Alan R. Hoskinson

University of Wisconsin-Madison, Madison, Wisconsin

Measurements and Simulations of Surface Dielectric 

Barrier Discharges Used as Plasma Actuators

Final Report 

NASA/CR—2012-217628

September 2012

Notice for Copyrighted Information 

This manuscript has been authored by an employee of the University of Wisconsin-Madison under Cooperative Agreement 

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Alan R. Hoskinson

University of Wisconsin-Madison, Madison, Wisconsin

Measurements and Simulations of Surface Dielectric 

Barrier Discharges Used as Plasma Actuators

Final Report 

NASA/CR—2012-217628

September 2012

Notice for Copyrighted Information 

This manuscript has been authored by an employee of the University of Wisconsin-Madison under Cooperative Agreement 

No. NNX07AB84A with the National Aeronautics and Space Administration and the Air Force Offi ce of Scientifi c Research 

(AFOSR) Grant No. FA9550–07–1–0025. The United States Government has a nonexclusive, irrevocable, worldwide license 

to prepare derivative works, publish or reproduce this manuscript, and allow others to do so, for United States Government 

purposes. Any publisher accepting this manuscript for publication acknowledges that the United States Government retains 

such a license in any published form of this manuscript. All other rights are retained by the copyright owner. 

National Aeronautics and

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at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by NASA technical management.

All of this work was conducted under the guidance of Dr. Noah Hershkowitz, my advisor for 5 years at the University of Wisconsin-Madison. He has always encouraged me to follow my interests, even when they lead to less familiar research territory 

and a semester as a teaching assistant. 

My research has also benefi ted over the years from numerous conversations with members of my doctoral committee: Dr. Carl 

Sovinec and Dr. Riccardo Bonazza from the Engineering Physics department, and Dr. Amy Wendt and Dr. John Scharer from 

the Electrical and Computer Engineering department. Several others have also provided me with technical advice and support. 

Dr. David Ashpis of the NASA Glenn Research Center provided numerous suggestions regarding pressure measurements in 

low-speed gas fl ows. Dr. Lutfi Oksuz, an associate professor at Süleyman Demiral University in Turkey and a visiting scientist 

at UW, assisted with the acquisition of the nanosecond-scale optical imaging data. Dr. Jon Ahn, a visiting professor from Sejong 

University in South Korea, provided helpful advice regarding the setup of the differential pressure sensor. Funding for this work 

was provided by NASA through cooperative agreement NNX07AB84A, monitored by Dr. David Ashpis, and by the Air Force 

Offi ce of Scientifi c Research (AFOSR) through Grant no. FA9550–07–1–0025, monitored by Dr. Robert Barker.

My studies have been supported indirectly by numerous others. Over the years I have had many helpful and interesting discussions with the other graduate students in my lab, particularly Ben Longmier, Scott Baalrud, Dongsoo Lee, and Young-Chul Kim. 

My undergraduate thesis advisor, Dr. Sven Bilén of Penn State, has continued to show interest in my progress and provide career

advice in the years since I worked with him.

Copyright © 2009 by Alan Robert Hoskinson

Acknowledgments
Abstract

Asymmetric surface dielectric barrier discharges (DBDs) have shown promise for use as aerodynamic actuators which can prevent flow separation from airfoils in low-Reynolds number gas

flow. Our DBDs used a symmetric triangular high voltage waveform to generate plasma in atmospheric pressure air. Plasma forms and decays in many nanosecond-scale microdischarges during

each millisecond-scale half cycle of the applied voltage, and the device induces a time-averaged

force on the nearby air. Time-averaged measurements indicated that the induced force of a singlebarrier actuator design (one electrode insulated from the plasma) can be increased exponentially

above the results of previous studies by decreasing both the length and thickness of the electrode

exposed to the plasma. This increased force may allow these devices to control flow separation

in a wider range of flow environments. Experiments using an intensified digital camera to examine the plasma on time scales of a few nanoseconds up to the applied voltage period showed

that, in addition to the previously-observed filamentary and jet-like plasma structures, discharges

with very thin exposed electrodes exhibited a weak but constant plasma immediately adjacent to

those electrodes. In double-barrier actuators (both electrodes insulated), decreasing the diameter

of the narrower electrode lead to increasing forces, and recorded images showed the simultaneous existence of both filamentary and jet-like plasma structures. The development and application

of a time-dependent, two-dimensional fluid plasma model has aided in undestanding the detailed

physics of surface DBDs at all time scales. For simulated single-barrier discharges, the model

qualitatively reproduced the filamentary and jet-like microdischarge structures. The model was

somewhat successful in reproducing the observed characteristics of double-barrier actuators. For

both actuator geometries, the model indicated that the majority of the forces induced on the neutral

gas occur in between microdischarges as the plasmas decay.

NASA/CR—2012-217628 i

Table of Contents

Page

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Applications of Surface Dielectric Barrier Discharges . . . . . . . . . . . . . . . . 1

1.2 Motivation of the Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Summary of Contributions by the Current Work . . . . . . . . . . . . . . . . . . . 4

2 Background and Review of Previous Research . . . . . . . . . . . . . . . . . . . . . 6

2.1 Collisional Plasma Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The Highly-Collisional Regime . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 The Drift-Diffusion Approximation . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Electrohydrodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Ionization and Attachment . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 High Pressure Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 DC Discharge Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Dielectric Barrier Discharges . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Active Flow Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Basic Fluid Dynamics Concepts . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Applications of Active Flow Control . . . . . . . . . . . . . . . . . . . . . 22

2.3.3 Active Flow Control Technologies . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Experimental Observations of DBD Plasma Actuators . . . . . . . . . . . . . . . . 25

2.4.1 Flow Control Using DBD Actuators . . . . . . . . . . . . . . . . . . . . . 26

2.4.2 DBD Actuator Operation in Quiescent Gas . . . . . . . . . . . . . . . . . 27

2.4.3 Effects of Actuator Geometry . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Computational Modeling of DBD Plasma Actuators . . . . . . . . . . . . . . . . . 32

2.5.1 Approaches to Plasma Modeling . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.2 Characteristics of Single Microdischarges . . . . . . . . . . . . . . . . . . 35

2.5.3 Repeated Microdischarges . . . . . . . . . . . . . . . . . . . . . . . . . . 38

NASA/CR—2012-217628 iii
Page

3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 DBD Actuator Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.1 Actuator Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.2 Actuator Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.3 Applied Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Force and Power Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 Electronic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.2 Stagnation Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.3 Discharge Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Optical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.1 Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.2 Plasma Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Experimental Force Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 Discharge Uniformity and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Single-Barrier Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.1 Effects of Cylindrical Exposed Electrode Diameter . . . . . . . . . . . . . 63

4.2.2 Effects of Rectangular Exposed Electrode Thickness . . . . . . . . . . . . 66

4.2.3 Comparison to Previous Experiments . . . . . . . . . . . . . . . . . . . . 68

4.2.4 Variation with Exposed Electrode Length . . . . . . . . . . . . . . . . . . 69

4.3 Double-Barrier Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Comparison of Balance and Stagnation Probe Techniques . . . . . . . . . . . . . . 74

5 Optical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.1 Observed Microdischarge Structures . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Single-Barrier Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Average Microdischarge Lengths . . . . . . . . . . . . . . . . . . . . . . 81

5.2.2 Filamentary to Filament-Free Transition . . . . . . . . . . . . . . . . . . . 86

5.2.3 Microdischarge Formation and Propagation . . . . . . . . . . . . . . . . . 89

5.3 Double-Barrier Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Computational Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1.1 Modeled System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.1.3 Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.1.4 Range of Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

NASA/CR—2012-217628 iv
Page

6.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.1 Numerical Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.2 Scharfetter-Gummel Discretization . . . . . . . . . . . . . . . . . . . . . 108

6.2.3 Poisson’s Equation Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2.4 Continuity Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2.5 Adaptive Time-Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2.7 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Measured Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 Description of Simulated Discharges . . . . . . . . . . . . . . . . . . . . . . . . . 119

7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.1 Simulated Microdischarge Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.1.1 Single-Barrier Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.1.2 Double-Barrier Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2 Effects of System Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3 Full-Scale Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.3.1 Single-Barrier Actuators with Circular Exposed Electrodes . . . . . . . . . 133

7.3.2 Single-Barrier Actuators with Thin Rectangular Exposed Electrodes . . . . 137

7.3.3 Double-Barrier Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.4 Comparisons to Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 141

7.4.1 Single-Barrier Microdischarge Structures . . . . . . . . . . . . . . . . . . 142

7.4.2 Double-Barrier Microdischarge Structures . . . . . . . . . . . . . . . . . . 146

7.4.3 Time-Averaged Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Appendices

Appendix A: Actuator Construction Procedure . . . . . . . . . . . . . . . . . . . . . . . 156

Appendix B: Details of the Computational Model . . . . . . . . . . . . . . . . . . . . . . 161

B.1 The Poisson’s Equation Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.2 The Continuity Equation Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Appendix C: List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

NASA/CR—2012-217628 v
List of Figures

Figure Page

1.1 Conceptual diagram of a DBD plasma actuator . . . . . . . . . . . . . . . . . . . . . 2

2.1 Volumetric DBD plasma in air at 10 Torr between two disc-shaped electrodes . . . . . 7

2.2 Operating regimes of a DC discharge . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Common DBD configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Images of the diffuse and filamentary modes in a parallel-plate DBD . . . . . . . . . . 17

2.5 Discharge current for a surface DBD operating in the filamentary mode . . . . . . . . 17

2.6 Two-dimensional control “volume” . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Boundary layer, flow separation, and wake of an airfoil . . . . . . . . . . . . . . . . . 21

2.8 Typical blade structure in the low-pressure stage of a jet turbine . . . . . . . . . . . . 23

2.9 A synthetic jet actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.10 A plasma synthetic jet actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.11 Demonstration of flow re-attachment on an airfoil . . . . . . . . . . . . . . . . . . . . 27

2.12 Example images of filamentary and jet-like microdischarges . . . . . . . . . . . . . . 29

2.13 Single-barrier DBD actuator dimensions examined in previous research . . . . . . . . 31

2.14 Simulated microdischarges for an actuator with a linearly-decreasing voltage . . . . . 35

2.15 Simulated streamer-like plasma for a positive bias . . . . . . . . . . . . . . . . . . . . 37

2.16 Simulated plasma for a negative bias . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 Schematics of standard single-barrier actuators . . . . . . . . . . . . . . . . . . . . . 44

NASA/CR—2012-217628 vi
Figure Page

3.2 Single-barrier actuator with side-by-side cylindrical exposed electrodes . . . . . . . . 45

3.3 Schematic of a double-barrier actuator . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Photographs of two single-barrier actuators . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Discharge circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.6 The shielded balance and actuator stand used for direct force measurements . . . . . . 49

3.7 Stagnation probe and control volume . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.8 Typical velocity data and skewed Gaussian fit . . . . . . . . . . . . . . . . . . . . . . 53

3.9 Typical charge and voltage data for a single-barrier actuator . . . . . . . . . . . . . . 54

3.10 Image field of view for a single-barrier actuator . . . . . . . . . . . . . . . . . . . . . 56

3.11 Image field of view for a double-barrier actuator . . . . . . . . . . . . . . . . . . . . 56

3.12 Discharge current during the positive- and negative-going half cycles . . . . . . . . . 58

4.1 Variation of the induced flow velocity in the z direction . . . . . . . . . . . . . . . . . 61

4.2 Run-to-run variation and conditioning in single- and double-barrier actuators . . . . . 62

4.3 Power dissipated by single-barrier actuators with cylindrical exposed electrodes . . . . 64

4.4 Measured forces for single-barrier actuators with cylindrical exposed electrodes . . . . 65

4.5 Power dissipated by single-barrier actuators with rectangular exposed electrodes . . . 66

4.6 Measured forces for single-barrier actuators with rectangular exposed electrodes . . . 67

4.7 Force efficiencies of single-barrier actuators with various exposed electrodes . . . . . 68

4.8 Normalized forces for actuators with varying electrode lengths . . . . . . . . . . . . . 70

4.9 Power dissipated by single-barrier actuators with varying length exposed electrodes . . 71

4.10 Measured forces for double-barrier actuators . . . . . . . . . . . . . . . . . . . . . . 72

4.11 Power dissipated by double-barrier actuators with various narrow electrode diameters . 73

NASA/CR—2012-217628 vii
Figure Page

4.12 Comparison of forces measured using the electronic balance and the stagnation probe . 74

4.13 Stagnation and static pressures measured near a single-barrier actuator . . . . . . . . . 75

5.1 Images of typical filamentary plasma structures . . . . . . . . . . . . . . . . . . . . . 78

5.2 Images of typical jet-like microdischarges . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Image of a typical filament-free plasma structure . . . . . . . . . . . . . . . . . . . . 79

5.4 Images of typical microdischarges in a double-barrier actuator . . . . . . . . . . . . . 80

5.5 “Backwards” filamentary and jet-like microdischarges in a single-barrier actuator . . . 81

5.6 Increasing single-barrier microdischarge length during each voltage half cycle . . . . . 82

5.7 Average light emission during the applied voltage cycle for a single-barrier actuator . . 83

5.8 Average light emission and edge fitting during both half cycles . . . . . . . . . . . . . 83

5.9 Average microdischarge lengths for two single-barrier actuators . . . . . . . . . . . . 85

5.10 Average microdischarge expansion rates for single-barrier actuators . . . . . . . . . . 86

5.11 Simultaneous filament-free and filamentary plasmas in a single-barrier actuator . . . . 87

5.12 Times and voltages of transition to fully filament-free modes . . . . . . . . . . . . . . 88

5.13 Discharge current showing the reduction in spikes during the filament-free period . . . 88

5.14 Nanosecond gate width images of filamentary microdischarges . . . . . . . . . . . . . 90

5.15 Increasing filamentary microdischarge length with increasing camera gate width . . . 90

5.16 Nanosecond gate width images of jet-like microdischarges . . . . . . . . . . . . . . . 91

5.17 Changing double-barrier microdischarge structure during each half cycle . . . . . . . 93

5.18 Average light emission during the applied voltage cycle for a double-barrier actuator . 94

5.19 Average microdischarge expansion rates for double-barrier actuators . . . . . . . . . . 94

6.1 A generic simulation geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

NASA/CR—2012-217628 viii
Figure Page

6.2 Mobility of nitrogen positive ions in nitrogen . . . . . . . . . . . . . . . . . . . . . . 101

6.3 Mobility of oxygen positive ions in oxygen . . . . . . . . . . . . . . . . . . . . . . . 101

6.4 Mobilities of electrons in nitrogen and oxygen . . . . . . . . . . . . . . . . . . . . . 102

6.5 Reduced ionization coefficients in both nitrogen and oxygen . . . . . . . . . . . . . . 103

6.6 Reduced attachment coefficient for oxygen . . . . . . . . . . . . . . . . . . . . . . . 104

6.7 The numerical grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.8 Dynamic time step adjustment during a single microdischarge event . . . . . . . . . . 113

6.9 The sliding threshold used to estimate new time-step sizes . . . . . . . . . . . . . . . 114

6.10 Schematics of the simulated single- and double-barrier actuators . . . . . . . . . . . . 119

6.11 Schematic of the simulated single-barrier actuator with a rectangular exposed electrode 121

7.1 Force induced by a simulated single-barrier actuator with a positive-going voltage . . 124

7.2 Simulated streamer-like microdischarge in a reduced-scale single-barrier actuator . . . 125

7.3 Simulated negative microdischarge in a reduced-scale single-barrier actuator . . . . . 126

7.4 Simulated positive microdischarge in a reduced-scale double-barrier actuator . . . . . 127

7.5 Simulated negative microdischarge in a reduced-scale double-barrier actuator . . . . . 128

7.6 Forces induced by simulated reduced- and full-scale single-barrier actuators . . . . . . 129

7.7 Simulated negative ion densities in single-barrier actuators . . . . . . . . . . . . . . . 130

7.8 Forces induced by simulated reduced- and full-scale double-barrier actuators . . . . . 131

7.9 Simulated negative ion densities in double-barrier actuators . . . . . . . . . . . . . . 132

7.10 Effects of exposed electrode diameter on forces for simulated single-barrier actuators . 133

7.11 Particle densities near simulated single-barrier actuators with negative-going voltages . 134

7.12 Time-averaged forces for full-scale single-barrier simulations . . . . . . . . . . . . . 135

NASA/CR—2012-217628 ix
Figure Page

7.13 Cycle-averaged forces induced by simulated single-barrier actuators . . . . . . . . . . 136

7.14 Simulated forces of single-barrier actuators with rectangular and circular electrodes . . 138

7.15 Effects of exposed electrode diameter on forces for simulated double-barrier actuators 139

7.16 Time-averaged forces for full-scale double-barrier simulations . . . . . . . . . . . . . 140

7.17 Cycle-averaged forces induced by simulated double-barrier actuators . . . . . . . . . 141

7.18 Simulation and imaging data for a single-barrier actuator with a positive-going voltage 144

7.19 Simulation and imaging data for a single-barrier actuator with a negative-going voltage 145

7.20 Simulation and imaging data for a double-barrier actuator with a positive-going voltage147

7.21 Simulation and imaging data for a double-barrier actuator with a negative-going voltage147

7.22 Comparison of simulated and measured forces for single-barrier actuators . . . . . . . 149

7.23 Comparison of simulated and measured forces for double-barrier actuators . . . . . . 151

8.1 Single-barrier actuator showing the exposed electrode length and width . . . . . . . . 153

B.1 Two functions which are linearized in the Poisson’s equation solver . . . . . . . . . . 163

NASA/CR—2012-217628 x
Chapter 1

Introduction

The ability to generate non-thermal plasmas at gas pressures on the order of one atmosphere

makes novel applications of plasma physics possible. This work describes the study and optimization of one method of high-pressure plasma generation, the surface dielectric barrier discharge,

for use as a method of active aerodynamic flow control. The results may also be useful for other

applications. We have conducted these investigations experimentally and by developing and applying a computational plasma model. This chapter briefly describes the use of such a discharge

for aerodynamics applications, the specific goals of this investigation, and summarizes the major

contributions of the work.

1.1 Applications of Surface Dielectric Barrier Discharges

A dielectric barrier discharge (DBD) consists of two electrodes with at least one layer of insulating material in between them. An oscillating voltage is applied to one of the electrodes, while

the other electrode is usually grounded. When the electric field in a gas region near the electrodes

exceeds the breakdown threshold, plasma is formed via an avalanching ionization process. The

presence of the insulating layer prevents the formation of arcs. In contrast to many other plasma

systems, in a DBD the plasma forms and is extinguished many times as the applied voltage oscillates. In this work we focus on surface discharges, in which the electrodes either sit on or

immediately underneath a dielectric plane. This type of discharge is discussed in more detail in

section 2.2.2.

NASA/CR—2012-217628 1
Volumetric DBDs are currently used in industry to generate ozone, treat polymer surfaces,

and as components in plasma televisions [1]. Surface DBDs are not as widely used, but have

been suggested for a variety of applications including both materials processing and modifying air

flow in jet turbines and over wing surfaces. Our work focuses on these latter applications, while

also studying the physics of the discharge, which may be of use to the former. The two areas of

application are discussed in sections 2.2.2 and 2.3. Surface DBDs with asymmetric configurations,

such as that in figure 1.1, affect gas flows by exerting a force on the neutral gas. In quiescent air,

this force generates a flow of neutral gas like that in figure 1.1.

induced

air flow

plasma

buried

electrode

exposed

electrode

Figure 1.1: Conceptual diagram of a DBD plasma actuator showing electrodes,

dielectric (black), and approximate direction of induced air flow

Work in recent years has shown that surface DBDs with one insulated electrode can reduce

or eliminate flow separation when placed on the leading edges and surfaces of airfoils . Flow

separation increases the drag and reduces the lift of airfoils. When used to limit separation of flows

from surfaces, the DBDs are commonly referred to as “plasma actuators” or “DBD actuators.”

The mechanism by which flow re-attachment occurs is only partially understood [2], but generally

increases as the actuator force increases [3]. Wind tunnel experiments have shown that plasma

actuators reduce or eliminate flow separation for background flow velocities . 20 m/s (Reynolds

numbers of . 1.6 × 105) [3–5]. The effectiveness depends on many factors including the location

and properties of the plasma actuator, the shape of the airfoil, the Reynolds number, the amount of

freestream turbulence, and the angle of attack of the airfoil relative to the background flow.

NASA/CR—2012-217628 2
1.2 Motivation of the Current Work

The most-studied potential application of plasma actuators is in jet turbine engines. In the lowpressure stage of commercial jet turbines, flow is fully attached during take-off and landing when

the Reynolds number of the flow is large. At cruising altitudes the Reynolds number decreases to

the order of 105 due in part to reductions in the gas density, and some flow separation occurs from

turbine blades [6, 7]. This separation leads to a several percent loss in turbine efficiency at these

altitudes [8]. While seemingly small, due to the large number of miles flown and fuel consumed by

commercial and military aircraft a few percent increase in fuel efficiency can have large economic

and environmental benefits.

Early attempts to use plasma actuators to eliminate flow separation in conditions like those

in low-pressure turbines have had only limited success [2, 5]. Increasing the force induced by

these actuators is expected to improve their performance in such environments, as well as opening

up new uses of plasma actuators. Our research works towards increasing the ability of plasma

actuators to cause flow re-attachment by examining the parameters which most strongly affect the

magnitude of the forces on the neutral gas.

Past studies of DBD plasma actuators have focused on single-barrier discharge geometries, in

which one of the electrodes is insulated and one is exposed to the plasma [e.g. 3]. In contrast,

studies examining surface DBDs for materials processing applications [9, 10] have used doublebarrier discharges, in which both of the electrodes are insulated. Comparing the two different

systems helps us determine the effects of geometry on the plasma and the induced forces.

The research presented in this work examines several questions about the relationships between

the discharge design, plasma dynamics, and the induced forces. These include:

• How does the electrode geometry affect the induced forces? What properties can be adjusted

to increase these forces?

• How does the plasma structure vary over the course of the applied voltage cycle?

• How does the plasma vary between single- and double-barrier devices?

NASA/CR—2012-217628 3
• Is a 2-dimensional fluid model sufficient to explain plasma behavior in a surface DBD?

1.3 Summary of Contributions by the Current Work

This work makes contributions to the experimental plasma actuator community by examining new device designs, validating force diagnostic techniques and making optical measurements

of plasma dynamics. It makes contributions to the computational plasma actuator community by

modeling the plasma in double-barrier systems, showing the effects of finite electrode size, and

qualitatively validating the simulated plasma dynamics and time-averaged forces against experimental measurements. The major results, innovations, and demonstrations of new techniques of

our work are summarized here.

We have demonstrated that the force induced by single-barrier actuators can be dramatically

increased by using short, thin exposed electrodes instead of the more commonly-used planar electrodes. Previously only a weak dependence on the thickness of the exposed electrode had been

observed. The material of these electrodes does not affect actuator behavior. We have also conducted the first systematic measurements of the double-barrier actuator geometry, using one narrow and one wide electrode. While actuators using this geometry are less powerful and efficient

than single-barrier devices, they provide an additional system in which we can compare the experimental and simulated results. Double-barrier discharges are also more commonly employed

in processing applications, so our observations of these devices may be of use to that community.

Both electronic balances [e.g. 11] and stagnation probe measurements [e.g. 12] have been used by

other groups to determine the induced forces. We have made quantitative comparisons between

these techniques, showing what assumptions may be safely made when interpreting the stagnation

probe data. Detailed results of our force measurements of different actuator designs are included

in chapter 4.

Using an intensified digital camera, we have measured features of the plasma microdischarges

that occur on the actuator surface when the applied voltage is rising and falling. We have identified

a new plasma regime that only occurs in single-barrier actuators with very short and thin exposed

electrodes, corresponding to the parameter space of increased device efficiency. We have also

NASA/CR—2012-217628 4
examined the behavior of average microdischarges in actuators with a variety of geometries, and

characterized the propagation velocity and lifetime of microdischarges in a single-barrier actuator.

These measurements can be used for comparisons to computational model results. The recorded

images are displayed and discussed in chapter 5.

We have developed a new 2-dimensional fluid model, partially based on the published work

of Bouef et al. [13, 14]. Our implementation has two features not included in existing codes:

an adaptive time-stepping algorithm to resolve short-time scale phenomenon without greatly increasing computation time, and the ability to model systems with arbitrary electrode shapes and

positions. The code is described in chapter 6. Using this code, we have simulated actuator systems

with the same dimensions as our experimental devices. The simulated microdischarges qualitatively resemble those recorded using the digital camera. We have run the first simulations of

actuators using the double-barrier geometry. For both geometries, the simulation indicates that

the time-averaged force occurs primarily between microdischarges as the plasma decays. These

results of our simulation runs are detailed in chapter 7.

NASA/CR—2012-217628 5
Chapter 2

Background and Review of Previous Research

The plasma actuators discussed in this work fall into the class of plasma discharges known as

surface dielectric barrier discharges (DBDs). These discharges often, as is the case in this work,

operate at or near atmospheric pressure. At these pressures, the behaviors of plasmas are strongly

influenced by collisions. The physics of discharges operating in this regime are summarized in

sections 2.1 and 2.2

Many studies of DBD actuators have been conducted since the devices were first described in

2000 by Roth et al. [15]. The general field of active flow control, of which DBD actuator research

is a subset, is briefly reviewed in section 2.3. Section 2.4 reviews the experimental measurements

that have been previously conducted regarding plasma actuator performance. It includes flow

measurements near actuators operating in both quiescent air and with background flow velocities,

as well as observations of the plasmas themselves. Section 2.5 summarizes existing efforts to

model important characteristics of the plasma actuator system.

We note that a significant portion of the research conducted on plasma actuator technology is

disseminated via the conference proceedings of several meetings of the American Institute of Aeronautics and Astronautics (AIAA). Unfortunately, much of this work is not subsequently published

in peer-reviewed journals. We cite such AIAA papers in this chapter only when no peer-reviewed

versions are available.

NASA/CR—2012-217628 6
2.1 Collisional Plasma Theory

A variety of gas discharges operate at or near atmospheric pressure including the surface DBDs

discussed in this work, parallel-plate DBDs such as the one pictured in figure 2.1, some microwave

discharges, and some of the DC discharges discussed in section 2.2.1. The frequency of collisions

in plasmas determine the transport properties, which are central to any theoretical or computational

attempts to describe a plasma system. In the analysis presented in this section, we consider plasma

with singly-charged ion populations.

exposed electrode

insulated electrode

Figure 2.1: Volumetric DBD plasma in air at 10 Torr between two disc-shaped

electrodes

2.1.1 The Highly-Collisional Regime

While in general high-pressure plasmas may be expected to be highly collisional, verifying

this for a particular system requires some analysis. Whether collisional or collisionless approximations are valid for a given species depends on whether the energy relaxation length for that

species is shorter or longer, respectively, than the appropriate scale length for the phenomenon of

interest [16]. Different species or different phenomenon in the same plasma may have different

collisionalities.

For the phenomena of interest here, the relevant scale length is the Debye length,

λD =



0Te

e

2ne

1/2

, (2.1)

the characteristic length for changes in the electrostatic potential in the plasma. Here Te is the

electron temperature in electron Volts, e is the fundamental charge, and ne is the electron density.

Low-temperature plasmas such as those in DBDs, glow discharges, and other devices have only a

NASA/CR—2012-217628 7
small fraction of the available particles ionized. Consequently, collisions between charged particles

are relatively rare and charged particle-neutral collisions dominate the dynamics.

For elastic ion-neutral collisions, both particles are of comparable mass so significant energy

transfer is possible between particles. The ion energy relaxation length is therefore approximately

equal to the mean free path for ion-neutral collisions. Due to the large mass difference between

electrons and neutral particles, little energy is transferred in elastic collisions between them, so

many collisions are necessary for electron energy equilibration. The energy relaxation length for

electrons experiencing only elastic collisions with neutrals is approximately

λ,e ≈ λel,e(mg/2me)

1/2

, (2.2)

where λel,e is the mean free path for elastic electron-neutral collisions, while mg and me are the

masses of neutral gas molecules and electrons, respectively [16].

When inelastic collisions are significant, the electron energy relaxation length will decrease. At

larger average electron energies the contributions of inelastic collisions may dominate. For atomic

gases the inelastic processes include ionization, electron attachment, and electronic excitation. In

molecular gases the list also includes rotational excitation, vibrational excitation, and numerous

dissociation processes [17]. In this case, the electron energy relaxation length is approximately

equal to the effective mean free path for all inelastic processes:

λ,e ≈ λin,e. (2.3)

The mean free path for each inelastic process depends on the details of the cross section for the

process as well as on the electron energy distribution function.

2.1.2 The Drift-Diffusion Approximation

In a non-magnetized, partially-ionized, collisional plasma, the bulk forces on each particle

species are caused by the electric field, pressure gradients, and a drag force which we assume to

be proportional to the average particle velocity. For a charged-particle species α, the momentum

NASA/CR—2012-217628 8
balance equation is:

mαnα



∂~vα

∂t + ~vα · ∇~vα



= ±enαE~ − ∇pα − mαnανmα~vα, (2.4)

where ~v is the fluid velocity, E~ is the local electric field, p is the pressure of the species under

consideration, and νm is the effective particle-neutral momentum transfer frequency. In using this

momentum transfer frequency, we are implicitly assuming that the effects of charged particlecharged particle collisions are negligible relative to charged particle-neutral collisions. At high

pressures, like those being considered in this work, the inertial term ~v · ∇~v is small relative to the

collisional term and can be neglected [17].

At high pressures particles will experience many collisions and reach steady-state flow velocities quickly. The time taken for this process is roughly equal to the inverse of the effective momentum transfer frequency. This frequency depends on the particle and neutral gas species, and

increases for larger neutral densities and electric fields. Based on published mobility data [18, 19],

we estimate that for large electric fields at atmospheric pressure, reasonable times to reach steadystate are 10−11 s for ions and 10−13 s for electrons. Since we are not concerned with plasma

motion or oscillation at or below these time scales, it is reasonable to neglect the time derivative in

equation 2.4.

Using p = nT with the particle temperature T is assumed to be constant, the particle flux Γ

can be written as

~Γα ≡ nα~vα = ±nαμαE~ − Dα∇nα, (2.5)

where the sign taken is the same as that on the particle charge [17]. Here we have also defined a

charged-particle mobility,

μα =

e

mανmα

, (2.6)

and a diffusion coefficient,

Dα =

Tα

mανmα

=

μαTα

e

. (2.7)

The two terms on the right-hand side of equation (2.5) are the electric field-induced particle drift

and the diffusive flux, leading to the description of this equation as the drift-diffusion approximation. It allows a concise description of charged-particle fluxes in highly-collisional systems.

NASA/CR—2012-217628 9
2.1.3 Electrohydrodynamic Forces

As described in section 2.1.2, charged particles moving in an electric field will experience

collisions predominantly with neutral gas particles. These collisions transfer directed momentum

to the neutral gas, generating a force on the neutral gas. In the field of DBD plasma actuators this

force is often called the electrohydrodynamic (EHD) force, while in corona discharge the same

effect is known as the electric wind [13, 20]. Researchers examining plasma actuators generally

ascribe their behavior to this force. Here we derive a mathematical description of the force for a

highly-collisional plasma containing electrons and singly-charged positive and negative ions. It

generally follows the development given by Boeuf and Pitchford [13].

In a system where the background flow velocity is subsonic (or zero, as in the experiments

and simulations of this work), the neutral gas molecules are relatively cold and have negligible

directed velocity compared to the velocities of the charged particles. Consequently we neglect any

momentum transfer from the neutrals to plasma particles. The EHD force density acting on the

neutral gas is then equal to the frictional drag on the plasma particles, summed over all particles

species:

~fEHD =

X

α

nαmα~vανmα,

This expression is obtained by noting that each plasma particle, on average, transfers a momentum

mα~uα to the neutral gas once per effective collision period 1/νmα. Using the definition of charged

particle mobility in equation (2.6) and assuming the plasma contains one species each of positive

ions (p), electrons (e), and negative ions (n), this becomes

~fEHD =

enp~vp

μp

−

ene~ve

μe

−

enn~vn

μn

=

e

~Γp

μp

−

e

~Γe

μe

−

e

~Γn

μn

.

By applying the drift-diffusion approximation for the fluxes given by equation (2.5), we can expand

this as

~fEHD = e(np − ne − nn)E~ − Tp∇np − Te∇ne − Tn∇nn. (2.8)

We note that the force on the neutral gas does not depend at all on particle mobilities. The physical

explanation of this is that while massive particles with lower mobilities transfer more momentum

NASA/CR—2012-217628 10
per collision, this is exactly balanced by the larger number of collisions experienced by lighter,

higher-mobility particles.

In systems like DBDs where the applied voltages are in the kilovolt range, the electric field

terms will dominate over the density gradient terms [13]. This allows us to neglect the latter terms

and simplify equation (2.8) into simply the net charge density times the electric field:

~fEHD ≈ e(np − ne − nn)E. ~ (2.9)

We apply equation (2.9) to calculate the induced force in our simulations, as described in section 6.3.

2.1.4 Ionization and Attachment

Electron-positive ion pairs may be generated by electron-impact ionization. In high-pressure

plasmas, electron impact ionization is most commonly modeled by using an ionization coefficient

α. In an electropositive gas, the rate at which new electron-ion pairs are created by ionization is

given [17] by

Siz = αΓe. (2.10)

By making an analogy to reaction rate equations, the dependence of the ionization coefficient

on the local electric field can be reasonably approximated as

α ≈

const

λ,e

exp 

−Eiz

eEλ,e

where Eiz is the ionization energy of the gas. This equation is valid for large electric fields in

which the energy electrons acquire between collisions, approximately eEλ,e, exceeds the electron

temperature Te. In situations where the energy relaxation length is inversely proportional to the

gas density, the ionization coefficient can be simply modeled using this equation. That relation

breaks down when the collision cross-sections vary with the electric field. For oxygen that occurs

when the reduced electric field E/p exceeds about 104 V m−1 Torr−1, where p is the gas pressure.

We note that often (and perhaps more appropriately from a perspective of determining particle

energy between collisions) the reduced electric field is specified as E/ng, where ng is the number

NASA/CR—2012-217628 11
density of the neutral gas, instead of E/p. Since much published data regarding physical parameters are given as functions of E/p for gases at room temperature, however, we use that definition

of the reduced electric field in this work.

In an electronegative gas, an electron attachment coefficient η is defined similarly to the ionization coefficient. The rate at which negative ions are created by electron attachment is given

by

Satt = ηΓe. (2.11)

For each electron removed from the plasma, a negative ion is created.

2.2 High Pressure Discharges

The plasmas discussed in this work exist at atmospheric pressure and have a low ionization

fraction. Because of these features, the plasma dynamics are dominated by collisions. The basic

equations describing such a system were reviewed in section 2.1. Additional physics specific to

DBDs are summarized in section 2.2.2. Since descriptions of DBD physics includes analogies and

references to high pressure DC discharges, relevant aspects of those DC systems are described in

section 2.2.1.

2.2.1 DC Discharge Regimes

Due to the maturity of research on high-collisionality DC discharge systems, groups studying

DBD plasmas often make analogies to DC discharge phenomena even though DBDs are powered

by oscillating voltages. For example, the DBD literature at times refers to the diffuse mode of a

DBD as an “atmospheric pressure glow discharge” (APGD), even though glow discharges do not

form and extinguish repeatedly as the plasmas do in DBDs. Only the DC discharge types and

mechanisms most relevant to DBD research are briefly discussed here. Figure 2.2 describes many

of the equilibrium states of a DC discharge between a cathode and anode. It also roughly describes

the breakdown regimes through which a discharge progresses as it moves from turn-on (no current)

to its equilibrium state.

NASA/CR—2012-217628 12
"Dark" Discharges Glow Discharges Arcs

Townsend Regime

Corona VB

Background Ionization

Glow-to-Arc

Transition

Normal Glow

Abnormal

Glow

Thermal

Arc

NonThermal

Arc

Voltage

Discharge Current (A)

10-10 10 10 10 10 10 10 10 -8 -6 -4 -2 0 2 4

Range of physics 

analogous to DBD

behavior

Figure 2.2: Operating regimes of a DC electric discharge tube, adapted from Roth

[21]. VB is the breakdown voltage of the discharge system.

In the Townsend regime, plasma formation relies on an external source of electrons. Depending

on the system under consideration, these electrons may come from thermionic or field emission

from the cathode, from the minimal background ionization of the gas, or from other sources.

Because these “seed” electrons must come from outside the plasma, the discharge is considered

non-self sustained. The strength of a discharge in the Townsend regime is controlled in part by

the rate at which these electrons are produced. A discharge relying on background ionization, for

instance, draws a barely measurable current [22].

If a strong electric field exists in the region where electrons are produced, some of the electrons

will gain enough energy to ionize neutral particles. Ionization produces new free electrons, which

may then be accelerated and cause additional ionization. This process is known as an electron

avalanche. A single avalanche ends when the electrons are either blocked by a material surface or

enter a region where the electric field is too weak to provide enough energy for ionization. In the

Townsend regime, the space charge density is generally too low to significantly alter the electric

field from its vacuum distribution.

NASA/CR—2012-217628 13
Due to the high mobility of electrons relative to ions, avalanches leave behind a cloud of positive charge. In situations where that space charge is large enough to modify the electric field, the

positive space charge shifts the region of strong electric field away from the cathode. If the shift

uncovers new free electrons, new avalanches occur and the process repeats. The result is a growing

narrow column of space charge, often appearing jagged like a spark or lightning. Such a plasma

column is commonly called a streamer. The streamer continues propagating until the region of

strong electric field is too weak to support significant ionization, too few free electrons exist in the

new region of strong electric field, or the streamer reaches the anode. In many situations the new

free electrons are provided by photoionization caused by photons emitted during the avalanche

phases [23]. The streamer breakdown regime occurs most frequently when the region of concentrated electric field is near the anode. It may not exist in some discharge configurations, such as

the electric discharge tube on which figure 2.2 is based.

In addition to background ionization and photoionization, free electrons may be supplied by

secondary emission caused by ion bombardment of the cathode surface. The flux of secondary

electrons from a surface is given by

Γ

SE

e = −γΓp. (2.12)

Here γ is the secondary electron emission coefficient of the surface, which in general is a function

of the surface material and condition, the bombarding ion species, and the energy and angular

distributions of the bombarding ions [17]. The negative sign indicates that the flux of secondary

electrons is away from the surface when the flux of positive ions is towards it. Secondary electrons

may be emitted from dielectric surfaces as well as electrodes, although the surface potential will

change due to this emission. The secondary emisison coefficients may be comparable to or larger

than those for electrodes. Surface bombardment by neutral metastable particles and high-energy

electrons may also cause secondary electron emission, but we do not consider these processes in

this work.

If, on average, at least one secondary electron is emitted due to the bombardment of all the ions

created in one avalanche, the discharge is self-sustained as it no longer requires an external source

of electrons. In this case, the plasma density quickly increases and the resulting system is termed

NASA/CR—2012-217628 14
a glow discharge. The threshold for transition from the Townsend to the glow regime is given by

the requirement for electron replacement [24]:

γ



exp Z xmax

0

(α − η) dx − 1



≥ 1. (2.13)

Here α and η are the ionization and electron attachment coefficients of the gas described in section 2.1.4. The α, η, and γ coefficients are often referred to as the first, second, and third Townsend

coefficients, respectively. Both α and η vary with the local reduced electric field. The integral in

equation (2.13) is evaluated over a region 0 < x < xmax in which the net electron production is

positive, i.e. α(x) − η(x) > 0.

The term “corona discharge” is used in the literature to refer to several different mechanisms

of plasma formation near an electrode with a small radius of curvature. The biased electrode, most

commonly a needle-like point or a thin wire, produces a strongly inhomogeneous electric field

distribution around it. The physics governing the discharge depend on the polarity of the applied

bias, the distribution of the electric field, and the properties of the gas medium. In this work we will

distinguish between the types by their discharge mechanisms, and refer to Townsend, streamer, or

glow coronas. The corona regime noted in figure 2.2 includes only Townsend coronas.

When the sharper electrode is used as the cathode, the plasma is commonly called a negative

corona. Depending in part on the electric field strength, the discharge may operate as either a

Townsend or a glow corona, with at least some of the electrons supplied by secondary emission

from the cathode. The plasma is largely confined to a narrow region around the cathode where the

electric field is strong enough to cause ionization. Outside of this space, electrons and negative

ions drifting to the anode provide current continuity while positive ions are essentially absent.

When the sharper electrode is used as the anode, the plasma is commonly called a positive

corona. Due to the weaker electric field near cathode compared to the negative corona, field emission and secondary emission from the cathode are generally not important. Unless an additional

source of electrons is available, the positive Townsend corona is quite weak. Increasing the applied

voltage causes a transition into a streamer corona regime. If the field at the cathode becomes large

enough to support secondary emission, the plasma will transition into a glow corona [22].

NASA/CR—2012-217628 15
2.2.2 Dielectric Barrier Discharges

We use analogies to the DC discharges described above to describe the operation of dielectric

barrier discharges, which are the focus of this work. A dielectric barrier discharge (DBD) is

conventionally described as a system in which a plasma forms between two electrodes, at least

one of which is insulated from the plasma, with an alternating high voltage applied between the

electrodes. In the absence of a dielectric layer, the plasma would quickly transition to a thermal arc

discharge. In a DBD the plasma deposits charge on the dielectric layer, shielding out the electric

field in the gas region and preventing this transition. DBDs differ strongly from DC discharges

due to their time-varying nature. The plasma in a DBD forms and extinguishes at least once, and

often many times, during each half cycle of the oscillating applied voltage.

(a) (b) (c) (d)

Figure 2.3: Common DBD configurations, showing electrodes, dielectric materials

(in black), and the approximate plasma regions (inside dashed outlines). The drawings show (a) a single-barrier parallel-plate discharge, (b) a single-barrier concentric

discharge, (c) a single-barrier surface discharge and (d) a double-barrier surface discharge, also called a coplanar discharge.

DBDs are operated at neutral gas pressures from tens of Torr to atmospheric pressure or higher,

with applied voltages amplitudes typically in the kilovolt range. Applied voltage frequencies typically range from 50 Hz [e.g. 25–27] to tens of kilohertz [e.g. 28, 29], but some experiments have

been carried out at even higher frequencies [30]. The most common discharge configurations

are shown in figure 2.3. Parallel-plate designs, like figure 2.3(a) are often employed in materials

processing applications. The concentric cylinder design of figure 2.3(b) was designed by Werner

von Siemens in 1857 to generate ozone, and is still widely used for that purpose today [1]. This

work focuses on single- and double-barrier surface discharges, shown in figures 2.3(c) and 2.3(d)

respectively.

NASA/CR—2012-217628 16
(a) (b)

Figure 2.4: Images of (a) a diffuse discharge in neon and (b) a filamentary discharge in argon between parallel-plate electrodes at the top and bottom of each

image. From Trunec et al. [34].

Volume DBDs have been shown to operate in at least two distinct modes: diffuse and filamentary. In a diffuse discharge the plasma forms and is extinguished exactly once per half-cycle of the

applied voltage, accompanied by a single current pulse in each half cycle [31]. At each point in

time in this regime the plasma appears uniform in directions parallel to the discharge electrodes, as

in figure 2.4(a). In contrast, filamentary discharges consist of many short-lived, narrow channels of

plasma, as in figure 2.4(b) that form and quench on time scales of a few nanoseconds to hundreds

of nanoseconds. These streamer-like microdischarges are spread out in time, corresponding to

many short current pulses in a “spiky” discharge current waveform similar to the one in figure 2.5.

The microdischarges are often spread out in space as well, although in some circumstances they

may self-organize [32, 33].

0 0.5 1 1.5 2

−8

−4

0

4

8

Time (ms)

Applied voltage (kV)

0 0.5 1 1.5 2

−6

−3

 0

 3

 6

Discharge current (mA)

Voltage

Current

Figure 2.5: Discharge current for a surface DBD operating in the filamentary mode,

showing the many current spikes corresponding to individual microdischarges. The

DBD is driven by a 1 kHz sinusoidal voltage.

NASA/CR—2012-217628 17
Numerous factors influence which mode is present. These include the species, purity, pressure,

and flow rate of the gas, the discharge geometry and construction materials, and the shape, amplitude, and frequency of the applied voltage waveform. The primary way experimentalists achieve

diffuse modes is to add helium to the gas mixture [35], while the addition of oxygen tends to lead

to filamentary discharge modes [36]. Relatively large electrode gaps and voltages far above the

breakdown voltage tend to generate filamentary discharges.

The plasma dynamics specific to surface DBDs have not been studied in the same depth as

those in volume DBDs. The characteristics unique to these surface DBDs so far discovered are

discussed here. Korzec et al. [37] operated a single-barrier surface discharge in pure helium at

various pressures and found some characteristics shared with both filamentary and homogeneous

modes. The plasma had significant spatial visible inhomogeneity, suggesting the existence of

filaments, but also had electrical signals and light emission similar to a homogeneous discharge.

In our work, all of the discharges were operated in atmospheric pressure air at voltages well above

breakdown, and all of the discharges were filamentary in nature.

Gibalov and Pietsch [38] examined a single-barrier surface discharge and showed that the

plasma, as measured by deposited surface charge, extended much further away from the exposed

electrode after a single positive voltage pulse was applied to the exposed electrode than after a

single negative one. Enloe et al. [39] observed the light emission from a similar discharge driven

by a sinusoidal waveform, and observed that the plasma extended a similar distance both when

the applied voltage was increasing and decreasing. The rates of plasma expansion were noted to

be similar in the two half cycles. Due to the differences in applied voltages and methods of observation, these two studies are not directly comparable, but they are among the only experiments

to examine plasma expansion in surface DBDs. Gibalov and Pietsch [40, 41] developed a 2-D

simulation of the filamentary regime for double-barrier surface DBDs. Based on their numerical

experiments they concluded that photoemission of electrons from the dielectric surface is essential for the appearance of a streamer-like structure. Without photoemission, they observed only a

weaker, non-propagating plasma and significantly longer duration current pulse.

NASA/CR—2012-217628 18
DBDs have seen a surge of new ideas for and research into possible applications in the last 10 to

15 years. In addition to their historical (and current large-scale) use in the production of ozone [1],

volume DBDs have been used in plasma display panels [42] and studied for use in the destruction

of hazardous gases [43], surface cleaning [44], and etching of polymers [45]. Surface DBDs

have also been proposed for several materials processing applications, including the conversion of

cellophane surfaces into perfluorinated surfaces [9] and increasing the wettability of polypropylene

fabric [10]. The surface DBD configuration offers some advantages for materials processing, as

the surface being modified is not part of the discharge circuit. In contrast, in parallel-plate systems

the processed material generally serves as the dielectric layer.

2.3 Active Flow Control

Active flow control is generally defined as intentionally modifying a fluid flow field in a way

that is controllable during system operation. For external flow fields, characteristics that may be

desirable to modify include the laminar-to-turbulent transition, flow separation (or the location of

separation) from an airfoil surface, and the level of free-stream turbulence [46]. Before discussing

some of the specific applications of active flow control in section 2.3.2 and the various technologies

used to effect flow control section 2.3.3, a brief summary of relevant background material in fluid

dynamics is given in section 2.3.1.

2.3.1 Basic Fluid Dynamics Concepts

Systems including fluid motion can be analyzed by applying a control volume formulation,

in which physical laws are applied to a specified region of space. For a two-dimensional, nonaccelerating control “volume” of gas like the one in figure 2.6, Newton’s second law of motion can

be written as

F~

surf

L

+

F~

body

L

=

∂

∂t Z Z

area

~u ρg dA +

I

edge

~u ρg ~u · d

~`. (2.14)

F~

surf includes all forces acting on the edges of the control volume, while F~body includes all forces

acting inside the control volume. Surface forces commonly include pressure forces, shear forces,

NASA/CR—2012-217628 19
and the reaction forces of structures in contact with the fluid. The most commonly considered

body force is the weight of the fluid, but we will consider the electrohydrodynamic force described

in section 2.1.3. In equation (2.14), ~u is the fluid velocity of the gas, ρg is its mass density, and L

is a unit length perpendicular to the plane of figure 2.6.

Figure 2.6: Two-dimensional control “volume”

The pressures that contribute to the surface forces in equation (2.14) are thermodynamic pressures. In fluid dynamics these pressures are more commonly called static pressures, here denoted

as p. A pressure sensor whose opening faces into a flowing gas would measure a larger pressure,

known as the stagnation pressure. The stagnation pressure p0 is given by

p0 = p +

1

2

ρgu

2

, (2.15)

with the increase in pressure caused by the stopping of the flow as it enters the pressure sensor.

The static pressure in a flowing fluid can be measured by ensuring that the opening to the sensor

faces in a direction perpendicular to the flow velocity ~u.

A key parameter in determining the flow regime is the Reynolds number, which is a dimensionless ratio approximating the ratio of inertial forces to viscous forces. Flows with larger Reynolds

numbers tend to be more turbulent, while flows with smaller Reynolds numbers are more likely

to be laminar [47]. Determining the Reynolds number requires identifying a characteristic length.

For gas flow over airfoils, this length is commonly chosen to be the chord length of the airfoil,

`c [3]. The chord length is the length of a straight line connecting the leading and trailing edges

of an airfoil. These edges are identified in figure 2.7. For all of the discussions in this work, the

Reynolds number used is the chord Reynolds number,

Rec =

ρgu `c

μg

, (2.16)

NASA/CR—2012-217628 20
wake

boundary layer

location of flow separation

trailing edge

Figure 2.7: Diagrammatic representation of the boundary layer, flow separation,

and wake of an airfoil. Adapted from Fox and McDonald [47].

where μg is the dynamic viscosity of the neutral gas.

When fluid flows along a solid surface, the system is usually analyzed by splitting it into two

separate components: the free-stream flow far from the surface and a boundary layer near the

surface. Typically, when a flowing fluid encounters a surface the initial boundary layer will be

laminar, with smooth streamlines. At some point along the surface, the flow will transition from

laminar to turbulent. The location of this point depends on many factors including the shape of

the surface, the surface roughness, the level of turbulence in the free-stream flow, the free-stream

velocity, the fluid viscosity, and the pressure gradient along the surface.

Depending on the shape of the solid surface, the pressure in the boundary layer may decrease or

increase as the flow moves along the surface. The former case is referred to as a favorable pressure

gradient, as the pressure tends to reinforce the existing flow pattern. The latter case, in which

∂p/∂x > 0, is known as an adverse pressure gradient. The presence of an adverse pressure gradient

makes flow separation from the surface possible. Flow separation, pictured in figure 2.7, occurs

when the velocity in the boundary layer is reduced to (or below) zero. It significantly increases

the drag on a surface. While flow separation may occur for both laminar and turbulent boundary

layers, for given free-stream conditions a smaller adverse pressure gradient is required to cause

flow separation in laminar boundary layers. Turbulent boundary layers carry more momentum,

making it more difficult to slow the boundary flow down to zero velocity and cause separation.

NASA/CR—2012-217628 21
2.3.2 Applications of Active Flow Control

Plasma actuator technologies like DBD plasma actuators and those discussed in section 2.3.3

have been shown to be most effective in flows with relatively low Reynolds numbers (on the order

of 105 or smaller) [48]. Consequently the applications described in this section involve gas flows

with either small flow velocities or low gas densities. While many of the papers cited in this

section concern the use of DBD plasma actuators for specific applications, the other technologies

discussed in section 2.3.3 could also potentially be used. More in-depth reviews of the applications

of plasma actuators may be found in works by Corke and Post [48] and Moreau [46].

In the low-pressure stage of jet turbines, the flow over the turbine blades, like those shown

in figure 2.8, is fully attached during the conditions usually present during take-off and landing.

At cruise altitudes, however, the reduction in air density causes the Reynolds number to drop to

about 80,000 [8]. Due to the this drop, the flow becomes more laminar and may separate early

on the blades before reattaching further on. This feature is known as a “separation bubble,” [6]

and it causes noticeable turbine efficiency losses [8]. The reattachment of the flow occurs due

to a laminar-to-turbulent transition. If this transition occurs prior to the point of separation, the

separation bubble can be largely suppressed [7].

While passive flow control methods could increase the free-stream turbulence of the flow, these

would have detrimental effects during take-off and landing. Active flow control can also increase

the turbulence while still allowing the turbine to operate at peak efficiency [2, 5] for both high and

low Reynolds number flows by only operating the control device at cruise altitudes. Computational

fluid dynamics (CFD) simulations have demonstrated the viability of flow control techniques that

inject momentum into the flow both continuously and at reduced duty cycles [49]

Reducing flow detachment in the low-pressure stage of jet turbines may be particularly useful

in the context of high-altitude unmanned aerial vehicles (UAV). At high altitudes the reduction in

air density pushes the Reynolds number lower, and engineers desire to increase the blade spacing to

save on turbine weight [50], which worsens the adverse pressure gradient. Both of these increase

the likelihood of flow separation [51]. Mission profiles for such aircraft call for long-duration

missions, making fuel efficiency a high priority.

NASA/CR—2012-217628 22
gas flow

approximate

locations of flow

separation

Figure 2.8: Typical blade structure in the low-pressure stage of a jet turbine, also

noting typical points of flow detachment for low Reynolds number flows

In some situations, separation can cause the generation of flow instabilities in an object’s wake.

These instabilities are a significant source of aircraft noise. Thomas et al. [52] demonstrated that

active flow control can be used to significantly limit flow separation noise production caused by

flow over a cylinder mimicking a landing-gear strut. Active flow control has also been suggested

for use in limiting the noise caused by jet engine exhaust [53]. Other proposed uses include eliminating flow separation on the wings of slow-flying UAV’s [54] and improving lift on helicopter

blades [48].

2.3.3 Active Flow Control Technologies

The DBD plasma actuators discussed in this work represent only one method currently being

researched to effect flow control. While many techniques exist to modify flow fields, only the ones

which have effects similar to those of DBD actuators (and thus have been proposed for similar uses)

are reviewed here. These technologies include direct gas injection into the flow, synthetic jets using

piezoelectric actuators, and plasma synthetic jets. In order to be useful in a particular application,

NASA/CR—2012-217628 23
DBD actuator technology must advance to the point where there are significant advantages over

these other techniques.

Active flow control using gas injection functions simply by forcing gas out through small holes

in an airfoil surface. The injected gas jets can alter the flow field in several ways, such as by adding

momentum to the boundary layer or promoting the laminar-to-turbulent transition. McAuliffe and

Sjolander [55] used steady gas injection at about a 40◦angle to the surface of turbine-like blades,

and found the technique to be useful only when free-stream turbulence levels were low. In some

designs suction is used in addition to injection [56], while in others the gas injection may be pulsed

for improved efficiency [50]. Gas injection techniques have the advantage of being relatively

straightforward to model, but require gas lines and valves to be placed inside the airfoil structure.

Synthetic jets seek to eliminate the disadvantages of direct gas injection. These devices consist

of a small cavity embedded in the airfoil surface, as shown in figure 2.9, with a piezoelectric

surface on one side of the cavity. The piezoelectric actuator is driven at a frequency resonant with

the cavity, with the result that gas is drawn in from near the airfoil surface and then pushed away.

The cavities are often cylindrical, with numerous cavities spaced along the span of the airfoil,

but may also be designed using other shapes [57]. Synthetic jets have been show to reduce flow

detachment both experimentally [e.g. 58] and computationally [e.g. 59]. Synthetic jets require

significantly less hardware than direct gas injection, and can achieve large jet velocities. Due

to the dependence on a resonance mechanism, such actuators do require tight tolerances in the

fabrication of the resonant cavities.

Plasma synthetic jets are a fusion between synthetic jets and DBD actuators. These devices are

DBDs, and may be either annular or linear [60]. These two geometries are pictured in figure 2.10,

with the annular device having rotational symmetry and the linear device having uniformity in

the direction perpendicular to the plane of figure 2.10. Both geometries accelerate flow from

several directions toward the center of the device. The flows meet and turn upwards, yielding a

flow structure similar to a piezoelectric synthetic jet [61]. Plasma synthetic jets were developed

relatively recently, so their effectiveness for flow control has not been evaluated as extensively as

the other techniques described here.

NASA/CR—2012-217628 24
airfoil surface

piezoelectric

actuator

resonant

cavity

Figure 2.9: A synthetic jet actuator

plasma

Figure 2.10: A plasma synthetic jet actuator. The annular geometry has a rotational

axis of symmetry in the center of the figure, while the linear geometry is uniform in

the dimension perpendicular to the plane of the figure.

2.4 Experimental Observations of DBD Plasma Actuators

Since the DBD plasma actuator was first described by Roth et al. [15] in 2000, the field has

seen an increasing number of investigations into both the physics and applications of this device.

This section reviews some of the experiments conducted in this area, including descriptions of the

general device characteristics and the demonstration of their efficacy when used for reducing flow

detachment. It also describes specific experiments dealing with the effects of discharge geometry

and optical measurements of plasma behavior that have direct bearing on the current work. A more

comprehensive review of experimental plasma actuator research, including DC corona actuators in

addition to DBDs, was recently conducted by Moreau [46].

NASA/CR—2012-217628 25
The majority of DBD actuator research has examined single-barrier geometries similar to that

shown in figure 1.1, with both electrodes having rectangular cross sections. Most actuators are

driven by sinusoidal voltages applied to the exposed electrode, and are operated in atmospheric

pressure air. Unless stated otherwise, all of the experiments and actuators below had these characteristics. The general characteristics of this type of actuator are described in section 2.4.2. Effects

of altering the discharge geometry are dealt with specifically in section 2.4.3. The experiments described in section 2.4.1 were conducted in wind tunnels with non-zero background flow velocities,

while the remainder of section 2.4 deals with DBD operation in quiescent gas.

2.4.1 Flow Control Using DBD Actuators

Numerous experimental studies have examined the ability of the plasma actuator to reduce

drag and delay the onset of flow separation on airfoils. These studies have been conducted in wind

tunnels with various airfoil shapes, actuator positions, and flow conditions. Several of these studies

are summarized here to demonstrate the effectiveness of DBD actuators for active flow control.

Several groups have used DBD actuators to eliminate flow separation from airfoils, as shown

in figure 2.11. On the top surface of an airfoil, Roth [4] placed a series of actuators which induced

forces parallel to the background flow, and succeeded in causing flow re-attachment for some

angles of attack in flow speeds greater than 7 m/s. By placing an actuator on the leading edge

of the airfoil (which induced forces perpendicular to the top airfoil surface), Post and Corke [3]

achieved flow re-attachment for background velocities up to 30 m/s. Roupassov et al. [62] found

similar results using actuators arranged to generate forces perpendicular to the background flow

but parallel to the surface. These actuators were driven by positive voltage pulses rather than a

sinusoidal waveform. Taken together, these results suggest that the direction of the induced force

is not critical to the results. The flows likely re-attach due to a forced transition to a turbulent

boundary layer rather than to direct momentum injection into the boundary layer.

Huang et al. [5] used a DBD actuator inducing force parallel to the air flow to eliminate detachment on a cascade of turbine-like blades similar to that shown in figure 2.8. They examined

flow conditions with Reynolds numbers ranging from 104

to 105, and found the technique to be as

NASA/CR—2012-217628 26
Off On

(a) (b)

Figure 2.11: Demonstration of (a) flow separation on an airfoil and (b) flow reattachment when a plasma actuator on the leading edge is turned on. From Corke

and Post [48].

effective as a passive control technique, with the additional advantage of being able to turn off the

actuators at higher Reynolds numbers where flow separation is not problematic.

2.4.2 DBD Actuator Operation in Quiescent Gas

Much research, including that presented in chapters 4 and 5 of this document, has examined

the behavior of DBD actuators in quiescent air. Operating in air simplifies many experimental

details, allowing groups to examine a variety discharge parameters in order to optimize actuator

design. Designs are generally optimized for maximum induced force or maximum flow velocity

(which are roughly proportional to one another) in the direction parallel to the dielectric surface.

Maximizing the force should increase the effectiveness of the actuator for flow control purposes

regardless of whether the mechanism for this control is momentum injection into the boundary

layer or inducing a laminar-to-turbulent transition.

Pons et al. [63] were among the first groups to directly measure the flow induced by an actuator

operating in quiescent air. Using a stagnation probe technique they showed that the induced flow

was confined within about a centimeter of the surface. They also noted that the efficiency of

the actuator in transferring electrical energy into flow energy was on the order of 0.05%. Jukes

et al. [64] used a hot-wire anemometer with a pulsed actuator to verify that an induced velocity

was indeed due to some plasma-air interaction and not due to thermal effects alone, as had been

previously suggested.

NASA/CR—2012-217628 27
Not surprisingly, numerous researchers have demonstrated that the flows induced by DBD

actuators increase as the amplitude of the applied voltage signal increases. Enloe et al. [39] found

that both the maximum induced flow velocity and the discharge power increased proportionally

to the voltage amplitude raised to the 7/2 power, so the increased flow velocities are largely due

to increased power dissipation; the efficiencies of the actuators remain essentially unchanged.

Increasing the frequency of the applied voltage also increases the discharge power over at least the

range 200 Hz to 10 kHz [65–67]. Depending on the discharge geometry and the voltage amplitude,

however, the induced forces or flow velocities may increase linearly [67], saturate above about

1 kHz [66], or, for larger voltage amplitudes, decrease with increasing frequencies [65].

The composition of the neutral gas also plays a significant role. DBD actuator efficiency increases linearly with the fraction of oxygen in the gas [68]. Oxygen forms both positive and

negative ions more easily than nitrogen, and one or both of these properties likely contribute to the

increase. Flow measurements by Kim et al. [69] qualitatively point to the importance of negative

ion motion as a key driver of the induced force. We note that the net actuator force remains in the

same direction even as oxygen concentration is reduced to zero [68]. If negative ions play a large

role in generating the observed force, their action must be in same direction as that generated by

positive ion motion.

Measurements of induced forces and velocities provide relatively little information about the

plasma itself. To study plasma behavior, several groups have examined the light emitted from it.

We summarize their measurements here and compare them to our own in chapter 5. Enloe et al. [39,

70] focused a photomultiplier tube on narrow spatial regions to obtain temporally- and spatiallyresolved data describing the plasma light emission. Using this technique they observed that the

plasma increased in length as the applied voltage was rising, extinguished when dV/dt changed

sign, and formed anew and expanded again as the applied voltage decreased. The expansion rates

for both half cycles were identical, but increased as the applied voltage amplitude increased [70].

High-speed digital cameras have also been used to observe the plasma structure. Two groups

[71, 72] examined images with exposure times much longer than the duration of a microdischarge,

and noted that plasma structures differ significantly between the two half cycles of the applied

NASA/CR—2012-217628 28
voltage. When the applied voltage (to the exposed electrode) is increasing, the plasma appears in

narrow filaments like those in figure 2.12(a). When the applied voltage is decreasing, the plasma

appears in more diffuse jet-like structures like those seen in figure 2.12(b). Sasoh et al. [73]

recorded images with exposure times down to ten nanoseconds. They noted that the microdischarges appeared at random locations along the electrode width, and were randomly distributed

in time during periods when the applied voltage was large enough to cause ionization. They estimated the microdischarge lifetime when the applied voltage was positive-going to be on the order

of 10 ns.

(a) (b)

exposed electrode

buried electrode

exposed electrode

buried electrode

Figure 2.12: Example images of (a) filamentary and (b) jet-like microdischarges

in a single-barrier actuator

Several attempts have been made to measure the induced force or velocity as a function of

time to determine which portion of the voltage cycle. Forte et al. [74] measured the air velocity

near the downstream edge of an actuator using laser Doppler velocimetry (LDV) and found that

it oscillated at the applied voltage frequency. This indicates that the two halves of the applied

voltage cycle function differently from one another, but due to an unknown transit time of the air

between the plasma and the measurement location the relative actions of each half cycle could not

be determined.

Enloe et al. [68] used a laser-deflection technique to observe fluctuations in gas density. Their

measurements indicated that when the applied voltage was decreasing, a region of higher density

air formed near the plasma and moved downstream as the plasma quenched, leaving behind a

NASA/CR—2012-217628 29
region of lower density. This lower density slowly disappeared over the course of the positivegoing half cycle. The same group used a pendulum system to determine that the contribution of the

negative-going half cycle to the induced force dominates that of the positive-going half cycle by a

ratio as large as 97 to 3 [71]. Phase-locked particle image velocimetry measurements by Kim et al.

[69] qualitatively corroborated the importance of the negative-going half cycle. The dominance of

the negative-going half cycle, combined with the influence of oxygen concentration on the force

discussed above, suggest that negative ion acceleration when the applied voltage is decreasing

plays a central role in actuator performance. During the negative-going half cycle, the electric

field is directed towards the exposed electrode, so the EHD force described by equation (2.9) will

be in the measured direction only when acting on electrons and negative ions.

More recently, Enloe et al. [75] found that the induced force of a DBD actuator oscillates over

the course of a single half cycle of the applied voltage. When the voltage was positive-going the

force was initially positive, then became negative; the same oscillation was observed when the

voltage is negative-going. While the gross positive forces were comparable for both half cycles,

the negative force was much larger when the applied voltage was positive-going. Both half cycles

induced positive net forces, but the net force was much larger during the negative-going half cycle,

which is consistent with the results described above. The authors attributed the negative forces to

drag between the induced air flow and the dielectric surface [75].

All of these studies of single-barrier actuators with rectangular cross-section electrodes and

sinusoidal applied voltages point to several important, repeatable results. The efficiencies of the

actuators in transferring electrical energy into flow energy are quite small. While the total power

transferred can generally be increased by raising the amplitude or frequency of the applied voltage,

the efficiency is only strongly affected by changes in the electrode geometry (discussed below)

and the gas environment. That the removal of oxygen from the gas greatly decreases the actuator

efficiency strongly suggests that the formation of negative ions plays a large role in momentum

transfer. This idea is reinforced by experiments showing that the great majority of this momentum

transfer from plasma to gas occurs when the applied voltage is decreasing.

NASA/CR—2012-217628 30
2.4.3 Effects of Actuator Geometry

When used for active flow control, the gas environment of a DBD actuator is set by external

conditions. Improvements to actuator performance can be achieved by adjusting either the driving

voltage or the discharge geometry. Previous work in optimizing the device geometry is summarized here. The dimensions that have been adjusted in previous research are shown schematically

in figure 2.13

electrode gap

buried electrode

length

exposed electrode thickness

dielectric thickness

Figure 2.13: Single-barrier DBD actuator dimensions examined in previous research

Several variations to the exposed electrode diameter shape and location have been examined.

Enloe et al. [39] tested actuators with a range of exposed electrode thicknesses, using electrodes

with both rectangular and cylindrical cross sections. They found a linear increase in the actuator

efficiency (force / power) and no change in discharge power as the electrode thickness was reduced,

although we re-examine these findings in section 4.2.3. An exposed electrode constructed from a

thin mesh was used by Abe et al. [67], and yielded somewhat increased forces compared to rectangular cross-section exposed electrodes for the same applied voltages. Gibson et al. [76] used a

rectangular exposed electrode of constant thickness, but varied how deeply embedded it was in the

dielectric. They observed no plasma when the top of exposed electrode extended 0.5 mm or more

above the dielectric surface, and the largest induced force when the top of the exposed electrode

was 0.5 mm below the dielectric surface. All of their measured forces, however, were significantly

smaller than a “standard” actuator with the entire exposed electrode above the dielectric surface.

Increasing the length of the buried electrode increases the maximum observed flow velocity,

but this effect saturates above a certain value [66]. That value depends on the amplitude of the

applied voltage signal [39]. It is likely that the length of the buried electrode only affects actuator

NASA/CR—2012-217628 31
performance when it is short enough to restrict the expansion of the plasma along the dielectric

surface above it.

Forte et al. [66] examined actuators with different electrode gaps, finding a maximum in the

induced flow velocity in between 3 and 7 mm, although the 2 mm dielectric thickness used in those

tests was large compared to most other groups. The same group tested several different dielectric

thicknesses. For a constant applied voltage, actuators with thinner dielectric layers induced faster

air flows, but this dependence was removed when normalizing to the amount of power consumed

by each actuator [66].

2.5 Computational Modeling of DBD Plasma Actuators

The modeling of plasma actuators discussed in the literature varies between focusing entirely

on the aerodynamic effects of the actuator while ignoring the details of the plasma [e.g. 77] to detailed simulations of the plasma generation yielding only qualitative information about the induced

flow [e.g. 13]. Our computational model falls on the latter end of this continuum, so while a variety of approaches to actuator modeling are summarized in section 2.5.1, sections 2.5.2 and 2.5.3

deal exclusively with fluid models of only the plasma. As with the description of previous experimental work, the actuators examined here are exclusively single-barrier actuators with rectangular

cross-section exposed and buried electrodes. The described voltages are applied to the exposed

electrodes, while the buried electrodes are grounded. All of the described models include only two

spatial dimensions.

2.5.1 Approaches to Plasma Modeling

Computational studies of DBD actuators span a wide range of complexities, from highly simplified models used to couple forces into computational fluid dynamics (CFD) codes to firstprinciples models of the plasma dynamics. This range is briefly summarized here, roughly from

least to most focus on the plasma itself. More detail is given on the codes focused on modeling the

plasma dynamics, as that is the focus of the model used in this work.

NASA/CR—2012-217628 32
Among the most simplified plasma models are those like the one employed by Hall et al. [77],

in which a doublet, or a collocated flow source and sink, replaces the actuator in a CFD model.

An earlier model by Shyy et al. [78] approximated the plasma as a constant body force within

a triangular region defined by the region of strongest electric field, and used coupled this force

into CFD simulations. A lumped circuit model of the plasma by Orlov and Corke [79] captured

some of the temporal characteristics of the plasma, and has recently been applied as part of CFD

calculation [80]. The CFD calculations in all of these studies yielded flow patterns in qualitative

agreement with experimental velocity measurements.

Limited attempts have been made to use particle-in-cell (PIC) codes to model surface DBD

plasma dynamics. Font [81] modeled an actuator operating in pure nitrogen with constant biases

applied to the exposed electrode. He obtained microdischarge structures in some ways similar to

those described in section 2.5.2, with quick discharges forming and extinguishing when the voltage

on the exposed electrode was negative and longer discharges appearing when it was positive. The

plasma size appeared to be limited by the short buried electrode length chosen. More recently

Huerta and Ludeking [82] used a commercial PIC code to simulate a similar system, but were

limited to the initial stages of the ionization avalanche due to restrictions on the number of particles

allowed.

Several groups have used the collisional approximations described in section 2.1 to model the

DBD actuator plasma. These models solve the following system of equations. First Poisson’s

equation,

∇ · (∇φ) = −e (n+ − n−), (2.17)

is solved for the electrostatic potential φ. Here  is the permittivity of the medium, while n+

and n− are the total number densities of positively- and negatively-charged particles, respectively.

Continuity equations are then solved for each of the considered particle species:

∂nα

∂t = −∇ · Γα +

XSα −XRα, (2.18)

NASA/CR—2012-217628 33
where S and R are source and sink rates, respectively, for each of the modeled processes that

introduce or remove particles from the system. Drift-diffusion approximations, described by equation (2.5), are used to describe the particle fluxes. The models differ in which particle species are

considered, what source and sink processes for each species are included in the models, and the

numerical methods used to solve these equations.

Singh and Roy [83] used a finite element scheme to solve the system of equations for a singlebarrier DBD operating in helium, considering field-dependent ionization but no other sources of

electrons. By coupling the calculated EHD forces to a CFD code they modeled an induced flow

confined within a few millimeters of the actuator surface [84], similar to experimental data. In

the CFD computation, this force was sufficient to eliminate separation in background flows of

10 m/s [85]. The plasma model, however, indicated significant space charge density above the

exposed electrode and past the downstream end of the buried electrode [83, 84]. In contrast,

experiments have observed plasma only on the downstream edge of the exposed electrode, and

the plasma never extends significantly beyond the edge of the buried electrode [39]. Extending

the model to include the major positive and negative ion species present in air plasmas did not

qualitatively change their results [86].

Two groups have solved the above same system of equations using finite difference techniques,

although the exact methods have evolved in order to adapt the models to specific problems of interest. The groups, based out of the Universite Paul Sabatier in France and Princeton University in  ́

the United States, have both studied single-barrier actuators with thin (one grid cell thick) exposed

and buried electrodes. The sources and sinks included by the two groups are electron-impact ionization, recombination, secondary-electron emission, and dielectric charging [13, 87]. Recently

the Princeton group has incorporated photoionization as well [88]. Both groups use only limited

plasma chemistry. Initial studies by the French group included only electrons and a single species

of positive ions [13, 14, 89], while more recent work [90] and all of the Princeton results [87, 91]

add a single negative-ion species. The published work of the French group was used as the basis

for the model used in our research. The differences between their work and our own are noted

along with a detailed description of our model in chapter 6.

NASA/CR—2012-217628 34
Both groups observed the formation of short duration (5–100 ns) microdischarge events. If the

applied voltages increase or decrease with time, numerous microdischarges may occur, such as the

repeated nanosecond-width peaks in the electron density in figure 2.14. The properties of individual microdischarge events simulated by these two groups are described in section 2.5.2, while the

new phenomena that arise when repeated microdischarges occur are described in section 2.5.3.

30 35 40 45 50

Time (μs)

Electron density (a.u.)

−10

−9

−8

−7

−6

−5

Voltage (kV)

voltage n

e

Figure 2.14: Simulated microdischarges for a single-barrier DBD actuator with a

linearly decreasing applied voltage

2.5.2 Characteristics of Single Microdischarges

The physics of microdischarges depend strongly on the direction of the electric field in the gas

region. This field direction depends on the potential difference between the exposed electrode and

the dielectric surface. We consider the cases when the exposed electrode is at a higher and lower

potential than the dielectric surface separately below.

When the potential of the exposed electrode is higher than that of the dielectric surface, electrons and negative ions are accelerated into exposed electrode, while positive ions are pushed

towards the dielectric. With this polarity, two regimes are possible: a corona-like regime at lower

voltages and a streamer-like regime at higher voltages. These “positive” microdischarges are described here. Initially, electron avalanches occurs in a region of strong electric field, generally

next to the edge of the exposed electrode. The field strength, which is partially determined by

the electrode geometry, and the level of background ionization affect the density of the resulting

NASA/CR—2012-217628 35
plasma [88]. Positive ion flux from this plasma to the dielectric surface will supply secondary

electrons to the plasma, with the rate of secondary electron flux controlled by the incident ion flux,

and therefore the initial plasma density.

If the plasma generated through the initial avalanches is insufficiently dense, the secondary

electron flux to the plasma will be too small to lead to further increases in the plasma density.

In this case, the microdischarge behaves like a positive Townsend corona [14, 87, 89], and is

sometimes referred to as “corona-like”. Electrons drawn in from the background gain enough

energy to cause avalanches only very near the exposed electrode. The newly-born positive ions

are slowly pushed away from the exposed electrode. The net charge densities are not large enough

to significantly modify the potential contours, and the ion flux charges up the dielectric surface

slowly compared to the streamer-like microdischarges.

If the plasma is sufficiently dense, the secondary electrons from the dielectric surface will more

than replace electrons lost to the exposed electrode. These electrons cause new avalanches, further

increasing the plasma density. The charging of the dielectric surface by positive ion bombardment

and electron emission shifts the region of strongest electric field away from the exposed electrode.

The cycle of secondary electron emission, ionization, and dielectric charging repeats for a section

of dielectric farther from the exposed electrode. The result is a dense region of plasma, depicted in

figure 2.15, that propagates along the dielectric surface away from the exposed electrode [13]. It

continues propagating until the electric field within it is no longer sufficient to support significant

ionization. The charging of the dielectric surface by the microdischarge results in the potential

on that surface being raised to approximately the potential on the exposed electrode. The dense,

moving plasma shares many characteristics with the streamer discharge mechanism discussed in

section 2.2.1, with the “uncovered” electrons provided in this case by secondary emission. Because

of this similarity, these microdischarges are often simply referred to as streamers.

In 2-dimensional fluid models, the natural background electron density in air of approximately

107 m−3is insufficient lead to streamer formation at voltages comparable to those in experiments

unless photoionization is included in the model [88]. Since modeling photon transport is computationally expensive, most existing models avoid it by using elevated initial densities [e.g. 13] or

NASA/CR—2012-217628 36
Figure 2.15: Simulated streamer-like plasma formed with +1200 V applied to the

exposed electrode. The electrodes are shown in black, with the exposed electrode at

y = 50 μm and the buried electrode at y = 0. The dashed line shows the dielectric

surface and the contours represent equipotential surfaces.

by providing additional electrons by specifying a minimum background plasma density [e.g. 87].

Recently Likhanskii et al. [88] included an approximate model for photoionization. They found

that it both removed the necessity for an elevated background plasma density and increased the

propagation velocity of the streamer front by a factor of two to three.

When the potential of the exposed electrode is lower than that of the dielectric surface, positive ions are pulled into the exposed electrode while electrons and negative ions are accelerated to

the dielectric surface. Due to the differences in particle motion, the behaviors of these microdischarges are quite different from those described above. The “negative” microdischarge proceeds

as described here.

Initially, as with the positive microdischarge, electron avalanches occur in a region of strong

electric field. Again, this is usually near the edge of the exposed electrode. Positive ion bombardment of the exposed electrode provides secondary electrons which cause new avalanches,

increasing the plasma density. Unlike in the streamer-like microdischarge, the ion density remains

NASA/CR—2012-217628 37
highest in a region near the exposed electrode throughout the discharge [13]. This region bears

some resemblance to the negative Townsend and glow corona regimes described in section 2.2.1.

No photoionization is required for the negative microdischarge to occur, even with small levels

of initial plasma density [88]. This is likely due to the ready supply of secondary electrons from

positive-ion bombardment of the exposed electrode.

Electrons are pushed out of the dense plasma region by the electric field. This leaves behind

a cathode sheath-like structure, which supports a significant voltage drop across it. The electrons

stream towards the dielectric, either reaching it and charging its surface or attaching to an electronegative particle and forming a negative ion on the way [87]. Some electrons may start new

avalanches, although these will be relatively weak due to the lowered electric field strength away

from the exposed electrode. The charging of the dielectric adjusts the electric field so that it pushes

electrons (and negative ions) towards a location further away from the exposed electrode, as shown

in figure 2.16. Charge deposition on the dielectric surface reduces the electric field only to just below the level needed to support electron avalanches, rather than completely shielding it as is the

case for the positive microdischarge. Due to the high mobility of electrons, the lifetime of the

negative microdischarge is much shorter than that of the positive microdischarge.

In both positive and negative microdischarges, the plasma dynamics are largely controlled by

secondary electron emission caused by incident positive ion flux. With a positive voltage applied

to the exposed electrode, the positive ion flux is directed at the dielectric surface. Due to surface

charging, the point where this flux strikes the surface and causes electron emission changes with

time, and we observe a propagating plasma region. With a negative voltage, positive ions are

pullled into the exposed electrode. Since the electrode’s voltage is held constant, the source of

secondary electrons does not move and the region of dense plasma remains stationary.

2.5.3 Repeated Microdischarges

DBDs operated in a filamentary mode have many microdischarges during each half-cycle of the

applied voltage, as described in section 2.2.2. To accurately model the plasma, then, simulations

must examine the interactions of these events with each other. Since after each microdischarge

NASA/CR—2012-217628 38
Figure 2.16: Simulated plasma formed with -1200 V applied to the exposed electrode. The electrodes are shown in black, with the exposed electrode at y = 50 μm

and the buried electrode at y = 0. The dashed line shows the dielectric surface and

the contours represent equipotential surfaces.

the applied voltage is partially or entirely shielded by charge deposition on the dielectric surface,

generating repeated discharges requires a time-varying applied voltage. In this work, we use the

term “positive-going” to refer to a situation in which the voltage drop from the exposed electrode to

the buried electrode is increasing and positive microdischarges occur. We use the term “negativegoing” to refer to a situation in which the voltage drop from the exposed electrode to the buried

electrode is decreasing and negative microdischarges occur. Similar to section 2.5.2, we describe

the plasma with positive- and negative-going voltages separately.

With a positive-going voltage, actuators move through both corona-like and streamer-like

phases. Initially the voltage is relatively small and cannot support a streamer-like microdischarge

discharge. As it increases, eventually a threshold is reached and a streamer-like microdischarge

occurs [14, 89]. The rate at which the streamers occur is set by the ramp rate of the applied voltage. Generally the duration of a streamer-like event is on the order of 100 ns, while the coronalike phase persists for most of the time in between streamer-like microdischarges. Because of

NASA/CR—2012-217628 39
the long duration of the corona-like phase relative to the streamer-like phase, the majority of the

time-averaged EHD force induced by an actuator with a positive-going voltage occurs during the

corona-like phase, even though the absolute force is larger during the streamer-like phase [14].

Microdischarges with a positive-going voltage are relatively independent of one another. Each

streamer raises the potential on the dielectric surface to nearly the potential on the exposed electrode; the entire voltage drop needed to drive subsequent microdischarges must be supplied by

the increase in applied voltage since the last streamer. Electrons for the corona-like phase are,

however, supplied by free electrons remaining from the previous microdischarge, so increasing

the voltage ramp rate increases the initial electron density since the plasma from the previous microdischarge has had less time to decay. Consequently, the rate at which streamers occur increases

somewhat faster than linearly with the voltage ramp rate [14].

Since secondary electron emission plays a large role in streamer formation and propagation,

one might expect that adjusting the secondary emission coefficient of the dielectric surface would

have large effects on the discharge. Unfer et al. [92], however, noted that a reduction in this

coefficient in the model by a factor of 5000 had only modest effects, reducing the number of

streamer microdischarges in one half cycle of a sinusoidal applied voltage from 5 to 3 and the

total induced EHD force by about 10%. This finding is favorable for actuator modeling, since

secondary emission coefficients for dielectric surfaces are not well characterized and may depend

on the details of surface condition and other factors.

Qualitatively, microdischarges with a positive-going voltage are also independent of the gas

chemistry, although some quantitative effects exist. The addition of a negative ion species to the

model reduces the electron density during the decay phase immediately following a streamer [90].

In addition, the motion of negative ions towards the exposed electrode during the corona-like phase

cause a small EHD force in the opposite direction as the prevailing force [87]. The inclusion of

electron detachment from negative ions results in an extended corona-like phase and an accompanying increase in the time-averaged induced force [92].

The behavior of the plasma with a negative-going applied voltage depends strongly on whether

the model allows for the formation of negative ions via electron attachment (and the presence of an

NASA/CR—2012-217628 40
electronegative gas). We first summarize the plasma behavior in pure nitrogen (an electropositive

gas) before describing the changes that occur when negative ions are included.

The negative microdischarge described above depends partially on the background positive ion

concentration to start the discharge process, via secondary emission of electrons from the exposed

electrode. Ion motion to the exposed electrode is slow, due to the low mobility of ions, so the

positive ion density decays much more slowly after a microdischarge than does the electron density

for a positive microdischarge. In addition, in a negative microdischarge the charge deposition on

the dielectric surface only partially shields out the applied voltage. As a result, for the same rateof-change of applied voltage and a pure nitrogen background, negative microdischarges occur on

the order of 30 times more frequently than do positive streamers [14]. Beyond the use of the

positive ion density from the previous microdischarge to start a new one, each microdischarge in

nitrogen is relatively unaffected by the others.

When an electronegative gas (in these simulations, oxygen) is present, the large electron fluxes

through the regions of smaller electric fields away from the exposed electrode promote the formation of negative ions. For more information about attachment coefficients, see section 6.1. Because

of this process, ion densities during and after negative microdischarges are considerably larger than

in positive microdischarges.

Unlike in an electropositive gas, negative microdischarges in an electronegative gas interact

strongly with one another. Since the mobility of negative ions is much lower than that of electrons,

some of the negative ions created in one microdischarge will not reach the dielectric, and thus are

still present, by the start of the next microdischarge. Over the course of several microdischarges a

negative ion “cloud” builds up [87, 90]. The cloud is not solely composed of negative ions; positive

ions are also present, but in smaller concentrations. The action of the electric field on the negative

ion cloud in between breakdown events is the dominant cause of the EHD force with a negativegoing voltage. This force is in the same direction as the force observed for a positive-going voltage.

The contributions of the positive-going and negative-going forces to the time-averaged force are

comparable to one another for the discharge conditions that have been modeled [93].

NASA/CR—2012-217628 41
A few direct comparisons have been made between the plasma-focused simulations and experimental measurements. Orlov et al. [72] found qualitative agreement between the structure

of simulated microdischarges in nitrogen and optical imaging of plasma structures in air using

exposure times much longer than an individual microdischarge. We have performed similar comparisons, which are described in chapter 5. Boeuf et al. [93] simulated actuators with a variety of

different effective frequencies and voltage amplitudes, and found approximately the same trends

in the induced force as seen experimentally by Roth and Dai [65].

NASA/CR—2012-217628 42
Chapter 3

Experimental Methods

We have conducted several experimental measurements to characterize the behavior of DBD

plasma actuators with a variety of discharge geometries. These actuators are described in more

detail in section 3.1. In order to better determine the ideal actuator design to maximize the induced

force, we have measured the force. To ensure the accuracy of these results, we employed two

techniques: direct measurement using an electronic balance and a stagnation probe method. Highspeed imaging experiments were performed in order to observe the plasma dynamics on short time

scales not accessible by the force measurements.

All the experimental measurements in this work were conducted in atmospheric pressure air.

The air was simply room air in our laboratory. No attempts were made to control for daily or

seasonal fluctuations in air temperature, humidity, etc. In addition, as described in section 3.1.2 all

actuators were hand-made and undoubtedly varied slightly in quality. Based on our experimental

results, the effects of these variations were small relative to the effects of changing the actuator

geometry or the applied voltage. Since the exact level of these fluctuations are unknown, however,

they are not included in the uncertainty levels presented with the results of chapters 4 and 5.

3.1 DBD Actuator Description

We have studied a range of DBD plasma actuators, which are described in this section. The

geometric variations we have examined experimentally are described in section 3.1.1 along with

the reason each variation was selected. These actuators were all constructed by hand. The materials

NASA/CR—2012-217628 43
and methods used to build them are described in section 3.1.2. Finally, the circuitry and the applied

voltages we used to drive the actuators are described in section 3.1.3.

3.1.1 Actuator Geometries

In the experimental portion of this work, we have primarily examined single-barrier DBD actuators, while also performing some measurements on double-barrier actuators. We consider several

variations to the geometry of the single-barrier actuators in order to determine which parameters

are most important. Both single- and double-barrier geometries have been shown to induce air flow

in the nearby neutral gas [70]. All of the figures in this section are drawn such that the induced air

flow moves from left to right.

As described in section 2.2.2, single-barrier surface discharges include two electrodes, one of

which is insulated. We refer to the insulated electrode as the buried electrode, since it is placed

beneath a layer of dielectric, and the non-insulated electrode as the exposed electrode. In all the

single-barrier actuators used in this work, the dielectric thickness was 0.25 mm and the buried

electrode length was 15 mm. The actuators spanned 60 mm in the direction perpendicular to the

schematics of figure 3.1.

substrate

epoxy

1 mm

dielectric

15 mm

exposed electrode buried electrode

diameter

(a)

substrate

epoxy

1 mm

dielectric

15 mm

exposed electrode buried electrode

length

thickness

(b)

Figure 3.1: Schematics of single-barrier actuators with (a) a cylindrical exposed

electrode and (b) a rectangular cross-section exposed electrode

The experimental measurements in this work primarily examine the differences in actuator

behavior between devices with exposed electrodes of various sizes, shapes, and materials. Some

tests use cylindrical exposed electrodes, as shown in figure 3.1(a). The results of force measurements using this actuator design are described in sections 4.1, 4.2.1, 4.2.3, and 4.4. We have also

NASA/CR—2012-217628 44
studied single-barrier actuators with exposed electrodes of rectangular cross-section, shown in figure 3.1(b). The results of force measurements using these actuators are included in sections 4.2.2,

4.2.3, and 4.2.4. For both of these designs, the horizontal gap between the rightmost edge of the

exposed electrode and the leftmost edge of the buried electrode was 1 mm, as shown in figure 3.1.

Optical measurements of both designs are described in chapter 5.

To aid in examining the differences between actuators with rectangular and cylindrical exposed

electrodes, we have used actuators with several cylindrical electrodes placed side-by-side, as pictured in figure 3.2. This design allowed us to maintain a curved electrode edge while expanding

the effective length. Adjusting the number of cylinders controls this length. Cylinders were always

added or removed from the left-hand edge so that no gaps existed between the cylinders, and the

electrode gap was held constant at 1 mm. Force measurements of actuators with this design are

discussed in section 4.2.4.

substrate

epoxy

1 mm

dielectric

15 mm

exposed electrodes buried electrode

...

N, ..., 2, 1

Figure 3.2: Single-barrier actuator with side-by-side cylindrical exposed electrodes

We have also examined double-barrier DBD actuators, in which both electrodes are insulated.

To provide the geometric asymmetry required to induce a net force, all of the double-barrier actuators used one narrow electrode and one wide electrode. Both are placed immediately below

the dielectric layer. The double-barrier geometry is shown schematically in figure 3.3. Similar to

the single-barrier geometry, the dielectric thickness was 0.25 mm, the wide electrode length was

15 mm, and the devices spanned 60 mm in the direction perpendicular to the figure. The horizontal

gap betweeen the rightmost edge of narrow electrode and the leftmost edge of the wide electrode

was set to be 2 mm, as shown in figure 3.3. The gap was larger than the electrode gap in the

single-barrier geometry due to the high likelihood of arc formation through the epoxy observed in

double-barrier actuators with 1 mm electrode gaps.

NASA/CR—2012-217628 45
substrate

epoxy

2 mm

dielectric

15 mm

narrow electrode wide electrode

diameter

Figure 3.3: Schematic of a double-barrier actuator

In general, double-barrier actuators require larger applied voltages to generate plasma and

induce smaller forces than single-barrier actuators [94]. We include them in this study for a variety

of reasons. The double-barrier geometry has fewer possible sources of irregularities compared to

single-barrier discharges. Double-barrier actuators have no features above the surface to impede

the induced air flow, and the electrodes are not in physical contact with the plasma so their material

characteristics cannot affect the discharge. Such actuators may also find practical applications, as

the lack of an exposed electrode means they will be less prone to arcing to nearby materials and not

susceptible to degradation via dielectric deposition on an electrode. Most importantly for our work,

double-barrier actuators provide an additional system in which we can compare our experimental

data to the results of the computational model.

3.1.2 Actuator Construction

For all actuators used in this work, the dielectric was made from a 0.25 mm thick polyester

film. The relative dielectric constant of the film, d, was given by the manufacturer as 3.3. In

order to avoid plasma formation underneath the dielectric layer, we filled this space with Stycast

2651-40 epoxy, which had a similar relative dielectric constant of 3.8. A 10 mm thick acrylic layer

placed below the epoxy was used to mount the actuators, and is not believed to have played any

role in the discharge. The buried and wide electrodes were made from copper foil tape 0.04 mm

thick with a 0.05 mm acrylic adhesive. The tape was mounted to the dielectric film, so the adhesive

was between the film and the buried or wide electrodes in figures 3.1–3.3.

Cylindrical exposed and narrow electrodes in single- and double-barrier actuators were constructed of copper (Cu), tungsten (W), or stainless steel (SS). They consisted either of straightened

NASA/CR—2012-217628 46
wires (for smaller diameters) or straight rods (for larger diameters). Rectangular cross-section exposed electrodes in single-barrier actuators were made from either copper or stainless steel. The

0.04 mm thick copper electrodes used the same foil tape as the buried electrodes. Images of singlebarrier actuators with cylindrical and rectangular exposed electrodes are shown in figure 3.4.

Exposed electrode Exposed electrode

Buried

electrodes

Epoxy holding

down exposed

electrode ends

Acrylic backing

Dielectric film

Electrical

connection

(a) (b)

Figure 3.4: Photographs of single-barrier actuators with (a) a cylindrical exposed

electrode and (b) with a rectangular cross-section exposed electrode

We constructed actuators from the dielectric layer down. For a single-barrier actuator, the

buried electrode was affixed to the bottom of the dielectric. Several layers of the insulating epoxy

were then applied to build up thickness before attaching the acrylic backing. After flipping the

actuator over, an exposed electrode was placed on top of the dielectric film. The exposed electrode

was held in place with tape while applying epoxy to both ends. That epoxy is noted in figure 3.4.

The tape was removed prior to use. Construction of double-barrier actuators proceeeded similarly,

except that the narrow electrode was epoxied in place immediately after the wide electrode was

affixed. A step-by-step procedure describing actuator construction is included in appendix A.

3.1.3 Applied Voltage

We created the applied voltage waveforms using an arbitrary signal generator and routed the

output to a Trek 20/20B high voltage amplifier. The output of the amplifier was passed through

a two-stage low-pass filter to remove megahertz-frequency noise on the signal before being connected to the exposed electrode (for single barrier actuators) or the narrow electrode (for double

barrier actuators). The buried or wide electrodes were always grounded. A circuit diagram describing the connections is shown in figure 3.5. We used the resistor or capacitor in the ground

NASA/CR—2012-217628 47
Signal

generator

Discharge

HV cable

1 kΩ

0.12 μF

High voltage

amplifier

+

- Oscilloscope

3 kΩ 3 kΩ

Low-pass filter

50 pF50 pF

Over-voltage

protection

Figure 3.5: Discharge circuit diagram

path of the discharge to measure the current or accumulated charge, respectively, as described in

section 3.2.3.

In this study all force, flow, power, and optical emission measurements were made on actuators

driven by symmetric triangle waveforms oscillating at 1 kHz, except for the stability and uniformity experiments of section 4.1 which used 500 Hz sine waves. The triangular voltage shape

was chosen to simplify comparisons between the experimental measurements and the simulations.

1 kHz was the highest frequency that could be achieved by the HV amplifier for the range of voltage amplitudes used. With symmetric triangle voltages, plasma ignition occurred for single-barrier

discharges when the amplitude was between about 3 and 4 kV. For double-barrier actuators, ignition generally occurred between 10 and 11 kV amplitude signals. At voltage amplitudes above

13 kV, the dielectric film would degrade and allow arc formation, destroying the actuator.

3.2 Force and Power Measurements

Prior research has used several methods to measure the action of DBD actuators on the neutral

gas. One such method consists of simply mounting the actuator being tested on an electronic balance and directly measuring the force [e.g. 39]. A separate technique involves the measurement

of the stagnation pressure of the induced flow velocity [e.g. 95]. We have applied both of the balance and the stagnation pressure techniques, with the details described in sections 3.2.1 and 3.2.2

NASA/CR—2012-217628 48
respectively. Often in the literature actuator forces are reported in terms of the force per unit input

power. To make comparisons to these data, we also determined the input power by the method

described in section 3.2.3. All of the measurements described in this section are time-averaged,

and consider only forces parallel to the dielectric surface.

3.2.1 Electronic Balance

We have measured the forces induced by our plasma actuators using an Ohaus model PA313

electronic balance with 1 mg (equivalent to 10−5 N) resolution. While the sensitivity of the scale

is less than that allowed by the stagnation probe technique described in section 3.2.2, it provides

a direct measurement of the induced forces. The force measurement was made by mounting an

actuator on a stand with the air flow directed upwards. The balance then measures the downward

reaction force on the actuator as a positive force.

Figure 3.6: The shielded balance and actuator stand used for direct force measurements

To eliminate RF interference between the applied high voltage and the internal balance electronics, we covered the balance using a grounded copper mesh with a hole above the balance pan.

An insulating platform, on which the actuator stand stood, fit through this hole. The balance, stand,

and an actuator are shown in figure 3.6. Electrical connections were made using light 32 AWG

wire to eliminate additional forces due to sagging wires. The high voltage wire was held several

inches above the copper mesh to eliminate electrostatic attraction between the two conductors. An

NASA/CR—2012-217628 49
acrylic box 25 cm × 61 cm × 36 cm housed the balance, shielding, and actuator. The box blocked

ambient air currents from influencing the measurement and prevented the ozone generated during

DBD operation from escaping into the lab air.

Time variations in the induced forces could not be directly observed due to the tendency of the

balance reading to slowly drift during actuator operation. To eliminate the effects of this drift, after

conditioning (see section 4.1) and immediately prior to recording data the high voltage was turned

off, the balance zeroed, and the high voltage quickly turned back on.

In order to directly compare our results from the balance measurements with the stagnation

probe data, we converted the former into forces per unit length by dividing by the width of buried

or wide electrode (60 mm). This is a good, but not exact, estimate of the plasma width, which

varies slightly from actuator to actuator. Based on visual observation we estimate the plasma

width variation to be on the order of ±5%, leading to uncertainties comparable to the uncertainty

in the force measurement itself.

3.2.2 Stagnation Probe

We have measured induced gas velocities using a small movable stagnation probe, shown in

figure 3.7. To aid in positioning, eliminate ambient air currents, and prevent ozone from entering the lab air, stagnation probe measurements were conducted inside a sealed cylindrical metal

chamber. The chamber was 17 cm tall and 30 cm in diameter, with the actuator located near the

center of the chamber. The probe itself consisted of a glass capillary tube with a 0.4 mm inner

diameter and 0.2 mm wall thickness. Tube pressure was determined using a Dwyer model 607-01

differential pressure transmitter, with the low pressure side connected to a port on the bottom of

the chamber. This measurement technique is similar to that used successfully by Pons et al. [63].

In order to compare the stagnation probe and balance measurements, we converted the measured pressures into induced forces using the method derived below. The derivation is similar to

that performed by Baughn et al. [12], although our system has no background flow velocity. To analyze the system, we consider a rectangular control volume on the actuator surface which includes

the entire plasma volume, such as that shown in figure 3.7. The momentum balance, described by

NASA/CR—2012-217628 50
Figure 3.7: Coordinate system, with the control volume used in the velocity-toforce conversion indicated by a dashed rectangle (left), and an image of the stagnation probe near a single barrier actuator (right)

equation (2.14), can be written for the x direction of this system as

Fplasma

Lz

+

Fshear

Lz

= ρ

Z

right

u

2

x dy

| {z }

I

+ ρ

Z

top

uxuy dx

| {z }

II

− ρ

Z

left

u

2

x dy

| {z }

III

+

Z

right

p dy

| {z }

IV

−

Z

left

p dy

| {z }

V

. (3.1)

The integral subscripts indicate over which side of the control volume the integral is performed,

and Lz is a unit length in the z direction. The only body force considered here is the electrohydrodynamic force of the plasma on the air, Fplasma. The included surface forces are static pressure

forces on left and right surfaces and shear force, Fshear, acting between the dielectric layer and the

air flow. We note that since the shear force acts to slow the +x-directed flow, it will always be

negative.

For our initial analysis, we simplify equation (3.1) with several assumptions that limit the

data required to infer each actuator’s force. The effects of these assumptions are investigated in

section 4.4. First, we assume the static pressure of the gas is constant everywhere and equal to the

reference pressure at the chamber bottom:

p = pref.

As the chamber was open to room air prior to being sealed off to reduce measurement noise, pref

is approximately equal to atmospheric pressure for all measurements. The assumption of constant

static pressure allows us to neglect terms IV and V in equation (3.1), as they will cancel. It also

permits a simple conversion from the measured stagnation pressure to a velocity by rearranging

equation (2.15):

ux ≈

q

2(p0 − pref)/ρg. (3.2)

NASA/CR—2012-217628 51
Here p0 is the measured stagnation pressure and ρg is equal to 1.17 kg/m3

, the mass density of

air. Previous measurements by Enloe et al. [68] indicated that density fluctuations near plasma

actuators are generally smaller than 2% of the background, so the density may be reasonably

assumed to be constant. The assumption of a constant static pressure was implicitly made by Pons

et al. [63] in their measurements near a single-barrier DBD actuator, as they used equation (3.2) to

determine the velocity.

In addition to the assumption of constant static pressure, in our initial analysis we neglect the

momentum flux into the control volume through the left and top edges. With this assumption,

terms II and III in equation (3.1) can be neglected:

Z

top

uxuy dx ≈ 0 and Z

left

u

2

x dy ≈ 0.

In section 4.4 we show that term III makes a significant, although not dominant, contribution to

the total force.

Once these assumptions are made, equation (3.1) simplifies to

Fplasma

Lz

+

Fshear

Lz

≈ ρ

Z

right

u

2

x dy. (3.3)

In practice, we fit an analytic function to the observed velocities and use that in equation (3.3). We

used a skewed Gaussian,

ufit = u0

√

y exp "−



y − y0

L0

2

#

,

where u0, y0, and L0 are free parameters. This function was chosen both due to its excellent

empirical fits to the observed data and because it always satisfies the no-slip condition of zero flow

velocity at a material surface. An example fit is shown in figure 3.8.

Equation (3.3) is equivalent to the calculations by Baughn et al. [12] with the background

velocity reduced to zero. That study measured the absolute plasma force and so also subtracted

a shear force. In this work we compare the calculated forces to those measured directly with an

electronic balance; since the balance measurements inherently include the shear force, we leave it

on the left-hand side of (3.3). The stagnation pressures used for force estimates in chapter 4 were

measured at x = 8 mm. Our calculations therefore only include the shear force up to 8 mm, while

NASA/CR—2012-217628 52
0 2 4 6 8

0

0.5

1

1.5

y (mm)

x velocity (m/s)

data

fit

Figure 3.8: Typical velocity data and skewed Gaussian fit

the balance measurements include the shear force out to the edge of the dielectric at approximately

16 mm. We estimate the effect of this discrepancy in section 4.4.

3.2.3 Discharge Power

To determine the electrical power dissipated by the plasma actuators, we placed a 0.12 μF

capacitor in the ground path of the actuator, as shown in figure 3.5. Provided its capacitance is

much larger than the capacitance of the actuators (which were generally smaller than 10 pF), the

circuit characteristics are unaffected by its presence. The charge deposited in the actuator is equal

to that on the measurement capacitor, and can be determined by measuring the voltage across

the capacitor. Typical data are shown in figure 3.9(a). Plotting the charge against the applied

voltage yields a parallelogram-like shape, sometimes called a Lissajous figure, such as the one in

figure 3.9(b).

The average actuator current can be found by taking the time derivative of the charge on the

capacitor. The time-averaged power is then determined by integrating the product of the applied

voltage and the actuator current over an integer number of voltage cycles [63]. This is equivalent

to finding the area inside the parallelogram of figure 3.9(b). The total dissipated power includes

losses to dielectric heating, power required to generate the plasma, energy transfer to the neutral

flow, and a small amount of reflected power [65].

NASA/CR—2012-217628 53
0 0.5 1 1.5 2 2.5

−8

−4

0

4

8

Time (ms)

Applied voltage (kV)

−240

−120

0

120

240

Accumulated charge (nC)

voltage charge

(a)

−10 −5 0 5 10

−100

−50

0

50

100

Applied voltage (kV)

Accumulated charge (nC)

(b)

Figure 3.9: Typical charge and voltage data for a single-barrier actuator. The same

data are plotted (a) versus time and (b) as a Lissajous figure.

The discharge current can be determined by placing a resistor in ground path of the DBD. Discharge currents are typically “spiky,” like the current shown in figure 2.5. The individual current

spikes are so short that they are sometimes missed by a digital oscilloscope examining an entire applied voltage cycle. Consequently, integrating the product of the applied voltage and the observed

current is not as accurate as the charge-voltage method described above.

3.3 Optical Imaging

We conducted measurements of the visible light emission from the plasma using intensified

charge-coupled device (ICCD) cameras. This type of camera is capable of observing light emission

at time scales as short as 2 ns, allowing us to observe the structure of microdischarges as well as the

average plasma behavior on the time scales of the applied voltage. In all measurements of singlebarrier actuators, the actuators were powered by a 7 kV amplitude symmetric triangle waveform

at 1 kHz. For measurements of double-barrier actuators, the amplitude of the applied voltage was

increased to 12 kV.

NASA/CR—2012-217628 54
3.3.1 Imaging System

The observations of microdischarge formation and propagation in section 5.2.3 were performed

using an Andor iStar ICCD camera. All other measurements used a Princeton PI-MAX ICCD

camera. Both cameras have similar capabilities. The observed image in an ICCD camera is formed

by the following processes. Photons entering the camera strike a photocathode, releasing electrons.

These electrons are drawn into a microchannel plate (MCP). Electrons exiting the MCP strike a

phosphor, which emits photons that are collected by the CCD array. When biased, the MCP serves

as an electron multiplier, intensifying the signal. When unbiased, the MCP effectively blocks any

signal from reaching the CCD. The bias signal to the MCP then controls both the gain of the

camera and the time period during which collected light will reach the CCD array. The length of

this time period is known as the gate width of the intensifier, and is functionally equivalent to the

exposure time of a traditional film camera.

Due to the extremely short duration of individual microdischarges, described in section 5.2.3,

camera gating could not be triggered by the microdischarges themselves. Instead, gating was

controlled by a trigger signal from the waveform generator. Since the delay between the trigger

signal and the gating pulse could be set via software commands, we were able to observe the

plasma at any specified time during the applied voltage cycle.

All imaging experiments were conducting with the camera pointed at the dielectric surface;

this view is equivalent to looking downwards from above in figures 3.1 and 3.3. Prior to the

examination of the plasma in each actuator, we recorded a reference image of the actuator in

ambient light. Since images of the plasma were recorded in the dark, these reference images were

essential to determining the physical location of the plasma relative to the electrodes. Example

reference images are shown in figure 3.10 for a single-barrier actuator and figure 3.11 for a doublebarrier actuator.

Most camera images captured square areas approximately 11 mm by 11 mm, as shown in

figures 3.10 and 3.11. The short-gate width images of section 5.2.3 captured a slightly smaller

area of approximately 9 mm by 9 mm. In general, the field of view varies slightly from actuator

to actuator due to small variations in the distance between the camera and the actuator surface. In

NASA/CR—2012-217628 55
epoxy

1 mm

exposed electrode

buried electrode

diameter

11 mm

11 mm

1 mm

e

b

x

y

z

x

z

Figure 3.10: Image field of view for a single-barrier actuator

addition, the positions of the electrodes in the field of view vary slightly from actuator to actuator.

Each image shown in chapter 5 is accompanied by guide marks showing the positions of the

electrodes. For each image of a single-barrier actuator the exposed electrode position and length

is indicated by a line marked with an ‘e’, while the position of the buried electrode is indicated by

a ‘b’, as shown in figure 3.10. Similarly, for each image of a double-barrier actuator the position

and approximate diameter of the narrow electrode is indicated with an ‘n’, while the position of

the wide electrode is denoted with a ‘w’, as in figure 3.11.

epoxy

2 mm

narrow electrode

wide electrode

diameter

11 mm

11 mm

2 mm

n

w

x

y

z

x

z

Figure 3.11: Image field of view for a double-barrier actuator

NASA/CR—2012-217628 56
3.3.2 Plasma Imaging

Radiation emission from plasmas occurs when excited states of neutral molecules and ions

return to a lower energy state via photon emission. In millitorr-pressure plasmas photon emission

occurs via electric dipole radiation [17], while at atmospheric pressure this de-excitation is more

likely collision-induced. The excitation of ground-state particles may occur via either electron or

ion impact. Which particle dominates the excitation process is a complicated function of chargedparticle energy distributions and excitation cross-sections. Since the particle energy distribution

functions are dependent on the local electric fields, which may change rapidly, the light emission

described in chapter 5 should not be taken as corresponding to the density of a particular species.

Instead, we only take the observed images as loosely corresponding to the plasma structure. We use

these images to make qualitative comparisons to simulation results in section 7.4, and to observe

variations in actuator behavior on time scales not accessible to our force measurements.

As discussed in section 2.4.2 and shown experimentally in section 5.1, the plasma structure in

surface DBDs depends largely on the rate-of-change of the applied voltage, rather than its absolute

value. It is then logical to divide our applied symmetric triangle voltage into two halves: one

in which the voltage applied to the exposed or narrow electrode is rising, and one in which this

voltage is falling. We refer to these periods as the positive-going and negative-going half cycles,

respectively.

Microdischarges occur during only a portion of each of the half cycles. As discussed in section 2.5, microdischarges deposit charge on the dielectric surface to shield out the applied voltage.

When the voltage rate-of-change switches direction, the electric field in the gas region is too weak

to cause ionization until the applied voltage rises (or falls) to a level sufficient to supply the required field. Once the applied voltage reaches this critical level, microdischarges occur throughout

the remainder of the half cycle. These periods of inactivity and activity are visible in the discharge

current shown in figure 3.12.

Based on our observations using the cameras, microdischarges occur roughly randomly distributed in time throughout the active periods of each half cycle. Since camera gating was triggered based on the applied voltage cycle, individual image acquisitions only sometimes contained

NASA/CR—2012-217628 57
0 0.2 0.4 0.6 0.8

−8

−4

0

4

8

Time (ms)

Applied voltage (kV)

Inactive periods

Active periods

−2

−1

 0

 1

 2

Discharge current (mA)

voltage

current

Figure 3.12: Discharge current during the positive- and negative-going half cycles

of a single-barrier actuator, showing periods of activity and inactivity

light-emitting plasma. The fraction of acquired images containing plasma varied depending on

camera gate width, which half-cycle was under investigation, and the actuator geometry. As rough

averages, plasma was visible in approximately 50% of images at a gate width of 10 μs, while

finding a single microdischarge image at a gate width of 10 ns required acquiring and discarding

hundreds or thousands of empty images. The images presented in chapter 5 are representative of

images in which plasma was visible. Unless noted otherwise, the brightness of all those images

have been scaled to emphasize the observed plasma structures rather than to allow comparisons

between images.

In sections 5.2.1 and 5.3 we examine the average behavior of microdischarges during each half

cycle. We used the following method to perform this averaging. The applied voltage cycle was

divided into 100 segments, each 10 μs long. For each of those segments, we applied 100 identical

gating pulses to the camera, effectively integrating the light from 100 exposures (each of the same

segment of the cycle, but at a different time) onto a single image. This gave us 100 images with

tens to hundreds of microdischarges in each image. For each image, we then added together the

collected signal of each column of pixels. This is equivalent to compressing the image in the

direction of actuator uniformity, which is the z direction of figures 3.10 and 3.11. These steps

yield, for each segment of the applied voltage cycle, the average light emission as a function of

NASA/CR—2012-217628 58
x, the distance from the edge of the high voltage electrode. Examples of the results are shown in

figures 5.7, 5.8, and 5.18.

NASA/CR—2012-217628 59
Chapter 4

Experimental Force Measurements

We have performed force measurements on a variety of single- and double-barrier DBD plasma

actuators, as described in chapter 3. All of the measured forces and velocities described in this

chapter are in the the x direction in figure 3.7, that is the direction parallel to the dielectric surface

and perpendicular to the electrode edges. Section 4.1 examines the spatial uniformity and temporal stability of the induced air flow. Sections 4.2 and 4.3 describe our measurements of singleand double-barrier discharges, respectively. Finally, section 4.4 quantitatively compares the results from the electronic balance and stagnation probe measurements. Many of these results were

published in Hoskinson et al. [94].

4.1 Discharge Uniformity and Stability

The usefulness of our force measurements depends on the validity and repeatability of those

measurements. Since the stagnation probe technique yields a force per unit length, while the balance technique measures an absolute force, a valid comparison requires that the force is approximately uniform along the actuator width. Validity is further addressed in section 4.4 by comparing

the stagnation probe and electronic balance measurements. We address both the uniformity and

repeatability of the actuator force in this section.

The plasmas formed in plasma actuators, like most DBD air plasmas at atmospheric pressure,

are visibly non-uniform. The plasma forms numerous narrow filaments bridging between the

two electrodes. The induced velocities, however, are relatively uniform. We have scanned the

stagnation probe along the z direction (see figure 3.7) of two actuators, with the results shown in

NASA/CR—2012-217628 60
figure 4.1. At each point, we estimated the flow velocity by applying equation (3.2). The actuators

were a single-barrier actuator with a 0.25 mm diameter cylindrical exposed electrode and a doublebarrier actuator with a 1.0 mm diameter narrow electrode. For all measurements, the stagnation

probe was 8 mm away from the exposed or narrow electrode edge and 0.5 mm above the dielectric

surface.

0 10 20 30 40 50 60

0

0.5

1

1.5

z (mm)

Velocity (m/s)

double−barrier single−barrier

"hot spots"

Figure 4.1: Variation of the induced flow velocity in the z (uniform) direction. The

point z = 30 mm corresponds to the center of the 60 mm wide electrodes. The

stagnation probe was scanned in both directions, with the data points from each

direction having different shading.

At two locations along the single-barrier actuator, velocities were noticeably larger than the

average. These two locations corresponded to “hot spots” on the actuator; the plasma was visibly

brighter in these areas than along the rest of the device. We have previously created similar bright

filaments by intentionally introducing a small protrusion on the exposed electrode of a single barrier actuator. It is likely the hot spots are caused by small non-uniformities in the actuator materials

behaving in a similar fashion.

Neglecting the variation at such hot spots and at the actuator edges, the velocities near the

single-barrier actuator were always within ±20% of their values at the center. Double-barrier velocities showed similar uniformity: always within ±30% of their values at the center. This relative

homogeneity is reasonable, since the visible plasma non-uniformity is on the scale of 1 mm or

less, while the velocities were measured several millimeters away from the plasma. The stagnation

probe measurements discussed in the remainder of the work avoided examining velocities near

NASA/CR—2012-217628 61
“hot spots.” The additional force provided by these regions will be implicitly included in the balance measurements, but as long as they are not numerous this will have only a small effect on the

measured forces.

Early stagnation probe measurements showed large run-to-run variations, with stronger variations for double-barrier actuators. We observed that this variability diminished as the actuator

run time immediately prior to the measurement increased. Velocity profiles for both single- and

double-barrier actuators reached steady state after 15–45 minutes of run time, but the variability

returned if the actuators were left off overnight. This variability in velocity profiles is illustrated

in figures 4.2 for actuators driven by 500 Hz sinusoidal waveforms. In both sets of measurements,

run 2 was conducted 24 hours after run 1. While the four profiles of the single barrier actuator

show only relatively small differences, the peak velocity for the double barrier actuator increased

by almost 50% after conditioning for 90 minutes.

0 2 4 6

0

0.2

0.4

0.6

0.8

y (mm)

Velocity (m/s)

run 1, 0 min

run 2, 0 min

run 1, 90 min

run 2, 90 min

(a)

0 2 4 6

0

0.2

0.4

0.6

0.8

y (mm)

Velocity (m/s)

run 1, 90 min

run 2, 90 min

run 1, 0 min

run 2, 0 min

(b)

Figure 4.2: Run-to-run variation and conditioning in (a) a single-barrier actuator with a 0.40 mm cylindrical exposed electrode and (b) a double-barrier actuator

with a 0.25 mm narrow electrode. The times indicated how much run-time elapsed

immediately prior to the measurement.

The power dissipated in double-barrier actuators changed by less than 12% over the conditioning period, with the direction of the change appearing to be random. Single-barrier actuators

exhibited a slightly larger change in the power (less than 15%), but all actuators dissipated less

NASA/CR—2012-217628 62
power after conditioning. In order to eliminate this source of uncertainty, immediately prior to

all force and power measurements we ran each single-barrier actuator for at least 15 minutes, and

each double-barrier actuator for at least 30 minutes.

Shin and Raja [96] observed similar run-to-run variation and stabilization of the discharge current after conditioning on the same time scales for a parallel-plate DBD. They eliminated thermal

effects and the build-up of gas impurities as possible causes of the variation, which they ascribed

to plasma modification of the dielectric surface’s secondary electron emission coefficient. This

coefficient is likely to change upon exposure to room air, leading to the run-to-run variation.

4.2 Single-Barrier Actuators

Our force measurement efforts have focused on single-barrier actuators. The results of those

experiments are included in this section. As described in section 2.4, this geometry has been much

more widely studied by other groups, providing opportunities to compare to previous research,

which we do in section 4.2.3. We have discovered, as described in section 4.2.1, that when the

exposed electrode is cylindrical, the induced time-averaged force increased exponentially as the

diameter is decreased. The increase in force appears to be largely due to the reduction in the electrode size in the dimension parallel to the dielectric surface, rather than the reduction in electrode

thickness. Only a relatively small force increase is seen in actuators with rectangular cross-section

exposed electrodes if only the thickness, and not the length, is reduced.

4.2.1 Effects of Cylindrical Exposed Electrode Diameter

We have conducted force and power measurements on single-barrier actuators with cylindrical

exposed electrodes ranging from 0.03 mm to 1.0 mm in diameter, as described in section 3.1.

Force measurements were made using both the electronic balance, described in section 3.2.1, and

the stagnation probe technique, described in section 3.2.2. Discharge powers were determined as

described in section 3.2.3.

The measured discharge powers for these actuators were independent of the material used to

make the exposed electrode. For exposed electrode diameters above approximately 0.2 mm, the

NASA/CR—2012-217628 63
power was also independent of the diameter, as shown in figure 4.3. Below this size, however, the

dissipated power decreased with decreasing electrode diameter. In section 5.2.2 we show that this

is due to the reduced number of filamentary microdischarge events in actuators with these small

diameter electrodes. Fewer microdischarges corresponds to a reduced current draw, and therefore

less dissipated power. In the range examined, the dissipated power was approximately proportional

to the applied voltage amplitude for all actuators.

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

Exposed electrode diameter (mm)

Dissipated power (W)

8 kV − Cu 8 kV − W 8 kV − SS

7 kV − Cu 7 kV − W 7 kV − SS

6 kV − Cu 6 kV − W 6 kV − SS

Figure 4.3: Power dissipated by single-barrier actuators with cylindrical exposed

electrodes of various diameters and materials. The indicated voltages are the amplitudes of the applied symmetric triangle waveform.

In contrast to the power, the induced forces increased sharply as the exposed electrode diameter

decreased. This phenomenon is shown in figure 4.4 in both the electronic balance and the stagnation probe data. The indicated voltages in that figure are the amplitudes of the applied symmetric

triangle waveform, and the fit lines are described below. We also note that the material composition of the exposed electrode had no noticeable effect on the data; actuators with similarly-sized

electrodes induced approximately the same force.

It is likely that the major difference between electrode materials relevant to plasma formation

is their differing secondary electron emission coefficients, although the literature regarding these

for appropriate ions and energies is sparse. Based on simulation results, Boeuf et al. [93] predicted

that the value of the emission coefficient of the electrode (which, in their model, was set equal to

NASA/CR—2012-217628 64
0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

6

7

x 10−4

Exposed electrode diameter (mm)

Force (N)

8 kV − Cu

8 kV − W

8 kV − SS

7 kV − Cu

7 kV − W

7 kV − SS

6 kV − Cu

6 kV − W

6 kV − SS

(a)

0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

Exposed electrode diameter (mm)

Force linear density (N/m)

8 kV − Cu

8 kV − W

8 kV − SS

7 kV − Cu

7 kV − W

7 kV − SS

6 kV − Cu

6 kV − W

6 kV − SS

(b)

Figure 4.4: Measured (a) forces, determined with the electronic balance, and (b)

linear force densities, determined with the stagnation probe technique. All data are

for single-barrier actuators with cylindrical exposed electrodes.

the coefficient for the dielectric) has little effect on the force induced by the actuator, while being

a strong predictor of dissipated power. Our data suggest that changing the emission coefficient of

just the exposed electrode has little effect on the induced force or on the dissipated power.

The variation of the actuator force and linear force density with the diameter of the exposed

electrode can be reasonably approximated by an exponential function:

Ffit = F0 exp(−d/d0), (4.1)

where F0 and d0 are free parameters and d is the electrode diameter. We have fit equation (4.1)

to the data from both the balance and stagnation probe techniques, with the units of F0 chosen to

match the data. These fits are shown in figures 4.4(a) and (b). Fitted values of d0 for the electronic

balance measurements were 0.17 ± 0.02, 0.24 ± 0.04, and 0.36 ± 0.06 mm for the 6, 7, and 8 kV

tests, respectively. The values of d0 for the stagnation probe data were similar but slightly smaller,

at 0.14±0.01, 0.18±0.03, and 0.25±0.04 mm. Further quantitative comparisons between the two

measurement techniques are described in section 4.4. The increase in the value of d0 as the voltage

NASA/CR—2012-217628 65
increases indicate that equation 4.1 is a simple empirical fit, and does not completely separate the

dependencies of the force on electrode geometry and voltage.

4.2.2 Effects of Rectangular Exposed Electrode Thickness

We have also measured the forces induced by single-barrier actuators with rectangular crosssection exposed electrodes of varying thickness. This design was depicted in figure 3.1(b). All of

the actuators discussed here had exposed electrode lengths of 10 mm or greater; shorter exposed

electrodes are considered in section 4.2.4. These long rectangular exposed electrodes are similar

to the designs studied extensively by other groups, as discussed in section 2.4. For these actuators,

we have measured the forces only using the electronic balance.

The electrical power consumed by actuators with exposed electrode thicknesses ranging from

0.01 mm to 1.1 mm is plotted in figure 4.5. Since we have shown that the material of the exposed

electrode does not affect either force or power, we do not distinguish here between electrodes of

different composition. Outside of small variations at small thicknesses, the electrical power use

was essentially independent of electrode thickness. The power dissipation at each voltage level

was comparable to that of single-barrier actuators with thick cylindrical electrodes.

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

Exposed electrode thickness (mm)

Dissipated power (W)

8 kV 7 kV 6 kV

Figure 4.5: Power dissipated by single-barrier actuators with rectangular crosssection exposed electrodes of various diameters. The indicated voltages are the

amplitudes of the applied symmetric triangle waveform.

NASA/CR—2012-217628 66
At thicknesses greater than approximately 0.4 mm, the forces induced by actuators with rectangular exposed electrodes were similar to those induced by actuators with cylindrical electrodes

of the same thickness. While the forces of actuators with rectangular electrodes increased at thicknesses smaller than this, they did so relatively slowly, as shown in figure 4.6. For example, at the

same applied voltages an actuator with a 0.03 mm diameter cylindrical exposed electrode induced

forces 2 to 5 times larger than an actuator with a 0.03 mm thick rectangular exposed electrode.

The differences between the electrode shapes is examined in section 4.2.4.

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

x 10−4

Exposed electrode thickness (mm)

Force (N)

8 kV

7 kV

6 kV

Figure 4.6: Measured forces, determined with the electronic balance, of singlebarrier actuators with rectangular cross-section exposed electrodes. The indicated

voltages are the amplitudes of the applied symmetric triangle waveform.

In addition to the data included in figure 4.6, we measured the forces induced by an actuator

using an exposed electrode made from copper foil tape with a conductive adhesive (for a total

thickness of 0.09 mm). This device induced slightly larger forces than would be expected from

considering the data in figure 4.6 for reasons that are still unclear. We found that this design,

however, was abnormally sensitive to sub-millimeter variations in the electrode gap, making it

difficult to replace the actuator if it was damaged during operation. We have omitted the data from

this actuator.

NASA/CR—2012-217628 67
4.2.3 Comparison to Previous Experiments

Previous experiments examining the effects of electrode size on actuator performance examined force efficiency (force / power) rather than absolute force. The trends of force efficiency for

our measurements were similar to those for absolute force described above, since the powers were

similar for all actuators except those with very narrow cylindrical exposed electrodes. The force

efficiencies for single-barrier actuators with both cylindrical and rectangular exposed electrodes

are plotted in figure 4.7. The division by the discharge power largely, but not completely, removes

the dependence on the applied voltage amplitude in the range we have examined.

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

x 10−4

Exposed electrode thickness (mm)

Force efficiency (N/W)

8 kV − cylindrical

7 kV − cylindrical

6 kV − cylindrical

8 kV − rectangular

7 kV − rectangular

6 kV − rectangular

Figure 4.7: Force efficiencies of single-barrier actuators with cylindrical and long

rectangular exposed electrodes. The indicated voltages are the amplitudes of the

applied symmetric triangle waveform.

In section 2.4.3 we referenced the results of Enloe et al. [39], who found that the force efficiency of single-barrier actuators increased linearly with decreasing exposed electrode thickness

regardless of the shape of the electrode. They tested cylindrical electrodes with diameters between about 0.35 and 1.0 mm and rectangular electrodes with thicknesses between about 0.08 and

0.7 mm. Our results are consistent with these findings. Above 0.35 mm thickness, figure 4.7

shows that the force efficiencies of actuators with both cylindrical and rectangular electrodes were

similar. The force efficiencies depart from a linear dependence on electrode thickness only below

approximately 0.3 mm thickness for cylindrical electrodes and 0.05 mm for rectangular electrodes.

NASA/CR—2012-217628 68
Quantitative comparisons between the current work and previous studies by other groups have

to be made with care due to the differences in actuator geometry, applied voltages, and measurement techniques between groups. Enloe et al. [39] measured a maximum force efficiency

of 8 × 10−5 N/W. That measurement was made using an actuator with a 0.08 mm thick rectangular exposed electrode, and fits with our results in figure 4.7. We achieved a maximum force

efficiency using an actuator with a 25 μm diameter exposed electrode, and measured a value of

7.2 × 10−4 N/W, or nearly an order of magnitude larger than previous tests. Several other groups

have published data showing the force induced by their actuator designs, but have not provided

enough information for us to make direct comparisons.

Other groups have focused on measurements of the induced flow velocities in air instead of

the time-averaged forces. Pons et al. [63] measured a maximum induced flow velocity in air of

approximately 3.3 m/s using a discharge power per unit length of 47 W/m, but did not report any

total forces. Forte et al. [66] were able to induce larger flow velocities on the order of 6 m/s by

increasing the discharge power to approximately 170 W/m. Even larger powers on the order of

500 W/m were used by Roth and Dai [65] to achieve maximum flow velocities of approximately

5 m/s. Our most efficient actuator induced a maximum flow velocity of 3.2 m/s, but required only

16 W/m of electrical power to do so.

4.2.4 Variation with Exposed Electrode Length

As described in the preceding sections, single-barrier actuators with cylindrical and rectangular

exposed electrodes exhibited significant differences, even when the electrode thicknesses were

similar. To determine the source of those differences we have investigated a variety of actuator

designs, each examining one aspect of the electrode geometry. These tests looked at the height

of the exposed electrode above the dielectric surface, the sharpness of the electrode edge, and

the length of the electrode. Despite being dismissed by other researchers as having a negligible

influence on the actuator [39], only the electrode length was observed to have a large enough effect

to explain the difference between cylindrical and rectangular electrodes, as described below. To

simplify testing procedures, all force measurements described in this section were made using

NASA/CR—2012-217628 69
just the electronic balance, and all actuators were driven by 7 kV amplitude symmetric triangle

voltages.

We used the side-by-side cylindrical actuator design of figure 3.2 to approximate the length of

a (short) rectangular electrode while retaining the electrode edge profile of a cylindrical electrode.

We stacked eight 0.13 mm diameter stainless steel cylinders side-by-side for a maximum effective

length of just over 1 mm. After measuring the force using the electronic balance, we removed the

cylinder furthest from the buried electrode and repeated the measurement, continuing until only

one cylinder remained. As each cylinder was removed, the induced force increased, as shown in

figure 4.8.

We have also examined actuators with rectangular cross-section exposed electrodes of varying

lengths. As we reduced the length of the rectangular electrode, the force increased. Due to difficulties in accurately cutting narrow strips of metal, we have examined only a few actuators with

electrode lengths smaller than 1 mm. In order to compare the forces of actuators with different

electrode thicknesses, we divided each of the measured forces by the force that was induced by

an actuator with a long (>10 mm) rectangular electrode of the same thickness. In the case of the

0.13 mm cylinders, the appropriate reference force was extrapolated from existing data of actuators

with rectangular electrodes of similar thicknesses.

0 2 4 6 8 10

1

1.5

2

Exposed electrode length (mm)

Normalized force

side−by−side cylinders

0.04 mm thick rectangular electrodes

0.03 mm thick rectangular electrodes

Figure 4.8: Normalized forces for actuators with varying electrode lengths. The

measured forces were divided by the force for an actuator with a long rectangular

exposed electrode of the same thickness. The applied voltage amplitude was 7 kV.

NASA/CR—2012-217628 70
The results of these measurements are plotted in figure 4.8. They showed a smooth increase in

the normalized force as the electrode length was reduced. The rapid increase at electrode lengths

below 2 mm suggests that the differences between actuators with rectangular and cylindrical exposed electrodes are primarily due to differences in electrode length, rather than the exact shapes

of the electrode edges. For the actuators examined here, reducing the electrode length below approximately 3 mm had a much stronger effect than did reducing the electrode thickness.

The power dissipated by a single-barrier actuator was independent of the length of its exposed

electrode except for very short lengths. This is shown in figure 4.9, where the power use is effectively constant except for the single-cylinder actuator. For that device, the power dropped by 15%,

similar to the reduction in power seen for small diameter cylindrical electrodes in figure 4.3.

0 2 4 6 8 10

0

0.5

1

Exposed electrode length (mm)

Dissipated power (W)

side−by−side cylinders

0.04 mm thick rectangular electrodes

0.03 mm thick rectangular electrodes

Figure 4.9: Power dissipated by single-barrier actuators with exposed electrodes

of varying lengths. The applied voltage amplitude was 7 kV.

4.3 Double-Barrier Actuators

We have also conducted measurements of the forces induced by double barrier actuators. This

geometry was depicted in figure 3.3 and described in section 3.1.1. The force data were obtained

using the procedures described in sections 3.2.1 and 3.2.2. Because neither electrode is in contact

with the plasma for this geometry, we have not presented data for different materials separately.

No dependence of force or power on the narrow electrode material was observed.

NASA/CR—2012-217628 71
Data from the electronic balance and stagnation probe measurements are shown in figure 4.10.

The indicated voltages are the amplitudes of the applied symmetric triangle waveform, and the fit

lines are described below. Similar to the single barrier actuators, the induced force increased with

decreasing electrode diameter, although here it was the narrow electrode rather than an exposed

electrode that was varied.

0 0.2 0.4 0.6 0.8 1

0

2

4

6

8

x 10−5

Narrow electrode diameter (mm)

Force (N)

13 kV

12 kV

11 kV

(a)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

x 10−3

Narrow electrode diameter (mm)

Force linear density (N/m)

13 kV

12 kV

11 kV

(b)

Figure 4.10: Measured (a) forces, determined with the electronic balance, and (b)

linear force densities, determined with the stagnation probe technique. All data are

for double-barrier actuators with various diameter narrow electrodes.

The measured forces were small enough to approach the sensitivity limit of the balance, leading

to large relative uncertainties in the balance data. The absolute levels of the force were a factor of

5 to 10 smaller than those seen for single barrier actuators for the configurations studied here. For

actuators with narrow electrodes smaller than 0.1 mm in diameter, the dielectric layers tended to

quickly erode and allow arcing, so no data were recorded for these devices.

In contrast to the data from single barrier actuators, at large diameters the induced forces did not

decrease towards zero but rather asymptoted to a small but finite value. At larger values of narrow

electrode diameter, the electrode geometry does not become symmetric, which may explain the

non-zero force at large diameters. Rather, there is still an asymmetry in the shape of the electrode

cross sections (circular for the narrow electrode and rectangular for the wide electrode).

NASA/CR—2012-217628 72
To make reasonable approximations of the force behavior, instead of using equation (4.1) we

have fit

Ffit = F0 exp(−d/d0) + F1 (4.2)

to the data in figure 4.10. Here F0, F1, and d0 are free parameters. For the balance data, the

fitted values of d0 were 0.26 ± 0.25, 0.37 ± 0.22, and 0.29 ± 0.23 for the 11, 12, and 13 kV

tests, respectively. The values for the balance data have large confidence intervals due to the large

relative scatter of the measurements. The stagnation probe data were somewhat more accurate, and

had fitted d0 values of 0.26 ± 0.01, 0.27 ± 0.10, and 0.24 ± 0.11. Further quantitative comparisons

between the two measurement techniques are described in section 4.4.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Narrow electrode diameter (mm)

Dissipated power (W)

13 kV

12 kV

11 kV

Figure 4.11: Power dissipated by double-barrier actuators with various narrow

electrode diameters. Most actuators were tested several times, with the results of

each test plotted separately.

In order to produce a plasma, double barrier actuators required much higher applied voltages

than do the single barrier actuators. The rates of discharge events, as determined by both current

measurements and ICCD imaging, however, were much lower. The resulting reduced discharge

current lead to somewhat lower levels of power dissipation, shown in figure 4.11, for the range

of applied voltages considered here. The dissipated power decreased approximately linearly with

decreasing narrow electrode diameter. As with single-barrier actuators, actuators with small diameter electrodes were notably more efficient. The force efficiencies of double-barrier actuators

were, on average, a factor of about 2 smaller than those of single-barrier actuators.

NASA/CR—2012-217628 73
4.4 Comparison of Balance and Stagnation Probe Techniques

As described in sections 4.2.1 and 4.3, the same general trends in the induced forces with

electrode diameter have been observed using both the electronic balance and stagnation probe

measurement techniques. When the balance data were converted to forces per unit length, as described in section 3.2.1, they were consistently smaller than those calculated from stagnation probe

measurements using the assumptions of section 3.2.2. The two approaches are clearly proportional

to one another, however, as shown in figure 4.12. The stagnation probe data were approximately a

factor of two larger for all measurements.

0 0.005 0.01 0.015 0.02

0

0.002

0.004

0.006

0.008

0.01

0.012

Stagnation probe data (N/m)

Balance data (N/m)

y = x

y = 

1

/

2

 x

data

corrected data

Figure 4.12: Comparison of forces measured using the electronic balance and

stagnation probe. The circles were calculated using the using the assumptions of

section 3.2.2, while the corrected data is described in this section.

We have found that this difference is largely due to the assumptions made in converting the

measured stagnation pressures into induced forces. While each assumption made in section 3.2.2

has a relatively small effect on the calculated force, these effects add and lead to a significant

overestimation of the force when using the stagnation probe technique. To show this, we have

made additional detailed measurements around three actuators: single-barrier actuators driven at

7 kV with cylindrical exposed electrodes 0.03 and 0.11 mm in diameter, and a double-barrier

actuator driven at 12 kV with a narrow electrode 0.11 mm in diameter. The corrections provided

NASA/CR—2012-217628 74
by these data were sufficient to explain this observed difference between stagnation probe and

balance data.

We measured static pressures using a 2.5 mm diameter alumina tube sealed at one end, with a

1 mm diameter hole drilled 7 mm from the sealed end. The tube was oriented in the −x direction,

with the hole facing the z direction. Stagnation and static pressure measurements at three locations

near the single-barrier, d = 0.11 mm actuator are shown in figure 4.13. The data confirm that the

static pressure is constant downstream (+x direction) of the high voltage electrode, but show a

small non-zero static pressure upstream. The exposed electrode is considered to be at x = 0.

0 5 10 15 20

−0.1

0

0.1

Pressure (Pa)

0 2 4 6 8 10

0

0.5

1

1.5

Pressure (Pa)

0 2 4 6 8 10

0

0.5

1

1.5

y (mm)

Pressure (Pa)

stagnation pressure

static pressure

x = −4 mm

x = +16 mm

x = +8 mm

Figure 4.13: Stagnation and static pressures measured at x = −4 (top), +8 (middle), and +16 mm (bottom) for a single barrier actuator with d = 0.11 mm driven

at 7 kV

Using these data for the single-barrier actuator, we can evaluate the right-hand side of equation (3.1), except for term II. We use the measured static pressure in place of the reference pressure

in equation (3.2). Evaluating terms I and IV along the surface at x = 8 mm and terms III and V

at x = −4 mm, we find a linear force density of 4.2 × 10−3 N/m, or 68% of the original force

estimate for these conditions. If we repeat the calculation and instead evaluate terms I and IV at

NASA/CR—2012-217628 75
x = 16 mm, thereby including the shear force all the way to the actuator edge, we calculate a

linear force density of 3.5 × 10−3 N/m. This is 56% of the original estimate, and compares well

to the balance-measured value of 3.7 × 10−3 N/m. The largest correction to equation (3.3) for the

0.11 mm single-barrier actuator was the inclusion of term III, the flux of momentum into the left

side of the control volume. This term was equal to 35% of the uncorrected force estimate.

Corrections for the other two actuators yielded similar improvements in the quantitative agreement between the stagnation probe and balance measurements. Using stagnation and static pressures measured at x = −4 and x = +16 mm, the calculated linear force density for the 0.03 mm

single-barrier actuator was 9.2×10−3 N/m (72% of the original estimate), compared to the balancemeasured value of 7.8 × 10−3 N/m. The corrected force density for the 0.11 mm double-barrier

actuator was 6.4 × 10−4 N/m (37% of the original estimate), compared to the balance-measured

value of 8.3 × 10−4 N/m.

It is clear from the improved agreement between the two measurement techniques that care

must be taken to quantitatively determine forces from pressure measurements. Our estimate of

term III in equation (3.1) itself may be slightly in error: the static probe was large enough to

possibly disturb the gas flow in the region of measurement. Still, we believe including the term as

measured is likely to be more accurate than continuing to neglect it completely.

We have not estimated the effect of the flux of x momentum through the top surface (term II

in equation (3.1)), as we cannot accurately measure uy in that region. PIV measurements [60] and

calculations [97] suggest that on the top surface ux is much smaller than its values on the left- and

right-hand edges of the control volume and comparable to uy, because most of the acceleration

occurs inside the control volume closer to the surface. This behavior would make our neglect of

term II reasonable. The relatively close agreement between the corrected stagnation probe and

balance measurements also supports this conclusion.

NASA/CR—2012-217628 76
Chapter 5

Optical Imaging

Imaging with the ICCD camera allowed us to observe plasma behavior on the different time

scales of interest for DBD plasma actuators. Individual microdischarges exist for times on the

order of 5 to 100 ns. Images of microdischarges on these time scales are shown in section 5.2.3.

Since all of the applied voltages oscillated at 1 kHz, plasma variations with the applied voltage

level occurred at time scales on the order of 10 μs. We show our observations of these variations

in sections 5.2.1 and 5.3. In examining all of the acquired images, we noted that the plasma

structures could be grouped in three distinct categories, which we describe in section 5.1. Many

features of plasma structures have been qualitatively reproduced in our simulation results, which

are described in sections 7.1, 7.4.1, and 7.4.2. Additional observations of one of those structures,

the newly-discovered filament-free plasma regime, are included in section 5.2.2.

5.1 Observed Microdischarge Structures

In our imaging experiments, we have observed three distinct plasma structures. These structures arise in a variety of different actuator types as described in this section. We refer to these

structures as filamentary, jet-like, and filament-free structures. The filamentary and jet-like structures have been previously observed, as described in section 2.4.2. The filamentary structure is

occasionally referred to in the literature as a streamer. The jet-like structure is sometimes referred

to as a “diffuse” microdischarge; we avoid that terminology as it only appears diffuse relative to

NASA/CR—2012-217628 77
the filamentary structure. The filament-free structure has not been previously identified. This latter structure has some similarities to the corona-like plasma that has been observed in actuator

simulations, a corrspondence that is discussed in more detail in section 7.4.

In single-barrier actuators, filamentary microdischarges appeared during the positive-going

half cycle. Typical discharge structures of this type are shown in figure 5.1. They are on the

order of 0.1 mm thick, with numerous branches forming in directions parallel to the dielectric surface. The total microdischarge length varied with the applied voltage, as described in section 5.2.1.

The gate widths used to record the images in figure 5.1 were long relative to the lifetime of a filamentary microdischarge, so the pictured structures are fully developed.

e

b

e

b

(a) (b)

Figure 5.1: Images of typical filamentary plasma structures. These images were

recorded in single-barrier actuators and used gate widths of 10 μs.

During the negative-going half cycle in single-barrier actuators, jet-like plasma structures

formed and dissipated. Like the filamentary structures, they touched a single point on the exposed

electrode. Jet-like structures are so named because they resemble a diffusing gas jet: dense at the

point of origin, growing gradually weaker at greater distances. The jet-like plasma structures pictured in figures 5.2(a) and 5.2(b) show the range of variation in this type of plasma structure. Some

microdischarges, like the one in figure 5.2(a), are nearly symmetric and decrease in brightness uniformly with distance away from brightest spot. Others, such as the plasma shown in figure 5.2(b),

have a less symmetric structure, with a brighter region near the center of the plasma. This latter

variation tended to occur more frequently in single-barrier actuators with thicker exposed electrodes.

NASA/CR—2012-217628 78
e

b

e

b

(a) (b)

Figure 5.2: Images of typical jet-like plasma structures. These images were

recorded in single-barrier actuators and used gate widths of 10 μs.

In single-barrier actuators with thin cylindrical exposed electrodes (smaller than about 0.15 mm

in diameter), we observed a third discharge structure, which we call the filament-free plasma. This

structure appeared only during the positive-going half cycle, generally after an initial phase of

filamentary microdischarges. It appeared as a weakly glowing band of plasma on the edge of the

exposed electrode nearest the buried electrode, as shown in figure 5.3. Unlike the filamentary and

jet-like microdischarges described above, the filament-free plasma did not form and extinguish

multiple times during the course of a single half-cycle. Instead, during the portions of the voltage

cycle in which we observed it, the filament-free plasma was visible in every recorded image, at

gate widths as short as a few nanoseconds. We describe this plasma structure in more detail in

section 5.2.2.

e

b

Figure 5.3: Image of a typical filament-free plasma structure. This image is of a

single-barrier actuator, and used a gate width of 10 μs.

NASA/CR—2012-217628 79
Microdischarges in double-barrier systems were observed to consist simultaneously of both

filamentary and jet-like plasma structures. During the positive-going half cycle, a short jet-like

structure formed between the center of the electrode gap and the region above the narrow electrode,

while a much longer filamentary structure extended from a point in the electrode gap downwards

over the wide electrode. This type of microdischarge is shown in figure 5.4(a). The opposite type

of structure appeared during the negative-going half cycle, as shown in figure 5.4(b). In this case,

the filamentary structure existed between the center of the electrode gap and the surface above the

narrow electrode, while a longer jet-like plasma extended from the electrode gap over the wide

electrode.

n

w

n

w

(a) (b)

Figure 5.4: Images of typical microdischarges in a double-barrier actuator. The

microdischarge during (a) the positive-going half cycle consists of a short jet-like

structure and a long filamentary structure, while that during (b) the negative-going

half cycle is made up of a short filamentary structure and a long jet-like structure.

The existence of similar plasma structures in both single- and double-barrier actuators suggests

that plasma behavior does not depend exclusively on the electrode structure. Since both filamentary

and jet-like structures form in double-barrier systems, we conclude that the plasma structure is at

least in part determined by the direction of the local electric field relative to the dielectric surface.

5.2 Single-Barrier Actuators

In section 5.1 we described the different types of microdischarges present in a single-barrier

surface DBD. Here we examine several characteristics of microdischarge behavior. We note that,

NASA/CR—2012-217628 80
for single-barrier actuators with cylindrical exposed electrodes, microdischarges also occurred on

the “back” side of the electrodes, away from the buried electrode. Filamentary and jet-like microdischarges on this side of the exposed electrode are shown in figure 5.5. In the remainder of

this section we neglect these microdischarges, only considering the longer microdischarges which

extend from the exposed electrode edge towards the buried electrode.

e

b

e

b

(a) (b)

Figure 5.5: “Backwards” (a) filamentary microdischarge during the positive-going

half cycle and (b) jet-like microdischarges during the negative-going half cycle in a

single-barrier actuator. Several microdischarges appear in (b), with the backwards

discharges circled. Both images used gate widths of 10 μs.

5.2.1 Average Microdischarge Lengths

As each half-cycle of the applied voltage progressed, the observed filamentary and jet-like

microdischarges extended farther from the exposed electrode. This increase in microdischarge

length is evident from the early- and late-half cycle images shown in figure 5.6. We have observed

similar increases in all actuators we have tested.

To allow us to determine average microdischarge lengths, we have applied the accumulation

and averaging procedures described in section 3.3.2 to a variety of single-barrier actuators with

both cylindrical and rectangular exposed electrodes. The results for a single-barrier actuator with

a 0.11 mm diameter exposed electrode are shown in figure 5.7. The phases of the voltage cycle

during which filamentary, filament-free, and jet-like plasmas exist for this actuator are noted on

the figure. Figure 5.7 shares many characteristics with the results of Enloe et al. [39], who used

NASA/CR—2012-217628 81
e

b

e

b

(a) (b)

(c) (d)

Figure 5.6: Increasing single-barrier microdischarge length from (a) 50% (0 V) to

(b) 95% (6.3 kV) of the way through the positive-going half cycle and from (c) 50%

(0 V) to (d) 95% (−6.3 kV) of the way through the negative-going half cycle.

a movable photomultiplier tube to measure the light emission as a function of time at a variety

of x locations. They examined a single-barrier actuator with a rectangular exposed electrode, and

noted that the regions of emitted light grew at approximately the same rate during the positive- and

negative-going half cycles, and that this rate varied with the applied voltage amplitude [70].

In order to determine the growth or expansion rates for our own discharges, estimates of the

average microdischarge lengths are required at each point in the voltage cycle. To find these

lengths, we applied two fitting algorithms to the average light emission data. The first fitting

routine, which we refer to as the edge-based algorithm, initially checked the first derivative of the

data, identifying the peak with the largest value of x. It then fit a decaying exponential function

only to the light emission data at larger x values than the peak. Two example fits using this

algorithm are shown in figures 5.8(a) and 5.8(b). The second fitting routine, which we refer to

NASA/CR—2012-217628 82
Figure 5.7: Average light emission during the applied voltage cycle for a singlebarrier actuator with a 0.11 mm diameter exposed electrode. Each column of pixels

corresponds to the average light emission from a 10 μs period.

−2 0 2 4 6 8

x (mm)

Brightness (a.u.)

data

edge−based

+ area−based

+

(a)

−2 0 2 4 6 8

x (mm)

Brightness (a.u.)

data

edge−based

+ area−based

+

(b)

Figure 5.8: Average light emission and edge fitting during times in the (a) positivegoing and (b) negative-going half cycles. The + symbols indicate the x locations

defined as the average microdischarge edges by the fitting algorithms.

NASA/CR—2012-217628 83
as the area-based algorithm, integrated the light emission plotted from the edge of the exposed

electrode in the positive x direction. It defined the edge of the average microdischarge as the point

at which the integration has reached 95% of its final value. Example fits using this algorithm

are also shown in figures 5.8(a) and 5.8(b). The linear fits made by Enloe et al. [39, 70] to their

emission data were similar to our edge-based fits, with the edge threshold defined by the sensitivity

of their instruments.

The two edge-finding algorithms yielded similar, but not identical, results. The algorithms

depend on the precise shape of the emitted light curve, including how much noise is present, and

each returns questionable results for curves of certain shapes. We chose to present results from

both algorithms because these two methods use different features of fitted curves. The results from

these two techniques provide a reasonable estimate of which observed features are independent of

the chosen fitting technique.

From the fitting algorithms we obtained the average microdischarge length as a function of time

for each examined actuator. For most actuators, these lengths increased approximately linearly

for the majority of each half cycle of the applied voltage. The only exception to this was the

single-barrier actuator with the smallest diameter exposed electrode, which quickly transitioned to

a non-changing filament-free plasma. Lengths for two single-barrier actuators, determined using

the edge-based method, are shown in figure 5.9. The periods of linearly increasing microdischarge

lengths are evident in this figure. In each of the these periods we have made a linear fit to the data,

the slope of which is the rate at which the average microdischarge length increases.

These expansion rates for all tested single-barrier actuators are shown in figure 5.10 for both

fitting algorithms. While there were quantitative differences between the results from each fitting

algorithm, we did find some consistent qualitative features. For actuators with cylindrical exposed

electrodes, the expansion rates, shown in figures 5.10(a) and 5.10(c), were effectively independent

of diameter when that diameter was larger than about 0.15 mm. Below this diameter, the expansion rates for filamentary microdischarges increased, while those for jet-like microdischarges

decreased. This 0.15 mm threshold is similar to the diameter at which the measured force efficiencies of figure 4.7 began to increase sharply. The relatively large uncertainty for the actuator with

NASA/CR—2012-217628 84
0 0.2 0.4 0.6 0.8 1

0

2

4

6

Time (ms)

Microdischarge length (mm)

0.05 mm

0.64 mm

Figure 5.9: Average microdischarge lengths for single-barrier actuators determined

using the edge-based algorithm. Example fits to the linear regions, used to determine the expansion rates of figure 5.10, are shown for the actuator with the 0.64 mm

diameter exposed electrode.

the 0.03 mm diameter cylindrical electrode in figure 5.10(a) was due to the very short filamentary

period of this actuator (see section 5.2.2).

Previously we reported an expansion rate of about 5 m/s for jet-like microdischarges in an

actuator with a 0.3 mm diameter exposed electrode [98]. This low value was due to averaging fewer

exposures together and using only an edge-based fitting method. The difference between those

measurements and the more recent ones shown in figure 5.10 demonstrate that the observed rates

are dependent on the methods of observation and the fitting algorithms. Due to these dependences,

here we have focused only on consistent changes in the rates with electrode geometry.

Expansion rates for actuators with rectangular exposed electrodes, shown in figures 5.10(b)

and 5.10(d), behaved differently. The expansion rates increased slightly at very small thicknesses,

and this increase was similar between the filamentary and jet-like microdischarges. The rates did

not diverge like those for actuators with cylindrical exposed electrodes at small diameters. These

differences in behavior between actuators with cylindrical and rectangular exposed electrodes correlate with the variations in forces between the two geometries. This qualitative correlation suggests that further studies of the visible plasma structure will likely be useful in understanding the

induced EHD forces.

NASA/CR—2012-217628 85
0 0.2 0.4 0.6 0.8 1

0

5

10

15

Exposed electrode diameter (mm)

Expansion rate (m/s)

filamentary

jet−like

(a)

0 0.1 0.2 0.3 0.4 0.5

0

5

10

15

Exposed electrode thickness (mm)

Expansion rate (m/s)

filamentary

jet−like

(b)

0 0.2 0.4 0.6 0.8 1

0

5

10

Exposed electrode diameter (mm)

Expansion rate (mm)

filamentary

jet−like

(c)

0 0.1 0.2 0.3 0.4 0.5

0

5

10

Exposed electrode thickness (mm)

Expansion rate (m/s)

filamentary

jet−like

(d)

Figure 5.10: Rates at which the average microdischarges length increases for

single-barrier actuators. Lengths in the upper graphs were determined using the

edge-based algorithm for actuators with (a) cylindrical and (b) rectangular crosssection exposed electrodes. Lengths in the lower graphs were determined using the

area-based algorithm for (c) cylindrical and (d) rectangular cross-section exposed

electrodes.

5.2.2 Filamentary to Filament-Free Transition

The filament-free plasma identified in section 5.1 appeared only in single-barrier actuators with

narrow cylindrical electrodes. We observed the filament-free mode in actuators with exposed electrode diameters of 0.03, 0.05, 0.11, and 0.13 mm. The next largest electrode tested was 0.25 mm

in diameter, so the disappearance of the mode occurred as the electrode diameter increased above

approximately 0.2 mm.

The filament-free mode appeared gradually as the applied voltage rose during the positivegoing half cycle. In the initial stage of the positive-going cycle, the electric field is not strong

NASA/CR—2012-217628 86
enough to sustain a plasma, so no light is emitted. When the voltage rose sufficiently, filamentary

microdischarges began to occur. At some later point, the filament-free plasma appeared near the

exposed electrode, coexisting with the filamentary microdischarges, as in figure 5.11(a). As the

applied voltage continued to rise, filamentary microdischarges occurred less frequently, as illustrated in figures 5.11(b) and 5.11(c). Eventually the filamentary microdischarges ceased occurring,

and the “full” filament-free mode of figure 5.3 continued until the end of the positive-going half

cycle.

e

b

(a) (b) (c)

Figure 5.11: Simultaneous appearance of filament-free plasma and filamentary

microdischarges in a single-barrier actuator. Each image used a gate width of 10 μs,

but is the sum of 100 accumulated exposures. The images were recorded at (a)

0.31 ms, (b) 0.34 ms, and (c) 0.37 ms into the 0.50 ms long positive-going half

cycle of a single-barrier actuator with a 0.05 mm diameter exposed electrode.

The time at which the plasma transitioned to a completely filament-free mode was strongly dependent on the diameter of the exposed electrode. As shown in figure 5.12, the filament-free mode

occurred earlier in the half cycle for actuators with smaller exposed electrodes. The times at which

the filament-free mode initially appeared were more difficult to determine, as the dim filamentfree plasmas were sometimes obscured by the many filamentary microdischarges. Approximately,

however, the onset occurred 70 μs (∼2 kV) prior to the transition to a fully filament-free mode.

This gap between the initial appearance of the filament-free mode and the total disappearance of

filamentary microdischarges was independent of the exposed electrode diameter.

The transition to a filament-free mode is responsible for the reduction in dissipated power for

single-barrier actuators with small diameter cylindrical exposed electrodes. The actuators used less

NASA/CR—2012-217628 87
0 0.05 0.1 0.15

0

0.1

0.2

0.3

0.4

0.5

Exposed electrode diameter (mm)

Transition time (ms)

−7

−4.2

−1.4

1.4

4.2

7

Transition voltage (kV)

Figure 5.12: Times and voltages at which transition to fully filament-free modes

occur for single-barrier actuators with cylindrical exposed electrodes. 0.5 ms is the

end of the positive-going half cycle.

power, as shown in figure 4.3, when they spent a longer portion of the positive-going half cycle in

the filament-free mode. The current spikes seen during the filamentary period, as in figures 3.12

and 5.13, were largely suppressed during the filament-free mode, as shown in figure 5.13. This

lack of current spikes reduced the average discharge current and thus the dissipated power.

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

−8

−6

−4

−2

0

2

4

6

8

Time (ms)

Applied voltage (kV)

−2

−1

0

1

2

Discharge current (mA)

voltage

current

inactive

period

filament−free

period

filamentary

period

Figure 5.13: Discharge current for a single-barrier actuator with a 0.05 mm diameter cylindrical exposed electrode. The filament-free phase of the discharge had

notably fewer current spikes.

NASA/CR—2012-217628 88
5.2.3 Microdischarge Formation and Propagation

In addition to the examination of the plasma on time scales comparable to the applied voltage, as described above, we have also studied plasma behavior on the time scales of individual

microdischarges. Rather than examining different times during the applied voltage cycle, here we

fixed the examined time and varied the gate width of the image. While the collected images are of

different microdischarges, they can yield information about the time evolution of an average microdischarge. The images in this section are all of a single-barrier actuator with a 0.3 mm diameter

cylindrical exposed electrode, and a 7 kV amplitude symmetric triangle voltage at 1 kHz. We have

examined only times near the end of each half cycle. During the positive-going half cycle this was

just prior to the voltage reaching 7 kV; during the negative-going half cycle this was just prior to

the voltage reaching −7 kV. We have published the results of this section in Hoskinson et al. [98].

Since the actuator had a 0.3 mm diameter exposed electrode, only filamentary microdischarges

occurred during the positive-going half cycle. At the shortest gate width examined, 2 ns in figure 5.14(a), we observed a plasma region little more than a millimeter in length. As we increased

the gate width the plasma length increased, as shown in figures 5.14(a) through 5.14(e). This behavior suggests that filamentary microdischarges actually consist of a dense, bright plasma region

propagating along the dielectric surface. Since the camera integrates the light emitted from the

plasma, at longer gate widths the plasma has moved farther and thus appears longer.

By measuring the length of the bright plasma region as a function of camera gate width, as in

figure 5.15, we estimated the propagation velocity of a microdischarge during the positive-going

half cycle. The slope of the fitted line indicates a velocity of 3 × 104 m/s. If the microdischarge

began or ended during the image acquisition time the measured length will be too short, so we

believe the calculated velocity represents a lower bound on the actual propagation velocity. Based

on the saturation of the lengths above 100 ns gate widths, shown in figure 5.15, we estimate

100 ns to be an upper bound on the lifetime of a filamentary microdischarge in this actuator.

The propagation shown in figures 5.14 and 5.15 describes the motion of the plasma during a

microdischarge, in contrast to the much slower variations in the average plasma behavior over the

course of the applied voltage cycle which we described in section 5.2.1.

NASA/CR—2012-217628 89
e

b

(a) 2 ns (b) 5 ns (c) 15 ns

(d) 20 ns (e) 80 ns

e

b

Figure 5.14: Nanosecond gate width images of filamentary microdischarges. The

plasma length increases at larger gate widths. Gate widths are stated below each

image.

0 50 100

0

1

2

3

4

Gate width (ns)

Discharge length (mm)

1250

Figure 5.15: Plasma length increases with camera gate width during the positivegoing half cycle, suggesting plasma propagation. The dashed line shows the linear

fit used to determine the propagation velocity.

NASA/CR—2012-217628 90
In contrast to the filamentary microdischarges, the jet-like microdischarges during the negativegoing half cycle appeared to form quickly and remain in place. Figures 5.16(a) through 5.16(d)

show plasmas of essentially the same length, suggesting that the plasma did not grow or propagate

in the 5 to 1000 ns time frame. The images in figures 5.16(a) and 5.16(b), the shorter gate width

images, were much dimmer than those in figures 5.16(c) and 5.16(d). In contrast, we noted little

difference in the light intensity between images with gate widths of 20 ns and 1 μs. This suggests

that while the microdischarges form in a time on the order of a few nanoseconds, their lifetime

may be as long as 20 ns. If the microdischarge continued emitting light for a longer time we would

expect the longer gate-width images, which show total integrated light, to be brighter.

(a) 5 ns (b) 10 ns

e

b

(c) 20 ns

e

b

(d) 1000 ns

Figure 5.16: Nanosecond gate width images of jet-like microdischarges. The

plasma length remains approximately constant with camera gate width during the

negative-going half cycle. Gate widths are stated below each image.

NASA/CR—2012-217628 91
5.3 Double-Barrier Actuators

We have also examined the average behavior of microdischarges in double-barrier actuators.

These experiments were conducted on several double-barrier devices driven by 12 kV amplitude

symmetric triangle voltages oscillating at 1 kHz. As described in section 4.3, microdischarges occurred less frequently in double-barrier actuators than in the single-barrier geometry at the voltage

levels examined here. Nanosecond-scale gate width measurements were not attempted for these

actuators due to the rarity of microdischarges occurring during the exposure.

Due to the lower frequency of microdischarge occurrence, plasma appeared in a smaller fraction of captured images than when we examined single-barrier actuators. Despite this, in nearly

all of the 10 μs gate-width images in which at least one microdischarge appeared, a few to tens of

microdischarges were captured. This suggests that the distribution of microdischarges in time is

not strictly random. We hypothesize that the electrons from one microdischarge may provide the

necessary “seed” electrons in the gas for a microdischarge to occur in an adjacent space, leading to

multiple microdischarges occurring simultaneously or in rapid succession. The microdischarges

we observed in double-barrier actuators also exhibited more variation in the plasma structure from

image to image.

Similar to single-barrier devices, in double-barrier actuators we observed changes in the microdischarge structures as each half-cycle of the applied voltage progressed. Because the microdischarges in double-barrier actuators consisted simultaneously of both filamentary and jet-like

plasma structures, these changes are more complicated than simple increases in microdischarge

length. As the positive-going half cycle progressed, the filamentary structure extended further

away from the narrow electrode. Concurrently, the jet-like portion of the microdischarge increased

in length, not by extending past the narrow electrode, but by spanning a greater portion of the

electrode gap. Figures 5.17(a) and 5.17(b) show microdischarge structures early and late in the

positive-going half cycle, demonstrating both of these changes.

During the course of the negative-going half cycle, the average lengths of the jet-like portions

of the microdischarges increased. This can be clearly seen in figures 5.17(c) and 5.17(d), with the

NASA/CR—2012-217628 92
n

w

n

w

(a) (b)

(c) (d)

Figure 5.17: Changing double-barrier microdischarge structure from (a) 50% (0

V) to (b) 95% (10.8 kV) of the way through the positive-going half cycle and from

(c) 50% (0 V) to (d) 95% (−10.8 kV) of the way through the negative-going half

cycle.

jet-like structures extending further over the wide electrode later in the half cycle. The filamentary

portions of the microdischarges appeared to extend slightly further past the narrow electrode later

in the half cycle, but this was difficult to measure objectively. Unlike the positive-going half cycle,

we observed no variation in the position of the filamentary/jet-like transition during the negativegoing half cycle.

To obtain more quantitative measures of changes in the average microdischarge structure, we

performed the same averaging and length-finding procedures as in section 5.2.1. The average light

emission from a double-barrier actuator with a 0.13 mm diameter narrow electrode is shown in

figure 5.18. The increase in average microdischarge lengths is evident in this figure, although the

expansion was neither as rapid nor as uniform as in single-barrier actuators.

NASA/CR—2012-217628 93
Figure 5.18: Average light emission during the applied voltage cycle for a singlebarrier actuator with a 0.13 mm diameter narrow electrode. Each column of pixels

corresponds to the average light emission from a 10 μs period.

0 0.2 0.4 0.6 0.8 1

0

5

10

15

Narrow electrode diameter (mm)

Expansion rate (m/s)

positive−going

negative−going

(a)

0 0.2 0.4 0.6 0.8 1

0

2

4

6

Narrow electrode diameter (mm)

Expansion rate (m/s)

positive−going

negative−going

(b)

Figure 5.19: Rates at which the average microdischarge length increases for

double-barrier actuators, determined using (a) the edge-based algorithm and (b) the

area-based algorithm

NASA/CR—2012-217628 94
The edge- and area-based fitting algorithms also functioned effectively on the average light

emission curves for double-barrier actuators. Because both methods find only the positive-x

plasma edge, the measured plasma length was the length from the edge of the narrow electrode to

the plasma edge above the wide electrode. This length did not include any portion of the plasma

extending from the narrow electrode away from the wide electrode. Performing these fits and then

finding the slopes of the regions where the average microdischarge length is linearly increasing

yielded the expansion rates of figure 5.19. Unlike the single-barrier devices, we did not observe

any strong trends in the microdischarge expansion rates as the electrode diameter was varied. We

do note that the expansion rates for microdischarges during the negative-going half cycle were

always smaller than those during the positive-going cycle.

NASA/CR—2012-217628 95
Chapter 6

Computational Model Description

In order to gain additional insight into the plasma dynamics that occur on time and space scales

too short to measure experimentally, we have developed a computational model of the surface DBD

system. This model is based on the published work of Boeuf et al. [13, 14], but is an independent

implementation. In addition to the features of that model, we have added the ability to model

systems with electrodes of arbitrary shape and size. We have also, as described in section 6.2.5,

included a dynamically-adjustable time step.

Due to limitations in computational power, the model is two-dimensional. It assumes uniformity in the z direction of figures 3.7 and 3.10. The images of chapter 5 indicate that the plasma is

in fact non-uniform in this direction, which is rightly taken as a limitation of this model. Despite

this, two-dimensional plasma models by ourselves and other groups have succeeded in reproducing

numerous features of the plasma actuator system.

6.1 Physical Model

The simulation code written as part of this work is a two-dimensional fluid model. The model

is designed to simulate relatively slow processes, using Poisson’s equation to find the electrostatic

potential. Each particle species is each governed by a continuity equation. Electron-impact ionization, electron attachment, positive ion-electron recombination, and positive ion-negative ion

recombination processes are considered as the only sources and sinks. Particle transport is governed by the drift-diffusion approximation, which is valid for highly-collisional systems.

NASA/CR—2012-217628 96
6.1.1 Modeled System

The simulated system always includes a high voltage electrode, a grounded electrode, and a

rectangular dielectric layer, as in figure 6.1. The dielectric layer extends the full width of the

system, and always extends from the bottom of the domain (y = 0) to a specified height. The code

uses a single value for the dielectric constant inside of this dielectric layer, and assumes  = 0

everywhere outside of this material. The relative dielectric constant d for this layer was set equal

to 3.0 for all simulations described in this work. The electrodes may have any shape, size, and

position.

dielectric

HV electrode

grounded electrode

gas region

domain width

domain height

Figure 6.1: A generic simulation geometry including a high voltage electrode, a

grounded electrode, and a dielectric layer

The model includes only three particle species: positive ions, electrons, and negative ions.

All particles are assumed to be singly charged. A real plasma in air will contain many particle

species; the number of species here is limited to avoid sacrificing computational efficiency. Unfer

et al. [92] modeled a more detailed system including two positive ion species, one negative ion

species, electrons, and a metastable species, and found qualitatively similar behavior to the simplified chemistry. The major effect of the additional particle species was to allow the plasma to

expand further away from the exposed electrode in a single-barrier system. Our ion species are

intended to represent averages of the actual variety of ions present. The positive and negative ions

are treated as having properties equal to the respective weighted averages of the properties of the

major positive- and negative-ion species in a real plasma. This averaging process is described in

section 6.1.3. All of the simulations presented in this work used an air-like mixture of 600 Torr

(partial pressure) of nitrogen and 160 Torr of oxygen.

NASA/CR—2012-217628 97
6.1.2 Governing Equations

Poisson’s equation, given by equation (2.17), can be written for the system described above as

∇ · (∇φ) = −e(np − ne − nn) − δsσ. (6.1)

The subscripts p, e, and n indicate positive ions, electrons, and negative ions, respectively. The

surface charge density σ is evaluated only at the dielectric surface, as indicated by the delta function

δs. It is self-consistently calculated by time-integrating the charged particle flux to the surface.

Poisson’s equation is coupled to three continuity equations: one for each particle species, each

including the appropriate source and sink terms. The continuity equation was stated for a general

species α in equation (2.18), repeated here for reference:

∂nα

∂t = −∇ · Γα +

XSα −XRα, (2.18)

Here Γ is the particle flux, while S and R are source and sink terms for the particle species. The

sums are taken over all relevant processes for each species. In our simulation, positive ions are

created via electron-impact ionization, and removed by recombination with either electrons or

negative ions. New electrons are introduced via electron-impact ionization and secondary electron emission from surfaces, and removed by recombination with positive ions and attachment to

neutral particles. Negative ions are formed via electron attachment, and removed through positive

ion-negative ion recombination. Each of these processes is described further below. All particles

may also be lost to exposed material surfaces. No photoionization processes are included. Some

recent work suggests they may be required for streamer propagation during positive bias [91].

This requirement was avoided in these simulation runs by introducing an initial “seed” density of

1013 m−3electrons and the same density of positive ions, which is larger than the charged-particle

density in sea level air.

In all cases the particle transport is assumed to obey the drift-diffusion approximation given by

equation (2.5). That equation describes a vector flux; in the simulation’s algorithms, the continuity

equations are split into x and y equations. For example, the positive ion flux in the x-direction,

Γpx, is given by

Γpx = +μpExnp − Dp

∂np

∂x , (6.2)

NASA/CR—2012-217628 98
with similar expressions applied to the other directions and species. Here μ is the particle mobility,

Ex = −∂φ/∂x is the electric field strength in the x direction, and D is the particle diffusion

coefficient.

As described in section 2.2.1, electrons may be emitted from material surfaces via secondary

electron emission. In our simulation code, the total electron flux near a surface is given by

~Γe = ~Γ

bulk

e − γ~Γp, (6.3)

where γ is the secondary electron emission coefficient of surface. The bulk electron flux in equation (6.3) is simply that calculated using the appropriate drift-diffusion equation. The minus sign

indicates that secondary electron flux away from a surface is proportional to the ion flux into that

surface. Equation (6.3) is evaluated only for ions accelerated into a surface by the local electric

field. For positive ions this is when the electric field is directed into the surface. Ions diffusing

to the surface against the electric field are assumed to not have enough energy to “knock out”

electrons.

The simulation code accepts two constant values for secondary electron emission coefficient:

one for electrode surfaces, and one for the dielectric surface. For all results described in this

work, we used values of 0.05 secondary electrons per incident ion for both surfaces. This allowed

us to to compare our results more directly to those in the literature [e.g. 13], where the same

values were used. Actual coefficients are dependent on many factors including the energy of the

bombarding particle, the species of the bombarding particle, the surface temperature, and other

surface conditions. We were unable to find empirical data useful for parameter ranges relevant for

our system.

The rates of electron-impact ionization and electron attachment are calculated using equations (2.10) and (2.11), respectively, with the coefficients treated as functions of the local reduced

electric field:

Siz = α (E/p)







~Γe







and Satt = η (E/p)







~Γe





 .

The functional forms of both the ionization and attachment coefficients are described in section 6.1.3. Similar to secondary electron emission, both ionization and attachment are evaluated

NASA/CR—2012-217628 99
only where the electron flux is accelerated by the local electric field. Electrons diffusing against

the field are assumed to not have enough energy to attach to or ionize neutral particles.

Recombination rates, both between positive ions and electrons (pe) and between positive ions

and negative ions (pn), are treated as proportional to the local densities of the recombining species,

similar to the models of Tuszewski and White [99] and Boeuf and Pitchford [13]. For positive ions

and electrons,

Rpe = rpenpne,

where rpe is the relevant recombination coefficient. Positive ion-negative ion recombination is

handled similarly.

6.1.3 Physical Parameters

Many of the physical parameters used in the simulation code are in reality functions of particle

velocity or energy distribution functions. In order to limit the amount of computational effort

devoted to calculating these parameters, we have treated them as functions of the local reduced

electric field, E/p. This method is sometimes known as the local field approximation (LFA). In

the simulation, all parameters are calculated from analytic fits to published data. Where possible

the analytic functions have been chosen to follow known scaling laws; in most cases, however, the

functions are chosen purely for their good fit to the available data.

Particle mobilities are based on fits to empirical data from several sources [18, 19]. For positive

ions, the scaled mobilities, in m2 Torr V−1s

−1

, are approximated by

μp p = 0.05492 exp 

−E/p

6858 

+ 0.07509 exp 

−E/p

38175 

+ 0.0308 for N2 (6.4)

and

μp p = 0.06841 exp 

−E/p

59678 

+ 0.09194 exp 

−E/p

12763 

+ 0.0320 for O2, (6.5)

where p is the total gas pressure in Torr and E/p is the reduced total electric field in V m−1 Torr−1.

Mobilities are calculated separately for the x and y directions, using slightly different values for

NASA/CR—2012-217628 100
102

103104105

0

1

2

x 10−4

E / p (V / m / Torr)

Mobility (m

2 / V / s)

Siglo database

Basurto 2000

model

Figure 6.2: Mobility of N+

2

in N2 from two sources [18, 19] and equation (6.4), all

for atmospheric pressure nitrogen gas at room temperature

the total electric field as described in section 6.2.1. Positive ion mobilities in nitrogen are shown

in figure 6.2, while those in oxygen are in figure 6.3.

For electrons, we approximated the scaled mobilities by fitting to empirical data from the Siglo

database [19], using

μe p = 24.32 exp 

−E/p

1057 

+ 19.38 exp 

−E/p

23430 

+ 14.45 (6.6)

in N2 and

μe p = 173.1 exp 

−E/p

195.1



+ 36.19 exp 

−E/p

3134.4



+ 31.73 exp 

−E/p

18205 

+ 12.49 (6.7)

in O2. Electron mobilities in both nitrogen and oxygen are shown in figure 6.4. Few data are

available for the mobility of negative oxygen ions in any gas, so we use a scaled average mobility

102103104105

0

1

2

x 10−4

E / p (V / m / Torr)

Mobility (m

2 / V / s)

Siglo database

Basurto 2000

model

Figure 6.3: Mobility of O+

2

in O2 from two sources [18, 19] and equation (6.5), all

for atmospheric pressure oxygen gas at room temperature

NASA/CR—2012-217628 101
102

103104105

0

0.05

0.1

0.15

0.2

E / p (V / m / Torr)

Mobility (m

2 / V / s)

Siglo database − O2

model − O2

Siglo database − N2

model − N2

Figure 6.4: Mobilities of electrons in N2 and O2 [19], plotted with equations (6.6)

and (6.7). Both gases are assumed to be at atmospheric pressure and room temperature.

which is constant with respect to the electric field [19]:

μnp = 0.181125 m

2 V−1

s

−1

. (6.8)

For simulations with multiple gases present we determine effective mobilities by averaging the

calculated mobilities for each gas, weighting each value by the partial pressure of the parent gas:

μeff =

1

p

2



pN2 (μN2p) + pO2 (μO2p)



, (6.9)

where pN2 and pO2 are the partial pressures of nitrogen and oxygen respectively. This is not a true

effective mobility for ions, as the mobilities of the ions in their non-parent gas is not taken into

account. Perhaps more importantly, the proportion of positive ion belonging to each species is

unknown. For air-like mixtures, we would expect oxygen ions to be much more prevalent, as the

ionization coefficient for oxygen is larger than that for nitrogen. Despite these uncertainties, the

use of equation (6.9) likely introduces less error than the other simplifications we have employed.

The drift-diffusion equations also require knowledge of the diffusion coefficients for all particles. The simulation code applies the Einstein relation, equation 2.7, to determine these coefficients, using constant particle temperatures. In this work, we have chosen values appropriate for a

low-temperature non-equilibrium plasma, setting Te equal to 1.0 eV, and both ion temperatures to

room temperature: Tp = 0.025 eV and Tn = 0.025 eV.

NASA/CR—2012-217628 102
The ionization coefficient for N2 is approximated by

α

p

(N2) =







4.71 × 10−11 |E/p|

3

, |E/p| < 1.4 × 104 V m−1Torr−1

3.32 (|E/p| − 12500)1/2

, |E/p| > 1.4 × 104 V m−1Torr−1

, (6.10)

where α/p is in m−1 Torr−1, and E/p is in V m−1 Torr−1. This was obtained from a fit to the

experimental data compiled in several sources [19, 100]. The ionization coefficient for O2 has

been obtained similarly from a fit to the experimental data compiled in in the Siglo database [19],

using the same functional form as the N2 coefficient at large E/p:

α

p

(O2) =







1.17 × 10−10 |E/p|

3

, |E/p| < 1.1 × 104 V m−1Torr−1

0.0319|E/p| − 211, 1.1 × 104 < |E/p| < 2.1 × 104

6.32 (|E/p| − 16300)1/2

, |E/p| > 2.1 × 104 V m−1Torr−1

. (6.11)

The analytic approximations and experimental data for both gases are shown in figure 6.5.

103104105

0

1000

2000

E / p (V / m / Torr)

α / p (1 / m / Torr)

Siglo − O2

model − O2

Siglo − N2

Posin 1936 − N2

model − N2

Figure 6.5: Reduced ionization coefficients in both nitrogen and oxygen

When both gases are present, the ionization coefficients are averaged according to

αeff =



α

p

(N2)



p(N2) + 

α

p

(O2)



p(O2). (6.12)

Based on a fit to the simulated data compiled in the Siglo database [19], we use

η ≈

"

1.307 +

33200

|E/p|

exp 

− (ln |E/p| − 9.04)2

2.53 !# pO2,

NASA/CR—2012-217628 103
102

103104105

0

5

10

E / p (V / m / Torr)

η / pO2 (1 / m / Torr)

Siglo database

model

Figure 6.6: Reduced attachment coefficient for oxygen

to approximate the attachment coefficient in oxygen. Here η is in m−1and pO2 is in Torr. This fit

to the available attachment data is shown in figure 6.6. Nitrogen is an electropositive gas, so its

attachment coefficient is always zero.

For our model, a value of 10−13 m3/s is used for the recombination coefficients rpe and rpn

[14, 99]. Both coefficients are assumed to be constant.

6.1.4 Range of Validity

Many of the simplifications used in this model stem from assumptions of high collisionality and a neglect of self-induced magnetic fields. We discussed this highly-collisional regime in

section 2.1.1, and the simplifications that regime allows throughout section 2.1. The neglect of

magnetic field terms is discussed below.

The drift-diffusion equations and the local field approximation (LFA) require that charged particles are in equilibrium with the local electric fields. To satisfy this requirement, the plasma must

be highly collisional as described in section 2.1.1. In particular, particle energy relaxation must

occur on a scale shorter than the Debye length:

λ < λD (6.13)

The Debye length was given by equation 2.1. Boeuf [101] has suggested that when this criterion is

not satisfied, the effect of applying the LFA is to overstate the ionization rate in regions of strong

electric field and to understate it in the bulk plasma.

NASA/CR—2012-217628 104
To estimate the validity of our simulation runs, we have examined two simulated situations

in each of two simulation runs in detail. The simulations were of a positive-going voltage and a

negative-going voltage applied to a reduced-scale single-barrier actuator with a 25 μm diameter

exposed electrode, and we examined times both during and in between microdischarge events.

Details regarding the simulated system are included in section 6.4, while other results from these

runs are described in sections 7.1 and 7.2. Due to their low mobilities and short energy relaxation

lengths, the ions were always highly collisional, so we focus on the collisionalities of the electrons.

During microdischarge events for both applied voltages, electron densities peaked around

1021 m−3. The electric field strengths peaked around 8 × 107 V/m at approximately the same

locations as the maximum densities. Using the assigned electron temperature of 1 eV, the minimum Debye lengths were on the order of 2 × 10−7 m. In such dense, high-field regions, ionization

processes are the dominant mechanism of energy transfer. From equation (2.3), energy relaxation

lengths for electrons in such regions are equal to the ionization length, which can be calculated

using equations (6.10), (6.11), and (6.12). We found an ionization length on the order of 10−6 m,

making it notably longer than the Debye length. The ratios of energy relaxation lengths to Debye

lengths were approximately constant throughout the length of the plasma channel. Consequently,

our description of the plasma behavior during microdischarges is not strictly accurate. The overestimation of ionization in this region suggested by Boeuf [101] may not be as problematic as might

initially be expected, as it will help compensate for the lack of photo-ionization in the model,

which would primarily occur in regions of dense plasma.

During the build-up phases prior to microdischarge events for both applied voltages, the plasma

densities and electric field strengths were lower than their values during microdischarges. The

maximum electron densities were on the order of 5×1017 m−3, corresponding to Debye lengths on

the order of 10−5 m. The electric field strengths were on the order of 107 V/m. At this field strength,

both ionization and elastic collision processes are important. We determined the ionization length

as described above, and calculated the energy relaxation length due to elastic collisions from the

mobility using equations (2.6) and (2.2). The effective total energy relaxation length was on the

order of 6 × 10−6 m, or somewhat shorter than the Debye length. If excitation processes were

NASA/CR—2012-217628 105
included, the energy relaxation length would be even smaller. Consequently, we conclude that

electrons are highly collisional in between microdischarge events, and criterion (6.13) is satisfied

in these conditions.

Even when the above criterion is satisfied, the local field approximation (LFA) may still not

be strictly valid, particularly in the determination of the ionization coefficient. The distribution

function for electrons will, even in steady-state, be non-smooth. Certain energy ranges will be

depleted by inelastic excitation collisions and collisions with material boundaries. These effects

can only be completely accounted for with a kinetic treatment [102]. The LFA is employed in this

work due to its large advantages in computational efficiency.

The use of Poisson’s equation as the sole description of the electromagnetic physics imposes

an additional restriction on the simulation. In using it, we have explicitly ignored any magnetic

field effects. These effects become significant when the Lorentz force can change the direction of

a particle’s trajectory before it experiences a collision. Electrons will be more strongly affected by

a magnetic field than heavy ions, so we write the restriction for them:

ωce  νeg, (6.14)

where ωce = e|B~ |/me is the electron-cyclotron frequency in a magnetic field B~ , and νeg is the

electron-neutral collision frequency. Stated more concisely, magnetic field effects on transport can

be ignored when many collisions prevent a particle from gyrating in the magnetic field [17].

Since we assume no background magnetic field, and applied voltage varies relatively slowly

for all simulations described here, induction by plasma current is the dominant source of magnetic

fields. The various simulations described in chapter 7 had peak plasma currents between about

20 and 200 A/m, calculated as described in section 6.3. These currents flowed in thin channels of

plasma, so it is reasonable to describe them as sheet currents. Doing so allows us to calculate the

induced magnetic fields, which would range up to about 10−4 T. Corresponding electron-cyclotron

frequencies are of the order 107s

−1

. Using equation (2.6) and the data of section 6.1.3, we estimate

the minimum electron-neutral collision frequencies to be of the order 1012 s

−1

. Consequently,

condition (6.14) is well satisfied at all times during the simulations in this work. In real systems,

as shown in chapter 5, the plasma current sometimes constricts into narrow filaments, which may

NASA/CR—2012-217628 106
increase the local magnetic field. This behavior cannot, however, be captured in a 2-dimensional

simulation.

6.2 Numerical Methods

The simulation code first solves Poisson’s equation and then solves the continuity equations

using semi-implicit methods. The simulation domain is discretized in both space and time, with

the equations solved iteratively at each time step. The numerical methods used in the code are first

order accurate in both space and time. The solution methods generally follow those of Boeuf et al.

[14], with adaptations to allow arbitrary electrode geometries and a dynamic time-step size. We

have implemented the code primarily in MATLAB®, with several of the most computationallyintensive functions written in C.

This section uses the following notation. Subscripts i and j indicate that a quantity is evaluated

at the i

th grid index in the x direction and the jth grid index in the y direction. A superscript m

indicates that a quantity is evaluated at the mth time step of the simulation. The length of a time

step is ∆t, which is set by the dynamic time-stepping adjustment described in section 6.2.5.

6.2.1 Numerical Grid

The model imposes a Cartesian grid onto the simulation region described in section 6.1.1. The

intersection points of the grid are conventionally specified by integer indices i and j in the x and

y directions, respectively. For all of the simulations here, the grid spacing sizes ∆x and ∆y were

equal. They are, however, carried through separately in some of the derivations below and in

appendix B.

Different quantities are defined at different locations on the numerical grid, as shown in figure 6.7. All particle densities n are located at integer grid indices (i, j), as are the values of the

potential φ. The x-directed components of the particle fluxes and electric field are located at halfinteger indices in x and integer indices in y, i.e. at (i +

1

2

, j). Conversely, y-directed components

are located at (i, j +

1

2

): integer indices in x and half-integer indices in y.

NASA/CR—2012-217628 107
(i, j)

n , n , n , φ

(i + , j)

Γ , Γ , Γ , E

(i, j + )

Γ , Γ , Γ , E

p

px ex nx

py ey ny

e n

x

y

1

2

1

2

Figure 6.7: Portion of the numerical grid, noting the quantities defined at different

locations

Many of the physical parameters discussed in section 6.1.3 are functions of the electric field

magnitude. The magnitudes, which are required at each integer and half-integer grid point, are

calculated as







E~







i,j

=

r

1

4



Exi+ 1

2

,j + Exi− 1

2

,j2

+

1

4



Eyi,j+ 1

2

+ Eyi,j− 1

2

2







E~







i+ 1

2

,j

=

r

Exi+ 1

2

,j2

+

1

16



Eyi+1,j+ 1

2

+ Eyi+1,j− 1

2

+ Eyi,j+ 1

2

+ Eyi,j− 1

2

2

(6.15)







E~







i,j+ 1

2

=

r

1

16



Exi+ 1

2

,j+1 + Exi+ 1

2

,j + Exi− 1

2

,j+1 + Exi− 1

2

,j2

+



Eyi,j+ 1

2

2

.

Electrode and dielectric geometries are defined via their positions on the grid. Grid points

included in electrode definitions are treated as fully inside the electrode; the electrode surface lies

somewhere between the included point and the adjacent grid point. The dielectric layer is defined

by a single value: the y index of its horizontal surface. All grid points below that y index are inside

the dielectric, while all points above are in a gaseous region. Points at that index lie precisely on

the dielectric surface.

6.2.2 Scharfetter-Gummel Discretization

The numerical stability of both the Poisson’s equation solver and continuity equation solver

depend on the use of the Scharfetter-Gummel discretization of the particle fluxes [103]. This discretization is a type of upwinding scheme that is commonly employed in semiconductor physics

NASA/CR—2012-217628 108
simulations. While the scheme succeeds in removing the grid-based instabilities that would otherwise appear, it does so at the cost of reducing the algorithm spatial accuracy to first order and

allowing numerical diffusion at larger spatial step sizes. One derivation is described below.

The particle fluxes are described by the drift-diffusion approximation, given by equation (2.5).

Since the flux is defined at half-integer indices, we enforce the equation there. For transport in the

x direction,

Γx









i+ 1

2

,j

= ±μEx









i+ 1

2

,j

n − D

∂n

∂x









i+ 1

2

,j

,

with the plus sign taken for positive ions and the minus sign for electrons and negative ions. Since

the particle densities are defined at integer grid indices, the density at (i +

1

2

, j) is unknown. In

other situations this density would be obtained by averaging the values at i and i + 1, but here

this procedure leads to a numerically unstable algorithm except for small electric fields on fine

grids [103].

Instead, the above equation is solved as a first-order ordinary differential equation in n. In doing so we treat the flux, electric field, mobility, and diffusion coefficient as approximately constant,

relative to the density changes, in the region between i and i + 1. The solution, applying boundary

conditions of the “known” values of n at i and i + 1, is

ni+1,j =

(Γx)

i+ 1

2

,j

(μEx)

i+ 1

2

,j

+

"

npi,j −

(Γx)

i+ 1

2

,j

(μEx)

i+ 1

2

,j #

exp "±

(μEx)

i+ 1

2

,j

Di+ 1

2

,j

∆x

#

.

Solving for the flux in a symmetric form yields the Scharfetter-Gummel discretization:

Γx









i+ 1

2

,j

=

Di+ 1

2

,j

∆x

 z

i+ 1

2

,j

1 − e

z

i+ 1

2

,j 

ni+1,j +

Di+ 1

2

,j

∆x

 

z

i+ 1

2

,j

1 − e

−z

i+ 1

2

,j !

ni,j (6.16)

where

z

i+ 1

2

,j =

± (μEx)

i+ 1

2

,j

Di+ 1

2

,j

∆x =

±Exi+ 1

2

,j

T

∆x. (6.17)

Similar expressions exist for y-directed fluxes. We use expressions like equation (6.16) for the

particle fluxes in solutions to both Poisson’s equation and the continuity equations. No averaging

is required to find any of the values in the above equations with the grid defined as described in

section 6.2.1. As before, the plus sign is taken for positive ions and the minus sign for electrons and

NASA/CR—2012-217628 109
negative ions. In each case, the appropriate parameter values must be used for each species. We

note that equation (6.16) reduces to the correct limits, namely the diffusive flux for small electric

fields, using

limz→0



z

1 − e

z



= −1,

and the drift flux for large electric fields.

6.2.3 Poisson’s Equation Solver

Poisson’s equation, given in equation (6.1) for this system, requires knowledge of the charge

distribution. Finding the potential at a new time step m + 1 then requires knowledge of the charge

distribution at time m + 1. Since Poisson’s equation is solved prior to the continuity equations,

the charge at m + 1 is not known. A fully explicit solution would use the charge densities of the

previous time step. This technique, however, restricts the amount the charge is allowed to change

during a given time step, limiting the maximum time step to the dielectric relaxation time [104]:

∆t < 0/σp,

where σp is the total plasma conductivity (which depends on the particle densities and mobilities).

Since the time scales of interest in plasma actuators are generally many orders of magnitude larger

than the dielectric relaxation time, this restriction makes explicit simulations impractical for our

desired uses of the simulation. An alternate technique, used here, is to make a prediction of the

charge density at time m + 1. Using a first-order Taylor expansion of the densities at m + 1 allows

us to rewrite Poisson’s equation in the plasma bulk as

0∇2φ

m+1 = −e



n

m

p − n

m

e − n

m

n + ∆t



∂np

∂t −

∂ne

∂t −

∂nn

∂t  . (6.18)

The density time derivatives are obtained from the set of continuity equations, given by equation (2.18). Since that set conserves net charge, all of the source and sink terms exactly cancel,

leaving only the divergences of the fluxes:

∂np

∂t −

∂ne

∂t −

∂nn

∂t = −∇ · Γp + ∇ · Γe + ∇ · Γn. (6.19)

NASA/CR—2012-217628 110
Each of these fluxes is evaluated using the drift-diffusion equation, given by equation (6.2). The

solver routine uses the semi-implicit method suggested by Ventzek et al. [105], in which potential

is taken at time m+ 1 while all densities use the known values at time m. For the positive-ion flux,

this is written as

~Γpx ≈ +np

mμpE~

x

m+1

− Dp

∂

∂x (np

m),

with similar expressions for the other fluxes. Each of the flux equations is then discretized using

the Scharfetter-Gummel method, given by equation (6.16), and inserted into equation (6.19). The

result is then substituted back into equation (6.18), yielding an equation depending only on potentials at time step m + 1 and densities at time step m. That equation can then be discretized using

standard second-order finite difference approximations for the spatial derivatives. We rearrange it

into the form

φ

m+1

i,j = C1(i, j)φ

m+1

i+1,j + C2(i, j)φ

m+1

i−1,j + C3(i, j)φ

m+1

i,j+1 + C4(i, j)φ

m+1

i,j−1 + C5(i, j), (6.20)

where each C term depends only on quantities evaluated at time m. The derivation of equation (6.20) is described in greater detail in Appendix B.1, including corrections necessary at the

dielectric surface and to take secondary electron emission into account. We solve equation 6.20

using a successive over-relaxation method [106] with a relaxation parameter of 1.9. The iteration

is continued until the maximum change in the potential is less than a specified error value. The

chosen error values for each simulation run are given in table 6.1.

6.2.4 Continuity Solver

The particle continuity equations, given by equation (2.18), are solved after the solution of

Poisson’s equation. They can be solved at time m + 1 in a more straightforward manner than

Poisson’s equation, because the potential at time m + 1 is already known. The flux terms are directly evaluated at time m + 1 using the Scharfetter-Gummel discretization of the drift-diffusion

equation. The source and sink terms described in section 6.1.2 depend non-linearly on the particle

densities, and so cannot be directly included in an implicit solution scheme. Instead, they are evaluated using quantities from time step m, which is discussed further below. The general continuity

NASA/CR—2012-217628 111
equation evaluated at time m + 1, as implemented in the continuity solver, is

∂n

∂t









m+1

i,j

= −∇ · Γ

m+1 +

XS

m −

XR

m. (6.21)

The time derivative in equation (6.21) is evaluated using the first-order backwards difference

formula, so this method is first-order accurate in time. Once the flux expressions and source term

values are substituted into equation (6.21), and finite difference formulas have been applied to the

derivatives, it can be solved for the the particle density at m+1. This solution is described in detail

in Appendix B.2. For each particle species, we find an equation of the general form

nα

m+1

i,j = C1(i, j)nφm+1i+1,j + C2(i, j)n

m+1

i−1,j + C3(i, j)n

m+1

i,j+1 + C4(i, j)n

m+1

i,j−1 + C5(i, j). (6.22)

We solve equation (6.22) for each particle species using the Gauss-Seidel method [106], with the

process continuing until no relative change in a density value ∆n/n exceeds a specified error in a

single iteration. For all of the simulations described here, that error was 10−3.

Models for source and sink processes were described in section 6.1.2. While the evaluations

of secondary electron emission, positive ion-electron recombination, and positive ion-negative

ion recombination are straightforward, the calculations of ionization and attachment rates bear

additional comment. Both of these rates depend on the electron flux. Fluxes, as described in

section 6.2.1, are defined at half-integer spatial indices. Equation (6.21), however, requires the

ionization and attachment rates at integer spatial indices in order to update the particles densities.

To determine the appropriate fluxes for these rates, we simply average the nearby fluxes, using







~Γe







i,j

≈

vuut

1

4

 

Γex









i+ 1

2

,j

+ Γex









i− 1

2

,j!2

+

1

4

 

Γey









i,j+ 1

2

+ Γey









i,j− 1

2

!2

.

Since any averaging scheme can increase the effective numerical diffusion, this is not necessarily

the ideal solution. At present not enough work has been done analyzing the effects of this averaging

on simulation behavior to determine whether a different method would be more desirable.

NASA/CR—2012-217628 112
6.2.5 Adaptive Time-Stepping

The Poisson’s equation solver, as described in section B.1.1, uses first-order Taylor expansions

to linearize the expressions for particle fluxes. This expansion requires









∆E∆x

Te









 1 where









E∆x

Te









< 1 and

∆E

E

> −1 where









E∆x

Te









> 1.

(6.23)

Here we have used the electron temperature for T. While the accuracy of the Taylor expansions

strictly require that condition (6.23) be satisfied for each particle species, the mobility of electrons

is so high that electron motion affects the calculated potentials for much smaller electric fields

than do ion motions. In its computations, the solver routine uses an arbitrary value of 0.5 as the

threshold for acceptably small values of |∆E∆x/Te|. This value represents a compromise between

mathematical accuracy and computation speed.

In order to ensure that condition (6.23) is satisfied during each time step, while still maintaining reasonable computation times, we have implemented an adaptive time-stepping scheme. Our

simulation runs exhibit short periods of rapid change during microdischarges interspersed with

longer periods of plasma decay, as shown in figure 2.14. With an adjustable time-step size, the

simulation can take small steps when the plasma evolves quickly and large steps when the system

is relatively stable, as shown in figure 6.8.

132.75 133 133.25

Time (μs)

Current (a.u.)

10−11

10−10

10−9

10−8

∆t (s)

Figure 6.8: Dynamic time step adjustment during a single microdischarge event

NASA/CR—2012-217628 113
In the code, the appropriate time-step size to ensure that condition (6.23) is satisfied is estimated at the beginning of each simulation loop and later verified. The estimation is based on how

well condition (6.23) was satisfied during the previous time step. If the minimum values of either

∆E/E or −af |∆E∆x/Te| (where af is a user-set adjustment factor) are more negative than a set

threshold, ∆t is reduced. If the value is smaller than the threshold / 20, ∆t is increased. Immediately following the solution of the Poisson equation, compliance with condition (6.23) is checked

for the current time step. If it has been violated, ∆t is reduced and the solution is repeated. This

repetition reduces the computational efficiency, and is strongly affected by the chosen value of the

adjustment factor af . Based on several test runs, we chose adjustment factors of 1 for runs with

positive-going applied voltages and 2 for runs with negative-going voltages.

The threshold described above is chosen based on the current size of the time step. We have

chosen a functional form for this threshold of

threshold = −



∆t

∆tmin[1/(log10 ∆tmin−log10 ∆tmax)]

,

which is plotted in figure 6.9. The parameters ∆tmin and ∆tmax are described below. The form

was chosen to make ∆t “easier” to decrease when large time steps are being taken and “harder” to

decrease when ∆t is already relatively small.

The code restricts the range in which ∆t is allowed to vary to between specified values of

∆tmin and ∆tmax. If ∆t is allowed to increase too far, details of the plasma evolution can be lost

before the simulation has a chance to “slow down” by decreasing the time-step size. In the extreme

10−11 10−10 10−9 10−8

−1 

−0.8 

−0.6 

−0.4 

−0.2 

0 

∆t (s)

Threshold level

Figure 6.9: The sliding threshold used to estimate new time-step sizes

NASA/CR—2012-217628 114
case of an unbounded ∆t and a time period of stable plasma behavior, ∆t has been observed to

increase until the majority of the simulation occurs during a single time step. While maintaining

strict accuracy would require no minimum bound on ∆t, in practice this can result in long (and

unpredictable) computation times. A minimum bound yields shorter computation times by sacrificing some amount of accuracy. The maximum and minimum bounds used for each simulation

run are described in section 6.4.

6.2.6 Boundary Conditions

At the four edges of the simulation region, symmetry boundary conditions are applied to the

potential as well as to all particle densities. This condition on φ is equivalent to setting the component of the electric field perpendicular to the boundary to zero. The conditions on the densities are

equivalent to setting the diffusive flux of each species out of the system to zero. With no electric

field, the drift flux is also zero, so the combination of boundary conditions effectively prevents all

particles from leaving the system.

At grid cells inside the electrodes and dielectric all particle densities are set to zero. At grid

cells on the dielectric surface the particle densities are also set to zero, but a non-zero surface

charge density is permitted. At each time step, the surface charge density σ is calculated using

σ

m+1

i = σ

m

i + ∆t

 

−Γpy









m+1

i,js+ 1

2

+ Γey









m+1

i,js+ 1

2

+ Γny









m+1

i,js+ 1

2

!

, (6.24)

where the js +

1

2

index indicates that these are the fluxes just above the surface of the dielectric,

which is located at the y index js.

As described in section 6.1.2, secondary electron emission is considered from all surfaces.

The electrodes are allowed to have arbitrary geometries, with the flux of emitted electrons always assumed to be normal to the local surface. This calculation is described in more detail in

section B.1.3.

At grid cells inside the electrodes, the potential is set to a given value (either the applied voltage

or ground). At grid cells inside the dielectric, the potential is calculated normally, save with the

dielectric constant adjusted to 0d, where d is the relative coefficient of the dielectric material.

NASA/CR—2012-217628 115
The potential at the dielectric surface is calculated by treating the dielectric constant on the surface

is equal to 0(1 + d)/2 and applying Poisson’s equation. This calculation includes the surface

charge, and is described in more detail in section B.1.6.

6.2.7 Limitations

In addition to the restrictions discussed in section 6.1.4, the discretization of the spatial domain

adds additional restrictions and innacuracies to the model. In order to ensure that the drift-diffusion

approximation and the local field approximation are satisfied at every grid point, the simulations

should satisfy the restriction that

λ < ∆x.

This restriction ensures that each particle is in equilibrium with the applied field by the time it

passes the next grid point. A particle in a system which violated the restriction should in reality

increase in velocity and energy as it passed by each successive grid point until it finally reaches the

mobility-limited flow velocity of μE~ . In such a system, the drift-diffusion approximation would

yield larger fluxes at some grid points than would be predicted by a more accurate physical model.

In spite of this, decreasing the grid spacing does not reduce the global accuracy of the system.

Rather, the stated condition places a limit on the effective spatial resolution of the system: there is

no benefit to reducing ∆x below the energy relaxation length. Any real plasma structures at smaller

spatial scales would be removed by the application of the drift-diffusion approximation. Using

the energy relaxation lengths calculated in section 6.1.4, we see that this condition is marginally

satisfied in our full-scale simulations. The finer grid spacing of the reduced-scale simulations

provides slightly more resolution than is needed to resolve simulated plasma structures.

Where the electric field varies rapidly, strict accuracy requires

∆x < λD. (6.25)

While the solution to the Poisson equation on a spatial grid is approximately correct even when

this condition is not satisfied, it will spatially average peak electric fields. This will have relatively

little effect on the drift-diffusion approximation, which depends linearly on the electric field. Some

NASA/CR—2012-217628 116
of the source and and sink terms, however, depend non-linearly on the electric field. The additional

spatial averaging of the electric field will in general decrease the ionization constant and increase

particle particle mobilities, but how these changes will affect the source and sink rates will depend

on the details of the situation. As with restriction (6.13), we see that due to the large densities

during microdischarge events condition (6.25) is only satisfied between microdischarge events.

Considering criterion (6.25) as well as the validity criteria discussed in section 6.1.4, we see that

the model is generally accurate during the plasma build-up and decay phases in between microdischarge events, but loses some accuracy when simulating the strong electric fields and high plasma

densities present during the microdischarges.

The choice of numerical methods also affects the accuracy of the results. The use of the

Scharfetter-Gummel scheme for determining the particle fluxes adds non-physical diffusion to

the simulated system [92]. The additional diffusion likely reduces the density in the front of a

streamer-like microdischarge, among other effects. The use of solution algorithms that are first

order in both space and time also reduces the accuracy of the results. The use of dynamic timestepping helps to reduce the errors caused by large time steps. Boeuf et al. [14] examined the

effects of increasing the spatial step size, finding that error levels were acceptable at step sizes of

7 μm and smaller.

6.3 Measured Quantities

Typical simulation runs involve densities and potentials defined at about 5 × 105 grid points,

evaluated at each of about 106time steps. Recording all data at each time step would have required

prohibitive amounts of storage capacity. Instead, we primarily recorded and examined a small

number of spatially- or temporally-averaged quantities as described in this section.

As described in section 2.1.3, the plasma in a dielectric barrier discharge plasma actuator exerts

a force on the background gas. This force is calculated at each point using equation (2.9):

~fEHD ≈ e(np − ne − nn)E. ~ (2.9)

NASA/CR—2012-217628 117
Since the electric fields are defined at half-integer grid indices while the densities are defined at

integer grid indices, we shift the electric fields by averaging adjacent values. The EHD forces

in each direction are spatially-integrated into linear force densities and recorded as functions of

time. The results presented in chapter 7 deal with force densities in the x direction unless noted

otherwise.

The simulation code calculates the linear current density, in Amperes per meter, supplied by

an external circuit to each electrode in the simulation. Current to an electrode is calculated as the

sum of the direct particle current incident on the electrode and the change in the electrode surface

charge necessary to maintain the specified voltage. The former can be calculated directly from the

particle fluxes. The latter is determined by applying the 2-dimensional form of Gauss’s law:

I

E~ · d

~` = Q/Lz,

where Q/Lz is the linear charge density on the electrode, and the integral is taken around the

surface of the electrode. The change in Q/Lz over one time step is the capacitive portion of the

circuit current. If an electrode extends beyond the simulation boundaries, only the current to the

portions inside the simulated region is calculated. While in an ideal system the current to each of

the two electrodes would be identical, in the finite domain the two currents may differ slightly due

to the conditions imposed at domain boundaries. In our simulations, the current to the high voltage

electrode was always less noisy than that to the grounded electrode. All current data in chapter 7

are the currents to the high voltage electrode. We have benchmarked the current calculation to the

analytic current in a purely capacitive system.

In order to provide a rough picture of the plasma behavior over time, the code also periodically

determines the average particle density for each species and estimates the locations of the maximum particle densities. The average particle densities are smaller for larger simulation domains

and for systems with large exposed electrodes. Consequently, we will consider only trends in the

average density behaviors, and neglect differences in the absolute values. The positions of maximum particle densities are also recorded, and allow a rough estimate of the location of the front of

streamer-like microdischarges.

NASA/CR—2012-217628 118
6.4 Description of Simulated Discharges

For both single- and double-barrier geometries, we have conducted two series of simulations:

full scale and reduced scale. The dimensions used in the full-scale simulations were chosen to

be similar to those of the actuators we have studied experimentally. Reduced-scale systems used

spatial dimensions a factor of 10 smaller than the full-scale dimensions. The reduced-scale systems

were initially studied for their relatively short computation times, and have also provided insight

into the effects of changing the geometric scale as described in section 7.2.

The simulated single- and double-barrier geometries are shown in figure 6.10. For both geometries, we chose to specify a separation between the left-hand edge of the domain and the center of

the high voltage electrode rather than the length of the grounded electrode. This choice was made

to ensure sufficient separation between the high voltage electrode and the domain boundary to

avoid edge effects for all chosen electrode sizes. The dimensions and simulation parameters for

these series of simulations are given in table 6.1. The only significant difference between the experimental actuator dimensions and those of the simulated systems was the thickness of the buried

and wide electrodes. In the experiments, these electrodes were always 36 μm thick. In the simulations, these electrodes were originally specified to extend to the bottom of the simulation domain

for simplicity, and this geometry was maintained through new systems to allow easier comparisons with the older data. This lead to the use of the grounded electrode thicknesses specified in

table 6.1, which for the full-scale systems are significantly larger than in the experiments.

electrode gap

x

y

buried

electrode

thickness

dielectric

thickness

exposed electrode

diameter

left edge separation

(a)

electrode gap

x

y wide

electrode

thickness

dielectric

thickness

narrow electrode

 diameter

left edge separation

(b)

Figure 6.10: Schematics of the simulated (a) single-barrier and (b) double-barrier

actuators, illustrating the dimensions specified in table 6.1

NASA/CR—2012-217628 119
Single barrier Double barrier

reduced-scale full-scale reduced-scale full-scale

Domain width 2.0 mm 18 mm 2.0 mm 12 mm

Domain height 0.3 mm 1.5 mm 0.3 mm 2.0 mm

∆x = ∆y 4 μm 7 μm 4 μm 7 μm

∆tmax 10−8s 10−8s 10−8s 2 × 10−8s

∆tmin 10−11 s 10−11 s 10−11 s 10−11 s

Absolute allowed error in φ 0.1 V 1.0 V 0.1 V 1.0 V

Relative allowed error in n 10−3 10−3 10−3 10−3

Voltage ramp rate ±50 V/μs ±150 V/μs ±100 V/μs ±200 V/μs

Diameter range 10–100 μm 0.1–1 mm 10–100 μm 0.1–1 mm

Grounded electrode thickness 10 μm 0.25 mm 50 μm 0.50 mm

Dielectric thickness 25 μm 0.25 mm 25 μm 0.25 mm

Electrode gap 0.1 mm 1.0 mm 0.2 mm 2.0 mm

Left edge separation 50 μm 1.0 mm 0.5 mm 1.5 mm

Table 6.1: Simulation parameters for reduced- and full-scale simulations

All simulation runs were conducted with linearly increasing or decreasing voltages applied

to the exposed electrodes (for single-barrier systems) or the narrow electrodes (for double-barrier

systems). The other electrode in each system was grounded. Linear voltage ramps were chosen to mimic the two half-cycles of the symmetric-triangle waveform used in the experimental

measurements of chapters 4 and 5, and to allow comparisons to the simulations of other groups

[e.g. 14]. The voltage ramp rates listed in table 6.1 were chosen to ensure that numerous discharge

events occurred within a reasonable amount of computation time rather than to match experimental

voltages. This resulted in absolute voltage levels larger those used in experimental measurements.

In addition to the single-barrier actuators with cylindrical exposed electrodes described above,

we have also simulated a full-scale single-barrier device with a thin rectangular cross-section electrode. A schematic showing this actuator is presented in figure 6.11. With the exception of the

NASA/CR—2012-217628 120
exposed electrode, all system parameters were identical to those in table 6.1 for the full-scale

single-barrier actuators. The exposed electrode here was one grid cell thick and 1.5 mm long. The

thickness was chosen to mimic the thin exposed electrodes used by Boeuf et al. [14] and Likhanskii

et al. [87].

All simulations were run via the Condor batch queuing system [107] on idle Linux workstations

in the University of Wisconsin’s College of Engineering. Computation times for reduced-scale

simulation runs examining 200 μs of simulated time required computation times on the order of

one week. Full-scale runs encompassing 200–300 μs required from two to three months.

electrode gap

x

y

buried

electrode

thickness

dielectric

thickness

exposed

electrode length

Figure 6.11: Schematic of the simulated single-barrier actuator with a thin rectangular cross-section exposed electrode

NASA/CR—2012-217628 121
Chapter 7

Simulation Results

The two sets of simulation runs provided information about plasma density variations, induced

forces, and discharge currents during the simulated time periods. In all simulated systems, the

changing voltage lead to the appearance of short, intense microdischarge events when the plasma

density increased sharply, separated by periods of lower plasma density and minimal discharge

current. The character of these microdischarges is described in section 7.1. While the microdischarges themselves are similar in the reduced- and full-scale systems, the periods between these

events change significantly between system scales, as discussed in section 7.2. Detailed data from

the set of full-scale simulation runs are presented in section 7.3, while these data are compared to

the experimental measurements of chapters 4 and 5 in section 7.4.

Prior to beginning the simulation runs described in this chapter, we performed benchmark tests

using the same geometry and applied voltage as that used by Boeuf et al. [14]. The general behavior and induced force levels were quite similar between our runs and those in the literature

for both positive- and negative-going voltages. Our simulation, however, produced slightly more

microdischarge events with a positive-going voltage, and slightly more than half as many microdischarge events with a negative-going voltage, relative to the results of Boeuf et al. A likely

cause of these discrepancies is the difference in the choice of the model for the ionization coefficient, which was specified in the reference [14]. Test runs performed while implementing the

variable time-stepping algorithm in our code indicated that the time-step size, not specified in the

reference, also had moderate effects on the repetition rate of microdischarges. In a system similar to the single-barrier reduced-scale geometries, lowering the average time-step size roughly an

order of magnitude below 1 ns reduced the microdischarge repetition rate by approximately 10%

NASA/CR—2012-217628 122
for positive-going voltages, but left the negative-going microdischarge rate roughly unchanged.

Increasing the average time-step size by a factor of roughly 5 increased the repetition rate by

approximately 25% for positive-going voltages and decreased it for negative-going voltage by approximately 25% for negative-going voltages.

7.1 Simulated Microdischarge Behavior

As the amplitude of the voltage applied to the exposed electrode increases, microdischarges

form and eventually dissipate. An example of the resulting oscillation in the electron density was

shown in figure 2.14. Microdischarge frequency and intensity, as well as the plasma evolution

between microdischarges, change with geometric parameters as described in sections 7.2, 7.3.1,

and 7.3.3. This section describes the qualitative features that are independent of the exact device

geometry for both single- and double-barrier actuators. The reduced- and full-scale simulation

runs exhibited qualitatively similar microdischarge behaviors. All data shown in this section come

from the reduced-scale systems due to the relatively short computation times required to re-run

those simulations and capture density data during microdischarge events.

7.1.1 Single-Barrier Actuators

The general characteristics of microdischarges in single-barrier systems with non-thin electrodes were similar to those described in section 2.5.2 for systems with thin electrodes. When the

voltage of the exposed electrode was increasing relative to that of the buried electrode, we observed alternating corona-like phases and streamer-like microdischarges. In the corona-like phase,

electrons attracted to the exposed electrode cause ionization, but the resulting plasma is not sufficiently dense to alter the background electric field. The action of the electric field on the positive

ions in the plasma, however, can produce a significant induced force, as shown in figure 7.1.

When the applied voltage became large enough to support increased ionization, the plasma

density increased enough to alter the background field, resulting in a streamer-like microdischarge

like that pictured in figure 2.15. A more detailed picture of plasma evolution is given in figure 7.2. During this type of microdischarge, a tenuous plasma formed near the edge of the exposed

NASA/CR—2012-217628 123
0 20 40 60 80 100 120 140

0

0.2

0.4

0.6

Time (μs)

Force (N/m)

corona−like

phase 

streamer−like 

microdischarge

0 20 40 60 80 100 120 140

−4

 0

 4

 8

Voltage (kV)

voltage

force

Figure 7.1: Force induced by a simulated reduced-scale single-barrier actuator with

a 50 μm diameter exposed electrode. The voltage applied to the exposed electrode

was positive-going, and is also shown.

electrode. This is the corona-like phase. The plasma increased in density and arched over the

previously-charged region of the dielectric surface, reached the uncharged region of the surface,

and began propagating away from the exposed electrode. The propagation ceased either when the

electric field was no longer sufficient to sustain the dense plasma region, or when it ran into the

edge of the simulation domain. Since the majority of the induced force occurred during the coronalike phase, we do not expect that prematurely-stopped streamers will have a significant effect on

the time-averaged forces discussed in section 7.3.

When the voltage of the exposed electrode was decreasing relative to that of the buried electrode, we observed negative microdischarges similar to the one pictured in figure 2.16. The plasma

evolution during this type of microdischarge is shown in figure 7.3. Initially a region of dense

plasma formed near the exposed electrode edge. The electric field pushed electrons out of this

region and towards the dielectric surface, ionizing neutrals on their way and charging the surface

negatively. The next electrons moving through are then not be attracted to the negatively-charged

part of the dielectric, and moved to a point farther away. The ionization and charging lead to a narrow sliver of plasma that expanded along the surface of the dielectric. It expanded at a rate much

faster than the propagation of the streamer-like microdischarges (note the differing time scales in

figures 7.2(a) and 7.3(a)). This thin plasma column appearred as positively-charged because the

electrons were quickly lost to the surface. Throughout the process, the densest region of plasma

remained next to the exposed electrode.

NASA/CR—2012-217628 124
134.55 134.6 134.65

Time (μs)

Current (a.u.)

134.5676 μs

134.6129 

μs

134.6188 μs

134.6322 μs

−5 0 5 10 15 20

x 1018

(a) (b)

Figure 7.2: Simulated (a) discharge current and (b) plasma evolution during

a streamer-like microdischarge for a reduced-scale single-barrier actuator with a

positive-going voltage applied to the 50 μm diameter exposed electrode. The dielectric surface is noted by a dotted line, and the two electrodes are shown in black.

7.1.2 Double-Barrier Actuators

As with the single-barrier actuator simulations, in the double-barrier geometry some general

features of the microdischarges were similar in both reduced- and full-scale systems. The microdischarges in double-barrier systems share features with both the streamer-like and negative

microdischarges seen in simulated single-barrier systems, described in section 7.1.1. We did not

observe any equivalent to the corona-like plasma in double-barrier actuators.

NASA/CR—2012-217628 125
30.54 30.56 30.58

Time (μs)

Current (a.u.)

30.5511 

μs

30.5525 

μs

30.5567 μs

30.5663 μs

−5 0 5 10 15 20

x 1018

(a) (b)

Figure 7.3: Simulated (a) discharge current and (b) plasma evolution during a negative microdischarge for a reduced-scale single-barrier actuator with a negative-going

voltage applied to the 50 μm diameter exposed electrode. The dielectric surface is

noted by a dotted line, and the two electrodes are shown in black.

Figure 7.4 shows the plasma evolution in a microdischarge with a positive-going voltage applied to the narrow electrode. Plasma initially formed above the narrow electrode due to electronimpact ionization in the strong electric field there. The charging of the dielectric surface by electrons moved the region of strongest electric field towards the wide electrode. The dense plasma

moved in this direction, arching over the electrode gap and propagating along the dielectric surface. This propagating region was essentially identical to the streamer-like microdischarges seen

in single-barrier actuators. For the voltages tested here, the length of this microdischarge was always shorter in double-barrier systems. While the streamer-like structure formed and propagates,

a weak, thin channel of plasma expanded near the dielectric surface in the opposite direction (left

NASA/CR—2012-217628 126
28.6 28.65 28.7

Time (μs)

Current (a.u.)

28.6041 μs 28.6316 

μs

28.6369 μs

28.6857 μs

−5 0 5 10 15 20

x 1018

(a) (b)

Figure 7.4: Simulated (a) discharge current and (b) plasma evolution during a microdischarge event in a reduced-scale double-barrier actuator with a positive-going

voltage applied to the 50 μm narrow electrode. The dielectric surface is noted by a

dotted line, and the two electrodes are shown in black.

in figure 7.4). Its characteristics were similar to those of the negative microdischarge of figure 7.3,

although its progress was halted once it reached the far edge of the narrow electrode.

With a negative-going voltage applied to the narrow electrode, the microdischarges were similar to those a positive-going voltage except that they were “flipped” left to right. This type of

microdischarge is shown in figure 7.5. The plasma initially formed above the edge of the wide

electrode closest to the narrow electrode. A streamer-like plasma arched over the electrode gap,

reached the dielectric surface, and propagated until it reached the far edge of narrow electrode.

At this point, it began to weaken rapidly as the electric field strength drops off quickly. As this

was happening, a thin plasma channel formed over the wide electrode, similar to the negative

microdischarge pictured in figure 7.3(b) although significantly less dense.

NASA/CR—2012-217628 127
27.68 27.72 27.76

Time (μs)

Current (a.u.)

27.6790 

μs

27.6998 

μs

27.7097 μs

27.7516 

μs

−5 0 5 10 15 20

x 1018

(a) (b)

Figure 7.5: Simulated (a) discharge current and (b) plasma evolution during a

microdischarge event in a reduced-scale double-barrier actuator with a negativegoing voltage applied to the 50 μm narrow electrode. The dielectric surface is noted

by a dotted line, and the two electrodes are shown in black.

7.2 Effects of System Scale

Simulations of full-scale single-barrier actuators indicated that the behavior when a positivegoing ramp voltage was applied to the exposed electrode was largely the same as has been previously observed for smaller scale discharges [14, 87]. Several sharp spikes in the force parallel to

the dielectric surface appeared, corresponding to streamer-like microdischarge events, but the majority of the time-averaged force occurred between these events during the corona-like phases [14],

previously noted in figure 7.1. These phases, identified by the regions of steadily increasing force

between microdischarge events, were observed in both reduced- and full-scale systems, as shown

in figure 7.6(a). The most notable qualitative difference between the two systems was that in the

NASA/CR—2012-217628 128
0 50 100 150 200

0

0.2

0.4

0.6

0.8

Time (μs) for reduced−scale run

Force (N/m)

0 25 50 75 100

0

0.2

0.4

0.6

0.8

Time (μs) for full−scale run

reduced−scale

full−scale

(a)

50 75 100 125 150

−0.6 

−0.4 

−0.2 

0 

0.2 

0.4 

Time (μs)

Force (N/m)

reduced−scale

full−scale

(b)

Figure 7.6: Forces induced by simulated reduced- and full-scale single-barrier actuators with (a) positive-going and (b) negative-going voltages. The reduced- and

full-scale systems had exposed electrode diameters of 50 μm and 0.5 mm, respectively.

full-scale systems, following a streamer-like microdischarge, the force briefly became negative

(−x direction in figure 6.10) before returning to the corona-like phase.

When negative-going ramp voltages were applied to the exposed electrodes of single-barrier

discharges, the behaviors of the reduced- and full-scale systems differed strongly. Both systems

exhibited relatively frequent pulses of negative force corresponding to microdischarges, but these

pulses in the full-scale system tended to be grouped in time as shown in figure 7.6(b). While in

both systems the induced force became positive immediately following each pulse, in the fullscale systems this effect was much more pronounced. Other groups have pointed to the buildup of

negative-ion density as the cause of this positive force [87, 90]. This buildup is verified by the data

in figure 7.7, which show the increase in the average negative-ion density over the entire simulated

period for the full-scale system, but a stable average value (after several microdischarges) for the

reduced-scale system. We also note that the decay rate for the negative-ion density was much

smaller in the full-scale system.

In double-barrier systems with positive-going voltages applied to the narrow electrodes, the

full-scale actuator systems behaved similarly to the reduced-scale ones, as shown in figure 7.8(a).

NASA/CR—2012-217628 129
0 20 40 60 80

1012

1015

1018

Time (μs)

Negative ion density (a.u.)

reduced−scale

full−scale

Figure 7.7: Spatially-averaged negative ion densities for negative-going voltages

applied to simulated single-barrier actuators. The reduced- and full-scale systems

used exposed electrode diameters of 25 μm and 0.25 mm, respectively.

The x-directed force in both cases rose quickly during microdischarges then decayed at a slightly

slower rate. The majority of the time-integrated force occurred during this decay period. In fullscale systems after the force decayed they sometimes became briefly negative due to the production

of a significant negative ion population during the discharge event. In between microdischarges

the forces were negligible.

In contrast, with negative-going applied voltages in the double-barrier geometry, the behavior

of the induced forces changed significantly when the dimensions of the simulated system increase,

as shown in figure 7.8(b). In both systems the force exhibited a negative spike during a microdischarge event, but in the full-scale system the force became positive for a significant fraction of

the time in between microdischarges. This period of positive force indicates that, like in singlebarrier systems, a cloud of ions are generated that persist long after an intense microdischarge

event. This explanation is verified by examining the negative ion densities in the two systems,

shown in figure 7.9. In the full-scale system the negative-ion density decayed much more slowly

than in the reduced-scale system. The ratio of peak to minimum negative ion density (after several microdischarges) was also smaller in the full-scale system. The electric field “pushing” the

larger negative-ion densities present in the full-scale system between discharge events explains the

positive forces seen for the negative-going voltage in figure 7.8(b).

NASA/CR—2012-217628 130
0 20 40 60 80 100

−0.2

0

0.2

0.4

0.6

Time (μs) for reduced−scale run

Force (N/m)

0 50 100 150 200 250

−0.2

0

0.2

0.4

0.6

Time (μs) for full−scale run

reduced−scale full−scale

(a)

0 20 40 60 80 100

−0.4 

−0.2 

0

Time (μs) for reduced−scale run

Force (N/m)

0 50 100 150 200 250

Time (μs) for full−scale run

reduced−scale

full−scale

(b)

Figure 7.8: Forces induced by simulated reduced- and full-scale double-barrier

actuators with (a) positive-going and (b) negative-going voltages. The reduced- and

full-scale systems had narrow electrode diameters of 50 μm and 0.5 mm, respectively.

In both single- and double-barrier actuator simulations we observed much more rapid decay

in the negative ion populations in reduced-scale systems relative to full-scale systems, shown in

figures 7.7 and 7.9. The rates of both positive- and negative-ion decay were more rapid in reducedscale systems regardless of the applied voltage. The lower plasma decay rates in full-scale systems

likely explains the significant forces seen immediately following microdischarges.

For runs with negative-going applied voltages, the positions of maximum negative-ion density

were, on average, significantly farther above the dielectric in full-scale systems. This suggests that

the ion cloud that forms with a linearly-decreasing voltage is significantly thinner (in the vertical

dimension) in reduced-scale systems. Whether this is due to the different electrode geometry itself

or simply that the cloud is constrained by the smaller simulation domain size is not yet clear. A

thinner cloud, however, means that on average the negative ions are closer to the dielectric surface.

In the case of a negative-going applied voltage they would be attracted to the surface and quickly

lost from the system, with this process occurring more quickly the closer the negative ions are

to the surface. We hypothesize that this collection by the dielectric surface is the dominant loss

NASA/CR—2012-217628 131
0 50 100 150 200

1012

1015

1018

Time (μs)

Negative ion density (a.u.)

reduced−scale

full−scale

Figure 7.9: Spatially-averaged negative ion densities for negative-going voltages

applied to simulated double-barrier actuators. The reduced- and full-scale systems

used narrow electrode diameters of 50 μm and 0.50 mm, respectively.

mechanism in reduced-scale systems. The only other negative-ion loss mechanism in the model is

positive ion-negative ion recombination, which should be similar in the two systems.

With positive-going applied voltages, the plasma column formed at approximately the same

thicknesses in both reduced- and full-scale systems. Positive ions may be lost to the dielectric surface, but negative ions are often born far from the exposed electrode or above the narrow electrode.

Consequently, positive ion-negative ion recombination will likely be the dominant loss mechanism

for negative ions in these runs. While the decay rates for negative-ion densities were still faster for

the reduced-scale systems, possibly due to locally higher densities, the difference was not nearly

as strong as when comparing negative-going runs.

7.3 Full-Scale Simulations

As described in section 6.4, the full-scale simulations examined plasma actuator systems with

dimensions nearly identical to those of the devices studied experimentally. As in chapter 4, we

separate the results based on the shape and location of the high voltage electrode. We consider

single-barrier systems with cylindrical exposed electrodes, single-barrier systems with thin rectangular electrodes, and double-barrier systems in their own sections below.

NASA/CR—2012-217628 132
7.3.1 Single-Barrier Actuators with Circular Exposed Electrodes

Our experimental measurements showed that single-barrier actuators with short, thin exposed

electrodes induced greater forces. This type of electrode was simplest to obtain by using thin

cylindrical wires. Consequently we have focused our two-dimensional simulations on examining

single-barrier discharges which used circular exposed electrodes of various diameters.

For the case of a linearly-increasing applied voltage, the primary differences between the forces

induced by the different actuators occurred during the corona-like phase. The force induced during

this phase was significantly larger for devices with smaller diameter exposed electrodes, as shown

in figure 7.10(a). In addition, the periods of negative force immediately following each microdischarge were shorter for actuators with smaller electrodes. We did not observe any strong trend in

the positive microdischarge repetition rate as the electrode diameter changed.

0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

Time (μs)

Force (N/m)

0.25 mm

0.75 mm

(a)

60 80 100 120

−0.8 

−0.6 

−0.4 

−0.2 

0

0.2

Time (μs)

Force (N/m)

0.25 mm

0.75 mm

(b)

Figure 7.10: Forces for simulated single-barrier actuators with the indicated exposed electrode diameters for (a) positive-going applied voltages and (b) negativegoing applied voltages

Simulations with negative-going applied voltages also showed quantitative and qualitative dependencies on the exposed electrode diameter. Both the average rate at which microdischarges

occurred and the level of induced force in between these events dropped markedly as the exposed

electrode diameter increased. The discharge events, corresponding to oscillations in the electron

density, are shown for two actuators in figure 7.11. While both devices exhibited rapid discharges

NASA/CR—2012-217628 133
at certain times, the actuators with larger exposed electrodes also experienced long periods of inactivity. These inactive periods allowed the plasma to decay to relatively low densities, as shown

in figure 7.11, reducing the induced force. While the first microdischarge immediately following

an inactive period was stronger (in the sense of higher plasma density) than other microdischarges,

over time this did not compensate for the loss of induced force during the inactive periods. The

fraction of time spent in inactive periods increased as the exposed electrode diameter increased,

although the exact mechanism for this effect is still unknown.

40 60 80 100

108

1012

1016

Time (μs)

Particle density (a.u.)

40 60 80 100

Time (μs)

negative ion density electron density

0.25 mm diameter 0.75 mm diameter

(a) (b)

Figure 7.11: Charged-particle densities near simulated single-barrier actuators

with negative-going voltages applied to exposed electrodes (a) 0.25 mm in diameter and (b) 0.75 mm in diameter. Positive ion densities were nearly identical to the

negative ion densities in all cases.

As described in section 6.4, we ran single-barrier simulations with the applied voltage changing

at rates of ±150 V/μs. When a positive-going voltage was applied to the exposed electrode, we

began the simulation runs at +3.0 kV, well below the level needed to induce a microdischarge.

With negative-going voltages, however, −3.0 kV was large enough to immediately generate a

strong, non-physical microdischarge. To remove this effect, the simulations with negative-going

voltages were started at −1.5 kV.

The time-averaged forces that resulted from these runs are shown in figure 7.12. The timeaveraged forces for all simulated devices increased throughout the simulated time, with the rates

NASA/CR—2012-217628 134
of growth being particularly strong for negative-going applied voltages. Boeuf et al. [93] previously noted that domain widths shorter than the plasma length caused the time-averaged forces

for negative-going voltages to reach a steady value. Since we observed increasing forces, as in

figure 7.12(b), we conclude that the domain was wide enough to not influence plasma evolution.

0 20 40 60 80 100

0

0.02

0.04

0.06

0.08

0.1

0.12

Time (μs)

Time−averaged force (N/m)

(a)

0 20 40 60 80 100

0

0.02

0.04

0.06

0.08

0.1

Time (μs)

Time−averaged force (N/m)

0.10 mm

0.25 mm

0.50 mm

0.75 mm

1.00 mm

(b)

Figure 7.12: Time-averaged forces (averaged from time 0 to time t) for full-scale

single-barrier simulations using (a) positive-going voltages and (b) negative-going

voltages. Both plots use the legend in plot (b).

By making two reasonable assumptions, we can consider a positive-going run and a negativegoing run taken together as one full cycle of an applied symmetric triangle waveform. First, since

the runs were conducted separately, we must assume that the plasma from one half cycle has

almost completely decayed by the beginning of the next half cycle. If this is the case, there will

be essentially no interaction between the half cycles except via charge deposition on the dielectric

surface. Second, we assume that the most important parameter in determining the electric field in

the gas region is the voltage drop between the exposed electrode and the dielectric surface, rather

than the voltage drop between the two electrodes. If the second assumption is valid, a system with

−5 kV applied to the exposed electrode and charge deposited on the dielectric such that the surface

potential is −5 kV is essentially identical to a system with 0 V applied to the exposed electrode

and no charge on the dielectric surface. This assumption allows us to treat the runs as half cycles

even though all run voltages start near 0 V and either increase or decrease.

NASA/CR—2012-217628 135
Applying these assumptions, we take a 110 μs-long positive-going run and a 120 μs-long

negative-going run together as a full cycle. The equivalent applied voltage signal would be a

9.8 kV amplitude symmetric triangle wave oscillating at 3.8 kHz. Both the amplitude and the frequency are somewhat larger than the values we tested experimentally, due to our desire to shorten

computation times. Simulating systems with voltages similar to those in the experiments would

have required significantly increased computation times.

Using combined positive- and negative-going simulation runs as an effective voltage cycle, we

estimated time-averaged forces for each actuator design. We did this by integrating the EHD force

during each simulation run and dividing by the effective simulated time. We assumed that prior

to each simulated time period there was a ramp from 0 V to the starting voltage during which

no significant momentum transfer occurred. The resulting cycle-averaged forces are shown in

figure 7.13. The time-averaged forces for both positive- and negative-going voltages increased

sharply at diameters below 0.5 mm for the reasons described above. The forces for both positiveand negative-going applied voltages were similar for all tested actuators.

In the full-scale simulations, we note that the dominant sources of the induced forces were

during the build-up period prior to a microdischarge for a positive-going voltage, or during the

decay phases following microdischarges for negative-going voltages. Due to their extremely short

0 0.2 0.4 0.6 0.8 1

0

0.02

0.04

0.06

0.08

0.1

0.12

Exposed electrode diameter (mm)

Time−averaged force (N/m)

F

avg

 −150 V/μs

F

avg

 cycle

F

avg

 +150 V/μs

Figure 7.13: Cycle-averaged forces induced by simulated single-barrier actuator

with cylindrical exposed electrodes. Results are shown for the positive-going half

cycle, the negative-going half cycle, and the entire cycle for the effective voltage

described in the text.

NASA/CR—2012-217628 136
duration, the microdischarges themselves directly contributed little to the time-averaged forces.

Their importance to the system is to either provide a seed plasma for the next corona-like phase

(with a positive-going voltage) or to increase the plasma density prior to a decay phase (for a

negative-going voltage).

Since the choice of effective voltage amplitude and frequency was essentially arbitrary, we

consider the results of other choices by examining figure 7.12. We can estimate averaged forces

for effective voltages of higher frequencies and smaller amplitudes by examining the values of

the time-averaged forces at times prior to the ends of the simulation. We conclude that while the

exact values of the time-averaged force vary slightly with the chosen time, the general conclusion

of increasing force with decreasing exposed electrode holds at all observed times for the chosen

voltage ramp rate.

7.3.2 Single-Barrier Actuators with Thin Rectangular Exposed Electrodes

As described in section 2.5, many other groups have modeled single-barrier actuators using

rectangular-cross section exposed electrodes with thicknesses of a single spatial step size. Here

we compare our full-scale single-barrier simulation runs using a circular cross-section exposed

electrode with a 0.10 mm to runs using a rectangular exposed electrode. The rectangular electrode

geometry was depicted in figure 6.11.

The actuator with the thin rectangular exposed electrode behaved quite similarly to the actuator

with the smallest examined circular electrode diameter. The forces for positive- and negative-going

voltages are shown in figures 7.14(a) and 7.14(b). With a positive-going voltage, the actuator with a

rectangular electrode induced comparable forces during the corona-like phase to the actuator with

a 0.10 mm diameter circular electrode. The rate at which streamer-like microdischarges occur,

however, is approximately the same in the two systems.

With a negative-going voltage, the actuator with a rectangular electrode excited more frequent

microdischarges than did the actuator with a circular electrode, as seen in figure 7.14(b). The

device with the rectangular electrode also had no long inactive periods, unlike our simulations

of actuators with relatively thick exposed electrodes. These behaviors follow the same trend as

NASA/CR—2012-217628 137
40 60 80 100 120 140 160

0

0.5

1

1.5

Time (μs)

Force (N/m)

circular, 0.10 mm

thin rectangular

(a)

10 20 30 40 50 60

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Time (μs)

Force (N/m)

circular, 0.10 mm

thin rectangular

(b)

Figure 7.14: Comparison of the forces for simulated single-barrier actuators with

rectangular- and circular-cross section exposed electrodes. Results are shown for

(a) positive-going and (b) negative-going applied voltages.

observed for various diameter circular electrodes in section 7.3.1, where actuators with thinner exposed electrodes excited more frequent microdischarges and had shorter periods of inactivity. The

rectangular electrode is effectively 7 μm thick. The rapid microdischarges lead to relatively large

time-averaged plasma densities, which in turn contributed to the increasing force in figure 7.14(b).

This force behavior is qualitatively the same as that observed by Boeuf et al. [93] for a similar

geometry.

7.3.3 Double-Barrier Actuators

Double-barrier DBDs with the geometry shown in figure 6.1(b) also induced forces in the

positive-x direction when both positive- and negative-going voltages were applied to the narrow

electrodes. Compared to the single-barrier geometry, however, runs simulating double-barrier systems changed relatively little as the high voltage electrode diameter varied. The forces induced

during runs with positive- and negative-going applied voltages are shown in figure 7.15 for two

narrow electrode diameters. For all simulation runs, the peak levels of induced force are comparable to those seen for single-barrier systems, but the average levels were an order of magnitude

smaller due to lower microdischarge repetition rates.

NASA/CR—2012-217628 138
0 50 100 150 200 250

0

0.2

0.4

0.6

0.8

Time (μs)

Force (N/m)

0.25 mm 0.75 mm

(a)

0 50 100 150 200 250

−0.6

−0.4

−0.2

0

Time (μs)

Force (N/m)

0.25 mm

0.75 mm

(b)

Figure 7.15: Forces for simulated double-barrier actuators with the indicated narrow electrode diameters for (a) positive-going applied voltages and (b) negativegoing applied voltages

With a positive-going applied voltage, neither the microdischarge frequency or intensity change

significantly as the narrow electrode size was increased. Indeed, as shown in figure 7.15(a), not

even the timing of microdischarge occurrence varied with narrow electrode diameter. The only

systematic change in the behavior of the induced force was in a short, weak negative force immediately following microdischarge events. This force became more negative as the narrow electrode

diameter increased.

In runs applying negative-going voltages, we also saw relatively little change in plasma behavior with the size of the narrow electrodes. As shown in figure 7.15(b), the peak positive force

immediately following a microdischarge increased slowly with increasing narrow electrode diameter. These larger peaks appear to be associated with the generation of higher average plasma

densities during negative microdischarges in systems with larger narrow electrodes.

When integrated over several microdischarges, these small variations did have some effects

on the time-averaged forces. These time-averaged forces are shown in figure 7.16 as functions

of the simulated time. Due to the significant positive forces exerted during the decay periods

immediately following microdischarges during negative-going runs, the time-averaged forces for

negative-going voltages were nearly an order of magnitude larger than those for positive-going

NASA/CR—2012-217628 139
0 50 100 150 200 250

0

1

2

3

4

x 10−3

Time (μs)

Time−averaged force (N/m)

(a)

0 50 100 150 200 250

0

0.01

0.02

0.03

Time (μs)

Time−averaged force (N/m)

1.00 mm

0.75 mm

0.50 mm

0.25 mm

0.10 mm

(b)

Figure 7.16: Time-averaged forces (averaged from time 0 to time t) for full-scale

double-barrier simulations using (a) positive-going voltages and (b) negative-going

voltages. Both plots use the legend in plot (b).

runs. As in the single-barrier simulations, with a negative-going voltage the time-averaged force

increased throughout the simulated time period, as shown in figure 7.16(b).

By making assumptions similar to those described in section 7.3.1, we can treat the combination of a positive-going run and a negative-going run as a full cycle of a symmetric triangle voltage

waveform. We used 275 μs-long runs for both positive- and negative-going voltages. The runs

used starting voltages of ±6 kV. As in the analysis of single-barrier simulation results, we assume

a time period prior to each run in which the voltage moves from 0 V to the starting voltage and

no significant force exists. Positive- and negative-going runs may then be considered together as

simulating one cycle of a 1.6 kHz symmetric triangle waveform with an amplitude of 30.5 kV. The

resulting time-averaged forces are shown in figure 7.17. Considering the results of figure 7.16, the

exact choice of simulated time did not effect the resulting trends, as the differences between the

forces induced by different geometries remained consistent throughout the simulated time period.

As with the single-barrier data, we have only considered a single magnitude for the voltage rate of

change.

In general, the cycle-averaged forces increased slowly as the narrow electrode diameter increased. This trend is, in one sense, the opposite of that observed in the single-barrier simulation

NASA/CR—2012-217628 140
0 0.2 0.4 0.6 0.8 1

0

0.01

0.02

0.03

0.04

Narrow electrode diameter (mm)

Time−averaged force (N/m)

F

avg

 −200 V/μs

F

avg

 cycle

F

avg

 +200 V/μs

Figure 7.17: Cycle-averaged forces induced by simulated double-barrier actuator

with cylindrical narrow electrodes. Results are shown for the positive-going half

cycle, the negative-going half cycle, and the entire cycle for the effective voltage

described in the text.

runs. As we reduce the diameter of the high voltage electrodes in both cases, in the absence of

plasma the electric fields near these electrodes increase. The insulation of this electrode in the

double-barrier system, however, clearly leads to significantly different consequences of this increased field strength.

The cycle-averaged forces for our double-barrier simulations showed significantly less variation with electrode diameter than did the single-barrier simulations. The cycle-averaged forces for

the double-barrier runs decreased by about 50% as the narrow electrode diameter was decreased

from 0.1 to 1.0 mm. In comparison, the cycle-averaged forces for single-barrier simulations increased by 350% as the exposed electrode diameter was decreased through the same size range.

7.4 Comparisons to Experimental Results

Except in ideal circumstances, making comparisons between computational models and experimental results requires caution. In our situation, we do not expect the results of the simulations

to quantitatively match the experiments due to the simplifications required to improve computational efficiency, but of limited physical validity. In particular, the code models two-dimensional

systems, despite the obviously three-dimensional nature of the microdischarges shown in chapter 5, and employs a highly simplified plasma chemistry. In addition, empirical data for secondary

NASA/CR—2012-217628 141
electron emission coefficients were not available for the relevant ion species, energies, and materials, forcing us to choose relatively arbitrary values. While inaccuracies in these coefficients

may not strongly affect the total induced forces, they may change the intensities of the microdischarges [92], and therefore the microdischarge structure. To reduce computation times, we have

also used larger voltage rates-of-change than were present in the experimental measurements.

Despite these caveats, we believe it is important to make connections between the computational model and the experiments in order to suggest possible directions for future development in

both areas. To date, only very few of these comparisons have been made by other groups [72, 93].

Due to the noted differences between the simulations and the experimental system, we restrict

ourselves to making mostly qualitative comparisons. We focus on the observed microdischarge

structures in sections 7.4.1 and 7.4.2, and the trends of time-averaged force with changing electrode dimensions in section 7.4.3.

In all of the simulation runs, some general features of the experiments were reproduced. As

the voltage in each run increased or decreased, we observed repeated microdischarge events. Each

of these microdischarges corresponded to a spike in the simulated discharge current, as observed

in the experiments. In addition, all of the simulations predicted a time-averaged force in the +x

direction, the same direction as measured experimentally using the electron balance and stagnation

probe techniques. We focus on more detailed similarities and differences below.

7.4.1 Single-Barrier Microdischarge Structures

The simulation runs provided information about the distributions of charged-particle densities.

The ICCD imaging, in contrast, recorded light emission from both neutral and charged particles

that became excited due to collisions with charged particles. In making comparisons, then, we

must assume that the emitted light is roughly proportional to the number of collisions, which in

turn is roughly proportional to the plasma density. We discussed this assumption in more detail

in section 3.3.2. The camera experiments viewed the actuator from above; they imaged the x-z

plane as shown in figures 3.10 and 3.11. The simulations modeled the x-y plane, and assumed the

system to be uniform in the z direction. Consequently, in both this section and in section 7.4.2 we

NASA/CR—2012-217628 142
focus on plasma variations in the x direction, which is along the dielectric surface moving away

from the edge of the exposed or narrow electrode. Similar comparisons between instantaneous

plasma structure and ICCD images have been made by Orlov et al. [72]. Here we also discuss

the observed microdischarge lengths, plasma regimes, and microdischarge durations. As in section 7.1, the figures in this section and section 7.4.2 use data from the reduced-scale simulation

runs due to the relative difficulty in recording plasma density distributions at specific times during

a microdischarge in the full-scale runs.

The single-barrier simulations reproduced many features of the microdischarges seen experimentally with ICCD imaging. With a positive-going applied voltage, a short, dense region of

plasma formed near the exposed electrode and propagated down the dielectric surface. This process has been called a streamer-like microdischarge, and was described in more detail in section 7.1.1. The plasma during such a microdischarge is shown for a reduced-scale system in figure 7.18(a). The ‘e’ and ‘b’ notations in both figures 7.18(a) and 7.18(b) correspond to the exposed

and buried electrode locations. We also observed the same general behavior experimentally in filamentary microdischarges, as shown in a 15 ns gate width image in figure 7.18(b) and described

in section 5.2.3. The reduced-scale simulations also captured the existence of a significant region

of dense plasma immediately adjacent to the exposed electrode, as seen in both figures 7.18(a)

and 7.18(b).

In full-scale simulation runs, using the position of maximum plasma density diagnostic described in section 6.3, we determined that the streamer-like microdischarges extended essentially

to the edge of the simulation domain. This is approximately equivalent to 17 mm from the exposed

electrode edge. Typical experimentally-observed filaments such as those in figures 5.1 and 5.6(b)

extended from 5 to 7 mm. We believe the differences in microdischarge lengths are due to two

factors. First, the simulations used larger applied voltages than did the experiments. The imaging experiments of section 5.2.1 demonstrated increases in length as the applied voltage rose, so

we expect that at voltages as large as those in the simulation runs we would observe longer filaments. Second, the imaged filaments are not straight, but rather jagged. The total path length

NASA/CR—2012-217628 143
(a)

e

b

(b)

Figure 7.18: Comparison of (a) simulated densities and (b) an ICCD image

recorded for a single-barrier actuator with a positive-going voltage. The simulation was of a reduced-scale system, and includes the same data as figure 7.2. The

15 ns gate-width image is the same as shown in figure 5.14(c).

of a microdischarge will in general be larger than the distance of its end from the exposed electrode. In contrast, the two-dimensional nature of the computational model forces the simulated

microdischarge to propagate directly away from the exposed electrode.

While we observed comparable filamentary/streamer-like plasma structures in the imaging experiments and the simulations, the simulations did not reproduce a filament-free regime. We observed this regime experimentally only at the very end of the positive-going half cycle for exposed

electrode diameters on the order of 0.1 mm. While the simulations used larger applied voltages,

the smallest simulated exposed electrode was 0.10 mm in diameter, close to the threshold for the

appearance of the filament-free regime. Additional simulations with smaller exposed electrode

sizes would help determine whether the model includes the necessary physics to reproduce this

regime.

The single-barrier simulations also reproduced many features of the microdischarges seen experimentally in actuators with negative-going applied voltages. A comparison of a reduced-scale

microdischarge and an experimentally-observed image are shown in figures 7.19(a) and 7.19(b),

NASA/CR—2012-217628 144
(a)

e

b

(b)

Figure 7.19: Comparison of (a) simulated densities and (b) an ICCD image

recorded for a single-barrier actuator with a negative-going voltage. The simulation was of a reduced-scale system, and includes the same data as figure 7.3. The

image used a 10 μs gate width, and is of a single-barrier actuator with a 0.40 mm

diameter exposed electrode.

respectively. The computational model predicted a dense region of plasma immediately adjacent

to the exposed electrode edge, with a sparser plasma extending along the dielectric surface. This

jet-like plasma structure was observed experimentally as well. The horizontal spread of the plasma

in figure 7.19(b) could not be reproduced by the model due to its confinement to two spatial dimensions.

We have estimated the duration of microdischarges in the full-scale simulation runs by examining the widths of the current pulses. For positive-going voltages, typical streamer-like microdischarge durations were approximately 50 ns, with no significant variation observed for different

exposed electrode diameters. For negative-going voltages, the current pulse widths varied with

the exposed electrode diameter. For the run with a 0.10 mm electrode, typical microdischarge

durations were approximately 5 ns. For the run with a 1.00 mm electrode the durations varied

strongly, with larger current pulses lasting approximately 35 ns and smaller pulses lasting only a

few nanoseconds.

NASA/CR—2012-217628 145
A similar analysis of current pulse duration could not be used experimentally due to significant parasitic capacitance in the measurement circuit. We estimated microdischarge lifetime for a

single-barrier actuator with a 0.30 mm diameter electrode using the ICCD camera in section 5.2.3.

There we estimated an upper bound of 100 ns for the lifetime of a filamentary microdischarge,

and an upper bound of 20 ns for a jet-like microdischarge. These values are comparable to those

seen in the simulation, although we note that these are only single points of comparison, requiring

confirmation before they can be stated as firm evidence of model accuracy.

7.4.2 Double-Barrier Microdischarge Structures

The simulation runs of double-barrier actuators reproduced some of the plasma structures seen

experimentally with the ICCD camera. During the positive-going half cycle, the experimental

images showed both jet-like and filamentary plasma structures, as pictured in figure 7.20(b). The

reduced-scale simulations, from which the example data in figure 7.20(a) were taken, do show a

dense propagating plasma similar to the streamer-like microdischarge seen in single-barrier runs.

Since we described strong similarities between the simulated streamer-like microdischarges and

the experimentally-observed filamentary microdischarges in section 7.4.1, it is reasonable to make

the same comparison here. While the experimental image also contains a bright jet-like structure,

the simulations predicted only a relatively weak plasma near the narrow electrode. This plasma

did tend to be thin, similar to the simulated negative microdischarge described in section 7.1.1, but

was much weaker than seen in single-barrier systems.

As noted in section 7.1.2, a negative-going applied voltage applied to simulated double-barrier

actuators resulted in plasma behavior flipped from that when a positive-going voltage is used. The

simulation, shown in figure 7.21(a), showed a dense propagating region the stopped just past the

−x side of the narrow electrode, and thin, weak plasma channel above the buried electrode. This

latter feature is more easily visible in figure 7.5. The camera image, shown in figure 7.21(b), was

also reversed from the positive-going case. It includes short filamentary structures above the narrow electrode and jet-like structures above the wide electrode. While the simulation has reproduced

NASA/CR—2012-217628 146
(a)

n

w

(b)

Figure 7.20: Comparison of (a) simulated densities and (b) an ICCD image

recorded for a double-barrier actuator with a positive-going voltage. The simulation was of a reduced-scale system, and includes the same data as figure 7.4. The

image used a 10 μs gate width, and is of a double-barrier actuator with a 0.40 mm

diameter narrow electrode.

(a)

n

w

(b)

Figure 7.21: Comparison of (a) simulated densities and (b) an ICCD image

recorded for a double-barrier actuator with a negative-going voltage. The simulation was of a reduced-scale system, and includes the same data as figure 7.5. The

image used a 10 μs gate width, and is of a double-barrier actuator with a 0.40 mm

diameter narrow electrode.

NASA/CR—2012-217628 147
both observed structures, as with a positive-going voltage the simulated plasma corresponding to

the jet-like structure was quite weak compared to the brightness of the camera image.

For both positive- and negative-going applied voltages, the simulation appears to under-predict

the plasma in the regions where we experimentally observed jet-like plasma structures. The major

simplifications made in the model—two-dimensionality, limited plasma chemistry, and lack of

photoionization and photoemission—are also present in single-barrier simulations, in which we

do see a negative microdischarge with many similarities to the jet-like structure. The exact reason

for the discrepancy in double-barrier microdischarge behavior between the experiments and the

simulations remains unclear.

7.4.3 Time-Averaged Forces

As with our examinations of microdischarge structure, comparisons of simulated and measured time-averaged forces require careful considerations of the differences between the physical

and modeled systems. In particular we note that the effective voltages and amplitudes used in the

simulations are larger than for the experimentally-applied voltage waveforms. While in general

we expect that the larger voltage amplitudes and frequencies will increase the simulated force, this

has only been proven for single-barrier actuators [93]. We also note that our discussions of the effective voltage waveforms in the simulations include the same assumptions stated in section 7.3.1,

primarily that the two half-cycles would not interact with each other if modeled sequentially.

As discussed in section 7.3.1, the set of full-scale simulation runs allowed us to calculate a

cycle-averaged forces for single-barrier actuators. The cycle was an effective voltage waveform

oscillating at 3.8 kHz with an amplitude of 9.8 kV. We can directly compare these simulated forces

to the experimentally-measured forces described in section 4.2. Figure 7.22 shows the simulated

and measured forces together as functions of the exposed electrode diameter. Qualitatively the two

sets of data show the same general trend of increasing at smaller diameters.

The two force scales in figure 7.22 indicate that the simulated forces were an order of magnitude larger than those measured experimentally. This discrepancy likely has two major sources.

First, as noted above, both the effective amplitude and frequency are significantly larger for the

NASA/CR—2012-217628 148
0 0.2 0.4 0.6 0.8 1

0

0.002

0.004

0.006

0.008

0.010

0.012

Exposed electrode diameter (mm)

Experimental force (N/m)

0

0.025

0.050

0.075

0.100

0.125

0.150

Simulated force (N/m)

experimental balance data (8 kV)

full−scale simulation data

Figure 7.22: Comparison of simulated and experimentally-measured forces for

single-barrier actuators. The simulated force data are the cycle-averaged forces of

figure 7.13. The experimental values were measured using the electronic balance

for a 13 kV amplitude voltage, and are the same data as in figure 4.4.

simulations than for the experiments. As discussed in section 2.4.2, the induced forces generally increase as both voltage amplitude and frequency are raised in this range. Second, the twodimensional nature of the computational model tends to overestimate the induced forces. The

model assumes that all properties, including the induced force, are uniform in the third dimension.

As we saw in chapter 5, however, the plasma is generally confined to narrow channels in this third

(z) dimension. While the plasma may diffuse in the z direction as it decays (and the plasma may

induce a significant force during this decay period), it seems unlikely that the induced force will

be uniform in this direction even in when averaged over time. Since the experimentally-measured

forces are averaged in the z dimension, they include both regions of plasma and regions of neutral

air, reducing the apparent force.

While the induced forces for simulations and measurements of single-barrier actuators with

cylindrical electrodes showed the same qualitative trends with exposed electrode diameter, this

agreement breaks down when we consider a rectangular exposed electrode. Averaged over the

same effective voltage as the other single-barrier simulations, the rectangular electrode actuator

discussed in section 7.3.2 induced approximately the same force as did the system with a 0.10 mm

diameter exposed electrode. This result is in stark contrast with our experimental measurements

NASA/CR—2012-217628 149
of sections 4.2.1 and 4.2.2, which showed that actuators with long, thin rectangular exposed electrodes induced significantly lower forces than did those with cylindrical electrodes. The absence of

the filament-free plasma in the simulations may be responsible for the lack of separation between

the devices with cylindrical and circular electrodes. This regime was present in the experimental

actuators in which the greatest difference was observed between actuators with the two electrode

geometries.

In section 7.3.1 we noted that the simulated time-averaged forces for positive- and negativegoing voltages were of comparable sizes. While the extremely closeness of these values, shown

in figure 7.13, were the product of the particular simulation durations we chose, the fact that both

half cycles contributed significantly to the cycle-averaged force was not. Comparable values of the

induced force for positive- and negative-going voltages have also been predicted by the simulations

of Boeuf et al. [93]. These findings match up with the experiments of Enloe et al. [75], who found

experimentally that the gross positive forces induced during each half cycle of the applied voltage

were similar in both sign and magnitude. We note that our reduced-scale simulations predicted

a negative net force with negative-going applied voltages for larger electrode diameters, which

would contradict the cited experimental measurements. The hypothesis by Enloe et al. [75] that

the smaller net force during the positive-going half cycle was caused by drag between the induced

gas flow and the dielectric surface would explain the differences between the simulation data and

the experimental results of Forte et al. [66]. The latter group found significant induced gas flow

only during one half cycle of the applied voltage, but made their measurement far downstream of

the actuator, after drag could have reduced the positive half cycle-induced flow down to negligible

velocities.

In contrast to the single-barrier actuator data, the simulations and experimental measurements

of time-averaged forces for double-barrier actuators showed opposite behaviors with changing narrow electrode diameter. As shown in figure 7.23, the simulation predicted an decreasing induced

force as the narrow electrode diameter decreased, while the experiments showed the opposite trend.

Although neither trend was particularly strong compared to the changes seen for single-barrier devices, the discrepancy is clear and may indicate that an important physical mechanism is missing

NASA/CR—2012-217628 150
from the model. The model did, however, correctly predict that time-averaged forces induced by

double-barrier systems were an order of magnitude smaller than those of single-barrier actuators.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x 10−3

Narrow electrode diameter (mm)

Experimental force (N/m)

0

0.003

0.006

0.009

0.012

0.015

experimental balance data (13 kV)

Simulated force (N/m)

full−scale simulation data

Figure 7.23: Comparison of simulated and experimentally-measured forces for

double-barrier actuators. The simulated force data are the cycle-averaged forces of

figure 7.17. The experimental values were measured using the electronic balance

for an 8 kV amplitude voltage, and are the same data as in figure 4.10.

NASA/CR—2012-217628 151
Chapter 8

Conclusion

We have conducted a variety of physical and computational experiments in order to optimize

and better understand surface dielectric barrier discharges for use as aerodynamic actuators. These

discharges use oscillating high voltages to excite plasmas in ambient air, and a dielectric layer between two electrodes to suppress the formation of arcs. We have examined both the time-averaged

and nanosecond scale behaviors of the actuator systems. In our time-averaged experiments we

have made dramatic increases in the induced force, which is expected to translate into improved

ability to reduce or eliminate flow separation on airfoils in low-Reynolds number gas flows. Optical imaging of the plasma on short time scales showed plasma structures that varied with both

discharge geometry and the slope of the applied voltage. We have qualitatively validated many

results of our newly-developed computational model against the experimental data at both time

scales, giving us a more detailed picture of the plasma physics.

Our time-averaged force measurements of single-barrier plasma actuators showed that the material of the exposed electrode did not significantly affect either the induced air flow or the dissipated electrical power of the system. In contrast, simultaneously reducing the thickness and length

of the exposed electrode, shown in figure 8.1, lead to exponential increases in the induced force,

while simultaneously decreasing the required power. In our experiments the ideal design of thin,

short electrodes was most easily obtained using thin wire electrodes. The fastest growth in the induced forces was observed as we reduced the electrode diameters below approximately 0.15 mm.

We have validated our results by using two measurement techniques: direct force measurements

using an electronic balance and flow measurement using a stagnation probe.

NASA/CR—2012-217628 152
exposed electrode buried electrode

length

thickness

Figure 8.1: Single-barrier actuator showing the exposed electrode length and width

We have also conducted the first systematic force measurements of double-barrier discharges,

in which both electrodes are insulated, used as plasma actuators. Our design used one narrow

cylindrical electrode and one wider electrode with a rectangular cross-section, both placed underneath a dielectric film. The force induced by such actuators was directed from the narrow towards

the wide electrode, and increased as the narrow electrode diameter was reduced.

In addition to our time-averaged force measurements, we have examined plasma dynamics

on time scales ranging from a few nanoseconds to the applied voltage period of 1 ms using an

intensified digital camera. Previous studies have noted that the plasma in single-barrier surface

discharges forms in filamentary structures when the voltage applied to the exposed electrode is

increasing, and in more diffuse jet-like structures when it is decreasing [71]. We have characterized

for the first time, the formation and lifetime for both of these types of microdischarges. Each

filamentary microdischarge consists of a small, bright region of plasma that propagates along the

dielectric surface. They appear filamentary because the plasma moves along a jagged path, and the

camera integrates the emitted light. Such microdischarges required on the order of 100 ns to form

for the parameters of our system. In contrast, the jet-like microdischarges appeared to form and

decay in time periods shorter than 20 ns.

As we reduced the exposed electrode diameter below the same electrode at which we began

to see large increases in the induced force, we also observed a suppression of the filamentary

microdischarges and the appearance of a constant weak glow near the exposed electrode. This

“filament-free” regime occurred only during the positive-going half cycle of the voltage. Unlike

other microdischarges, it did not grow and decay within a hundred nanoseconds, but remained

steady and visible until the end of the positive-going half cycle. The correlation between the

NASA/CR—2012-217628 153
suppression of filamentary microdischarges and the strong increase in the induced force suggests

that the filaments do not efficiently transfer momentum to the neutral gas.

The microdischarges in double-barrier actuators appeared to include both filamentary and jetlike structures. With a positive-going voltage applied to the narrow electrode, we observed a long

filamentary structure and a shorter jet-like structure in the opposite direction. With a negativegoing voltage, we saw long jet-like and short filamentary structures arrayed with the directions

switched. These results suggest that the form of the microdischarge depends in part on the direction

of the local electric field relative to the dielectric surface.

The experimental measurements allow us to make new recommendations for the design of

the future single-barrier plasma actuator systems. Being able to the choose the electrode material

based on its mechanical and electrical characteristics rather than its secondary electron emission

coefficient will ease the task of designers adapting plasma actuators for use in harsh environments

such as jet turbines. While designing for maximum efficiency would dictate using the thinnest and

shortest achievable exposed electrode, the designer will inevitably need to consider other factors.

As the electrode size is reduced, the increasing resistance of the electrode may lead to drops in

voltage along the width of the electrode, which in turn would cause non-uniform performance.

Of perhaps greater concern will be the erosion of such a small electrode. Plasmas, particularly

in systems including high voltages and corrosive gases like oxygen, can quickly sputter or etch

surfaces. In our experiments, we observed discoloration and roughening of our polymer dielectric

after only tens of minutes of operating time, although we did not study this in detail.

To examine details of the discharge physics not accessible to our experimental techniques,

we have developed a two-dimensional fluid model of the system. The model solves Poisson’s

equation, a set of continuity equations, and a set of drift-diffusion equations to determine the timedependent plasma behavior, applying many of the same approximations included in other models

in the literature. In addition to the abilities of these other models, our code also allows simulation

domains with arbitrary electrode shapes and sizes, and dynamically adjusts the time-step size to

maintain numerical accuracy while reducing computation times. We have applied the model to

examine simulated plasma variations on time scales from several nanoseconds to hundreds of

NASA/CR—2012-217628 154
microseconds, averaging data on the latter scales to approximate time-averaged values. The model

successfully reproduced many of the actuator behaviors observed experimentally, and represents a

first step towards a detailed understanding of the important physics in the discharge system.

Our simulations of single-barrier systems predicted the occurrence of discrete microdischarge

events. With a positive-going voltage applied to the exposed electrode, the plasma formed in a

small, dense region which propagated along the dielectric surface. With a negative-going voltage, the plasma formed and decayed more quickly, with the densest region remaining next to the

exposed electrode. These simulated structures strongly resembled the microdischarge structures

we observed experimentally. The effective time-averaged forces of experimental-scale simulations

roughly paralleled the increase in force at reduced exposed electrode diameters seen experimentally, but the code incorrectly predicted that a long rectangular exposed electrode would induce

similar levels of force. Simulated microdischarges in double-barrier systems showed somewhat

less resemblance to the experimentally observed structures. They did result in discrete microdischarges and a time-averaged force in the same direction as we measured experimentally. In both

system geometries, the simulation indicated that the dominant sources of time-averaged force acting on the gas occurred in between microdischarge events, as the plasma decayed.

Our experimental work has shown that the efficiency of a single-barrier plasma actuator can

be increased by a factor of 9 over previous studies by using a short and thin exposed electrode.

Imaging of the plasma suggests that some of this increase may be due to the appearance of a

previously unknown filament-free plasma regime that does not occur when using larger electrodes.

Qualitative comparisons between the force and imaging experiments and our simulation results

indicate that the fluid model incorporates much of the relevant physics of this type of discharge

system. We expect that the significantly larger forces induced by the plasma actuators considered

in this work will be able to limit flow separation in a wider range of background flow velocities than

previous designs, paving the way to reduce drag and inefficiencies in a variety of aerodynamics

applications.

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Appendix A: Actuator Construction Procedure

This appendix describes in detail the procedure used to construct the DBD plasma actuators

investigated in chapters 4 and 5. All of the accompanying photographs show the construction of

a single-barrier device with a cylindrical exposed electrode, and the text includes the adjustments

that would be needed to construct a different type of actuator.

1. Cut a section of the dielectric material and tape down the edges on a flat, level surface

Irregularities in the surface of the working area can lead to bumps in the actuator surface.

For the actuators in this work, a metal bar seen in each of the photographs below was used.

The surface was leveled to prevent epoxy from flowing to one side. The 0.25 mm thick

polyester film used as the dielectric material in this work was shipped on a roll. Because of

this, the cut section was curved, as can be seen in the photograph below. In order to ensure

a flat dielectric layer we placed film with the edges curved upwards, then taped the edges of

the film to the working surface.

2. Trace an outline of an actuator, cut a buried or wide electrode, and place it on the dielectric

Using an existing actuator, we traced an outline onto the dielectric film. In this work, all

actuators were 4 cm long by 7 cm wide. The outline served as a guide while placing the

electrodes and applying epoxy. All actuators in this work used copper foil tape with a nonconductive acrylic adhesive as the buried electrode (for single-barrier discharges) or the

wide electrode (for double-barrier discharges). All of these electrodes were 6 cm wide and

NASA/CR—2012-217628 156
1.5 cm long. The cut edge was always placed along the outside boundary, to ensure the

smoother factory-cut edge was closer to the other electrode. The tape firmly pressed down

to eliminate air bubbles and to prevent the inviscid epoxy from “leaking” between the tape

and the dielectric film.

3. Affix a wire to the electrode

To connect this electrode to the external circuit, a wire is required. A piece of copper tape

with a conductive adhesive was applied to connect the wire, both mechanically and electrically, to the electrode. As before, the tape was pressed down firmly to prevent epoxy from

seeping in between the wire and the electrode.

4. Affix a narrow electrode and wire (double-barrier only)

When constructing a double-barrier actuator, we affixed the narrow electrode to the dielectric

film using a procedure similar to that described in steps 8–10 below.

NASA/CR—2012-217628 157
5. Apply layers of insulating epoxy

We mixed the Stycast 2651-40 epoxy resin and Catalyst 9, both produced by Emerson &

Cuming, in a suitable container, and applied a layer to dielectric film inside the outline. After

allowing the epoxy to set, the application process was repeated until the epoxy layers were

thick enough to completely cover the portions of the wire(s) inside the outlined area. This

particular epoxy was chosen for its high dielectric strength, a dielectric constant similar to the

film, and for its relatively low viscosity. The low viscosity is required to avoid the formation

of air bubbles inside the epoxy. Plasma can form inside these air bubbles (increasing the

apparent power use), and the smaller effective thickness of epoxy reduces the total dielectric

strength.

6. Cut, tap, and attach the substrate

For the substrate, we cut a rectangle of clear, 1 cm thick acrylic the same size as the outline.

In one corner (that will be placed away from the electrodes), we drilled and tapped a hole

for a 1/4-20 screw. This hole was used for mechanical mounting the actuator. Onto the

existing epoxy, we applied a final thin layer of epoxy and placed the substrate on top. We

used weights (two bolts in the photograph here) placed next to the substrate to ensure it did

not slide out of place while the epoxy was setting.

NASA/CR—2012-217628 158
7. Flip over the actuator and remove excess dielectric material

After removing the tape holding the dielectric to the working surface, we flipped the actuator

over and trimmed the excess dielectric film using scissors. Some visible de-bonding between

the epoxy and dielectric occurred near the device edges. As long as the space between the

electrodes (or what will be the area between the electrodes) did not de-bond, this did not

affect actuator operation.

8. Affix a wire to the desired exposed electrode (single-barrier only)

The electrode in the photos here was a stripped single-stranded copper wire, placed next to

the actuator. We chose a connecting wire (white in the photos here) and soldered it to one

end of the electrode. When using tungsten or stainless steel exposed electrode, which cannot

be soldered to copper, we first spot-welded a small piece of gold-plated nickel wire to the

electrode, then soldered the nickel wire to the connecting wire.

NASA/CR—2012-217628 159
9. Tape the electrode at the desired location, and apply epoxy to the ends (single-barrier only)

We placed the exposed electrode on the surface at the desired distance (1 mm for all singlebarrier actuators described in this work) from the visible edge of the buried electrode. We

used strips of transparent tape to hold the exposed electrode in place. With the exposed

electrode held at its final position, we applied an insulating epoxy to the ends of the electrode.

Since this epoxy was required solely for structural purposes, bubbles were not a concern.

Consequently, we used a much more viscous epoxy (usually Hysol 0151, manufactured by

Loctite) than used underneath the dielectric layer.

10. Remove tape, trim wires, attach crimp-on connectors

Prior to operation, we removed the transparent tape, as shown in the photograph below. To

avoid wire sagging during balance measurements, we trimmed the connecting wires down to

less than 1 cm. Crimp-on connectors were then attached to allow connections to the external

circuitry. Examples of finished actuators were also shown in figure 3.4.

NASA/CR—2012-217628 160
Appendix B: Details of the Computational Model

B.1 The Poisson’s Equation Solver

The Poisson equation evaluated at time step m+1 using projected densities is given by equation

(6.18), repeated here for clarity:

0∇2φ

m+1 = −e



n

m

p − n

m

e − n

m

n + ∆t



∂np

∂t −

∂ne

∂t −

∂nn

∂t  .

The term in brackets is the projected charge density at time step m+1 based on the known densities

at time m. At the dielectric surface there will be an additional contribution due to accumulated

surface charge.

To work towards a numerical solution to this equation, we first apply the second-order accurate

finite difference approximations to the spatial derivatives. Assuming ∆x = ∆y for simplicity, we

get

φ

m+1

i+1,j + φ

m+1

i−1,j + φ

m+1

i,j+1 + φ

m+1

i,j−1 − 4φ

m+1

i,j =

−

e∆x

2

0



n

m

p − n

m

e − n

m

n



−

e∆t∆x

2

0



∂np

∂t −

∂ne

∂t −

∂nn

∂t 

. (B.1)

The time-derivatives of the densities in equation (B.1) are evaluated using the continuity equations. As described in section 6.2.3, the sources and sink they will cancel out in equation (B.1) and

we can write

∂np

∂t −

∂ne

∂t −

∂ne

∂t = −∇ · Γp + ∇ · Γe + ∇ · Γn

= −

∂Γpx

∂x −

∂Γpy

∂y +

∂Γex

∂x +

∂Γey

∂y +

∂Γnx

∂x +

∂Γny

∂y ,

where Γ is a particle flux.

The remaining task, then, is to evaluate of the particle flux derivatives in terms of the electrostatic potential. When doing this, we will use the semi-implicit method suggested by Ventzek et

al. [105], using densities evaluated at time m and potentials at time m + 1. The following sections

describe how each of these flux terms are evaluated.

NASA/CR—2012-217628 161
B.1.1 Positive Ion Fluxes

We consider the x-directed positive-ion flux first in detail, then describe the other particle fluxes

by analogy. Using the drift-diffusion approximation with the semi-implicit scheme described in

section 6.2.3,

Γpx ≈ +enp

mμpEx

m+1 − eμpTp

∂np

∂x

m

.

To avoid a grid-based instability we use the Scharfetter-Gummel discretization (see section 6.2.2):

Γpx









i+ 1

2

,j

≈

μpTp

∆x

 

z

m+1

i+ 1

2

,j

1 − e

z

m+1

i+ 1

2

,j

np

m

i+1,j +

z

m+1

i+ 1

2

,j

1 − e

−z

m+1

i+ 1

2

,j

np

m

i,j!

where

z

m+1

i+ 1

2

,j

≡

Ex

m+1

i+ 1

2

,j∆x

Tp

≈

−φ

m+1

i+1,j + φ

m+1

i,j

Tp

(B.2)

for positive ions. Similar expressions can be found for the other directions and species.

In order to linearize the flux, we write z

m+1 = zm + ∆z, and assume that ∆z, the change in z

during a given time step, is small for all particle species. “Small” is quantified below. This allows

us to expand the flux expressions in Taylor series and retain only the first two terms:

z + ∆z

1 − e

z+∆z

=

z

1 − e

z

+ ∆z



1

1 − e

z

+

zez

(1 − e

z

)

2



+ · · ·

≈ (z + ∆z)



1 − e

z + zez

(1 − e

z

)

2



−

z

2

e

z

(1 − e

z

)

2

= (z + ∆z) a + b

where

a =

1 − e

z + zez

(1 − e

z

)

2

(B.3)

b =

−z

2

e

z

(1 − e

z

)

2

. (B.4)

Similarly,

z + ∆z

1 − e

−z−∆z

=

z

1 − e

−z

+ ∆z



1

1 − e

−z

−

ze−z

(1 − e

−z

)

2



+ · · ·

≈ (z + ∆z)



1 − e

z − ze−z

(1 − e

−z

)

2



+

z

2

e

−z

(1 − e

−z

)

2

= (z + ∆z) c + d.

NASA/CR—2012-217628 162
where

c =

1 − e

−z − ze−z

(1 − e

−z

)

2

(B.5)

d =

z

2

e

−z

(1 − e

−z

)

2

. (B.6)

Subscripts on a, b, c, and d will indicate which z values to use. A subscript p indicates to use a

positive electric field and the positive ion temperature in the definition of z. Subscripts of halfinteger indices in i will indicate the use of x-directed electric fields in z.

Accuracy for linear expansions of functions of z normally requires |∆z|  1, but for this

particular case that restriction is significantly loosened. Both z/(1 − e

z

) and z/(1 − e

−z

) are

approximately linear except near z = 0, as seen in figure B.1. If |z| > 1, the linearizations are

reasonable provided ∆z/z > −1, i.e the electric field does not change sign during a single time

step. If |z| < 1, the stronger condition |∆z|  1 is required for strict accuracy. These conditions

are used to set the size of the simulation time step, as described in section 6.2.5.

−10 −5 0 5 10

−10 

−5 

0

5

10

z

f(z)

z / (1 − ez)

z / (1 − e−z)

Figure B.1: Two functions of z which are linearized during the derivation of the

Poisson’s equation solver algorithm

An additional complication exists when implementing equations (B.3)–(B.6) in the actual code:

all of these functions have removable singularities at z = 0. Consequently, when evaluating

these expressions near the singularity we employ Taylor expansions of a, b, c, and d for small

z. Deviations from the true limiting values of these functions are smaller than machine round-off

errors.

NASA/CR—2012-217628 163
Now we can approximate the linearized spatial derivative of the x-directed positive ion flux as

∂Γpx

∂x









i,j

≈

1

∆x

Tp

∆x



+ μpi+ 1

2

,j 

z

m+1

i+ 1

2

,japi+ 1

2

,j + bpi+ 1

2

,j

np

m

i+1,j

+ μpi+ 1

2

,j 

z

m+1

i+ 1

2

,j cpi+ 1

2

,j + dpi+ 1

2

,j

np

m

i,j

− μpi− 1

2

,j 

z

m+1

i− 1

2

,japi− 1

2

,j + bpi− 1

2

,j

np

m

i,j

− μpi− 1

2

,j 

z

m+1

i− 1

2

,j cpi− 1

2

,j + dpi− 1

2

,j

np

m

i−1,j

(B.7)

with z

m+1 = z + ∆z. Evaluating the zm+1 terms using equation (B.2), our approximation for the

flux derivative becomes

∂Γpx

∂x









i,j

≈

1

∆x

2



+ μpi+ 1

2

,japi+ 1

2

,j 

−φ

m+1

i+1,j + φ

m+1

i,j 

np

m

i+1,j

+ μpi+ 1

2

,jTpbpi+ 1

2

,jnp

m

i+1,j

+ μpi+ 1

2

,j cpi+ 1

2

,j 

−φ

m+1

i+1,j + φ

m+1

i,j 

np

m

i,j

+ μpi+ 1

2

,jTpdpi+ 1

2

,jnp

m

i,j

− μpi− 1

2

,japi− 1

2

,j 

−φ

m+1

i,j + φ

m+1

i−1,j

np

m

i,j

− μpi− 1

2

,jTpbpi− 1

2

,jnp

m

i,j

− μpi− 1

2

,j cpi− 1

2

,j 

−φ

m+1

i,j + φ

m+1

i−1,j

np

m

i−1,j

− μpi− 1

2

,jTpdpi− 1

2

,jnp

m

i−1,j

.

NASA/CR—2012-217628 164
Rearranging, we get

∂Γpx

∂x









i,j

≈ − φ

m+1

i+1,j

μpi+ 1

2

,j

∆x

2



api+ 1

2

,jnp

m

i+1,j + cpi+ 1

2

,jnp

m

i,j

− φ

m+1

i−1,j

μpi− 1

2

,j

∆x

2



api− 1

2

,jnp

m

i,j + cpi− 1

2

,jnp

m

i−1,j

+ φ

m+1

i,j μpi+ 1

2

,j

∆x

2



api+ 1

2

,jnp

m

i+1,j + cpi+ 1

2

,jnp

m

i,j

+

μpi− 1

2

,j

∆x

2



api− 1

2

,jnp

m

i,j + cpi− 1

2

,jnp

m

i−1,j



+

μpi+ 1

2

,jTp

∆x

2

bpi+ 1

2

,jnp

m

i+ 1

2

,j +

μpi+ 1

2

,jTp

∆x

2

dpi+ 1

2

,jnp

m

i,j

−

μpi− 1

2

,jTp

∆x

2

bpi− 1

2

,jnp

m

i,j −

μpi− 1

2

,jTp

∆x

2

dpi− 1

2

,jnp

m

i−1,j .

Based on the above approximation, we define some new terms,

Apx









i+ 1

2

,j

≡

e∆t

0

μpi+ 1

2

,j 

api+ 1

2

,jnp

m

i+1,j + cpi+ 1

2

,jnp

m

i,j

(B.8)

Bpx









i,j

≡

e∆tTp

0



μpi+ 1

2

,j bpi+ 1

2

,jnp

m

i+1,j − μpi− 1

2

,jdpi− 1

2

,jnp

m

i−1,j

+



μpi+ 1

2

,jdpi+ 1

2

,j − μpi− 1

2

,j bpi− 1

2

,j

np

m

i,j

. (B.9)

We note that all of the quanities in the A and B terms are evaluated at time step m, and so will not

vary during the iterative solution of the Poisson equation for time step m + 1. Now we can fully

describe the spatial derivative of the positive ion flux:

−

e∆t∆x

2

0

 

−

∂Γpx

∂x









i,j!

≈ −φ

m+1

i+1,jApx









i+ 1

2

,j

− φ

m+1

i−1,jApx









i− 1

2

,j

+ φ

m+1

i,j 

Apx









i+ 1

2

,j

+ Apx









i− 1

2

,j!

− Bpx









i,j

.

NASA/CR—2012-217628 165
Analogously, we define

Apy









i+ 1

2

,j

≡

e∆tμpi,j+ 1

2

0



api,j+ 1

2

np

m

i,j+1 + cpi,j+ 1

2

np

m

i,j

(B.10)

Bpy









i,j

≡

e∆tTp

0



μpi,j+ 1

2

bpi,j+ 1

2

np

m

i,j+1 + μpi,j+ 1

2

dpi,j+ 1

2

np

m

i,j

− μpi,j− 1

2

bpi,j− 1

2

np

m

i,j − μpi,j− 1

2

dpi,j− 1

2

np

m

i,j−1



(B.11)

for the y-directed flux of positive ions.

B.1.2 Electron Fluxes

The electron flux approximations are similar, except that secondary electron emission may be

present near material surfaces:

Γex ≈ −ene

mμeEx

m+1 − eμeTe

∂ne

m

∂x − γΓpx,

where γ is the secondary electron emission coefficient, which is zero in the gas region. The effects

of secondary emission are dealt with separately in the following section; only the change in the

bulk electron density is considered for the remainder of this section.

The Scharfetter-Gummel discretization for electron flux is nearly identical to that for ion flux:

Γex









i+ 1

2

,j

≈

μeTe

∆x

 

z

m+1

i+ 1

2

,j

1 − e

z

m+1

i+ 1

2

,j

ne

m

i+1,j +

z

m+1

i+ 1

2

,j

1 − e

−z

m+1

i+ 1

2

,j

ne

m

i,j!

except that now

z

i+ 1

2

,j ≡

−Exi+ 1

2

,j∆x

Te

=

+φi+1,j − φi,j

Te

for electrons.

Using the same approximations for a small variation in z and the definitions of equations (B.3)

to (B.6) (except now using the definition of z for electrons), we can define several terms that will

NASA/CR—2012-217628 166
simplify the resulting expressions:

Aex









i+ 1

2

,j

≡

e∆tμei+ 1

2

,j

0



aei+ 1

2

,jne

m

i+1,j + cei+ 1

2

,jne

m

i,j

(B.12)

Bex









i,j

≡

e∆tTe

0



μei+ 1

2

,j bei+ 1

2

,jne

m

i+1,j + μei+ 1

2

,jdei+ 1

2

,jne

m

i,j

− μei− 1

2

,j bei− 1

2

,jne

m

i,j − μei− 1

2

,jdei− 1

2

,jne

m

i−1,j

, (B.13)

so that

−

e∆t∆x

2

0

 

+

∂Γex

∂x









i,j!

≈ −φ

m+1

i+1,jAex









i+ 1

2

,j

− φ

m+1

i−1,jAex









i− 1

2

,j

+ φ

m+1

i,j 

Aex









i+ 1

2

,j

+ Aex









i− 1

2

,j!

− Bex









i,j

.

The analogous definitions are

Aey









i,j+ 1

2

≡

e∆tμei,j+ 1

2

0



aei,j+ 1

2

ne

m

i,j+1 + cei,j+ 1

2

ne

m

i,j

(B.14)

and

Bey









i,j

≡

e∆tTe

0



μei,j+ 1

2

bei,j+ 1

2

ne

m

i,j+1 + μei,j+ 1

2

dei,j+ 1

2

ne

m

i,j

− μei,j− 1

2

bei,j− 1

2

ne

m

i,j − μei,j− 1

2

dei,j− 1

2

ne

m

i,j−1



. (B.15)

for the y-directed electron flux.

B.1.3 Secondary Electron Flux

We will consider the x-directed flux of secondary electrons first. Secondary electron flux in

this direction must come from vertical surfaces. We assume that only positive ions may cause

secondary electron emission. The secondary electron flux is then given by

Γ

SE

ex









i+ 1

2

,j

= −



γi+1,j + γi,jΓpx









i+ 1

2

,j

.

NASA/CR—2012-217628 167
The two values of γ are added to account for a surface existing at either i or i + 1. The only time

both values will be non-zero will be inside of a material, where Γpx itself must be zero.

To determine the change in electron density, we need the derivative of this flux. Using the

constants defined in section B.1.1, this is

∂Γ

SE

ex

∂x









i,j

≈

Tp

(∆x)

2



−



γi+1,j + γi,j

μpi+ 1

2

,j 

z

m+1

i+ 1

2

,japi+ 1

2

,j + bpi+ 1

2

,j

np

m

i+1,j

+



z

m+1

i+ 1

2

,j cpi+ 1

2

,j + dpi+ 1

2

,j

np

m

i,j

+



γi,j + γi−1,jμpi− 1

2

,j 

z

m+1

i− 1

2

,japi− 1

2

,j + bpi− 1

2

,j

np

m

i,j

+



z

m+1

i− 1

2

,j cpi− 1

2

,j + dpi− 1

2

,j

np

m

i−1,j

We note that this is identical to equation (B.7), except with each mobility multiplied by a

negative secondary electron emission coefficient. Consequently we can construct constants for the

extra change in electron density due to secondary electron emission by analogy:

A

SE

ex









i+ 1

2

,j

≡

e∆t

0

μpi+ 1

2

,j 

γi+1,j + γi,j



api+ 1

2

,jnp

m

i+1,j + cpi+ 1

2

,jnp

m

i,j

(B.16)

and

B

SE

ex









i,j

≡

e∆tTp

0



+



γi+1,j + γi,jμpi+ 1

2

,j bpi+ 1

2

,jnp

m

i+1,j

−



γi,j + γi−1,j

μpi− 1

2

,jdpi− 1

2

,jnp

m

i−1,j

+



+



γi+1,j + γi,jμpi+ 1

2

,jdpi+ 1

2

,j

−



γi,j + γi−1,j

μpi− 1

2

,j bpi− 1

2

,j

np

m

i,j

(B.17)

in the x-direction. Then

−

e∆t∆x

2

0

 

+

∂Γ

SE

ex

∂x









i,j!

≈ − φ

m+1

i+1,jA

SE









i+ 1

2

,j

− φ

m+1

i−1,jA

SE









i− 1

2

,j

+ φ

m+1

i,j 

A

SE









i+ 1

2

,j

+ A

SE









i− 1

2

,j!

− B

SE









i,j

.

(B.18)

NASA/CR—2012-217628 168
Similarly,

A

SE

ey









i,j+ 1

2

≡

e∆t

0

μpi,j+ 1

2



γi,j+1 + γi,j



api,j+ 1

2

np

m

i,j+1 + cpi,j+ 1

2

np

m

i,j

(B.19)

and

B

SE

ey









i,j

≡

e∆tTp

0



+



γi,j+1 + γi,jμpi,j+ 1

2

bpi,j+ 1

2

np

m

i+1,j

−



γi,j + γi,j−1



μpi,j− 1

2

dpi,j− 1

2

np

m

i,j−1

+



+



γi,j+1 + γi,jμpi,j+ 1

2

dpi,j+ 1

2

−



γi,j + γi,j−1



μpi,j− 1

2

bpi,j− 1

2



np

m

i,j

(B.20)

for the y-direction.

B.1.4 Negative Ion Fluxes

Since the negative ions have a negative charge, their flux can be described in an exact analogy

to the electron flux. So

−

e∆t∆x

2

0



+

∂Γnx

∂t  







i,j

≈ −φ

m+1

i+1,jAnx









i+ 1

2

,j

− φ

m+1

i−1,jAnx









i− 1

2

,j

+ φ

m+1

i,j 

Anx









i+ 1

2

,j

+ Anx









i− 1

2

,j!

− Bnx









i,j

,

where

Anx









i+ 1

2

,j

≡

e∆tμni+ 1

2

,j

0



ani+ 1

2

,jnn

m

i+1,j + cni+ 1

2

,jnn

m

i,j

(B.21)

Bnx









i,j

≡

e∆tTn

0



μni+ 1

2

,j bni+ 1

2

,jnn

m

i+1,j + μni+ 1

2

,jdni+ 1

2

,jnn

m

i,j

− μni− 1

2

,j bni− 1

2

,jnn

m

i,j − μni− 1

2

,jdni− 1

2

,jnn

m

i−1,j

. (B.22)

In the calculations of an, bn, cn, and dn, the definition

z

i+ 1

2

,j = −Ex∆x/Tn

NASA/CR—2012-217628 169
is used. Similarly,

Any









i,j+ 1

2

≡

n∆tμni,j+ 1

2

0



ani,j+ 1

2

nn

m

i,j+1 + cni,j+ 1

2

nn

m

i,j

(B.23)

Bey









i,j

≡

e∆tTn

0



μni,j+ 1

2

bni,j+ 1

2

nn

m

i,j+1 + μni,j+ 1

2

dni,j+ 1

2

nn

m

i,j

− μni,j− 1

2

bni,j− 1

2

nn

m

i,j − μni,j− 1

2

dni,j− 1

2

nn

m

i,j−1



. (B.24)

for the y-directed negative ion flux.

NASA/CR—2012-217628 170
B.1.5 Modified Poisson Equation

Inserting our approximations for all of the flux derivatives into equation (B.1) yields

φ

m+1

i+1,j+φ

m+1

i−1,j + φ

m+1

i,j+1 + φ

m+1

i,j−1 − 4φ

m+1

i,j =

−

e∆x

2

0



n

m

p − n

m

e − n

m

n



+ Bpx









i,j

+ Bpy









i,j

− Bex









i,j

− Bey









i,j

− B

SE

ex









i,j

− B

SE

ey









i,j

− Bnx









i,j

− Bny









i,j

− φ

m+1

i+1,j 

Apx









i+ 1

2

,j

+ Aex









i+ 1

2

,j

+ A

SE

ex









i+ 1

2

,j

+ Anx









i+ 1

2

,j!

− φ

m+1

i−1,j 

Apx









i− 1

2

,j

+ Aex









i− 1

2

,j

+ A

SE

ex









i− 1

2

,j

+ Anx









i− 1

2

,j!

− φ

m+1

i,j+1 

Apy









i,j+ 1

2

+ Aey









i,j+ 1

2

+ A

SE

ey









i,j+ 1

2

+ Any









i,j+ 1

2

!

− φ

m+1

i,j−1

 

Apy









i,j− 1

2

+ Aey









i,j− 1

2

+ A

SE

ey









i,j− 1

2

+ Any









i,j− 1

2

!

+ φ

m+1

i,j 

Apx









i+ 1

2

,j

+ Apx









i− 1

2

,j

+ Apy









i,j+ 1

2

+ Apy









i,j− 1

2

+ Aex









i+ 1

2

,j

+ Aex









i− 1

2

,j

+ Aey









i,j+ 1

2

+ Aex









i,j− 1

2

+ A

SE

ex









i+ 1

2

,j

+ A

SE

ex









i− 1

2

,j

+ A

SE

ey









i,j+ 1

2

+ A

SE

ex









i,j− 1

2

+ Anx









i+ 1

2

,j

+ Anx









i− 1

2

,j

+ Any









i,j+ 1

2

+ Anx









i,j− 1

2



.

(B.25)

NASA/CR—2012-217628 171
This leads us to define

A ≡ 4 + Apx









i+ 1

2

,j

+ Apx









i− 1

2

,j

+ Apy









i,j+ 1

2

+ Apy









i,j− 1

2

+ Aex









i+ 1

2

,j

+ Aex









i− 1

2

,j

+ Aey









i,j+ 1

2

+ Aey









i,j− 1

2

+ A

SE

ex









i+ 1

2

,j

+ A

SE

ex









i− 1

2

,j

+ A

SE

ey









i,j+ 1

2

+ A

SE

ey









i,j− 1

2

+ Anx









i+ 1

2

,j

+ Anx









i− 1

2

,j

+ Any









i,j+ 1

2

+ Any









i,j− 1

2

and the update terms actually used in the successive over-relaxation algorithm:

C1 ≡

 

1 + Apx









i+ 1

2

,j

+ Aex









i+ 1

2

,j

+ A

SE

ex









i+ 1

2

,j

+ Anx









i+ 1

2

,j!

A (B.26)

C2 ≡

 

1 + Apx









i− 1

2

,j

+ Aex









i− 1

2

,j

+ A

SE

ex









i− 1

2

,j

+ Anx









i− 1

2

,j!

A (B.27)

C3 ≡

 

1 + Apy









i,j+ 1

2

+ Aey









i,j+ 1

2

+ A

SE

ey









i,j+ 1

2

+ Any









i,j+ 1

2

!

A (B.28)

C4 ≡

 

1 + Apy









i− 1

2

,j

+ Aey









i,j− 1

2

+ A

SE

ey









i,j− 1

2

+ Any









i,j− 1

2

!

A (B.29)

C5 ≡



+

e∆x

2

0



n

m

p − n

m

e − n

m

n



− Bpx









i,j

− Bpy









i,j

+ Bex









i,j

+ Bey









i,j

+ B

SE

ex









i,j

+ B

SE

ey









i,j

+ Bnx









i,j

+ Bny









i,j

A. (B.30)

Using our newly defined terms in equation (B.25) yields the final update expression for the bulk

plasma and bulk dielectric regions:

φ

m+1

i,j = C1(i, j)φ

m+1

i+1,j + C2(i, j)φ

m+1

i−1,j + C3(i, j)φ

m+1

i,j+1 + C4(i, j)φ

m+1

i,j−1 + C5(i, j). (B.31)

Inside electrodes, the potential must equal the specified voltage, so we set C1 = C2 = C3 =

C4 = 0 there, and set C5 equal to the voltage.

NASA/CR—2012-217628 172
B.1.6 Corrections at the Dielectric Surface

At the dielectric surface, the electrostatic potential is strongly affected both by the presence

of surface charge and the change in dielectric constant. In addition, many of the terms described

above are zero either at or just below a dielectric surface, so the update terms may be simplified

there.

Currently, the simulation code requires that there be only a single, rectangular dielectric region

extending from the bottom of the simulation domain to the grid index j = js. It fills the entire

x-width of the domain. The relative dielectric constant beneath the surface is d, and precisely on

the surface we take it to be (1 + d)/2. All particle densities are set to zero on the surface. The

derivation below reflects this geometry.

Consider just the left-hand side of Poisson’s equation, evaluated at the dielectric surface. Applying the standard finite difference approximations, the LHS becomes

−∇ · E~



= −0



1 + d

2

∂Ex

∂x +

∂

∂y (rEy)



= 0



1 + d

2∆x

2





φi+1,j + φi−1,j − 2φi,j

+



0

∆y

2

 

φi,j+1 + dφi,j−1 − 2

1 + d

2

φi,j.

Assuming ∆x = ∆y, we can further simplify this to

0

∆x

2



−4

1 + d

2

φi,j +

1 + d

2

φi+1,j +

1 + d

2

φi−1,j + φi,j+1 + dφi,j−1



.

At the dielectric surface the particle densities are all zero, but the right-hand side is not zero.

The flux of charge to the surface during all previous time-steps is accounted for in the surface

charge. The flux of charge to the surface during the current time-step is accounted for by the nonzero values of some of the flux derivatives. Since there are no particles on the surface, only the

y-derivatives of the particle fluxes will be non-zero. Accounting for these features, the RHS of

Poisson’s equation is

−σ

∆x

− e∆t

 

−

∂Γpy

∂y +

∂Γey

∂y +

∂Γ

SE

ey

∂y +

∂Γny

∂y !

NASA/CR—2012-217628 173
.

From our earlier description of the bulk plasma,

−e∆t∆x

2

0



−

∂Γpy

∂y  







i,js

= −φi,js+1Apy









i,js+ 1

2

− φi,js−1Apy









i,js− 1

2

+ φi,js Apy









i,js+ 1

2

+ Apy









i,js− 1

2

!

− Bpy









i,js

,

but due to the absence of particles on and under the dielectric surface,

Apy









i,js− 1

2

= Bpy









i,js

= 0.

The other particle species can be examined in a similar fashion. From the analysis of both rightand left-hand sides of the equation, we now define a new constant on the dielectric surface:

Ai,js = 4 

1 + d

2



+



Apy + Aey + A

SE

py + Any









i,js+ 1

2

. (B.32)

Now we may define new update constants similar to those discussed previously:

C1(i, js) = 

1 + d

2



/Ai,js (B.33)

C2(i, js) = 

1 + d

2



/Ai,js (B.34)

C3(i, js) = 1 + Apy









i,js+ 1

2

+ Aey









i,js+ 1

2

+ A

SE

py









i,js+ 1

2

+ Any









i,js+ 1

2

!

/Ai,js (B.35)

C4(i, js) = d/Ai,js (B.36)

C5(i, js) = σ

m∆x

0

/Ai,js. (B.37)

B.2 The Continuity Equation Solver

We will consider the case of electrons in detail, then state the algorithms for positive and

negative ions by analogy.

NASA/CR—2012-217628 174
B.2.1 Electrons

The continuity equation for electrons is given by

∂ne

∂t = −∇ · ~Γe + Siz − Satt − Rpe (B.38)

where Γe is the electron flux, Siz is the total rate of ionization, Satt is the total rate of attachment,

and Rpe is the total rate of positive ion-electron recombination. In an explicit scheme, the righthand side of equation (B.38) is evaluated at time step m. In a completely implicit scheme, it is

evaluated at m + 1. Here we use a semi-implicit scheme where the flux divergence is evaluated at

m + 1 and all other terms at m. This allows us to obtain the stability of an implicit scheme while

still maintaining much of the simplicity (and fast code execution) of an explicit scheme.

The electron flux consists of the bulk flux plus the flux of secondary electrons emitted from

material surfaces. For simplicity, we will group the divergence of the secondary electron flux with

the sources and sinks. Our semi-implicit scheme is then written as

∂ne

∂t = (−∇ · Γe)









m+1

+



−∇ · Γ

SE

e + Siz − Satt − Rpe









m

.

Begin by considering the bulk electron flux. As in the Poisson solver, we use the ScharfetterGummel discretization (see section 6.2.2) for the flux:

Γex









m+1

i+ 1

2

,j

=

μei+ 1

2

,jTe

∆x

 

z

m+1

i+ 1

2

,j

1 − e

z

m+1

i+ 1

2

,j

ne

m+1

i+1,j +

z

m+1

i+ 1

2

,j

1 − e

−z

m+1

i+ 1

2

,j

ne

m+1

i,j !

.

The values of z at m + 1 are already known, as the Poisson solver executes prior to the continuity

solver at each time step. To simplify the flux expressions, we define

Z1ex(z) = μeTe

∆x

z

1 − e

z

and Z2ex(z) = μeTe

∆x

z

1 − e

−z

,

where the subscript ex indicates to use the definition of z appropriate for x-directed electron flux.

With these definitions, we can write a simplified expression for the discretized spatial derivative of

the flux.

∂Γex

∂x









m+1

i,j

≈

1

∆x

Z1ex









m+1

i+ 1

2

,j

ne

m+1

i+1,j +

1

∆x

 

Z2ex









m+1

i+ 1

2

,j

+ Z1ex









m+1

i− 1

2

,j!

ne

m+1

i,j

−

1

∆x

Z2ex









m+1

i− 1

2

,j

ne

m+1

i−1,j .

NASA/CR—2012-217628 175
To simplify our final update expression for ne we define

Ae ≡

∆t

∆x

 

Z2ex









m+1

i+ 1

2

,j

− Z1ex









m+1

i− 1

2

,j!

+

∆t

∆y

 

Z2ey









m+1

i,j+ 1

2

− Z1ey









m+1

i,j− 1

2

!

.

Using this, we then define

C1e(i, j) ≡

 

−

∆t

∆x

Z1ex









m+1

i+ 1

2

,j!

(1 + Ae) (B.39)

C2e(i, j) ≡

 

+

∆t

∆x

Z2ex









m+1

i− 1

2

,j!

(1 + Ae) (B.40)

C3e(i, j) ≡

 

−

∆t

∆y

Z1ey









m+1

i,j+ 1

2

!

(1 + Ae) (B.41)

C4e(i, j) ≡

 

+

∆t

∆y

Z2ey









m+1

i,j− 1

2

!

(1 + Ae) (B.42)

C5e(i, j) ≡

h

ne

m

i,j + ∆t



−∇ · Γ

SE

e + Siz − Satt − Rpem

i,ji

(1 + Ae) (B.43)

so that

ne

m+1

i,j = C1ene

m+1

i+1,j + C2ene

m+1

i−1,j + C3ene

m+1

i,j+1 + C4ene

m+1

i,j−1 + C5e. (B.44)

Equation (B.44) is solved iteratively using the Gauss-Seidel method. For other particle species, all

terms except C5 can be calculated by direct analogy.

The ionization, attachment, and recombination rates in C5e are calculated as described in section 6.1.2. The flux of secondary electrons exists only near material boundaries, and is calculated

using

Γ

SE

ex









i+ 1

2

,j

=







−



γi+1,j + γi,jΓpx









i+ 1

2

,j

, Ex · Γpx > 0

0, Ex · Γpx < 0

.

The separate cases exist because we assume that any ions diffusing into the surface, rather than

accelerated by an electric field, will not have enough energy to knock off any electrons. The ycomponent of the secondary electron flux is computed similarly. The divergence of the flux is

determined using the same standard finite difference formulas used elsewhere in this derivation.

NASA/CR—2012-217628 176
B.2.2 Positive Ions

Positive ions are generated via electron-impact ionization, and lost through recombination both

with electrons and with negative ions. By analogy to the equations in section B.2.1, we write

C1p(i, j) ≡

 

−

∆t

∆x

Z1px









m+1

i+ 1

2

,j!

(1 + Ap) (B.45)

C2p(i, j) ≡

 

+

∆t

∆x

Z2px









m+1

i− 1

2

,j!

(1 + Ap) (B.46)

C3p(i, j) ≡

 

−

∆t

∆y

Z1py









m+1

i,j+ 1

2

!

(1 + Ap) (B.47)

C4p(i, j) ≡

 

+

∆t

∆y

Z2py









m+1

i,j− 1

2

!

(1 + Ap) (B.48)

C5p(i, j) ≡

h

np

m

i,j + ∆t(Siz − Rpe − Rpn)

m

i,ji

(1 + Ap). (B.49)

B.2.3 Negative Ions

Negative ions are created via electron attachment to neutral molecules, and lost only through

recombination with positive ions. Then for negative ions

C1n(i, j) ≡

 

−

∆t

∆x

Z1nx









m+1

i+ 1

2

,j!

(1 + An) (B.50)

C2n(i, j) ≡

 

+

∆t

∆x

Z2nx









m+1

i− 1

2

,j!

(1 + An) (B.51)

C3n(i, j) ≡

 

−

∆t

∆y

Z1ny









m+1

i,j+ 1

2

!

(1 + An) (B.52)

C4n(i, j) ≡

 

+

∆t

∆y

Z2ny









m+1

i,j− 1

2

!

(1 + An) (B.53)

C5n(i, j) ≡

h

nn

m

i,j + ∆t(Satt − Rpn)

m

i,ji

(1 + An). (B.54)

NASA/CR—2012-217628 177
Appendix C: List of Symbols

Symbols

A Constant parameter in the Poisson’s or continuity equation solvers

a, b, c, d Constant parameters in the Poisson’s equation solver

B Constant parameters in the Poisson’s equation solver

B~ Magnetic field (T)

af Adjustment factor used in determining ∆t

C1–C5 Constant parameters in the Poisson’s or continuity equation solvers

d Electrode diameter (m)

d0 Force or linear force density fit parameter (m)

E~ Electric field (V/m)

E/p Total reduced electric field (V m−1 Torr−1)

Eiz Ionization energy (eV)

e Fundamental charge, 1.602 × 10−19 C

F~

body Body force acting in a control volume (N)

Ffit Force or linear force density fit parameter (N or N/m)

Fplasma Plasma body force (N)

Fshear Shear force (N)

F~

surf Surface force acting on a control volume (N)

F0 Force or linear force density fit parameter (m)

F1 Force or linear force density fit parameter (m)

~fEHD Electro-hydrodynamic force density (N/m3

)

i, j Compuational grid indices in the x and y directions, respectively

js y grid index of the dielectric surface

L Unit length (m)

L0 Gas flow velocity fit parameter (m)

`c Chord length of an airfoil (m)

m Particle mass (kg)

NASA/CR—2012-217628 178
n Particle number density (m−3

)

p Pressure (Pa)

pref Reference pressure (Pa)

p0 Gas stagnation pressure (Pa)

Q Charge on an electrode (C)

R Sink rate (m−3s

−1

)

Rec Chord Reynolds number

r Recombination coefficient (m3/ s)

S Source rate (m−3s

−1

)

T Particle temperature (eV)

~u Gas flow velocity (m/s)

ufit Skewed Gaussian fit to the measured gas velocity (m/s)

u0 Gas flow velocity fit parameter (m/s)

V Applied voltage (V)

VB Breakdown voltage (V)

~v Charged-particle fluid velocity (m/s)

xmax Extent of corona discharge region (m)

y0 Gas flow velocity fit parameter (m)

Z1, Z2 Constant parameters in the continuity equation solver

z Non-dimensional electric field in the Scharfetter-Gummel discretization

α Ionization coefficient (m−1)

Γ Particle flux (m−2s

−1

)

γ Secondary electron emission coefficient

∆x x-spacing of the computational grid (m)

∆y y-spacing of the computational grid (m)

∆t Temporal step size (s)

 Electrical permittivity of a medium (F/m)

0 Permittivity of free space, 8.854 × 10−12 F/m

d Relative permittivity of the dielectric material

η Electron attachment coefficient (m−1)

NASA/CR—2012-217628 179
λ Mean free path (m)

λD Debye Length (m)

λel Mean free path for elastic collisions (m)

λin Mean free path for inelastic collisions (m)

λ Energy relaxation length (m)

μ Particle mobility (m2 V

−1

s

−1

)

μg Dynamic viscosity of the neutral gas (kg m−1s

−1

)

ν Collision frequency (s−1)

νm Effective momentum transfer frequency (s−1)

ρ Mass density (kg m−3)

σ Free surface charge on a dielectric (C / m2)

σp Plasma conductivity (S / m)

φ Electrostatic potential (V)

ωc Cyclotron frequency (s−1)

NASA/CR—2012-217628 180
Symbol subscripts

att due to electron attachment

bulk away from material surfaces

eff effective value of a quantity

diel of a dielectric

e of electrons

elec of an electrode

g of the neutral gas

iz due to electron-impact ionization

i, j at the i

th and jth grid indices

max, min maximum and minimum allowed values

N2 of nitrogen gas

n of negative ions

O2 of oxygen gas

p of positive ions

x, y, z in the x, y, or z directions, respectively

α of a generic charged-particle species

+ of all positively-charged particles

− of all negatively-charged particles

Symbol superscripts

m computational time index

SE due to secondary electron emission

NASA/CR—2012-217628 181
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Measurements and Simulations of Surface Dielectric Barrier Discharges Used as Plasma 

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This report was submitted as a dissertation in partial fulfillment of the requirements for the degree Doctor of Philosophy for Nuclear 

Engineering and Engineering Physics to the University of Wisconsin-Madison, Madison, Wisconsin, June 2009. Grant technical monitor, 

David E. Ashpis, Aeropropulsion Division, Glenn Research Center, organization code RTTO, ashpis@nasa.gov 

14. ABSTRACT

This report is a Ph.D. dissertation performed under NRA cooperative agreement and submitted as part of the final report. Asymmetric 

surface dielectric barrier discharges (DBDs) have shown promise for use as aerodynamic actuators for active flow control. In this project we 

studied DBD actuators experimentally and numerically. Our DBDs used a symmetric triangular high voltage waveform to generate plasma 

in atmospheric pressure air. Time-averaged measurements indicated that the induced force of a single barrier actuator design (one electrode 

insulated from the plasma) can be increased exponentially above the results of previous studies by decreasing both the length and thickness 

of the electrode exposed to the plasma. This increased force may allow these devices to control flow separation in a wider range of flow 

environments. Experiments using an intensified digital camera to examine the plasma on time scales of a few nanoseconds showed that, in 

addition to the previously-observed filamentary and jet-like plasma structures, discharges with very thin exposed electrodes exhibited a 

weak but constant plasma immediately adjacent to those electrodes. In double-barrier actuators (both electrodes insulated), decreasing the 

diameter of the narrower electrode lead to increasing forces, and recorded images showed the simultaneous existence of both filamentary 

and jet-like plasma structures. The development and application of a time-dependent, two-dimensional computational fluid plasma model 

has aided in understanding the detailed physics of surface DBDs at all-time scales. For simulated single-barrier discharges, the model 

qualitatively reproduced the filamentary and jet-like microdischarge structures. The model was somewhat successful in reproducing the 

observed characteristics of double-barrier actuators. For both actuator geometries, the model indicated that the majority of the forces 

induced on the neutral gas occur in between microdischarges as the plasmas decay. 

15. SUBJECT TERMS

Plasma; Dielectric barrier discharge; DBD; Flow control; High voltage 

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