source Figure 9-2: Electron and ion number densities, electron temperature and electric potential distribution for the case of no electron source. The snap shot is taken 20ns after the pulse start. The box size is 1mm horizontally and 0.5 mm vertically. A positive Gaussian pulse with 50ns FWHM (full width at half maximum) and 3kV amplitude is applied to the upper electrode and the lower electrode is considered grounded Figure 9-3: Electron and ion number densities, electron temperature and electric potential distribution for the case of uniform background electron number density. The snapshots are taken at (a) 20ns and (b) 30ns after the pulse start Figure 9-4: Electron number densities during the pulse. The color-plots are taken each 5ns, starting from 10ns. The emission current from the ferroelectric is A/cm Figure 9-5: Electron number densities, difference between electron and ion number densities, electric field and electric potential distribution for the case of A/cm emission current. (a) corresponds to the real scale geometry and (b) is zoomed region near the streamer head. The color-plots are taken 30ns after the pulse start xiii

14 Figure 9-6: The color-plots of the electron densities at 30ns after the pulse start for different values of emission current from the ferroelectric Figure 10-1: Effect of electron-electron interaction time on the drilling depth dependence of the laser pulse energy Figure 10-2: Drilling depth vs laser pulse energy in stainless steel for different number of single pulses. Laser wavelength is 532nm and the Gaussian beam radius at 1/e 2 level is w 0 =9.5 µm. The values of drilling depth per one laser pulse averaged over all experimental data for a given pulse energy are marked as average. The computed values of drilling depth are marked comp Figure 10-3: Drilling depth vs laser pulse energy in stainless steel for different number of single pulses. Laser wavelength is 355 nm and the Gaussian beam radius at 1/e 2 level is w 0 =6.3 µm. The values of drilling depth per one laser pulse averaged over all experimental data for a given pulse energy are marked as average. The computed values of drilling depth are marked computed Figure 10-4: Drilling depth vs laser pulse energy in copper for different number of single pulses. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =9.5 µm. The values of drilling depth per one laser pulse averaged over all experimental data for a given pulse energy are marked as average. The computed values of drilling depth assuming are marked computed Figure 10-5: Drilling depth vs laser pulse energy in copper for different number of single pulses. Laser wavelength is 355 nm and the Gaussian beam radius at 1/e level is w 0 =6.3 µm. The values of drilling depth per one laser pulse averaged over all experimental data for a given pulse energy are marked as average. The computed values of drilling depth are marked computed Figure 10-6: Drilling depth vs laser pulse energy in titanium for different number of single pulses. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =9.5 µm Figure 10-7: Drilling depth vs laser pulse energy in titanium for different number of single pulses. Laser wavelength is 355 nm and the Gaussian beam radius at 1/e level is w 0 =6.3 µm. The values of drilling depth per one laser pulse averaged over all experimental data for a given pulse energy are marked as average. The computed values of drilling depth are marked comp Figure 10-8: Energy deposition in copper for different pulse energies. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Laser duration is 10 ps Figure 10-9: Energy deposition in copper for pulse durations. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj xiv

15 Figure 10-10: Evolution of electron and lattice temperature at the copper surface. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj. Pulse duration is 10ps Figure 10-11: Evolution of drilling and melting depth of copper. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj. Pulse duration is 10ps Figure 10-12: Speed of material removal front (copper). Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj. Pulse duration is 10ps Figure 10-13: Evolution of electron and lattice temperature at the copper surface. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj. Pulse duration is 10ns Figure 10-14: Evolution of drilling and melting depth of copper. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj. Pulse duration is 10ns Figure 10-15: Speed of material removal front (copper). Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj. Pulse duration is 10ns Figure 10-16: Drilling and melting depth (copper) as a function of laser pulse duration. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 5 μj Figure 10-17: Drilling and melting depth (copper) as a function of laser pulse duration. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 20 μj Figure 10-18: Drilling and melting depth (copper) as a function of laser pulse duration. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 50 μj Figure 10-19: Drilling and melting depth (copper) as a function of laser pulse duration. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 100 μj Figure 10-20: Drilling and melting depth (copper) as a function of laser pulse duration. Laser wavelength is 532 nm and the Gaussian beam radius at 1/e level is w 0 =6.5 µm. Pulse energy is 200 μj Figure 11-1: Proposed configuration for modeling of DBD plasma actuator using adaptive meshes xv

16 xvi LIST OF TABLES Table 1: Characteristic scales for the DBD operation...20

17 xvii ACKNOWLEDGEMENTS First of all, I would like to thank my thesis advisor, Dr. Vladimir Semak. He was not only an outstanding scientist, guiding and assisting me in my research, but also an exceptional mentor throughout all obstacles on the path to this PhD dissertation. I would also like to thank Dr. Mikhail Shneider, my informal co-advisor, for teaching me ALL aspects of physics, mathematics, numerical methods etc, and for providing me priceless help during my entire period of graduate study. Without these two extraordinary people I cannot imagine I could complete this thesis. I am grateful to my other committee members, Dr. Akhlesh Lakhtakia, Dr. Albert Segall, Dr. Victor Pasko and Dr. Deborah Levin for service on my doctoral committee and extremely useful suggestions on the thesis organization and writing style. I thank my co-authors and collaborators, Dr. Sergey Macheret, Richard Miles and my friend Dmitry Opaits for all our discussion and for all your thoughts on my research. Dr. Macheret was an inspirer of a large part of my research. Dmitry s experimental results allowed discovering new perspective in my field of study. I would like to acknowledge many researchers around the world for fruitful discussions and exceptional help. I am very grateful to Dr. Jon Poggie, who managed to teach me a whole course of parallel computing during couple of days of his vacations, to Dr. Anne Bourdon and Dr. Victor Pasko for an outstanding tutorial on the photoionization problem and to Dr. Igor Kaganovich for our collaborative work on ferroelectric plasma sources. I thank Dr. John Schmissier, Dr. Jon Poggie, Dr. Joe Silkey, Dr. Deborah Kirby, Dr. Sarabjit Mehta, Dr. Dan Gregoire, Dr. David Ashpis, Dr. Richard McNitt not only for supporting my research efforts, but also for discussing the results and sharing their ideas on the directions of the research.

18 xviii Finally, I would like to thank my entire beloved family, my wife Anastasia, my parents Vladimir and Nadezhda, my sister Marianna, for supporting me during my graduate study. I would not be able to handle it without you. At last but not at least I would like to thank The Boeing Company, NASA, Air Force Office of Scientific Research, Wright-Patterson Air Force Research Laboratory, HRL Laboratories, PennState Applied Research Laboratory and Princeton University for the financial support.

19 1 Chapter 1 Introduction Plasma is conventionally defined as the fourth state of matter. As temperature rises, matter transforms from solid to liquid state at melting point, then to gaseous state at boiling point, and, finally, at temperatures of thousands degrees Kelvin, molecules and atoms dissociate into ions and electrons and matter transforms to plasma. However, the accurate definition of plasmas is much broader. The word plasma is used to describe a wide variety of macroscopically neutral substances containing many interacting free electrons and ionized atoms or molecules, which exhibit collective behavior due to the long-range coulomb forces (Bittencourt, 2004, p. 1). According to this definition, any neutral substance with relatively free electrons can be treated as a plasma. For example, matter does not need to be heated to the temperatures of thousands degrees Kelvin to order to become ionized. A high electric field can produce ionization of different gases at room temperature. The ions and neutrals of this weakly-ionized gas will remain at room temperature. Only electrons are heated to the temperature of the order of ten thousand degrees Kelvin in the presence of an electric field. Another example of plasmas is metal. Metal can be considered as ultra-dense gas of bound ions and almost free electrons. One of the main differences between the description of the plasmas and three other states of matter is the number of species. In order to describe gas or liquid flow, it is necessary to consider an interaction between molecules or atoms within the flow and

20 2 external impacts. On the contrary, the description of a plasma needs separate consideration of each kind of charged species and, for a lot of problems, neutral species. Despite the fact that the first comprehensive study of plasma properties and plasma phenomena was carried out by Langmuir (Langmuir, 1928) only 80 years ago, a number of devices, designed based on plasmas, became a part of our life. Fluorescent lamps and plasma-display-panel television sets are just several examples of such type of devices. A number of microelectronics components, such as computer chips, are produced using plasma-processing technology (Lieberman & Lichtenberg, 2005). A technology for ultraviolet lasers design is based on gas discharges (Elliot, 2005). The range of the developed and potential applications shows the necessity of comprehensive plasma description. Typically, plasma technologies utilize two approaches. The first one is based on phase transitions between plasmas and other states of matter. The second one is based on an interaction between the plasmas and other states of matter. However, both approaches use the same phenomena for plasma manipulation. Since a plasma is a substance, consisting of charged particles, its dynamics can be defined by externally applied electric or/and magnetic fields. Electric fields are usually used for generation, acceleration and heating of the charged particles. Magnetic fields are used for plasma confinement for fusion. This thesis analyzes the application of plasmas in high electric fields for flow separation control and ultra-short pulse laser drilling.

21 3 Dielectric Barrier Discharge (DBD) Plasma Actuators for Flow Separation Control An Overview of General Concepts of Plasma Aerodynamics Let us consider plasmas for aerospace applications. Electrohydrodynamic (EHD) and magnetohydrodynamic (MHD) devices have shown a potential for internal and external flow control as well as for power extraction (Myrabo & Raizer, 1994), (Kimmel, Hayes, Menart, & Shang, 2004), (Poggie, 2004), (Shang, Kimmel, Hayes, Tyler, & Menart, 2005), (Shang & Surzhikov, 2005), (Menart, Shang, Atzbach, Magoteaux, Slagel, & Bilheimer, 2005), (Leonov, Yarantsev, & Kuriachy, 2005), (Nishihara, Jiang, Rich, Lempert, Adamovich, & Gogineni, 2005), (Nishihara, Rich, Lempert, & Adamovich, 2006), (Miles, et al., 2006), (Macheret, 2006), (Zaidi, Smith, Macheret, & Miles, 2006), (Roth, Sherman, & Wilkinson, 2000), (Roth, 2003). In general, plasma can induce a body force on gas, heat gas or produce radicals for combustion. Body forces can be exerted on charged species (electrons and ions) by electric and magnetic fields and coupled to bulk gas by collisions (Miles, et al., 2006), (Roth, 2003), (Roth, Sherman, & Wilkinson, 2000). These forces can then be used to extract power from the flow or to control the flow. Both body forces and heating can be used to modify the flow (Myrabo & Raizer, 1994). There is also an increasing interest in utilization of the near-surface plasmas with and without magnetic fields in order to control both laminar and turbulent boundary layers, shock-boundary layer interactions, and, in particular, flow separation. Scientists at Air Force Research Lab studied dc surface plasmas in Mach 5 tunnel

22 4 (Kimmel, Hayes, Menart, & Shang, 2004), (Poggie, 2004), (Shang, Kimmel, Hayes, Tyler, & Menart, 2005), (Shang & Surzhikov, 2005), (Menart, Shang, Atzbach, Magoteaux, Slagel, & Bilheimer, 2005). Leonov et al. investigated flow control with high-current quasi-equilibrium surface arcs (Leonov, Yarantsev, & Kuriachy, 2005). Nishihara et al. demonstrated MHD effects on a supersonic turbulent boundary layer and flow deceleration by Lorentz force (Nishihara, Jiang, Rich, Lempert, Adamovich, & Gogineni, 2005), (Nishihara, Rich, Lempert, & Adamovich, 2006). A group at Princeton University proposed and demonstrated cold (nonequilibrium) constricted surface discharges ( snowplow arcs ) that are magnetically driven downstream or upstream at high velocity and impart momentum to the boundary layer (Miles, et al., 2006), (Macheret, 2006), (Zaidi, Smith, Macheret, & Miles, 2006). All those studies were devoted to high-speed, supersonic or hypersonic, flow control. Description of the DBD Plasma Actuator and its Application for Flow Separation Control One of the challenging aerodynamic problems is flow separation control. Flow separation occurs on the wings of an airplane at high angles of attack 1 (Anderson, Introduction to Flight, 2004, pp ). It decreases the lift of the wing leading to an airplane stall. Recently, the possibility of an application of the dielectric barrier discharge (DBD) plasma actuators for flow separation control and near-surface wind generation 1 Angle between the chord line of an airplane wing and the airplane velocity

23 5 (Roth, 2003), (Roth, Sherman, & Wilkinson, 2000) has garnered significant attention. The scheme of the DBD plasma actuator is shown in Fig. 1-1 (Likhanskii, Shneider, Macheret, & Miles, 2007). It consists of two electrodes separated by a dielectric. If a voltage is applied between the electrodes, the plasma is generated near the edge of the upper (exposed) electrode and induces gas flow. The characteristic size of the DBD plasma actuator is of the order of a millimeter in vertical direction and several centimeters in the horizontal (streamwise) direction. Figure 1-1: Scheme of the DBD plasma actuator. The first successful experiments on the flow separation control using the DBD plasma actuators were conducted by Reece Roth (Roth, 2003), (Roth, Sherman, & Wilkinson, 2000). In those experiments a series of the DBDs were mounted on the airfoil. Then, the airfoil was placed inside the wind tunnel with 2 m/s free stream air velocity. At low angles of attack, air flew around the airfoil, i.e. flow streamlines followed the airfoil profile. When the angle of attack was increased to 12 deg, the flow was separated from the airfoil. Figure 1-2 shows flow visualization around the airfoil at angles of attack corresponding to the flow separation. When the DBDs were turned on, the flow was

24 6 reattached, i.e. the flow separation was suppressed. This effect was observed in a limited range of angles of attack. At a 16-deg angle of attack, the DBDs could not reattach the flow. Figure 1-2: Demonstration of flow reattachment by the DBD plasma actuator (Roth, 2003). The flow separation control using the DBD plasma actuator is a promising concept. It significantly differs from the conventional methods of flow separation control based on the use of mechanical devices interacting with the flow in the separation region. The conventional flow control is achieved using artificial roughness of the surface (Lebeau, Reasor, & Jacob, 2007), (Stephani & Goldstein, 2007). Another method for flow control is use of wall jets (Amitay & Glezer, 2006). (Nishri & Wygnanski, 1998), (Naveh, Seifert, Tumin, & Wygnanski, 1998), (Greenblatt & Wygnanski, 2000) proposed to use an oscillating membrane for wall jet generation and consequent flow separation suppression. There are several disadvantages associated with the mechanical devices. Usually, rough surfaces decrease the aerodynamic characteristics of an airfoil. Some

25 7 mechanical devices have moving parts, and their response times can be quite long. The DBD plasma actuator does not have the above-mentioned weaknesses. The absence of moving parts makes DBD a durable device. The control, based on the DBD, is active, meaning that the DBD can be switched on or off or tuned to the particular conditions. The response time (milliseconds) is very short as compared to the mechanical actuators (seconds). The choice of the driving voltage provides significant tuning capabilities. Besides the described advantages over the mechanical actuation mechanisms, a conventional DBD, driven by AC voltage (Roth, Sherman, & Wilkinson, 2000), has a significant limitation flow separation can be suppressed at relatively low free stream flow velocities up to several tens of meters per second, and no effect was observed at higher velocities. This restriction limits DBD potential applications, such as flow separation control at commercial airplanes and turbine engines, or for hypersonic applications. The main goal for the DBD study is to understand the physics of the induced gas flow and to explore DBD potential capabilities, such as maximum induced flow velocities, maximum thrust, thermal energy deposition characteristics, for a various range of aerospace applications. During past 10 years the DBD plasma actuators were widely investigated both experimentally and numerically. Experimental Study of the DBD Plasma Actuators Experimental DBD studies are usually devoted to a development of an application base for the DBD and to understanding of the physics, underlying DBD operation. A

26 8 series of the wind tunnel experiments were conducted (Post & Corke, 2004), (Roupassov, Nikipelov, Nudnova, & Starikovskii, 2008), (Sidorenko, et al., 2007) for an investigation of the potential applications. The dependences of the flow pattern, lift force, and pressure distribution at the airfoil surface on the DBD mounting position, applied voltage profile, and DBD operation at different angles of attacks were studied. The DBD mounting position should coincide with the separation point on the airfoil for a certain angle of attack for the most efficient flow reattachment. Sidorenko et al. demonstrated a suppression of flow separation by the DBD, driven by high-voltage repetitive nanosecond pulses (tens of kilovolts amplitude and tens of nanoseconds width), at 110 m/s free stream velocities (Sidorenko, et al., 2007). The goal of another experimental approach was to understand the physics of the DBD operation. In those experiments the DBDs were usually investigated in quiescent air in order to eliminate an influence of the free stream flow on the DBD induced flow. Hall et al. used the particle imaging velocimetry (PIV) (Hall, Jumper, Corke, & McLaughlin, 2005) technique to quantitatively describe the DBD induced flow. The DBD plasma actuator was placed inside a gas chamber filled with PIV particles. The PIV particles were irradiated by a laser, and a fast CCD camera tracked their motion. When the DBD was turned on, the PIV particles were assumed to follow the induced flow stream lines. Figure 1-3 shows the typical profile of the induced gas velocity restored from the PIV measurements. The DBD is located on the horizontal surface between points -25 and -15 in Figure 1-3. The arrows show the directions and the relative magnitudes of the induced gas velocities.

27 9 Figure 1-3: Induced gas velocity pattern restored from PIV measurements (Hall, Jumper, Corke, & McLaughlin, 2005). Despite the wide use of the PIV technique, for flow visualization in aerodynamics, its applicability to the flow in presence of gas discharge is questionable. The PIV particles can be charged by the DBD plasma, and their motion can be induced not only by interaction with the gas flow, but also by the electric field in the DBD configuration. PIV measurements are intrusive, i.e. they have an impact on the DBD induced flow. The direct thrust measurements (Opaits, et al., 2008), (Enloe, et al., 2004) are examples of non-intrusive techniques. In those experiments the DBDs were set on electronic scales, which were placed in a Faraday cage in order to avoid the influence of the electromagnetic waves generated by the DBD. Based on the thrust measurements, the empirical law for the DBD thrust as a function of peak voltage of the applied sine signal was obtained. It was also demonstrated that there is an optimum operating frequency for the maximum thrust at particular conditions. However, the direct thrust measurement is also a questionable experimental technique due to the presence of viscous effects. Since

28 10 the DBD operates within a boundary layer, a portion of the DBD induced momentum of the gas can be lost to the aerodynamic surface. Therefore, direct thrust measurements would show the momentum of the induced gas flow rather than the momentum embedded by the DBD. The described experimental techniques provided an insight into the DBD operation. Nevertheless, there is still a necessity to develop a nonintrusive diagnostic technique to restore the whole flow pattern. Numerical Study of the DBD Plasma Actuators and Motivation for the Present Work Despite successfully demonstrated flow separation control using DBD plasma actuators and an apparent simplicity of the DBD design, a fundamental understanding of the underlying physics of the DBD operation is still incomplete. It is accepted that the experimentally observed near-surface gas acceleration to several meters per second velocity is due to the ions that move in electric field and transfer momentum to the gas molecules. However, not much was established beyond this simple idea. Challenges in modeling and understanding of the DBD plasma actuators stem from the physics of the problem. Generation, decay, drift, and diffusion of the electrons and both positive and negative ions must be correctly described by a consistent, comprehensive, physically-based model and resolved by an accurate numerical scheme. Since it is the electric space charge that is responsible for the net force acting on the gas, the Poisson equation for the electric potential must be fully coupled with the other

29 11 equations. Various processes occurring on both dielectric and metallic surfaces, such as electron attachment to the dielectric surface and secondary electron emission from both metallic and dielectric surfaces, must be included. Additional complexity for the modeling is relatively high air density at which the DBD actuators operate (approximately sea-level atmospheric density for ground tests). Typical commercial numerical codes for plasma simulations were developed for the rarified gas with pressures of the order of several Torrs. High air density reduces both time and length scales relevant to the problem. For example, avalanche ionization and sheath formation occur on a sub-nanosecond time scale, and sharp gradients of electric field and net charge density occur on a micron length scale. At the same time, the model should also be able to compute the plasma dynamics on macroscopic time (milliseconds) and length (millimeters or centimeters) scales. Because of this complexity, several initial modeling efforts (Roy & Gaitonde, 2005), (Roy, 2005), (Singh & Roy, 2005), (Boeuf & Pitchford, 2005), (Orlov & Corke, 2005) did not capture some essential features of dynamics of the DBD plasma actuators. The first attempts on DBD modeling were done in (Roy, 2005), (Roy & Gaitonde, 2005), (Singh & Roy, 2005). In those papers the DBD, driven by the radio-frequency (RF) sinusoidal voltage, was studied. The modeling was performed in helium-like gas at 300 Torr, however, the processes of surface charge accumulation, recombination and secondary emission, which are essential for the plasma discharges, were not taken into account. The simulations qualitatively explained that the force in the positive half-cycle is due to the downstream positive ion motion. Later, the developed plasma model was

30 12 coupled with hydrodynamic model for the description of the induced gas flow (Roy & Gaitonde, 2006), (Roy, Singh, & Gaitonde, 2006), (Singh, Roy, & Gaitonde, 2006). The first comprehensive plasma model for the DBD study in nitrogen was developed by Boeuf et al. (Boeuf & Pitchford, 2005). Positive DC voltage was applied between the electrodes in the DBD configuration in order to analyze plasma generation. The plasma was generated in streamer-like form. (Boeuf, Lagmich, Unfer, Callegari, & Pitchford, 2007) performed the first experimental scales simulations for the DBD, driven by the sine voltage, in air. It was concluded that the main contribution to the force, acting on the gas, is between the breakdown and the DBD downstream directed force in the negative half-cycle of the DBD operation is due to the negative ions drift. These results are in good agreement with ones obtained earlier in (Likhanskii, Shneider, Macheret, & Miles, 2006). Unfer et al. (Unfer, Boeuf, Rogier, & Thivet, 2008) coupled the developed plasma modeling for the DBD simulations in air with the Navier-Stokes solver and obtained the experimentally observed induced gas velocity profiles. Orlov et al. developed the first empirical model for the description of the DBD operation (Orlov & Corke, 2005). Later, a difference between discharge properties in negative and positive half-cycles of the sinusoidal voltage were investigated both numerically and experimentally. It was observed that the discharge developed in streamer form in the positive half-cycle of DBD operation and in diffusive form in the negative half-cycle of DBD operation. Finally, Moreau published an excellent review on major experimental and some of the numerical efforts on the study of the DBD operation (Moreau, 2007).

31 13 Despite the intensive both experimental and numerical study of the DBD plasma actuators, only a part of experimentally observed phenomena was explained, and no potential ways of the DBD improvement were proposed. Therefore, it is necessary to develop a consistent, comprehensive, and physically-based model that will describe the operation of the DBD actuators and proposed optimized DBD configuration. This thesis describes the successful and significant progress in the development of such model. Study of the Ultra-Short Pulse Laser Drilling of Metals and Motivation for the Present Work Another phenomenon, which is based on an interaction between high electric field with the charged particle and considered in this thesis, is ultra short laser pulse drilling. Laser drilling is being widely used as a manufacturing technique for a number of industrial applications (Semak & Matsunawa, 1997), (Bado, Clark, & Said, 2009), (Liu, Du, & Mourou, 1997), (Semak, Thomas, & Campbell, 2004), (Semak, Thomas, & Campbell, 2004), (Semak & Schiano, 2007), (Semak, Campbell, & Thomas, 2006), (Semak, Thomas, & Campbell, 2005), (Ivanov & Zhigilei, 2003), (Schafer, Urbassek, & Zhigilei, 2002), (Anisimov, Kapeliovich, & Perelman, 1974). The advantages of laser drilling over other drilling techniques, such as mechanical and chemical drilling, are the possibility of creating micron and submicron scale holes and fast drilling rates. The drawback of laser drilling is large amount of the produced melt. Therefore, there is a need

32 14 to develop a laser drilling technique with minimum melt generation while maintaining fast drilling rates. The theoretical investigation of laser-material interaction (Semak & Matsunawa, 1997), (Semak, Campbell, & Thomas, 2006), (Semak, Thomas, & Campbell, 2005), (Ivanov & Zhigilei, 2003), (Schafer, Urbassek, & Zhigilei, 2002), (Anisimov, Kapeliovich, & Perelman, 1974) demonstrated that there are two major processes responsible for the material removal: evaporation and melt ejection due to the recoil pressure. At relatively low temperatures (below the temperature of evaporation), the melt ejection is the dominant process for the material removal. When the temperature rises, the material removal is due to the evaporation. Recently, a significant interest was caused by a possibility of using ultra-short, ultra-high-intensity laser pulses in order to decrease the melt production and to enhance the evaporative drilling. The idea of this approach followed from theoretical and numerical investigation of laser-material interaction. When a laser irradiates metal surface, the laser radiation is absorbed by the electrons mainly within the skin layer of metal. During tens of picosecond time scale, the electrons transfer their energy to the lattice causing material temperature increase. Then, on the nanosecond time scale, the thermal energy is transferred into the bulk metal leading to the metal melting. The ultrashort, ultra-high-intensity pulses were expected to resolve this problem. It was assumed that, if the metal was irradiated by the femto- or picosecond high-intensity pulses, the metal would be heated within a skin layer depth above the critical temperature, and the material removal would be only due to evaporation.

33 15 However, recent experimental work (Campbell, Likhanskii, & Semak) showed substantially different results from those predicted by currently existing models. In those experiments different metals (aluminum, stainless steel, titanium, and nickel) were irradiated by a picosecond laser. The depth of the obtained holes was of the order of a micron unlike the predicted tens of nanometers drilling depth ; and with several microns thick melt was observed around the crater. Therefore, there is a clear need to develop an approach for accurate description of the experimentally observed phenomena and address the issues of possible improvements of laser drilling. Scientific Contributions The purpose of this Ph.D. dissertation is: develop a complete, consistent, comprehensive physical model for the description of the DBD plasma actuator; provide a detailed description of the physics of the DBD operation; propose novel optimal configurations of the DBD plasma actuator; make a developed model flexible to wide range of gas discharge applications, not limited to the modeling of the DBD plasma actuator; analyze the limitations of the DBD plasma actuators; investigate the ultra-short high-intensity pulse laser drilling of metals.

34 16 The major scientific contributions, resulting from this dissertation work, are summarized below: Development of complete, consistent, comprehensive physical model and both single- and multi-processor numerical models for the description of the DBD plasma actuator; Explanation of the force generation in the DBD plasma actuator configuration: The major contribution to the force in the DBD operation was shown to be between the breakdowns; The unidirectional force in the conventional DBD plasma actuator, driven by sinusoidal voltage, is due to the positive ions pushing in the positive half-cycle and negative ions in the negative halfcycle; An attraction of positive ions to the exposed electrode in the negative half-cycle and an attraction of negative ions to the exposed electrode in positive half-cycle are major sources of inefficiency for the conventional DBD plasma actuator; Proposal of novel DBD plasma actuator configuration, consisting of nanosecond pulses plus bias, for the improvement of the DBD operation; Resolution of the reverse breakdown in the case of applied negative nanosecond pulses and positive DC bias; Investigation of streamer-like solution for the plasma generation by the positive nanosecond pulses;

35 17 Demonstration of the significant dependence of streamer characteristic on the initial and background plasma number densities in the absence of photoionization; Detailed description of streamer formation and its propagation in the presence of photoionization in the DBD configuration; Development and implementation of the concept of modified time step, which allows simulating plasmas with 3-order of magnitude increased computational speed without loss of accuracy. This model can be used for a numerical description of a wide range of gas discharges in electronegative gases; Description of plasma generation by the ferroelectric plasma sources. It was demonstrated that the plasma generation is significantly influenced by the ferroelectric emission current; Development of the hydrodynamic model for the DBD induced flow for the description of the experimental investigations of the DBD plasma actuator. By using this model and general concepts of plasma physics, the limitation of the DBD operation was analyzed. It was demonstrated that significant portion of DBD induced momentum of the gas is lost to the surface due to the viscosity; Investigation of the possibility of an increase of the maximum DBD induced flow velocity by using a series of the DBD actuators;

36 18 Development of 1D physical model for the description of ultra-short highintensity pulse laser drilling; Explanation of the experimentally observed melt production and ultradeep laser drilling; Description of the drilling and melting of the metal via pico- and nanosecond high-intensity laser pulses; Analysis of the optimum conditions for pulse laser drilling. The results presented in this dissertation were published in (Likhanskii, Shneider, Macheret, & Miles, 2006), (Likhanskii, Shneider, Macheret, & Miles, 2007), (Likhanskii, Shneider, Macheret, & Miles, 2008), (Likhanskii, Semak, Shneider, Opaits, Macheret, & Miles, 2008), (Likhanskii, Semak, Shneider, Opaits, Macheret, & Miles, 2009), (Opaits, Shneider, Miles, Likhanskii, & Macheret, Surface Charge in Dielectric Barrier Discharge Plasma Actuators, 2008), (Opaits, Shneider, Miles, Likhanskii, & Macheret, Experimental Investigation of Dielectric Barrier Discharge Plasma Actuators Driven by Repetitive High-Voltage Nanosecond Pulses with DC or Low Frequency Sinusoidal Bias, 2008), (Opaits, et al., 2008).

37 19 Chapter 2 Physical Model of the DBD Plasma Actuator The comprehensive DBD modeling should rely on the physical model which tracks all essential plasma and gas phenomena. The DBD problem can be split into three parts: the discharge development, the momentum and thermal energy transfer from the charged species to the gas, and the induced gas motion. Let us consider the characteristic times of these processes and the characteristic scales based on the experimental observations and analytical estimates. First, let us consider geometrical scales. In the experiments (Roth, Sherman, & Wilkinson, 2000), (Opaits, et al., 2008) the visible plasma spreads several millimeters along the dielectric. The particle image velocimetry (PIV) measurements (Roth, 2003) indicate the induced gas motion several centimeters downstream from the visible plasma range. Second, let us consider time scales. Based on the current-voltage characteristics of the DBD (Post & Corke, 2004), the characteristic discharge development time is nanoseconds. The time of coupling between the charged particles and the gas molecules can be estimated by the characteristic plasma length and the ion drift velocities in gas. Since the plasma length (or visible region of the plasma glow) is of the order of millimeters and the ion drift velocities are of the order of 1 km/s, the coupling time is several microseconds. Finally, the induced gas velocity, which is tens of meters per

38 20 second, leads to the characteristic flow time scales of the order of ten milliseconds (Table 1). Based on the above discussion, it is reasonable to split the problem into two parts. First, the discharge development and coupling with the flow should be considered. The second is the consideration of the induced gas motion modeling using the plasma-flow coupling characteristics as an input. Table 1: Characteristic scales for the DBD operation Characteristic Time Characteristic Length Discharge Development 1-10 ns 1-5 mm Plasma-Flow Coupling μs 1-5 mm Induced Flow 1-10 ms 1-30 cm Plasma Model In order to completely describe the experimentally-observed DBD phenomena, the DBD modeling was performed in air. The weakly-ionized air plasma was modeled as a four-fluid mixture: neutral molecules, electrons, and positive and negative ions. The presence of negative ions in air (unlike the models for inert gases or pure nitrogen) is quite important for understanding of the DBD physics, as is shown in Chapter 3. The motion of the charged particles was considered in the drift-diffusion approximation (Boeuf & Pitchford, 2005). The number densities of electrons, positive and negative ions,

39 and neutrals are denoted as n,,, and e n+ n n 21 respectively, and fluxes of electrons, positive and negative ions as,, respectively. The rates of ionization, electron-ion and ion-ion recombination are denoted by α, β and β ii respectively. The rates of electron two-body dissociative detachment from negative oxygen ions ( ) and three-body attachment to neutral oxygen ( ) are denoted by k d and ν a respectively. Electron and ion diffusion coefficients and mobilities are D e, D i and μ e, μ i correspondingly. The thermal diffusion rate is k t. The electron temperature is T e. The Boltzmann constant is k. The continuity equations for electrons, positive and negative ions are then (Macheret, Shneider, & Miles, 2002), (Bourdon, Pasko, Liu, Celestin, Segur, & Marode, 2007): n t e + r Γ r 2 = α Γ r βn n + k nn ν n n + S, (1) e e e + d a e ph n t t + r Γ r = α Γ r n n + S, (2) + + e βnen+ βii r r n + Γ = β iin n kd nn +ν an 2 + where r Γ e r Γ r Γ = r r + r T n e ph, (3) e μ eene De ne kt Dene, (4) Te r r μ n, (5) + = i En+ Di + r r = μ En D n. (6) i i

40 Since the gas velocity is low in comparison with drift velocities of charged particles, the gas was considered stationary for plasma dynamics simulations. Ionization and recombination coefficients, as well as attachment, detachment rates, charged particles mobilities and electron temperature are considered as functions 22 r E / n, where E r is local electric field. The numerical values for rates of ionization, electron-ion recombination, ion-ion recombination, attachment, detachment and charged ion mobilities are standard and provided in (Macheret, Shneider, & Miles, 2002): 4 2 α = nkt ( E /( nkt ) 32.3), 1/cm for 44 < E /( nkt) < 176, V/(cm*torr), α = 15nkT exp( 365E /( nkt)), 1/cm for 100 < E /( nkt) < 800, V/(cm*torr), β = β ii (300 / Te = ) 1/ 2, cm 3 /s, 18 2 [ n(300 / ) ] 7 1/ (300 / T ) T, cm 3 /s, μ 2.76 /( nkt ), cm 2 /(V*s) for E /( nkt) < 49. 3, V/(cm*torr), i = μ ( /( nkt )) / E /( nkt ), cm 2 /(V*s) for E /( nkt) > 49.3, V/(cm*torr), i = k d = 8.6 *10 exp( ) *[1 exp( )], cm 3 /s, T T ( T e T ) ν a = 0.21* 1.4 *10 * exp * exp + Te T TeT, cm 6 /s. 2 ( ) T T e 0.78 * 1.07 *10 * * exp * exp Te T TeT Ion temperature is considered to be equal to gas temperature. Electron drift velocities V dr and electron temperature T e, as functions r E / n, are given in the tables in (Grigoriev &

41 Meilikhov, 1997, pp ). The electron mobilities are calculated using the following relation: 23 V dr μ e = r (7) E The diffusion coefficients are calculated using the Einstein relation D i = μ T, (8) i i where i denotes the species (electrons, ions). The boundary conditions at the electrode surface are, for the cathode: Γ = γ Γ, (9) en m + n and, for the anode: Γ +n = 0, (10) where n indicates the normal component. The boundary conditions at the dielectric surface are: Γ = γ Γ, if < 0 (E n is directed towards the dielectric) (11) en d + n E n and, Γ +n = 0, if > 0 (E n is directed from the dielectric). (12) E n Here, γ m and γ d are the effective secondary emission coefficients from metal and dielectric surfaces. The electric field and potential φ are related to the density of charged species according to the Poisson equation: r ε r ϕ) = e ( n + n n ), (13) ( e + E r = ϕ r, (14)

42 24 where e is a charge of an electron and ε is a dielectric permittivity of the media. The volumetric force acting on neutral gas and rate of thermal energy deposition Q can be generally expressed the following way (Boeuf & Pitchford, 2005): r F = e( n + n dn ( meμe dt e n e r ) E + dn m+ μ+ dt + + m dn r r μ ) E ( nekte + n+ kt dt + + n kt ) (15a) 2 Q = E e( μ ene + μ+ n+ + μ n ) (15b) For the typical conditions of the DBD operation, first term in expression in the RHS of Eq. (15a) will be much greater than two other terms (see Chapter 3 for discussion). The term S ph denotes a rate of photoionization. The classical model by Zheleznyak et al. (Zhelezniak, Mnatsakanyan, & Sizykh, 1982) provides the following expression for the photoionization: S ph r ( ) = V ' r I( ') g( R) dv', (16) 2 4πR r r r where R = ', functions I(r ') and g(r) are known functions of coordinates and plasma properties (Bourdon, Pasko, Liu, Celestin, Segur, & Marode, 2007). The solution of Eq. (16) is time consuming compared to the solution of Eqs (1)- (15). An inefficiency of the photoionization term description using the integral approach imposes strong limitations on the DBD modeling. An approach based on the solution of differential equations, rather than an integral one, was used for accurate photoionization modeling. This approach was initially proposed by Segur et al. (Segur, Bourdon, Marode, Bessieres, & Paillol, 2006) and Bourdon et al. (Bourdon, Pasko, Liu, Celestin, Segur, &

43 25 Marode, 2007) for general streamer modeling. This approach was compared with the integral approach (Bourdon, Pasko, Liu, Celestin, Segur, & Marode, 2007) and provided an excellent agreement. The idea of the approach was introduced by Larson et al. (Larsen, Thömmes, Klar, Seaïd, & Götz, 2002) for transitions from integral equations to differential ones. Based on (Bourdon, Pasko, Liu, Celestin, Segur, & Marode, 2007), (Segur, Bourdon, Marode, Bessieres, & Paillol, 2006), (Liu, Celestin, Bourdon, Pasko, Segur, & Marode, 2007), the following equations are numerically solved to find the photoionization term. The photoionization term is defined by: S r r ) = A p cψ ( ), (17) ph ( O2 0, j r j j where A1 = cm Torr, A2 = cm Torr and A = 0. cm Torr are constant coefficients; p O2 is a partial oxygen pressure in air (150 Torr at atmospheric conditions at ground level); and c is the speed of light in vacuum. r The functions Ψ ( ) are defined by: 0, j r Ψ γ φ γ φ 2 1, j 1 2, j 0, j ( r ) =, where = ( 1) n n 3 γ 2 γ γ. (18) r r Functions φ ), φ ( ) are found by the solution of the set of Helmholtz 1, j ( 2, j r equations: ( λ p ) λ p 2 2 r j O2 r j O2 ξnu ( r ) φ1, j ( ) φ1, j ( ) =, (19) 2 2 κ1 κ1 cτ u ( λ p ) λ p 2 2 r j O2 r j O2 ξnu ( r ) φ2, j ( ) φ2, j ( ) =, (20) 2 2 κ 2 κ 2 cτ u r r

44 n where κ n = + ( 1) and λ1 = cm Torr, λ2 = cm Torr, λ = 0. cm Torr The second factor on the right hand side of equations (19)-(20) is r nu ( ) ξ τ u r pq ν u ( ) r r = ξ r ν i ( ) ne ( ), (21) p + p ν ( ) q i where p is an ambient pressure, p q is a quenching pressure (p q = 30 Torr), ratio ν u ξ ν i = 0.06, and ionization frequency ν i is equal to: ν i = 0, for E/n<1.122*10 6 V/m, ν i = n * 10 p, for E/n>1.122*10 6 V/m, and where n is the neutral number density normalized to sea level and 2 3 [ log ( E / n) ] *[ log ( E / n) ] *[log ( E / p = * + n (Barrington-Leigh, 2001). Boundary conditions are: )] r r r r r φ ) n = λ p α φ ( ) λ p β φ ( ), (22) 1, j ( s j O2 1 1, j j O2 2 2, j r r r r r r φ ) n = λ p α φ ( ) λ p β φ ( ), (23) 2, j ( s j O2 2 2, j j O2 1 1, j r 5 n 5 n 6 6 where α n = (34 ( 1) 11 ) β n = (2 + ( 1) ) The solution of the set of six equations (19)-(20) with boundary conditions (22)- (23) using of expressions (17)-(18) provides a photoionization source term.

45 27 Flow Model The induced gas flow was modeled by solving a set of Navier-Stokes equations for viscous compressible fluid in two dimensions (Anderson, Tannehill, & Pletcher, 1984, pp ). Denote the gas density as ρ, velocity components in horizontal (x) and vertical (y) directions as u and υ, gas pressure as p, components of stress tensor as τ, τ, τ, total energy as E, components of thermal energy flux as q, q, xx xy yy t x y components of the volumetric force as as Q. F x, F y The continuity equation for gas density is:, and the rate of thermal energy deposition ρ ρu + t x ρυ + = 0 y (24) The momentum equations are: 2 ( ρu) ( ρu + p τ u xx) ( ρ υ τ xy + + ) t x y = F x (25) 2 ( ρυ) ( ρuυ τ xy ) ( ρυ + p τ yy + + ) t x y = F y (26) The energy equation is ( E ) (( Et + p) u uτ xx υτ xy + qx ) (( Et + p) υ uτ xy υ + + t x y + q ) t yy y = Q, (27) where 2 u υ 2 υ u u υ τ xx= μ 2, τ yy= μ 2, τ yy= μ +, (28) 3 x y 3 y x y x

46 28 q x T = κ, x q y T 2 2 = κ, E t = ( cυ T + ρ( u + υ )) (29) y Air viscosity μ, heat capacity Tannehill, & Pletcher, 1984, pp ): cυ, and thermoconductivity κ are taken from (Anderson, μ = *10 6 T 3 / 2 T 2 kg /( m * s), c = m 2 υ / s K and κ = 2.36c υ μ No slip boundary conditions were applied to the gas velocity on the surface (gas velocity on the surface was equal to zero). Summary The first step towards the consistent and comprehensive numerical modeling is a formulation of correct and efficient physical model. Based on the experimental observations of the DBD plasma actuator operation and qualitative understanding of the gas discharge phenomena, the DBD modeling was divided into plasma modeling and flow modeling. While previous plasma modeling efforts (Roy, 2005), (Roy & Gaitonde, 2005), (Singh & Roy, 2005), (Boeuf & Pitchford, 2005) managed to track only a part of physical processes, which are essential for the description of the DBD, the model, developed in Chapter 2, contains all necessary processes for the comprehensive DBD modeling. The plasma model takes into account processes such as ionization of the air, recombination of ions and electrons, attachment of electrons to oxygen, detachment of electrons from negative ions, drift and diffusion of charged particles in the electric field, charge build-up

47 29 and charge recombination on the dielectric surface, secondary electron emission, and the efficient photoionization model. Plasma model provides spatial and temporal resolution of volumetric force, acting on the bulk air, and rate of thermal energy deposition. The induced air flow is modeled solving a set of Navier-Stokes equations. The source terms for momentum and energy equations can be taken from the plasma model for complete DBD modeling.

48 30 Chapter 3 Low-Voltage Modeling of the DBD Plasma Actuator In a number of experiments (Post & Corke, 2004), (Hall, Jumper, Corke, & McLaughlin, 2005) and references therein, sinusoidal shape of the applied voltage was used as a baseline. In order to understand the physics of DBD plasmas and the nature of the force, exerted on the gas in those experiments, a series of computations with small scale geometry (several hundred microns) and high AC frequency (1000 khz) were performed. The small scales allow DBD simulations to be performed in a reasonable time while showing the essential physics. However, it is very important to compare the results with those obtained with larger plasma sizes in order to understand the scaling issues. Numerical Model For the simulations the following geometry was chosen. The problem is considered two-dimensional (Figure 3-1). The electrodes are considered infinitely thin. This is justified, since a number of experiments showed no considerable difference in DBD actuator performance (Post & Corke, 2004) when using slightly-protruding or buried electrodes. The sizes of electrode were from 0.1 mm to 1.5 mm, depending on plasma length at different voltage profiles. The dielectric thickness was equal to 300 microns. Without loss of generality the voltage was applied to exposed electrode while lower electrode was kept grounded. The grid size for the rectangular grid varied from 1 to

49 31 10 microns for different numerical experiments. The grid size for each numerical experiment was chosen according to the following principles: The grid size must allow correct resolution of the plasma phenomena all plasma processes must be resolved and verification of the obtained results on a finer grid must be obtained; The grid size should be maximized in order to reduce computational time. Note that the maximum grid size is independent on the size of the DBD. Figure 3-1: Scheme of the DBD plasma actuator. The second order accurate explicit MacCormack scheme (Anderson, Tannehill, & Pletcher, 1984, p. 119) was used for the DBD plasma dynamic modeling. This scheme was able to resolve all physical processes at relatively low voltages (up to 2 kv) or higher voltages (up to 4 kv) in negative half-cycle of the sinusoidal voltage. Using this scheme, the low voltage sinusoidal case and low voltage repetitive negative nanosecond pulses were simulated. Some wrinkles in the presented figures are due to the dispersion of MacCormack scheme. However, for higher voltages this scheme could not resolve some

50 plasma phenomena, such as streamer propagation and reverse breakdown, because the scheme dispersion led to the numerical instability 2. Scharfetter-Gummel (Gummel, 1964), (Scharfetter & Gummel, 1969) method resolved these problems and high voltage cases were modeled using this scheme (See Chapter 4). The successive over-relaxation method (Anderson, Tannehill, & Pletcher, 1984, pp ) was used to solve the Poisson equation. One of the difficulties in the plasma dynamic simulations is the choice of time step. In the plasma modeling two parameters determine it. First, the time step should allow the resolution of plasma formation and sheath dynamics (it includes ionization, recombination and diffusion processes and dielectric relaxation), and second, the time step should satisfy the Courant Friedrichs Lewy (CFL) condition (Anderson, Tannehill, & Pletcher, 1984, pp ). The combination of these two requirements leads to the time step of the order of picoseconds. In the modeling, the time step was recalculated on each time step according to the CFL condition and was from 1 to 10 picoseconds. The complexity of the stated problem - such as resolution of micron geometrical scales and picosecond time scale for a non-linear problem with particle generation, the necessity to compute micro- to milliseconds time intervals and the geometrical scales of the order of millimeters - sets limitations on the model for single-processor calculations. First, the model is two-dimensional. This fact does not allow resolving the 3D filament structures, observed in DBD experiments. Second, the sizes of the geometrical and time domains are limited. Usually, the size of the domain was 1 by 1 millimeters and the 2 Due to the complexity of the numerical problem (solution of a set of nonlinear differential equations coupled with parabolic equation) the analytical analysis of the stability conditions was impossible. Numerical stability of the scheme for each numerical experiment was verified by the behavior of the numerical solution. 32

51 33 computational time was several microseconds. Another issue is the range of applied voltages. It was observed during numerical experiments that as higher voltages are applied, the grid should shrink for numerical scheme stability. Note that halving the grid size leads to the eightfold increase of the computational time. Sinusoidal Voltage Modeling Let us consider the modeling of low-voltage sinusoidal signal. The applied voltage amplitude and frequency is 1.5kV and 1 MHz correspondingly. Initially, the air is considered to have 10 7 m -3 electrons and positive ions. This negligible amount of charged particles is present in air due to cosmic rays ionization ionization. Figure 3-2 shows the applied voltage and calculated total and displacement currents in DBD modeling with low sinusoidal voltage. During the negative half-cycle (the exposed electrode is cathode) of the sine voltage, the difference between the total and displacement currents is due to the gas breakdown. In the positive half-cycle (the exposed electrode is anode) the difference can be explained by the motion of both negative ions and the secondary electrons (emitted from the dielectric surface) to the exposed anode. Consider separately two half-cycles of sinusoidal-voltage DBD operation.

52 34 Figure 3-2: Applied voltage and calculated current for 1.5 kv, 1000 khz sinusoidal voltage. Solid red line shows the voltage shape, the dashed blue line shows the calculated total current, and the dotted blue line shows the displacement current. Negative half-cycle During the negative half-cycle (the exposed electrode is negative and the covered electrode is grounded), the breakdown starts when the instantaneous voltage reaches the threshold value. If the amplitude of applied voltage is not very high, the plasma development starts near the voltage peak, as shown in Fig. 3-3 (a, b). Electrons are generated at the cathode surface due to the secondary electron emission, and avalanche ionization then occurs in the strong electric field. Being light and fast, electrons separate from positive ions and attach to the dielectric surface, leaving behind a cloud of positive ions. Some electrons, i.e. those at the front and upper edges of the plasma, attach to

53 35 oxygen and form negative ions (Fig. 3-3b). The plasma thus propagates along the surface and negatively charges the surface during the propagation (Fig. 3-4). The propagation stops, and the current is cut off when the negative potential on the dielectric surface becomes sufficiently high, so that the difference in potential of the exposed electrode and the dielectric falls below the critical breakdown value. Increasing the applied voltage amplitude leads to an increase of the duration of this active phase and allows plasma to propagate farther along the surface. Under typical conditions, the active (avalanche) phase duration from several ns (for small-scale actuator) to tens of ns (for a large-scale device) is much shorter than the half-cycle of the applied voltage. Therefore, during most of the negative half-period, no avalanche ionization occurs and the plasma is decaying. During this part of the cycle the negative ions push the gas to the right (downstream), while the cloud of positive ions is attracted to the exposed electrode and pushes the gas upstream and down, toward the surface (Fig. 3-5). For lower voltage frequencies and larger voltage amplitudes a series of breakdowns will occur. The first breakdown will be similar to the described one. The second breakdown will occur once the electric field at the edge of the exposed electrode reaches the threshold condition. The time between the breakdowns will be defined by the slope of the applied voltage.

54 (a) V(t)/V a active phase Δ(phase/π) positive plasma cloud force (b) attached electrons positive plasma cloud force (c) attached electrons Figure 3-3: Negative half-cycle: A sine voltage waveform with short active phase; B illustration of processes during the active phase (avalanche breakdown of the gas, charging of the dielectric surface, and propagation of the plasma); C illustration of processes during the decaying phase (positive space charge, ions drift towards the surface).

55 37 (a) (b) (b) (d) Figure 3-4: Instantaneous distributions of electron number density (a), positive ion number density (b), negative ion number density (c), electric potential (d) and force, acting on neutral gas (e) during the ionization avalanche. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz). (e)

56 Figure 3-5: Instantaneous distributions of electron number density (a), positive ion number density (b), negative ion number density (c), electric potential (d) and force, acting on neutral gas (e) after the ionization avalanche in the negative half-cycle. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz). 38

57 39 Since the magnitude of negative potential on the dielectric surface is always somewhat lower than the peak negative potential on the exposed cathode, the positive ion cloud always moves towards the cathode, creating the backward-directed force on the gas and reducing the efficiency of generation of downstream-directed gas jet. Moreover, in the computations for a small-scale DBD actuator, this backward motion of ions was found to be strong enough to cause an overall negative (backward-directed) force during the cathode half-cycle. However, when the length scale of the plasma was increased (the streamwise length of the electrode was increased by a factor of 7, to several millimeters), the voltage amplitude increased to 3.5 kv, and the AC frequency decreased by a factor of 10, then reversal of the integrated force in the cathode phase was reached. It is due to the fact that the backward-directed force caused by the positive ions moving to the cathode was found to vary only slightly with the scaling, while the downstream-directed force caused by the negative ions increases with the increase in plasma size and voltage amplitude. Consequently, the integrated force is now directed from left to right (Fig. 3-6), as observed in the experiments (Post & Corke, 2004), (Hall, Jumper, Corke, & McLaughlin, 2005). Later, similar results were obtained in (Boeuf, Lagmich, Unfer, Callegari, & Pitchford, 2007). Three factors are especially important in understanding why the force on the gas is directed downstream even if the exposed electrode is at a negative potential: i) the charging of dielectric surface by the attached electrons, ii) the formation of negative ions and their motion, and iii) the relatively large scale of the DBD actuators. It is also important to note that the attraction of positive ions to the exposed electrode in the

58 40 negative half-cycle of the sinusoidal voltage significantly reduces (if not reverses) the overall average force that pushes the gas. This is a major source of inefficiency of the DBD plasma actuators. Figure 3-6: Average force during the negative half-cycle for sinusoidal applied voltage (amplitude 3.5 kv, frequency 100 khz). Positive Half-Cycle Two distinctly different regimes are possible in the positive half-cycle (Fig. 3-7). If the applied voltage frequency is low (lower than about 1-10 khz in millimeter or centimeter-scale plasmas) and voltage amplitude is high (above 5 kv), then, there will be a series of breakdown in positive-half cycle (Boeuf, Lagmich, Unfer, Callegari, & Pitchford, 2007). Almost all electrons at the dielectric surface will be removed due to recombination with ions after the first breakdowns. Therefore, plasma around the positive-exposed electrode can only exist as a positive corona, as shown in Fig. 3-7b. Electrons generated in the strong field, when the electrode potential is near its peak, drift towards the positive-exposed electrode (anode), leaving behind positive ions. The positive ions move in the opposite direction, i.e. upwards (away from the surface) and

59 41 along the dielectric surface, imparting momentum to the gas. Thus, the momentum imparted to the gas is in the same direction (away from the exposed electrode) as in the negative half-period. Note, that in this low-frequency, corona regime the cloud of positive ions near the tip of the exposed electrode partially shields the field and limits the current, stopping the avalanches. When the ions drift away, the shielding is removed, and the breakdown can start again. If the applied voltage frequency is high (according to the calculations, at least 100 khz) so that the electrons attached to the dielectric surface are not significantly destroyed by the end of negative half-period, then the electrons can be pulled from the surface in the positive half-cycle by secondary emission (impacts of high-energy ions and UV photons on the surface). These electrons can then generate avalanche ionization. The dielectric then plays the role of a cathode in this half-cycle, as illustrated in Fig. 3-7c. The motion of positive ions that imparts momentum to the gas is toward the surface and along it (Fig. 3-8). Note, that in the positive half-cycle, the negative ions move upstream, toward the exposed electrode, and impart an upstream-directed momentum to the gas, thus reducing the overall downstream-directed gas velocity.

60 active phase 0.5 (a) V(t)/V a Δ(phase/π) 1 2 (b) positive corona force Low frequency (c) force attached electrons High frequency Figure 3-7: Positive half-period: A sine voltage waveform with short active phase; B illustration of positive corona regime at low frequency (no charge on dielectric surface, electrons are drawn to the exposed electrodes, ion clouds expands upward and away from the exposed electrode, imparting momentum to the gas); C illustration of processes at high frequency (negative charge remaining on dielectric surface from the negative half-period, ions drift towards the surface and away from the exposed electrode, imparting momentum to the gas and recombining with electrons deposited on the dielectric surface).

61 43 (a) (b) (b) (c) Figure 3-8: Instantaneous distributions of electron number density (a), positive ion number density (b), negative ion number density (c), electric potential (d) and force, acting on neutral gas (e) in the positive half-cycle. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz). (d)

62 Summary of the Force Exerted on the Gas in the Case of Applied Sinusoidal Voltage 44 The average force acting on the gas for the low-voltage sine case is shown in Fig The first observation is that the gas experiences a strong force normal to the surface and towards it. This normal suction force, noted in (Post & Corke, 2004), (Hall, Jumper, Corke, & McLaughlin, 2005), is due primarily to the motion of a positivelycharged ion cloud to the exposed cathode in the negative half-cycle. In fact, this suction force turns out to be stronger than the average tangential force. However, the same positive-ion motion to the cathode also creates a strong upstream-directed force. This upstream force is even greater than the downstream-directed force created by the negative ions in the negative half-cycle if the device scale is small (less than 1 mm). In a largescale (at least several millimeters) device, computations showed that the negative ion generated downstream force becomes stronger than the upstream force due to positive ions, so that the overall force is directed downstream even in the negative half-cycle and the total force over the cycle is also in the downstream direction (see Fig. 3-6). Figure 3-9: Average force acting on neutral gas during the cycle. (Sinusoidal voltage, amplitude 1.5 kv, frequency 1000 khz).

63 45 However, the motion of positive ions to the negatively-biased exposed electrode remains a major source of inefficiency in generating a tangential component of the gas velocity. Note also that negative ions play diametrically opposing roles in the negative and positive half-cycles. In the negative half-cycle, the negative ions are primarily responsible for the downstream force, while in the positive half-cycle the negative ions move upstream and impede the downstream gas motion. The obtained results provide the first qualitative explanation of the DBD operation in atmospheric air with applied sine voltage. Overall, despite the dominant downstream directionality of the force, it is clear that a sinusoidal or near-sinusoidal voltage signal is an inefficient way of generation of a downstream-directed gas jet. In order to maximize the tangential force and the gas jet velocity, the inefficiencies discussed above should be eliminated or at least minimized. It is also worth mentioning, that since the major contribution to the force is between breakdowns, second term in expression (15a), which represents the force, acting on the gas, is much less than the first term. The gradient of electron pressure is also much less than the first term in expression (15a). Therefore, one can consider the force, acting on neutral gas, as a product of local space charge and local electric field.

64 46 Optimal Wave Form for Low Voltages: Repetitive Negative Short Pulses with DC Bias Physical Principles As shown in the discussion of the DBD actuator operation with sinusoidal voltage signal, the gas is pushed downstream due to the attraction of positive ions to the negatively charged dielectric surface in the positive half-cycle and due to negative ions in the negative half-cycle. The charging of the dielectric plays the key role in understanding and optimizing the process. In the computed DBD examples, this was accomplished by the removal of electrons from the plasma ionization wave onto the surface. However, in principle, other methods of charging the dielectric surface can be used. Indeed, to push the gas along the surface in the direction away from the exposed electrode, the motion of positive ions should be predominantly in that direction. For that to happen, the exposed electrode should be positively biased. However, dc current cannot just flow between neutral dielectric and the positive-exposed electrode. (The dielectric itself is needed in order to limit the current and thus to prevent transformation of the plasma into hot, constricted, unstable arc). To make the current flow possible in the regime when the exposed electrode is positive, electrons should be deposited onto the dielectric surface. That way, the electrons would be lifted from the surface by recombination with ions that drift and arrive at the surface and by secondary emission

65 47 (due to impact of energetic positive ions), and the operation would be similar to the cathode in glow discharge. Since the actuator operation will entail the removal of electrons from the surface, these surface-deposited electrons should be periodically replenished. The periodic replenishment of negative charge on the dielectric should be done very rapidly, to minimize the time of backward-directed positive ion motion. This can indeed be accomplished by periodic application (with repetition rate from tens to hundreds of khz) of very short (2-5 ns) strong negative voltage pulses. In these short pulses, the ionization avalanche developing from the exposed electrode would quickly reach the dielectric surface and would deposit electrons onto it, leaving behind the positive ions, so that the main plasma actuator operation could resume. Numerical Results The applied voltage and the corresponding calculated current in the circuit in the representative repetitive-pulse case is presented in Fig The breakdown occurs during the pulse and the peak in the current curve corresponds to the breakdown.

66 48 Figure 3-10: Current and voltage in a representative repetitive-pulse case in quasi-steady-state regime. The red line shows the voltage and the dashed line shows the current. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V). In the representative case the following parameters were chosen as follows: the Gaussian pulse width was 4 ns, the pulse amplitude was -1.5 kv, the dc bias was 0.5 kv, and the repetition rate was 500 khz (so that the interval between the pulses was 2 μs). The small-scale geometry was the same as in the sine-voltage cases. Figure 3-11 shows that after the pulse there is a very high concentration of positive ions near the exposed electrode. The electrons are attached to the dielectric surface, creating an effective cathode which attracts the positive cloud. Figures show the evolution of the positive cloud from the anode to the dielectric and the corresponding instantaneous potential distributions. The cloud moves to the right, neutralizing the surface charge and gradually dispersing. Figure 3-14 shows the average force acting on the gas between two

67 49 pulses. Comparison of Fig with Fig. 3-9 clearly shows that the waveform with short repetitive pulses eliminates the upstream-directed force that was the principal inefficiency source in the sine-voltage cases.

68 50 (a) (b) (c) Figure 13a Figure 3-11: Instantaneous distributions of electron number density (a) (The electron avalanche charges the dielectric surface during the pulse), positive ion number density (b) (The positive ion cloud is formed near the exposed electrode during the pulse), electric potential (c) and force, acting on neutral gas (d) at the peak pulse voltage. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V). (d)

69 51 (a) (b) (c) Figure 3-12: Instantaneous distributions of electron number density (a) (electron avalanche charged the dielectric by the end of the pulse), positive ion number density (b) (positive ions start to move to the dielectric surface), electric potential (c) and force, acting on neutral gas (d) immediately after the pulse. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V). (d)

70 52 (a) (b) (c) Figure 3-13: Instantaneous distributions of electron number density (a) (electrons are neutralized on the surface by positive ions), positive ion number density (b) (positive ions drift to the negatively charged dielectric surface), electric potential (c) and force, acting on neutral gas (d) 0.5 μs after the pulse. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V). (d)

71 53 Figure 3-14: Average force acting on neutral gas during one cycle. (Pulse voltage: amplitude -1.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 500 V). A simple physical model can describe the pushing action of plasma on the gas. Consider the gas in a region of length L (Fig. 3-15). A positive ion cloud with average ion number density n + and the width S is created near the exposed electrode. Figure 3-15: Simplified physical model (multiple sweeps by a porous piston ) describing the momentum transfer to the neutral gas in repetitive-pulse cases.

72 The moving ion cloud acts as a porous piston: it moves faster than the gas and accelerates the gas molecules that experience collisions with ions. Since at thermal 54 energies the ion-molecule collision cross section is approximately σ in cm 14 2 (Macheret, Shneider, & Miles, 2001), and the ion number density is typically n cm , the mean free path of the molecule with respect to collisions with ions is on the order of 1-10 meters - much longer than the mm width of the ion cloud. Therefore, only a small fraction of molecules experience collisions with ions during the single sweep of the ion cloud, which justifies the porous piston analogy. The force on a molecule inside the plasma column is: = ( ), (30) f k n M u u in where cm s is the ion-molecule momentum-transfer rate constant (Macheret, k in Shneider, & Miles, 2001), M is the ion mass (assumed equal to the molecule s mass), u + is the ion cloud (drift) velocity, and u is the gas velocity. Since the residence time of a molecule inside the ion cloud is approximately t S ( up u) a single sweep of the ion cloud is:, the gas velocity increment in Δ V = k n S 1 in +. (31) As the gas element in the boundary layer slowly moves along the wall, it experiences many hits by the consecutive rapidly-moving ion clouds. The number of these hits is approximately equal to L ( u% Δt), where u% is the average gas velocity, and Δt is the smaller of the interval between the pulses and the time of ion drift over the distance L. Denoting the initial gas velocity (i.e. the velocity at the entrance to the interaction region) as V 0 and the final (i.e. exit) gas velocity as V, one obtains:

73 55 L V V0 = ΔV1 1 ( V + V0 ) Δt 2 From Eqs. (31) and (32), one obtains:. (32) V V = in Lk n+ S, (33) Δt and assuming that initially the gas is at rest, V = 2 in Lk n+ S Δt. (34) In the computed small-scale case with repetitive (500 khz) short (4 ns FWHM) 2 kv pulses and 500 V dc bias, the approximate values of parameters in Eq. (34) are: L=0.3 mm, S=0.1 mm, Δ t = 2 10 s, and n + = 5 10 cm. Eq. (15) then gives the induced gas velocity of V=4 m/s. It is important to recognize that this velocity is similar to that commonly observed in the conventional DBD actuators, but it is achieved with considerably lower voltage amplitude and an order of magnitude smaller plasma size compared with those in the conventional DBD systems (Post & Corke, 2004), (Hall, Jumper, Corke, & McLaughlin, 2005). Summary The physics of the DBD plasma actuator, driven by low-voltage sinusoidal signal, was explained in detail. Three stages of the DBD operation are to be pointed out. The plasma is generated during the negative half-cycle at nanosecond time scales. During the breakdown, the electrons are deposited to the dielectric surface, leaving positive ion

74 56 cloud near the edge of the exposed electrode. During the negative half-cycle, positive ion drift towards the exposed electrode induces upstream directed force on the gas. The downstream directed force is generated by the positive ion drift in the positive half-cycle. Based on the described understanding of the DBD operation, the novel voltage profile repetitive nanosecond negative pulse superimposed on positive dc bias was proposed. The modeling showed an elimination of inefficiencies in the generation of the force, acting on the gas. During the nanosecond pulse, the plasma is generated, electrons are attached to dielectric surface, leaving positive ion cloud near the edge of the exposed electrode. Positive ion drift between the pulses induces the downstream directed force between the pulses. Both modeled cases sinusoidal voltage and pulses with dc bias demonstrated that the major contribution to the force, acting on the gas, is between the breakdowns, but not during the breakdowns. It was also shown for the first time that the downstream direction of the force on the gas in the negative half-cycle of the high-voltage sinusoidal signal is due to the drift of negative ions. In order to estimate the gas induced velocity by the DBD, driven by pulses and dc bias, an analogy with porous piston was developed. It was shown that low voltage pulses plus bias can generate near-surface gas jet with the velocities, comparable to sinusoidal voltage. Chapter 3 was mainly devoted to the low voltage modeling. The physics of DBD operation at higher voltages is considered in consecutive chapters.

75 57 Chapter 4 High-Voltage Modeling of the DBD Plasma Actuator Numerical Methods For low-voltage modeling (Chapter 3) second order accurate MacCormack scheme was used to simulate plasma dynamics. This model worked well and gave accurate results. Nevertheless, the transition to higher voltages could not be achieved because of the numerical instabilities. In order to overcome this problem, the flux corrected transport (FCT) (Anderson, Tannehill, & Pletcher, 1984, pp ) was added to the MacCormack scheme. The results obtained using FCT scheme are smoother. Figure 4-1 shows a comparison between time-averaged body forces on the neutral gas using these two methods after 3 pulses. The results obtained with and without FCT agree well: both produce similar downstream forces with only minor variations in the local force fields. Successive over-relaxation method was used to solve the Poisson equation (Anderson, Tannehill, & Pletcher, 1984, pp ). Figure 4-2 shows the voltage-current characteristics during the breakdown for the two numerical schemes. After the pulse, the current is very small. These characteristics obtained with and without the FCT also agree well. A grid convergence study showed some dependence of the results on the grid size. However, the qualitative results remained the same. Decreasing the mesh size results in plasma propagation somewhat farther downstream and thereby slightly increases the momentum transfer to the gas.

76 58 Figure 4-1: Time averaged force on the neutral gas. The left plot corresponds to computations with MacCormack method without FCT, and the right one to FCT method. (Pulse voltage: 1.5 kv amplitude, 4 ns FWHM, 500 khz repetition rate, and 500 V bias). Figure 4-2: Current and voltage in a representative repetitive-pulse case in quasi-steady-state regime. The red line shows the voltage, and the dashed line shows the current. The left plot corresponds to MacCormack method without FCT, and the right one to FCT method. (Pulse voltage: -1.5 kv amplitude, 4 ns FWHM, 500 khz repetition rate, and 500 V bias). \ Despite the fact that FCT provided smoother solutions, it still could not resolve high voltages. In order to overcome this difficulty, the Scharfetter-Gummel method (Gummel, 1964), (Scharfetter & Gummel, 1969) was used for flux calculation and Modified Euler method was used for solving continuity equations (Koenig, 1998, pp ).

77 59 High-Voltage Results Repetitive Negative Nanosecond Pulses plus Positive DC Bias Based on the numerical calculations for low voltages, it was proposed to use negative nanosecond pulses and positive dc bias to increase the effect on gas pushing in the DBD plasma actuator. The basic principle of this voltage performance was the generation of an ionization wave during the short pulse. When the pulse was applied, the electrons were created near the edge of electrode due to the high electric field in this region, then electrons moved along the electric field lines, ionizing the gas, and finally they attached to the dielectric surface, creating a virtual cathode. Since a large number of electrons were attached to the dielectric surface and the pulse time was too short to make positive ions move, a positive ion cloud was formed above the dielectric surface. After the pulse, this positive cloud was pushed from the electrode toward the dielectric surface by an applied positive dc bias and by the attraction of the electrons on the dielectric. The logical step to improve the performance of the DBD plasma actuator, driven by repetitive negative nanosecond pulses, is to increase the peak voltage of the pulses. This voltage increase should lead to an increase of the positive ion number density in the air after the pulse, and, therefore, an increase in the force acting on the air in downstream direction. In order to investigate this phenomenon, a numerical experiment was carried out. The following geometry was chosen: lengths of upper and lower electrodes were 0.15 and 0.8 mms, the computational domain was mm, the grid size was

78 60 µm. The dielectric thickness was 100 µm and there were no gaps in the horizontal direction between electrodes. When the pulse is applied to the upper electrode, the breakdown occurs and the electrons move to the dielectric, producing weakly ionized plasma and leaving a positive ion cloud behind them. However, after the voltage reaches its peak, a reverse breakdown occurs. This reverse breakdown looks like a cathode directed streamer (Fig ) and decreases the performance of the DBD plasma actuator. Figure 4-5 shows the average force on the gas after the pulse. There is a component near the edge of the electrode, directed upstream, however, the average force integrated over the volume is still directed downstream and is greater than in the case of low voltages. The positive effect on the integral force increase of applying higher voltage is due to the larger plasma size and, consequently, larger volume of gas is exposed to the DBD induced force. The quantitative comparison of these results will be given later in this Chapter.

79 Figure 4-3: Reverse breakdown. Applied voltage: negative pulses with positive bias. Peak voltage is kv, the bias is 0.5 kv, and the pulse FWHM is 4ns. The Figure shows the electrons and positive ions number densities, electric potential, and the instantaneous force on the gas at the moment corresponding to the peak voltage. 61

80 62 Figure 4-4: Reverse breakdown. Applied voltage is negative pulse with positive bias. Peak voltage is kv, the bias is 0.5 kv, and the pulse FWHM is 4 ns. The Figure shows the electrons and positive ions number densities, electric potential and the instantaneous force on the gas after the pulse. Figure 4-5: Average force acting on neutral gas during one cycle. (Pulse voltage: amplitude is -4.5 kv, FWHM is 4 ns, repetition rate is 500 khz, and the bias is 0.5 kv).

81 63 Repetitive Positive Nanosecond Pulses plus Positive DC Bias Another possible way of pushing gas by the DBD plasma actuator is by applying high voltage repetitive positive pulses plus positive dc bias. The physics of the downstream force acting on the gas in this case is different from the case with negative pulses. When a positive nanosecond pulse is applied to the upper electrode, at some voltage the electric field near the edge of the electrode is sufficient for the initiation of cathode-directed streamer propagation. During the pulse, the streamer propagates along the dielectric until the electric field at the leading edge (head) of the streamer is not sufficient to produce further ionization in front of it. The streamer stops at this point. From the theory of streamers and from the calculations it is well-known that the body of the streamer is quasi-neutral, but the head of the streamer carries a positive charge in order to displace the electric potential and produce the ionization in front of it. Therefore, after the pulse there will be a positive ion cloud in the front part of the decaying streamer. An applied dc bias after the pulse forces this cloud to move downstream and push the gas. Despite that the body of the streamer is quasineutral plasma, its effect on formation of force on the gas is negligible in comparison with the force from the positive ion cloud in front of the streamer. The reason is that the force is proportional to the difference between concentrations of positive and negative charge carriers. After the discharge, the electrons in the quasineutral streamer body are rapidly attached to the oxygen during the time scales of the order of tens of nanoseconds. Between the pulses, the drift of negative

82 ions to the upper electrode is compensated by the drift of positive ions downstream. Let us consider the formation of plasma in the case of positive pulses in detail. 64 The role of the Background Electron Number Density In order to correctly model the streamer propagation, one has to solve the problem of radiation transfer and calculate the number density of photoelectrons ahead of the streamer head. The number density of photoelectrons depends on the streamer and gas parameters. However, the problem of radiation transfer is quite complicated and time consuming for single processor modeling. The comprehensive model, which takes into account the photoionization, will be considered in Chapter 6. The simplified model considers the background charge in the air as having been created in the photo processes (Segur, Bourdon, Marode, Bessieres, & Paillol, 2006). Such simplified models were previously used for streamer modeling in (Liu & Pasko, 2004). There are two important parameters: initial and background charge number densities. The initial charge number density is the positive ion and electron number densities in the air at the beginning of calculations. Since the plasma formation is an exponential process, plasma cannot form without initial charges. However, the initial number density has to be small to have no influence on plasma formation. This fact leads to the problem of streamer formation if the voltage of the upper electrode is positive. In this case, the electrons will be sucked to the electrode during hundreds of picoseconds and the streamer will not be able to propagate. In reality, photoelectrons, which are formed ahead of the streamer head, resolve this problem. In the simplified model, photoelectrons are substituted by minimum background

83 65 electron number density. Minimum electron number density is kept in the computational domain to artificially account for the generation of the photoelectrons. This addition of electrons has negligible influence on the accuracy of the model of plasma-gas interaction because this minimum background electron number density is several orders of magnitude less than the plasma number density. The minimum electron number density should be chosen empirically to satisfy the experimental data on the streamer propagation speed and the propagation distance. The dependences of streamer propagation parameters on the minimum electron number density and the initial conditions were investigated. For this investigation, geometry similar the negative pulses simulation was chosen. Single positive Gaussian pulses with 4 ns FWHM and 3 kv voltage amplitude were applied to upper electrode. Two parameters were changed the initial plasma density and the minimum electron number density. If the initial and minimum electron number density is of the order of /m 3, the plasma cannot even appear. If the initial number density is very high (of the order of /m 3 ), the plasma forms, and a streamer propagates even without keeping minimum electron number density. When the initial plasma number density is /m 3, the minimum electron number density plays a critical role in the streamer formation. Without it, the electrons rapidly move to the electrode, and there will be no photoelectrons to sustain streamer propagation. Keeping minimum electron number density equal to /m 3 supplies the electrons for streamer propagation. Similar situation occurs when the initial plasma density and minimum electron number density drop to /m 3. Figure 4-6 shows a comparison of the streamer propagation speed at different initial plasma and minimum electron number densities. For the basic modeling the minimum electron number density was taken to be equal to /m 3.

84 Figure 4-7 shows the typical simulated streamer and Fig. 4-8 shows the time-average force on the gas distribution. 66 Figure 4-6: Streamer propagation speed in 3 different cases. Green line corresponds to the initial and minimum electron number density of /m 3, the purple line to /m 3, and the blue line to the initial electron number density of /m 3 and no minimum electron number density. Applied voltage waveform: positive pulses with positive bias. Peak voltage is 3 kv, the bias is 1kV, and the pulse FWHM is 4 ns.

85 Figure 4-7: Streamer-like ionization wave. Applied voltage waveform: positive pulses with positive bias. Peak voltage is 3 kv, the bias is 1kV, and the pulse FWHM is 4 ns. The Figure shows the electrons and positive ions number densities, electric potential and the instantaneous force on the gas. 67

86 68 Figure 4-8: Average force acting on neutral gas during one cycle. (Pulse voltage: amplitude is 3 kv, FWHM is 4 ns, the repetition rate is 500 khz, and the bias is 1 kv). It is worth mentioning that the keeping minimum electron number density is not essential for negative pulse modeling. In this case, the electrons appear on the electrode or dielectric due to secondary emission and then move along the electric field line producing the plasma. The reverse breakdown is also easily resolved, despite it is streamer like appearance. After the electron avalanche, there is plasma with an ionization fraction much greater than one due to photoionization. Comparison between High- and Low-Voltage Results As predicted in Chapter 3, higher voltages increase the DBD plasma actuator performance. Fig. 4-9 shows a comparison of i) low voltage repetitive negative nanosecond pulses with positive bias, ii) high voltage negative pulses with positive dc bias, and iii) repetitive positive nanosecond pulses with positive bias. The higher voltages

87 69 transfer three times more momentum to the gas in contrast to low ones. Note that positive pulses with amplitudes of 3 kv give the same effect as the negative pulses with amplitudes of 4.5 kv. Modeling high voltages also allows for the prediction of different ways of plasma formation in the DBD and different momentum transfer. For example, at low voltages the streamer does not propagate in the case of positive pulses, but at high voltages it does. The obtained results also give an important conclusion about the scaling predictions in numerical modeling. The DBD plasma actuators operate at different regimes not only at different voltage profiles, but at different voltage amplitudes. Further predictions of DBD performance at voltages of the order of kv require more extensive computational capabilities and the application of parallel computing methods. Figure 4-9: Momentum transferred to the gas. Blue and green lines correspond to the negative pulses with amplitudes -4.5 and -1.5 kv with positive bias of 0.5 kv, and the red line corresponds to the positive pulses with 3 kv amplitude and positive bias of 1 kv. FWHM for all pulses is 4 ns.

88 70 Summary Modeling of the DBD, driven by high-voltage negative pulses, demonstrated the DBD operation similar to the case of low-voltages. The major difference between two cases is the presence of the reverse breakdown for high voltages. The reverse breakdown creates a region of upstream directed force acting on the gas. However, the net force on the gas is in downstream direction. As the voltage increases, the momentum, transferred to the gas, also increases. Chapter 4 demonstrated the possibility of the downstream directed flow generation using repetitive positive nanosecond pulses plus positive dc bias. The physics of the plasma development is different from the case of negative pulses. The plasma is generated in the form of streamer. The downstream directed force, acting on the gas between the breakdowns is due to the drift of positive ions from the streamer head between the breakdowns. The concept of background plasma was introduced for the description of plasma generation in case of positive pulses. However, it is necessary to consider photoionization for quantitatively correct modeling of streamer propagation. Due to the computational cost of the DBD simulations it is necessary to developed more efficient numerical model of the DBD to simulate resolve photoionization problem. Chapter 5 describes a successful development of such model. The results of the DBD modeling with accurately resolved photoionization are presented in Chapter 6.

89 Chapter 5 Optimization of the Numerical and Physical Models for the DBD Plasma Actuator Simulations 71 Necessity for the DBD model optimization For further DBD optimization, it was necessary to develop a numerical model which could not only qualitatively explain the physics of the DBD, but also quantitatively agreed with the experimentally obtained data. However, the development of such a model rested on the resolution of several essential computational problems. One of the difficulties in the plasma dynamics simulations is the choice of a time step, described in Chapter 3. Another problem is the correct choice of the numerical grid size. The plasma processes are very sensitive to the local electric field, which in the DBD case varies at the characteristic scale on the order of 10 microns for the high applied voltages 3-4 kv. The necessity of the correct resolution of the plasma dynamics leads to a grid size of the order of several microns. However, the increase of the operating voltage leads to the further decrease of the grid size. It was observed that the glowing plasma propagates up to several millimeters on the surface and the non-glowing charges can spread even further (up to several centimeters) (Opaits, Shneider, Miles, Likhanskii, & Macheret, Surface Charge in Dielectric Barrier Discharge Plasma Actuators, 2008). Thus the modeling has to be done on a significant number of grid points (up to 10 3 x10 4 ) and must evolve through a large number of time steps (up to 10 9 ).

90 72 Numerical improvement parallel programming In Chapters 3-4 the single processor numerical model was used to simulate the physics of the DBD plasma actuator. The numerical results, which can be obtained using this model, are limited by the capabilities of a single processor computer for plasma dynamics simulation. The resolved geometrical scales (the computational domain is of the order of 1x1 mm), time scales (up to several microseconds) and moderate applied voltages (up to 3-4 kv) are not sufficient to quantitatively describe the experimentallyobserved phenomena. The characteristic time of one run on the single processor computer is of the order of one week. The large scale computations of the real experimental conditions are very time consuming. The only way to connect the numerical modeling to the experimental investigations is to develop a multiprocessor numerical model. Numerical Model Let us consider the necessary modifications in single processor code to make it multiprocessor one. The numerical model for plasma simulations in the DBD consists of two parts: the part which computes the plasma dynamics, and the part which solves the Poisson equation. Since these parts are solved separately using different types of numerical schemes and the Poisson Solver is only a subroutine for the plasma dynamics simulator, it is reasonable to split the problem of parallelization of the entire code into two parts. First, consider the Poisson Solver. In the single processor model, the Poisson

91 73 equation was solved using the Successful Over-Relaxation (SOR) method (Anderson, Tannehill, & Pletcher, 1984, pp ), which is the one of most powerful methods for solving elliptic equations on a single processor. However, this method could not be efficiently used in multi-processor calculations. The commonly-used numerical method for solving Poisson equation on parallel clusters is the Jacobi iterations on either 1D or 2D decompositions (Gropp, Lusk, & Skjellum, 1999, pp ). Consequently, a Poisson Solver for multi-processor computations using Jacobi iterations on a 2D decomposition was developed for the DBD modeling. The choice of the 2D decomposition instead of 1D one was motivated by the necessity to decrease time of data exchange between processors and, therefore, to increase the speed of the computations. The developed model involves an arbitrary number of processors for computations. The computational domain is considered to be rectangular and it is divided by four equal rectangular domains for each processor computations. The data exchange between the contiguous processors involves 5 point overlap (Fig. 5-1 shows an example for 4-processors). The second part of the parallel code computes the plasma dynamics. As in the case of Poisson solver, 2D decomposition with five-point overlap was used. The numerical methods, used for the single processor computations (Scharfetter-Gummel and Modified Euler methods (Gummel, 1964), (Scharfetter & Gummel, 1969), (Koenig, 1998, pp )) remain appropriate for the parallel computations. The major modification to this part of the code was the arrangement of the data exchange between the conjugate processors.

92 74 Processor 1 Overlapping Overlapping Processor 3 Points Processor 0 Points Processor 4 Figure 5-1: 2D decomposed numerical domain, used for the DBD simulations. Numerical Results Figure 5-2 shows the comparison of the time of computations between the initial single processor version and the parallel code. The computation was done on the same grid with the same parameters. Positive 3 kv pulses with 1 kv dc bias were applied. The graph can be subdivided into two parts one with low speed computations (first several tens of nanoseconds) and the rest. This difference is due to the different time consumption for Poisson Solver convergence at different stages of the discharge. Initially when the plasma is being generated in the form of a streamer, the potential is removed to the tip of the streamer, leading to the significant numbers of iterations in the Poisson Solver. Right after the pulse, the electric potential significantly changes due to the fast motion of electrons. However, after several tens of nanoseconds the electrons are attached to the oxygen, leaving an ion-ion plasma, or removed to either electrode or

93 dielectric in general. Relatively slow motion of ions slightly changes the electric potential at each time step, leading to the fast convergence. 75 Figure 5-2: Comparison of computational time between the original single processor and 4-processor parallel codes. Physical Improvement Modified Time Step Modeling DBD in air and understanding the DBD physics allow further optimization of the numerical model. For accuracy and stability of the modeling, the time step is defined by the CFL condition, i.e. particles cannot move more than a grid size during the time step. In other words, the faster velocities of the moving particles in the model correspond to smaller time steps. Electrons are responsible for the plasma generation during the breakdown stage. So the breakdown stage should be resolved with

94 76 the time step based on electron velocities. For the current grid size 2.5 microns and electron velocities ~ 10 6 m/s the time step is of the order of picoseconds. However, between the breakdowns (most of the time) the electrons are attached to the dielectric or oxygen and the plasma is ion-ion. The electron number density drops an order of magnitude ~10 ns after the breakdown, due to the attachment to oxygen. However, some portion of electrons remains in the plasma. These electrons have a negligible effect on DBD performance, but for numerical stability and accuracy the time step should still be defined by electron velocities. In the pulse-bias configuration the plasma generation and the gas pushing between the breakdowns are completely defined by the voltage profile. The breakdowns occur during the pulses. During the pulse it is necessary to compute plasma dynamics based on the electron times. After the pulse, during several tens of nanoseconds, the electrons are being attached to the oxygen, forming negative ions. After hundreds of nanoseconds the number density of electrons is negligible in comparison with the number density of negative ions. Based on these considerations a concept of modified time step was implemented. If the maximum electron number density becomes less than a chosen threshold of m -3, all electrons in the numerical domain are artificially attached to the oxygen and only the dynamics of ion-ion plasma is computed. The time step, computed based on ion-ion time, is three orders of magnitude greater than one computed based on electron time. In order to investigate the possibility of the modified time step implementation into the DBD problem, the following sample case was considered. Using the developed multiprocessor model the DBD with the applied 4 kv 4 ns positive pulse with 1 kv

95 77 positive dc bias was simulated. The chosen DBD geometry is described in the Chapter 4. The simulations ran on 32 processors. The first run involved the regular time step. The second one involved the modified time step. The threshold condition for electron number density was set to be m -3. Figure 5-3 shows the comparison between the integral momenta transferred to the gas using the regular and modified time steps. The results are in good agreement. However the momentum, computed based on modified time step, is less than one computed based on regular time step. The reason is that in reality only a portion of the electrons are attached to the oxygen. The other portion goes into the dielectric and electrode with velocities much higher than ion velocities. The neglect of this effect provides larger amount of negative charge in the gas than in reality. It leads to the decrease of the observed transferred momentum, since the positively charged species push the gas downstream and negative charged species push the gas upstream. Therefore, simulation of the DBD with positive pulses and positive dc bias using the modified time slightly underestimate the momentum transferred to the gas. Figure 5-3: Comparison of the momentum, transferred to the gas, computed using the regular and modified time steps.

96 78 Figure 5-4 shows the enhancement of the speed of computations with the modified time step. The resolution of the first hundred nanoseconds took approximately five hours of the computational time, and the resolution of the rest ~ 10 microseconds took just half an hour of the computational time. The implementation of the modified time step for the pulse configuration helps significantly decrease the computational time. The concept of the modified time step also provides an insight on the decrease of the force acting on the gas between pulses. In the single processor modeling just two microseconds after the pulse was considered. In the present calculations, the momentum transferred to the gas was investigated using the modified time step. Fig. 5-5 shows this dependence. There is a significant push of the gas during the breakdown, however the main momentum transfer occurs after the pulse. Between the pulses the behavior of the transferred momentum is almost linear with logarithm of time.

97 79 Figure 5-4: The computational time consumption for different stages of plasma modeling using the modified time step. Figure 5-5: Momentum, transferred to the gas (Pulse voltage: amplitude 4 kv, FWHM is 4 ns, and the bias is 1 kv).

98 80 Summary The limitations of the computational power of the single processor numerical model for the DBD plasma actuator had not allowed simulating all essential physical phenomena. In order to increase the capabilities of the DBD model, parallel numerical model was developed and the concept of modified time step was introduced. Parallel numerical model demonstrated a significant increase of the computational power for the DBD simulations. The use of four processors led to approximately 4 times increase of the computational speed. The developed model allowed simulating photoionization problem at 80 processors at reasonable time (see Chapter 6 for details). The concept of modified time step utilizes the physical phenomena, observed in previous DBD simulations (see Chapters 3 and 4). The impact of electrons in the DBD physics is crucial only during the breakdown and several tens of nanoseconds after it. Most of the time the plasma is ion-ion and the amount of electrons is negligible. However, accurate simulations of electron motion needs extremely short, picosecond time steps. The DBD model with the concept of modified time step uses picosecond time step, when the electrons are essential, and nanosecond time step, which is necessary for accurate simulation of ion motion, rest of the time. It leads to 3 order of magnitude increase of computational speed between the breakdowns without loss of accuracy. The DBD simulations with the concept of modified time step showed that the momentum transferred to the gas rises almost logarithmically with time.

99 81 Chapter 6 The Role of Photoionization in the Numerical Modeling of the DBD Plasma Actuator The importance of photoionization modeling for the DBD driven by repetitive positive nanosecond pulses was considered in Chapter 4. After the development of the parallel model it became feasible to include the correct photoionization into the numerical model and study the difference between the real photoionization and the concept of artificial electron background number density. Physical Model of the Photoionization In order to include the photoionization into the model it is necessary to compute term S ph using either integral approach (solution of equation (16)) or differential approach (solution of equations (17)-(23)). The solution of Eq. (16) is time consuming compared to the solution of equations (1)-(15). In order to compute the photoionization term at one grid point, one should integrate over the whole numerical domain. For example, if the computational domain is nxm points, the number of operations for the system (1)-(15) solution is approximately proportional to nxm (neglecting the convergence of Poisson Solver). However, one needs to compute the integral (16) n*m times for each of nxm points in order to get the

100 82 photoionization term. Therefore, the number of operations for the photoionization term is proportional to (n*m) 2. In the standard DBD modeling problems the numerical domain is usually of the order of 1000x100 points. Therefore, an introduction of the photoionization term, computed based on integral approach, requires an additional ~10 5 operations and makes problem solution inefficient. The solution of equations (17)-(23) is similar to the solution of Poisson equation for electric potential distribution. The number of operations is proportional to the nxm, which is much more efficient than for the case integral approach. Therefore, the differential approach was chosen for the photoionization modeling. Numerical Results For the photoionization modeling the following geometry was chosen. The sizes of exposed and insulated electrodes were 0.2 mm and 2.1 mm correspondingly. Dielectric thickness was 0.1 mm and relative dielectric permittivity was equal to 5 (Likhanskii, Semak, Shneider, Opaits, Macheret, & Miles, 2009). The electrodes were considered to be infinitely thin. The grid was uniform in both horizontal and vertical directions. The size of computational domain was 2.5x0.5mm. The simulations were carried out at Princeton Plasma Physics Laboratory (PPPL) Scientific Computing Cluster on 20 processors for the cases without photoionization and 80 processors for the cases with photoionization. The size of the numerical domain was limited by the maximum allowed

101 83 allocated RAM of 500 Mb for the simulation. The times for one pulse simulation with and without photoionization were about twelve and three hours correspondingly. Initial plasma number density for the simulations was chosen to be equal to 10 7 m -3. For the first simulations the positive pulse (FWHM is 4 ns, amplitude is 3 kv) was applied to the exposed electrode, while the insulated electrode was grounded. The pulse starts at 0 ns and reaches its maximum at 4 ns. Figure 6-1 shows the electron and positive ion number densities and electric potential distribution 7 ns after the start of the pulse. When the electric field at the edge of the exposed electrodes reaches the threshold value, the electron avalanche development starts in the region of high electric field (tens of microns around the electrode s edge). The maximum plasma number density reaches ~10 19 m -3 in the vicinity of the electrode. At the same, time the electrons from the bulk volume are sucked into the electrode, and the plasma generation stops due to the absence of the source of electrons in the air. In Chapter 4 the concept of minimum electron number density was used to compute the cathode-directed streamer propagation. Figure 6-2 shows the electron and positive ion number densities and electric potential distribution for the case of applying positive pulse (FWHM is 4 ns, amplitude is 3 kv) to the exposed electrode 7 ns after the pulse, but sustaining the minimum electron number density equal to 10 7 m -3.

102 84 Figure 6-1: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (3 kv amplitude, 4 ns FWHM) without photoionization or artificial plasma. Figure 6-2: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (3 kv amplitude, 4 ns FWHM) with 10 7 m -3 minimum electron number density.

103 85 In this case the electrons are artificially generated instead of arising from natural photoionization generation. The presence of the artificial source of electrons creates plasma with maximum number density ~10 22 m -3. Despite the observation of the initial stage of streamer generation, the streamer does not propagate along the dielectric. The reason is insufficient initial and minimum electron number densities. However, if the minimum electron number density is increased to m -3, the streamer can propagate. Figures show the streamer development 4 ns and 7 ns after the pulse application. Figure 6-3: Electron and positive ion number densities and electric potential distribution 4ns after the start of the pulse (3kV amplitude, 4ns FWHM) with m -3 minimum electron number density.

104 86 Figure 6-4: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (3 kv amplitude, 4 ns FWHM) with m -3 minimum electron number density. Using the concept of minimum electron number density it is possible to resolve partially correct streamer dynamics. However, the question of applicability of this concept to the simulations of the experimental condition arises. In order to use this concept it is necessary at least to calibrate the minimum electron number density for each set of experimental conditions based on the streamer propagation length. For example in the case of 3 kv the minimum electron number density 10 7 m -3 leads to the plasma generation only near the edge of exposed electrode whereas m -3 corresponds to the streamer propagation. However, if one increases the voltage, the streamer will propagate at 10 7 m -3. Figures show the streamer propagation for the positive 4 kv. Furthermore, the plasma structure in Figures is similar to the plasma structure in Figures , though the applied voltages are different. This is another illustration

105 of the dependence of the plasma simulation results on the minimum electron number density. 87 Figure 6-5: Electron and positive ion number densities and electric potential distribution 4ns after the start of the pulse (4 kv amplitude, 4 ns FWHM) with 10 7 m -3 minimum electron number density. The need to use plasma simulations to predict the DBD operation makes the application of minimum electron number density unfeasible. Therefore, there is a need to correctly resolve the photoionization phenomena. For the next set of simulations 4 ns, 3kV positive pulse was applied to the DBD in order to generate plasma. Figures show the initial stage of the streamer propagation in the zoomed region near the edge of the exposed electrode (marked as red rectangle). The lower boundary of computational domain corresponds to the DBD surface. Initial plasma density is set to 10 7 m -3 corresponding to the atmospheric conditions. When the applied voltage reaches approximately 1.5 kv, electrons in bulk volume, drifting to the edge of exposed electrode, start to ionize air.

106 88 Figure 6-6: Electron and positive ion number densities and electric potential distribution 7ns after the start of the pulse (4 kv amplitude, 4 ns FWHM) with 10 7 m -3 minimum electron number density. At the initial stage of the breakdown, the density of the generated plasma is relatively low, and electrons freely run into the electrode. Since the mobility of ions is 2-3 orders of magnitude less than the mobility of electrons, at nanosecond time scales ion motion can be neglected. At this stage of breakdown two processes need to be pointed out. First, the electrons not only ionize air, but also excite nitrogen, leading to the photons emission and consecutive oxygen photoionization. Electrons, generated by photoionization hundreds of microns from the edge of the exposed electrode, accelerate in the high electric field around the edge of exposed electrode and produce more ionization. They can be compared to the artificially generated electrons in the case of keeping minimum electron number density constant. Second, since the electrons are removed to the electrode, positive space charge is being accumulated near the edge of

107 89 exposed electrode. However, 2.5 ns after the start of the pulse the accumulated charge has negligible influence on the electric potential and electric field distribution (Fig. 6-7). At 2.6 ns after the start of the pulse (Fig. 6-8), significant charge is accumulated near the edge of the exposed electrode. This charge, which is associated with formation of plasma sheath, starts to shield the electric field of the electrode. The region of maximum electric field moves from the edge of the exposed electrode to the plasma sheath region. The plasma sheath plays the role of virtual electrode edge. At 2.7 ns after the start of the pulse (Fig. 6-9), the plasma sheath propagates along the dielectric surface. The physics of the sheath propagation is as follows. The electrons, generated in the region of high electric field, are sucked into the plasma sheath rather than to the exposed electrode. During this process, electrons leave positive charge behind and neutralize the positive ions of the sheath. Hence, preceding plasma sheath become quasineutral plasma, and the region, left by electrons, become consecutive plasma sheath. The region of quasinetral plasmas between the edge of the electrode and plasma sheath is called streamer body. Streamer body is highly conductive plasma. Figure 6-9 shows that the net charge density and electric field in the streamer body are relatively low. The region of the high electric field is removed from the electrode to the plasma sheath by the streamer body. The ionization occurs in the vicinity of plasma sheath region. The electrons in the bulk volume are generated by the photoionization. Figures show consecutive streamer propagation along the dielectric during the applied pulse.

108 90 (a) (d) (b) (e) (e) (c) (f) Figure 6-7: Electron (a) and positive ion (b) number densities, difference between number densities of positively and negatively charged species (c), rate of photoionization production (d), electric potential distribution (e) and electric field distribution (f) 2.5 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters.

109 Figure 6-8: Electron and positive ion number densities, difference between number densities of positively and negatively charged species, rate of photoionization production, electric potential distribution and electric field distribution 2.6 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters. 91

110 Figure 6-9: Electron and positive ion number densities, difference between number densities of positively and negatively charged species, rate of photoionization production, electric potential distribution and electric field distribution 2.7 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters. 92

111 Figure 6-10: Electron and positive ion number densities, difference between number densities of positively and negatively charged species, rate of photoionization production, electric potential distribution and electric field distribution 2.9 ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters. 93

112 Figure 6-11: Electron and positive ion number densities, rate of photoionization production and electric potential distribution 3ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters. 94

113 Figure 6-12: Electron and positive ion number densities, rate of photoionization production and electric potential distribution 5ns after the start of the pulse (3 kv amplitude, 4 ns FWHM). Distances along both horizontal and vertical axes are measured in meters. 95

114 96 Figure 6-13: Electron and positive ion number densities, rate of photoionization production and electric potential distribution 7ns after the start of the pulse (3kV amplitude, 4ns FWHM). Distances along both horizontal and vertical axes are measured in meters. Let us compare the streamer propagation for the case of minimum electron number density (case A) and photoionization (case B). The qualitative physics of streamer development is similar the electrons in front of the streamer head (artificial electrons or photoelectrons) are accelerated in the electric field, generating plasma around streamer s head. However, there are several differences. The first difference is the plasma distribution. In case A, some electrons diffuse out of the streamer and produce plasma number density ~10 16 m -3 approximately 100 μm above the streamer body. This cloud is produced after the streamer propagation and it has a negligible effect on streamer development. In case B, the photoionization creates plasma of the density of ~10 19 m -3

115 97 approximately 300 μm around the streamer. This plasma is conductive. Therefore, the electric potential partially penetrates into the region above the conventional streamer body, reducing the electric field at the streamer head. For example, the maximum electric field at the streamer head for 3 kv pulses computed with m -3 minimum electron number density is ~7.5*10 7 V/m and the maximum electric field for 3 kv pulses computed with photoionization is ~5*10 7 V/m. The streamer still propagates with the velocities of the order of 10 6 m/s due to the much larger number densities of electrons in front of the streamer head for the case of photoionization than in case of minimum electron number density. Figure 6-14 compares the speed of streamer propagation for the cases of minimum electron number density and photoionization at different conditions. The results of streamer propagation velocities with the photoionization are similar to ones measured experimentally by Roupassov et al (Roupassov, Nikipelov, Nudnova, & Starikovskii, 2008) for the cathode-directed streamer propagation.

116 98 1.2E+06 Streamer Propagation Speed, m/s 1.0E E E E E+05 3kV, END 1E+11 3kV, photoion. 4kV, END 1E+7 4kV, photoion. 7kV, photoion. 0.0E Time, ns 6 8 Figure 6-14: Computed streamer propagation velocities for the DBD driven by 4ns positive pulses for voltage amplitudes 3 kv,4 kv and 7 kv based on the concept of minimum electron number density and photoionization Summary An introduction of the efficient photionization model led to the first DBD modeling, which takes into account all essential physical phenomena for the DBD operation. The first quantitatively correct streamer modeling for the DBD geometry was performed. The physics of the streamer initiation and propagation was considered in detail. The qualitative characteristics of streamer propagation agreed with one, considered in Chapter 4 for the concept of minimum artificial plasma density. However, the quantitative values of plasma parameters are different. For example, the maximum electric field at the streamer head and streamer thicknesses were different for those cases.

117 The importance of those plasma characteristics of the streamer will be discussed in Chapter 8. 99

118 100 Chapter 7 Application of the developed model for the description of the experiments Numerical Modeling of the DBD Induced Flow Opaits et al. developed the Schlieren technique, which is totally non-intrusive, for the diagnostic of DBD-induced flows in quiescent air (Opaits, Shneider, Miles, Likhanskii, & Macheret, Experimental Investigation of Dielectric Barrier Discharge Plasma Actuators Driven by Repetitive High-Voltage Nanosecond Pulses with DC or Low Frequency Sinusoidal Bias, 2008). Since the discharge slightly heats the air, it is possible to visualize the induced laminar wall jet. The principal feature of our technique, distinguishing it from other studies, is modulation of the voltage waveform applied to the discharge so that the plasma actuator is operated in pulse-burst mode. In the burst mode, clearly distinguishable separate vortices are created by each burst. The advantage of the proposed technique is the ability to visualize 2-D, laminar, low speed, small, nonstationary plasma-induced jets. An advantage of studies with quiescent air is that it is possible to notice some features of the induced flow which otherwise would be washed out by an external flow. More detailed description of the technique and the experimental results can be found in (Opaits, Shneider, Miles, Likhanskii, & Macheret, Experimental

119 101 Investigation of Dielectric Barrier Discharge Plasma Actuators Driven by Repetitive High-Voltage Nanosecond Pulses with DC or Low Frequency Sinusoidal Bias, 2008). This technique provides the coupling between the theoretical investigation of the plasma-gas interaction and the experiment. The plasma model provides the rate of thermal energy deposition and the force exerted on the gas. The output parameters from the plasma calculations can then be used as an input for the time-accurate numerical computational fluid dynamic CFD model based on 2D Navier-Stokes equations. This model predicts similar vortical structures and allows us to restore the entire flow pattern by matching the Schlieren images of a single vortex. The 2D Navier-Stokes equations were solved using second order accurate MacCormack scheme (Anderson, Tannehill, & Pletcher, 1984, p. 119). The computational domain was 5x2 cm. The time step was determined by the CFL condition. The modeling was performed for quiescent air at initial temperature 300 K. The dielectric surface temperature 1 mm downstream from the exposed electrode was considered to be 350 K. Several cases were studied. The plasma region was considered to be 500x100 µm. The force on the gas was considered to act only in horizontal direction downstream and to be uniformly distributed in the interaction region. In the simulations, the force magnitude and rate of thermal energy deposition in the interaction region were varied.

120 102 Figure 7-1: Air density and velocity distribution 14 milliseconds after the burst start. Consider the flow induced by the DBD plasma actuator in pulse-burst mode. When the DBD is on, the gas in the interaction region is being heated and receives momentum in the downstream direction (from left to right). The motion of the gas generates pressure gradient in the vicinity of the interaction region. The gas is being sucked in that region from the left and from above, creating an upstream vorticity. At the same time another vortex is generated by the induced gas jet (Figure 7-1). There are important differences between the two vortices. One of them is the sign of vorticity. The vorticities of upstream and downstream vortices are negative and positive, respectively. Another principal difference is the gas density in these vortices. The upstream vortex involves motion of the quiescent air at room temperature, thus its temperature is constant. In contrast, the downstream vortex is generated due to the motion of heated gas jet. Thus, it is characterized by well-defined temperature and density profiles. In the Schlieren experiments on DBD induced flow, only density gradients can be visualized, i.e. the downstream vortex is observed and the upstream one is not detected. Comparison

121 103 between the experimentally observed vortices and the simulated ones is shown in Figure 7-2. These results can be important in theoretical investigations of the flow separation control using DBD the gas flow will interact not only with the observed downstream vortex, but also with the hidden upstream one. Figure 7-2: Comparison between experimentally observed and simulated Schlieren images for DBD induced flow. The images are 20 mm by 10 mm each and correspond to 7, 14, and 21 milliseconds after the burst start. In (Likhanskii, Shneider, Macheret, & Miles, 2007) the integral momentum transferred to the gas by means of positive and negative nanosecond pulses with dc bias, as well as the rate of thermal energy deposition in the interaction region were computed. Those data were extracted and were used it as an input to the Navier-Stokes solver. The absence of the saturation effects due to surface charging was assumed in the modeling. The pulser was considered to operate at 500 khz repetition rate. The computed vortex propagation velocity as a function of distance from the exposed electrode is shown in Fig Taking the results for momentum transfer from (Likhanskii, Shneider, Macheret, & Miles, 2007) for positive 3 kv pulses (4 ns FWHM) with 1 kv dc bias and distributing it equally in the interaction region, the force in that region is 1.4*10 5 N/m 3. The numerical results for this calculation are between the aqua and blue curves in Fig The other curves correspond to the different force magnitudes in the interaction region at constant

122 104 rate of thermal energy deposition. It can be clearly seen that the translational vortex velocity significantly drops with the propagation along the surface. This fact is very important, since using the Schlieren technique in the experiments one can extract the density gradients at several temporal points and thus determine the vortex propagation speed. The developed model thus allows inferring the induced flow velocity right at the edge of the plasma and the force magnitude in the interaction region by matching the experimental pictures with numerically obtained results. The effect of gas heting was also investigated; however, the results showed that the vortex size has almost no dependence on the rate of thermal energy deposition in the interaction region. Only the temperature of the gas within the vortex varies. Note that the gas temperature cannot be measured by the Schlieren technique, and additional measurements with other techniques should be used to provide an additional insight into the plasma-flow interaction. 6 5 Vortex propagation velocity, m/s f=2e+5, max v=7m/s f=4e+5, max v=9.7m/s f=6e+5, max v=11.8m/s f=1e+5, max v=5.1m/s f=0.5e+5, max v=3.5m/s Distance from plasma-flow interaction region, cm Figure 7-3: Vortex propagation velocity as a function of the distance from DBD at different forces in the interaction region. The maximum velocities of the induced flow at the vicinity of interaction region are designated as max v.

123 105 Numerical Modeling of the Surface Charge Accumulation on the Dielectric Based on the Experimental Measurements of the Dielectric Surface Potential The recent experimental results (Opaits, Shneider, Miles, Likhanskii, & Macheret, Surface Charge in Dielectric Barrier Discharge Plasma Actuators, 2008) showed the performance of DBD operated with both positive and negative nanosecond pulses imposed on dc bias decreases as time goes by and reaches the saturation. One of the possible explanations of this phenomenon is the accumulation of the surface charge on the dielectric. In order to check this assumption, the surface electric potential was measured under various conditions after the DBD was turned off. The charge was seen to persist for many minutes. A detailed description of these experiments is provided in (Opaits, Shneider, Miles, Likhanskii, & Macheret, Surface Charge in Dielectric Barrier Discharge Plasma Actuators, 2008). In order to interpret the experimental data, the numerical model based 2D Poisson solver was developed. The DBD plasma actuator consisted of two electrodes of length 2.5 cm, separated by the 100 micron Kapton tape. The electrodes were assumed to be infinitely thin. The size of computational domain was 2.5x1 mm. The computational domain included the whole lower electrode and 0.5 cm of the upper electrode. The reason for the neglecting the remaining part of the upper electrode is its insignificant contribution in the electric potential on the surface. Since the electric potential distribution was measured experimentally after the DBD was turned off and the

124 recombination time of the plasma is of the order of microseconds, the gas above the dielectric was considered uncharged. 106 Figure 7-4: Calculated 2D electric potential distribution based on the experimental data for the electric potential on the dielectric surface right after the voltage applied to DBD was turned off (Negative 4 kv pulses, positive 2 kv dc voltage). The computation started with no charge on the dielectric surface. Then the surface charge was iterated to obtain the experimentally observed distribution of the electric potential on the surface. As a result, the distribution of the charge on the dielectric surface was obtained. Having the experimentally measured potential, 2D electric potential distribution and the surface charge for each temporal measurement were reproduced. Figure 7-4 shows the calculated 2D electric potential distribution for the case of 3 kv negative pulses, 2 kv positive dc bias right after the applied voltage was turned off. Figures and Figures show the experimentally measured distribution of the electric potential and the numerically calculated surface charge for the cases of 3 kv negative pulses, 2 kv negative bias and 3 kv negative pulses, 2 kv positive bias.

125 107 Figure 7-5: Experimentally measured voltage on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, negative 2kV dc voltage). Figure 7-6: Calculated charge density on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, negative 2kV dc voltage).

126 108 Figure 7-7: Experimentally measured voltage on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, positive 2kV dc voltage). Figure 7-8: Calculated charge density on the dielectric surface at different moments of time after the DBD was turned off (Negative 4kV pulses, positive 2kV dc voltage). Before interpreting the obtained surface charge distribution, let us consider the possible decay processes which can occur on the dielectric surface after the applied

127 109 voltage is turned off. A charged particle, either positive or negative, on the surface can drift in the external electric field created by the rest of the surface charges. Another mechanism of motion is surface diffusion due to the presence of surface charge density gradients. And the last process is the detachment from the surface, which can be caused by gas molecules hitting the surface. The motion of the particle will be described by the following continuity equation: σ r r + Γ = S, t r r eσ r Γ = D σ + ϕ kt (35) Here σ is the surface charge density in Coulomb/m 2, Γ is the two-dimensional charge flux, S represents sources and sinks, D is the surface diffusion coefficient, T is the surface temperature, ϕ is the local electric potential, and e is the charge of electron. The first term in the expression for the charge flux represents the diffusion and the second one represents the drift in the tangential electric field. The set of equations (35) should be coupled with the Poisson equation for local electric field distribution. Let us estimate the impact of each term in the surface charge behavior. First, consider the coefficient of diffusivity. For the 2D diffusion (Oura, Lifshits, Saranin, Zotov, & Katayama, 2003, pp ): D = ( 1 4) a 2 ν, (36a) ( ( kt )) ν = ν 0 exp E diff (36b)

128 where 110 a is the length of the hop, ν is the frequency of the hops, ν 0 is the frequency of the oscillations of the charged particle in the potential well on the surface, E diff is the height of the potential well (diffusion activation energy). There is a lack of experimental or theoretical data for the hopping diffusion of the positive or negative oxygen ions on the Kapton surface. However, for estimations the experimental data by Pedersen et al. (Pedersen, et al., 2000) on the study of hopping diffusion of nitrogen atoms on the surface of Fe(100) were considered. The data were obtained using the STM by following the nitrogen atoms. The diffusion activation energy was 0.92eV and the hopping frequency was about 1 mhz for room E diff temperature. It is reasonable to suppose that the diffusion activation energy for ions on the dielectric surface should be greater, since the charge-induced charge interaction is stronger than the dipole-dipole interaction of the nitrogen atom on the surface of Fe. So the activation energy for the diffusion of electric charges on the dielectric surface should be of the order of 1 ev or even greater. Thus one can estimate that the charged particle diffuses on the dielectric surface about several Angstroms during the time scales ~ 10 min, and the estimated coefficient of diffusivity is of the order of m 2 /s. The first term in the second equation of the system (35) can be evaluated as the characteristic drop of the surface charge number density over the characteristic distance. As it can be observed in Fig. 7-5 the characteristic length of charge number density change is of the order of 1 mm. Thus the term σ can be evaluated as ~ 10 3 σ / m. Consider the second term. The characteristic tangential electric field on the dielectric surface is ~ V/m. The temperature of the dielectric surface is ~ K. Thus

129 the second term will be 111. Comparing the first and second terms, one can see that during at least the first several minutes after the applied voltage is turned off, drift will dominate over diffusion min ~ 10 7 σ / m In order to analyze the detachment of the surface charge, let us return to Fig In the case of dominant drift one should expect the relative conservation of the surface charge to the right side from the peak density (on the left side the charges can be neutralized on the electrode). However, Fig. 7-6 shows almost uniform decay of the charge number density. This decay corresponds primarily to the surface charge detachment. The rate of the detachment, evaluated based on the data in Fig. 7-6, is 1. Despite the obtaining the rate of detachment of charged particles from the dielectric, the mechanism of the detachment is still unclear. It can be caused either by interaction with the gas molecules or can be self-detachment due to the presence of the vertical component of electric field. The calculated surface charge distribution showed that the surface charge is mainly located ~5-20 millimeters downstream. However, during the breakdown, the visible plasma propagates just a couple of millimeters along the surface in streamer form. After the breakdown, as the streamer decays, the electrons rapidly attach to oxygen molecules, forming the negative ions (~10ns time scale). This means that the electrons cannot be the primary charge carriers for surface charging under these experimental conditions. The negative ions are attached to the surface. This statement is also proved by the observation of the decay rate of surface charge of both negative and positive signs (Figs. 7-6, 7-8). Both experiments showed the same rate of surface charge decay,

130 112 suggesting that the surface charges in both experiments are of the similar nature, i.e. ions. For attached electrons one could expect different rate of surface charge decay. Figures show the measured electric potential distribution and the calculated surface charge number density after different number of applied pulses. With the increase of number of applied pulses the surface charge reaches saturation. Figure 7-9: Experimentally measured voltage on the dielectric surface after 1, 10 and 100 applied pulses (Negative 4kV pulses, positive 2kV dc voltage).

131 113 Figure 7-10: Calculated charge density on the dielectric surface after 1, 10 and 100 applied pulses (Negative 4kV pulses, positive 2kV dc voltage). Summary One of the goals for the DBD description was a development of nonintrusive diagnostics of the DBD induced flow. Opaits et al (Opaits, Shneider, Miles, Likhanskii, & Macheret, Experimental Investigation of Dielectric Barrier Discharge Plasma Actuators Driven by Repetitive High-Voltage Nanosecond Pulses with DC or Low Frequency Sinusoidal Bias, 2008) developed Schlieren technique to visualize DBD induced flow. In order to get a complete pattern of the DBD induced flow, a set of simulations of using the developed flow model was conducted. Besides a good agreement between the numerical and experimental data, simulations also demonstrated that the gas flow velocity right after the DBD flow interaction region is approximately an order of

132 114 magnitude larger than the speed of vortex propagation. Once the DBD is turne on, it induces not only the downstream propagated vortex, but also an upstream one. Since the gas in the upstream propagating vortex is not heated by DBD, it is not observed on Schlieren images. Opaits et al. (Opaits, Shneider, Miles, Likhanskii, & Macheret, Surface Charge in Dielectric Barrier Discharge Plasma Actuators, 2008) also demonstrated that for the case of DBD, driven by pulses and dc bias, the velocity of the induced flow drops with time. The reason for it is surface charge, which accumulates on the dielectric surface after hundreds of pulses. Based on the experimental measurements of the voltage distribution on the dielectric surface, the density of surface charge was computed using 2D Poisson Solver. The physics of surface evolution was discussed in detail.

133 115 Chapter 8 Limitations of the DBD Induced Flow Velocity Recent growth of interest to the DBD plasma actuators is primarily caused by the possibility of their potential applications for active transonic and hypersonic flow control. Despite extensive research efforts were undertaken by groups around the world over past decade (Boeuf & Pitchford, 2005), (Hall, Jumper, Corke, & McLaughlin, 2005), (Borghi, Carraro, Cristofolini, & Neretti, 2008), (Enloe, et al., 2004), (Post & Corke, 2004), (Singh & Roy, 2005), (Opaits, Shneider, Miles, Likhanskii, & Macheret, Experimental Investigation of Dielectric Barrier Discharge Plasma Actuators Driven by Repetitive High-Voltage Nanosecond Pulses with DC or Low Frequency Sinusoidal Bias, 2008), the DBD configuration, suitable for high speed applications was not obtained. The following Chapter analyzes the limitations for the DBD induced flow velocity. Induction of the Inviscid Gas Flow by the Gas Discharge Consider one dimensional interaction between the gas discharge and inviscid gas flow. Volumetric force, acting on the gas in the discharge region, is, (37)

134 where e is the absolute value of an electron charge, Δn is the difference between number 116 densities of positively and negatively charged species, and is an applied electric field. The Poisson equation for an electric field is:, (38) where is dielectric permittivity of free space. Combining equations (37) and (38) one gets. (39) Equation (39) represents the gradient of the electrostatic pressure. According to the momentum equation, (40) where ρ is gas density is gas velocity before the DBD region and is gas velocity after DBD region. The force is integrated along the discharge area. Taking into account equation (39), one gets: (41) Therefore, induced gas velocity is (42) In the absence of external flow, DBD induces flow with the velocity:

135 . (43) 117 Let us estimate maximum DBD induced flow velocity. Maximum electric field in the gas discharge in atmospheric air is the cathode sheath electric field (or the electric field at the streamer head) (Raizer, 1991, pp ). The magnitude of the cathode sheath electric field is 3*10 7 V/m. Since air density at atmospheric conditions at room temperature is 1.2 kg/m 3, the maximum single DBD induced flow velocity is. (44) Study of the Viscosity Effects on the DBD Induced Flow In order to analyze the above described estimate the following numerical experiment was conducted. The DBD plasma actuator was placed in quiescent air. The physical domain was 3.5 cm by 1 cm. The numerical size of the uniform grid was 25 microns in horizontal direction and 10 microns in vertical direction. The interaction of the DBD plasma actuator with the air was modeled by applying constant volumetric force 1 cm from the left boundary of the physical domain in the region 1 mm by 100 microns. The force magnitude was chosen to be equal to. The rate of thermal energy deposition was equal to Figures 8-1 shows DBD induced flow 1, 4 and 8 milliseconds after the DBD was turned on. The structure of the flow is the same to one reported in (Roth, Sherman, & Wilkinson, 2000).

136 118 Figure 8-1: Single DBD induced flow 1, 4 and 8 ms after the DBD is turned on. Color plots represent gas density. Arrows represent the distribution of the induced gas velocity. Figure 8-2 shows a distribution of the horizontal component of the flow velocity in the vicinity of the DBD flow interaction region. The gas is being accelerated in the DBD - flow interaction region. The flow has maximum velocity right after the interaction region.

137 119 Figure 8-2: Distribution of horizontal component of the flow velocity in the vicinity of DBD flow interaction region 1, 4 and 8 ms after the DBD is turned on. Black rectangular shows the interaction region. Figure 8-3 shows the distribution of the horizontal component of gas velocity along the vertical axis right after the DBD flow interaction region. Obtained results are in good agreement with experimental measurements (Roth, Sherman, & Wilkinson, 2000). Figure 8-3: Distribution of the horizontal component of gas velocity along the vertical axis right after the DBD flow interaction region (1.1 cm from the left boundary of the physical domain).

138 120 Let us analyze the validity of the estimations of the maximum induced flow velocity. In the numerical experiment the electrostatic force (39) was changed by constant force, acting on the gas. In this case equation (41) can be rewritten as:, (41 ) where and. According to (41 ), the induced gas velocity after DBD flow interaction region is equal to. (42 ) However, Figure 8-4 shows that the maximum DBD induced flow velocity is equal to 9.65 m/s. Figure 8-4: Time evolution of the maximum DBD induced gas velocity for a single DBD. The discrepancy between analytical estimate and numerical results is due to the viscosity. Figure 8-5 shows the comparison of the momentum, transferred to the gas by

139 the DBD, and momentum of the induced flow. Approximately 80% of the DBD transferred momentum is lost to the wall due to the gas viscosity. 121 Figure 8-5: Time evolution of the momentum, transferred to the gas by the DBD, (purple) and momentum of the flow (blue). Since 80% of the DBD transferred momentum is lost, the effective force, acting on the gas, in expression (42 ) is 5 times less than, i.e.. Thus, the gas induced velocity should be equal to, which is in good agreement with numerically measured value. Therefore, one can conclude that maximum DBD induced flow velocity will be: in quiescent air and (45)

140 in the presence of external flow (46) 122 where <1 is a coefficient which represents viscous effects. Or in the case of constant applied force in quiescent air, and (45 ) in the presence of external flow (46 ) For the above described numerical experiment,. On the Possibility of Increase of the Induced Flow Velocity by a Series of DBD Plasma Actuators In order to increase the DBD induced flow velocity, several research groups (Roth, Sherman, & Wilkinson, 2000), (Borghi, Carraro, Cristofolini, & Neretti, 2008) proposed to use a series of the DBD plasma actuators. The idea of this approach is the following. Each single DBD plasma actuator can induce gas flow with several m/s gas velocities. If a series of DBDs is mounted on the aerodynamic surface, each upstream DBD should incorporate to the gas flow of the downstream DBD. However, the potential of such approach was unclear.

141 123 In order to investigate this approach single DBD and series of two and three DBDs were simulated. Single DBD was simulated with parameters, described in previous section. For series of DBDs each additional DBD was places 5 mm downstream the preceding one. The comparison of the DBD induced flow for each case 1, 4 and 8 milliseconds after the DBD is on is demonstrated in Figs. 8-1, 8-6, 8-7. Figure 8-6: Flow, induce by a series of 2 DBDs, 1, 4 and 8 ms after the DBDs were turned on. Color plots represent gas density. Arrows represent the distribution of the induced gas velocity.

142 124 Figure 8-7: Flow, induce by a series of 3 DBDs, 1, 4 and 8 ms after the DBDs were turned on. Color plots represent gas density. Arrows represent the distribution of the induced gas velocity. Figure 8-8: Time evolution of the maximum DBD induced gas velocity for a single DBD (green) and series of 2 (red) and 3 (blue) DBDs.

143 125 Figure 8-8 shows the evolution of the maximum DBD induced flow velocity for each case. When steady-state condition is reached, the maximum gas velocities for a single DBD and series of two and three DBDs are 9.65 m/s, m/s and 11.6 m/s correspondingly. Let us analyze this scaling by comparing DBD induced velocities from a single DBD and series of two DBDs. In a case of a single DBD, the DBD induced flow velocity is defined by applied volumetric force and viscous effects. However, for series of DBDs it is necessary to take into account the flow, induced by the upstream DBD, in order to describe the flow after the downstream DBD. From single DBD simulations, the DBD induced gas velocity 5mm downstream (the location of the second DBD for the case of series of two DBDs) is approximately 4m/s. Therefore, according to the expression (10 ), the DBD induced gas velocity should be equal to m/s, which is in good agreement with the results of simulations. Summary Theoretical limitations of the maximum DBD induced flow velocity were analyzed. It was shown that the viscous effects cause a loss of substantial amount of the DBD induced flow momentum. Viscous effects can be reduced if the DBD-flow interaction region is increased in vertical direction. A possibility of using series of the DBD plasma actuators for increase of the maximum induced flow velocity was analyzed. It was demonstrated that series of the

144 DBDs do increase the maximum induced gas velocity. However, such configuration is not feasible to achieve substantial increase in the induce gas velocity. 126

145 127 Chapter 9 Application of the Developed Model for the Description of Plasma Generation by the Ferroelectric Plasma Sources Introduction Recently the possibility of heavy ion fusion (HIF) was broadly studied (Dunaevsky, Krasik, Felsteiner, & Dorfman, 1999) and references therein. The idea of HIF is to bombard an inertial confinement fusion target with an intense heavy ion beam. One of the engineering challenges for the HIF is the ability to focus this beam. The problem arises due to the defocusing effect of the large electric charge, accumulated inside the beam. Groups from Lawrence Berkeley National Laboratory and Princeton Plasma Physics Laboratory suggested solving this problem by creating low-density plasma inside the experimental a vacuum chamber (Roy, et al., 2004). The electrons from the created plasma are sucked into the positively charged ion beam, reducing the space charge and, consequently, the defocusing effect. The problem of plasma generation in vacuum chamber is rather complex. The standard gas discharges are not applicable because of very low gas density in chamber. One of the possibilities to generate plasma in particular case is to use the ferroelectric plasma source (FPS) (Dunaevsky, Krasik, Felsteiner, & Dorfman, 1999). The design of the FPS is similar to the DBD design. FPS consists of a substrate electrode and electrode mesh, separated by a ferroelectric material.

146 128 If the voltage is applied between the electrodes, the plasma ignites. A group from Technion (Dunaevsky, Krasik, Felsteiner, & Dorfman, 1999), (Rosenman, Shur, Krasik, & Dunaevsky, 2000) widely studied the FPS experimentally. The relevant result of this work is the possibility of plasma generation in vacuum chambers allowing the use of FPS for plasma growth in HIF experiments. Description of the Physical Model The modeled configuration of the FPS is presented in Fig It consists of two electrodes separated by a ferroelectric layer. The lengths of upper and lower electrodes are 0.15 mm and 0.7 mm, respectfully. The ferroelectric layer width is 0.2 mm. The relative dielectric permittivity is chosen to be equal to 100. The electrodes are considered to be infinitely thin. A positive Gaussian Voltage pulse with τ=50 ns FWHM and 3 kv amplitude is applied to the upper electrode; the lower electrode is grounded. Breakdown starts at approximately 2.5 kv, t=15 ns. Figure 9-1: Scheme of ferroelectric plasma source.

147 129 The physical model is similar to one used for the DBD modeling. The modeled gas is air at room temperature and 250 Torr pressure. The voltage applied to the upper electrode results in weakly-ionized plasma generation near the edge of electrode which subsequently spreads above the dielectric. In the experiments on FPS are carried out in vacuum chamber. However, there is approximately 100 micron desorbed gas layer above the FPS. In order to take into account this effect the ionization was considered only in the layer of gas 100 microns above the surface. The main difference with the DBD model is the boundary conditions. The boundary conditions at the electrode surface are: G = - G, if E 0, (47) en g m + n n < G = 0, if E 0, (48) + n n > The boundary conditions at the dielectric surface are: G = - G + G, if E 0, (49) g en d + n emission n < G = 0, if E 0. (50) + n n > Here, γ m and γ d are the effective secondary electron emission coefficients from metal and dielectric, with numerical values 0.1 and 0.01 used in computations, correspondingly; index n denotes normal component of electric field and fluxes. G emission denotes the flux of emitted electrons from the ferroelectric due to the applied electric field. G emission was assumed constant but a number of calculations were carried out for different G emission.

148 130 Physics of Plasma Generation Typical results of simulations are shown in Figs Figure 9-2 shows an example of simulation when the emission flux from cathode is not taken into account. Figure 9-2: Electron and ion number densities, electron temperature and electric potential distribution for the case of no electron source. The snap shot is taken 20ns after the pulse start. The box size is 1mm horizontally and 0.5 mm vertically. A positive Gaussian pulse with 50ns FWHM (full width at half maximum) and 3kV amplitude is applied to the upper electrode and the lower electrode is considered grounded. Figure 9-2 corresponds to the plasma generation 20 ns after voltage application with the initial electron and ion number densities of 10 5 m -3 in the square region 50x50 μm around the edge of electrode. When the voltage is applied between electrodes, the electrons produce ionization near the edge of electrode, but plasma does not spread

149 131 further along the ferroelectric surface. This situation is similar to the simulation of the DBD plasma actuator in the absence of photoionization. The difference is that emission current from the ferroelectric surface is the source of the electrons to support plasma generation for the FPS. For the next set of numerical experiments the minimum background plasma density 10 5 m -3 was kept everywhere in the computational domain. Figures 9-3a and 9-3b correspond to the plasma generation 20 and 30 ns after the pulse is applied. In this case, plasma develops in a streamer-like form. The electric field is strongest at the streamer head and the ionization occurs mostly around it. As plasma develops, it acts as virtual anode with highest electric field around its edges and ionization occurs in the region of the highest electric field. The process of plasma formation and anode potential propagation is repeated during the time of streamer propagation along the ferroelectric. For the final set of numerical experiments the possibility of the electrons, emitted by the ferroelectric, to be the source for ionization without any background plasma present initially was studied. The simulations were performed using the above described geometry. The emission currents were considered to be uniform and independent on the electric field. The magnitude of emission currents varied from A/cm 2 to 10-7 A/cm 2. At low currents (<10-12 A/cm 2 ) the results are similar to the case absence of background plasma. The number of emitted electrons is not sufficient to ionize the gas during the pulse. However, the increase of the emission current from the ferroelectric surface to A/cm 2 leads to plasma generation and spreading along the ferroelectric surface.

150 132 (a) (b) Figure 9-3: Electron and ion number densities, electron temperature and electric potential distribution for the case of uniform background electron number density. The snapshots are taken at (a) 20ns and (b) 30ns after the pulse start.

151 133 Figure 9-4: Electron number densities during the pulse. The color-plots are taken each 5ns, starting from 10ns. The emission current from the ferroelectric is A/cm 2. Figure 9-4 shows the time-dependent plasma propagation along the ferroelectric surface. The color-plots represent the electron density in logarithmic scale. Initially there is no plasma above the surface. When the positive pulse is applied to the upper electrode, the electron emission from the ferroelectric surface is initiated. At the beginning of the pulse the voltage is relatively low and is not sufficient to start the ionization process (breakdown). In approximately 15 ns upper electrode voltage reaches 2 kv and the electric field at its tip becomes sufficient to produce breakdown. During later evolution of the discharge a streamer forms and propagates along the ferroelectric.

152 134 (a) (b) Figure 9-5: Electron number densities, difference between electron and ion number densities, electric field and electric potential distribution for the case of A/cm emission current. (a) corresponds to the real scale geometry and (b) is zoomed region near the streamer head. The color-plots are taken 30ns after the pulse start.

153 The detailed dynamics of the streamer propagation is shown in Fig The color-plot corresponds the moment of time 30 ns after the pulse start. Figure 9-5a shows the electron number density, departure from the quasi-neutrality (space charge density), the electric field and electric potential. Figure 9-5b shows a zoomed region near the streamer head. The observed electron number density in plasma is of the order of m -3 and the main part of the streamer is quasineutral. The streamer head on the other hand is essentially nonneutral, carrying a significant positive charge. It acts like a tip of the electrode, producing the strong electric field, and consequently the ionization around it. The electrons, emitted from the ferroelectric surface drift in the electric field from right to left in the figure. The avalanche ionization starts when the electrons reach the region of strong electric field. The generated electrons drift to the streamer head, whereas slow positive ions barely move on these time scales and accumulate. When these electrons reach the streamer head, their concentration is sufficient to neutralize it. The former streamer head becomes the prolongation of streamer channel and the positive ions, left by generated electrons, become the streamer head. Besides the described streamer propagation, an additional ionization between streamer channel and ferroelectric surface takes place 30 ns after pulse starts (Fig. 9-5). At this moment, the streamer potential is close to the anode one (Fig. 9-5a). The configuration is similar to the 1D discharge between the electrode and ferroelectric surface (the streamer acts as a virtual electrode). Ferroelectric fills approximately 2/3 of the gap between the electrodes. The rest 1/3 is air. Since the dielectric permittivity of the ferroelectric surface is much higher than one of air, the potential of about 3 kv drops in 7 the 100 micron thick air gap. The estimated electric field in the gap is 3 10 V/m and it is 135

154 136 in a good agreement with the computed one (Fig. 9-5a). This electric field is sufficient to initiate the discharge, starting from the ferroelectric surface due to electron emission current. During the next 15 ns the gap between the ferroelectric surface and the streamer is ionized to m -3. Figure 9-6 compares results of simulations for different emission currents. The value of emission current strongly influences the plasma propagation and development. Figure 9-6: The color-plots of the electron densities at 30ns after the pulse start for different values of emission current from the ferroelectric. Summary The model, developed for the DBD plasma actuator simulations, was used to describe the physics of the plasma generation by the ferroelectric plasma sources. It was demonstrated that the plasma propagates in streamer form, as it was observed experimentally. In contrast to the DBD plasma actuator, the source of electrons for