ABSTRACT
Title of dissertation: ONE-DIMENSIONAL ANALYTICAL MODEL
DEVELOPMENT OF A
PLASMA-BASED ACTUATOR
Sarah Haack Popkin, Doctor of Philosophy, 2014
Dissertation directed by: Professor Alison Flatau
Department of Aerospace Engineering
This dissertation provides a method for modeling the complex, multi-physics,
multi-dimensional processes associated with a plasma-based flow control actuator,
also known as the SparkJet, by using a one-dimensional analytical model derived
from the Euler and thermodynamic equations, under varying assumptions. This
model is compared to CFD simulations and experimental data to verify and/or
modify the model where simplifying assumptions poorly represent the real actuator.
The model was exercised to explore high-frequency actuation and methods of im-
proving actuator performance. Using peak jet momentum as a performance metric,
the model shows that a typical SparkJet design (1 mm orifice diameter, 84.8 mm3
cavity volume and 0.5 J energy input) operated over a range of frequencies from
1 Hz to 10 kHz shows a decrease in peak momentum corresponding to an actuation
cutoff frequency of 800 Hz. The model results show that the cutoff frequency is
primarily a function of orifice diameter and cavity volume.
To further verify model accuracy, experimental testing was performed involv-
ing time-dependent, cavity pressure and arc power measurements as a function of
orifice diameter, cavity volume, input energy and electrode gap. The cavity pres-
sure measurements showed that pressure-based efficiency ranges from 20% to 40%.
The arc power measurements exposed the deficiency in assuming instantaneous en-
ergy deposition and a calorically perfect gas and also showed that arc efficiency was
approximately 80%. Additional comparisons between the pressure-based modeling
and experimental results show that the model captures the actuator dependence on
orifice diameter, cavity volume, and input energy but over-estimates the duration
of the jet flow during Stage 2. The likely cause of the disagreement is an inaccurate
representation of thermal heat transfer related to convective heat transfer or heat
loss to the electrodes.
ONE-DIMENSIONAL ANALYTICAL MODEL DEVELOPMENT
OF A PLASMA-BASED ACTUATOR
by
Sarah Haack Popkin
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014
Advisory Committee:
Professor Alison Flatau, Chair/Advisor
Professor James Baeder
Professor Christopher Cadou
Professor Dianne O’Leary
Professor Norman Wereley
Copyright by
Sarah Haack Popkin
2014
Dedication
To my past, present and future family.
ii
Acknowledgments
I owe my deepest gratitude to all the people who have made this dissertation
possible. My relationship with my advisor, Dr. Flatau, was established more than
10 years ago and I had no idea how valuable that relationship would be. Thank
you, Dr. Flatau, for always being my advocate and guiding me through both my
Master’s and PhD and reading and editing both my thesis and this dissertation! I
would also like to thank the committee members for helping finalize this dissertation
to be a truly beneficial body of work for the flow control community. This work has
been financially supported by AFOSR under supervision of Drs. John Schmisseur
and Doug Smith through JHU/APL. Without the AFOSR support, accomplishment
of this research would have been much more difficult. I would also like to thank
PCB for helping me find the right pressure sensor for my volatile application and
understand the techniques for protecting it in the high-temperature environment.
At JHU/APL, several people have been involved with this project. I am very
grateful for the support and technical advice from Bo Cybyk, Bruce Land, Trent
Taylor, John Teehan and Dave VanWie. These colleagues have helped guide the
SparkJet research and helped me understand the broader picture in the field of
arcs, sparks, and plasma. Bruce, I would especially like to thank you for all of the
extra hours you spent designing the SparkJet circuitry and making sure I stayed safe
around high voltages and high current. Bo, I would also like to thank you for acting
as my advisor at JHU/APL and advocating my efforts to pursue this dissertation.
I would like to thank my colleagues at JHU/APL who have helped me along
iii
the way providing technical and friendly advice for the last six years. In addition,
pursuing this dissertation would have been impossible without the encouragement
from my supervisors despite working full-time at JHU/APL. To my family and
friends, thank you for never doubting my abilities and always asking me how I was
doing to let me know you were thinking of me.
Finally, to my husband, Russell Popkin. I am so grateful that you put up with
me despite the hundreds of hours spent away from you and the tears of frustration
and doubt. In my times of need, I had you there to steer me in the right direction
but also let me know you supported whichever direction I chose. I am so lucky to
have you by my side.
iv
Contents
List of Tables vii
List of Figures viii
Nomenclature xiii
1 Introduction 1
1.1 High-Speed Flow Control Applications . . . . . . . . . . . . . . . . . 2
1.1.1 Flow Separation . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Boundary Layer Transition . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Supersonic Open Cavity Flow . . . . . . . . . . . . . . . . . . 5
1.1.4 Jet Impingement / Jet Noise . . . . . . . . . . . . . . . . . . . 7
1.2 High-Momentum Flow Control Actuators . . . . . . . . . . . . . . . . 9
1.2.1 Passive Flow Control . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Active Flow Control . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Dissertation Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 SparkJet Actuator 23
2.1 Device Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Stage 1 - Arc Discharge . . . . . . . . . . . . . . . . . . . . . 24
2.1.2 Stage 2 - Jet Flow . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.3 Stage 3 - Refresh . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Analytical Modeling . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Computational Modeling . . . . . . . . . . . . . . . . . . . . . 31
2.2.3 Experimental Characterization . . . . . . . . . . . . . . . . . . 34
3 1-D Analytical Model 41
3.1 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Stage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Stage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 Stage 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.4 Thermal Modeling . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Single Cycle Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Simulating High-Frequency Actuation . . . . . . . . . . . . . . . . . . 71
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Experimental Details and Data Acquisition 79
4.1 SparkJet Actuator Device . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 SparkJet Power Supplies . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 External Supplies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5 Cavity Pressure Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 97
v
4.5.1 Signal Post-Processing . . . . . . . . . . . . . . . . . . . . . . 104
4.6 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Pressure-Based Results and Analysis 112
5.1 Basic Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1 Variation with Input Energy . . . . . . . . . . . . . . . . . . . 116
5.2.2 Variation with Cavity Volume . . . . . . . . . . . . . . . . . . 118
5.2.3 Variation with Orifice Diameter . . . . . . . . . . . . . . . . . 119
5.2.4 Variation with Electrode Gap . . . . . . . . . . . . . . . . . . 121
5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4 Comparison to 1-D Model . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Power-Based Results and Analysis 132
6.1 Arc Power Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2 Pseudo-Series Trigger Results . . . . . . . . . . . . . . . . . . . . . . 138
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Conclusions and Future Work 142
7.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Appendix 148
Bibliography 161
vi
List of Tables
3.1 Design parameters for numerical model used for comparison to CFD
results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
vii
List of Figures
1.1 Schlieren image of shock-induced flow separation on the upper surface
of an airfoil in transonic flow [1]. . . . . . . . . . . . . . . . . . . . . . 3
1.2 Sketch of flow over a flat plate through transition from a laminar
to turbulent boundary layer. The regions of flow from 1-6 are 1)
stable laminar flow, 2) unstable Tollmien-Schlichting waves, 3) three-
dimensional waves and vortex formation (Λ-structures), 4) vortex de-
cay, 5) formation of turbulent spots, and 6) fully turbulent flow, re-
spectively [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 F-22 Raptor flying with open bomb bay doors [5]. . . . . . . . . . . . 6
1.4 Demonstration of flow separation control in supersonic flow using
passive vortex generators [13]. . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Picture of surface porosity demonstrating the hole size and distribu-
tion for delaying boundary layer transition in hypersonic flow [14]. . . 20
1.6 Effect of a notched nozzle exit on the sound pressure levels versus
Strouhal number (Sr) associated with screech tones [10]. . . . . . . . 20
1.7 Design requirements and output for unsteady REM actuation. . . . . 21
1.8 Sketch of the COMPACT [28]. . . . . . . . . . . . . . . . . . . . . . . 21
1.9 Effect of nanosecond discharge on the coefficient of pressure (Cp)
distribution versus the percentage of the chord length (x/b, %) over
an airfoil in transonic flow [32]. . . . . . . . . . . . . . . . . . . . . . 22
2.1 Basic schematic of the three stages of SparkJet operation. . . . . . . 24
2.2 Microschlieren images at two time delays after Stage 1 initiation show-
ing the a) blast wave and b) turbulent jet formation in the early
portion of Stage 2 [42]. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 SparkJet flow velocity at the orifice based on a simplified, one-dimensional,
analytical model developed by Grossman, et al. [40]. . . . . . . . . . 30
2.4 First CFD simulation of SparkJet interaction with a Mach 3 crossflow
over a flat plate [45]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Variation in velocities determined from schlieren images for the pulsed
plasma jet for various capacitive energy depositions [41]. . . . . . . . 37
2.6 Average velocity field contours and vectors for the 25 µF discharge
at 30 µs, 50 µs, 70 µs and 90 µs delay times [41]. . . . . . . . . . . . 38
3.1 Sketch of the assumed control volume where the larger volume rep-
resents the cavity volume, the smaller volume represents the orifice
volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Electrical representation of the thermal heat transfer process for the
SparkJet actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Sketch of the thermal heat transfer modeled using finite difference
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Temperature profile used to compare the combined thermal resistance
and capacitance method and the discretization method . . . . . . . . 59
viii
3.5 Wall temperature response to the notional temperature profile using
the combined thermal capacitive and resistive model. . . . . . . . . . 60
3.6 Wall temperature response to the notional temperature profile simu-
lating 1 kHz actuation over 50 cycles. . . . . . . . . . . . . . . . . . . 60
3.7 Time and space dependent thermal response of the Macor material
to the notional temperature profile applied to the internal wall of the
Macor during a single cycle. . . . . . . . . . . . . . . . . . . . . . . . 62
3.8 Time and space dependent thermal response of the Macor material
to the notional cavity air temperature profile applied at a frequency
of 1 kHz over 50 cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9 Simulation of the time-dependent variation in the cavity pressure,
cavity temperature, cavity density, and the velocity through the ori-
fice during early portion of the SparkJet cycle. . . . . . . . . . . . . . 67
3.10 Simulation of the time-dependent variation in cavity pressure, cavity
temperature, cavity density, and the velocity through the orifice for
the entire SparkJet cycle. Note that cavity temperature, pressure,
and density have returned to original ambient conditions as given in
Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.11 Structured, axi-symmetric grid representation of the SparkJet inter-
nal geometry with grid axis units in millimeters. The red section
represents the grid cells raised to an elevated temperature and pres-
sure to represent the energy deposition in Stage 1. . . . . . . . . . . . 68
3.12 Time-dependent CFD simulation of the volume-averaged cavity pres-
sure, temperature, and density and area-averaged jet velocity through
the orifice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.13 Comparison of the cavity pressure, temperature, and density and
orifice velocity versus time based on the CFD simulations and the
simplified numerical model. . . . . . . . . . . . . . . . . . . . . . . . 70
3.14 Simulation of the time dependent variation in cavity pressure, tem-
perature, and density and the velocity through the orifice for three
SparkJet cycles at 100 Hz. . . . . . . . . . . . . . . . . . . . . . . . . 71
3.15 Simulation of the momentum through the SparkJet orifice as a func-
tion of time when operated at 100, 1000 and 5000 Hz. . . . . . . . . . 73
3.16 Magnitude plot of the normalized, steady-state momentum through-
put from the SparkJet actuator operated from 1 Hz to 10 kHz for ori-
fice diameters of 0.4, 1.0 and 2.0 mm, QC = 0.45 J, and v = 84.8 mm3
as predicted by the 1-D model. . . . . . . . . . . . . . . . . . . . . . 75
3.17 Magnitude plot of the normalized, steady-state momentum through-
put from the SparkJet actuator operated from 1 Hz to 10 kHz for
energy deposition values of 0.24, 0.45 and 0.90 J, do = 1 mm, and
v = 84.8 mm3 as predicted by the 1-D model. . . . . . . . . . . . . . 76
3.18 Magnitude plot of the normalized, steady-state momentum through-
put from the SparkJet actuator operated from 1 Hz to 10 kHz for cav-
ity volumes of 42.4, 84.8 and 169.6 mm3, QC = 0.45 J, and do = 1 mm
as predicted by the 1-D model. . . . . . . . . . . . . . . . . . . . . . 77
ix
4.1 Diagram showing the basic electrical connections and equipment used
to acquire experimental data and power the SparkJet power supplies. 80
4.2 Drawing of the SparkJet Macor lid with a 1 mm orifice diameter and
84.8 mm3 cavity volume. . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Drawing of the base component of the SparkJet for a cavity volume
of 84.8 mm3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Photograph of the SparkJet cavities and lids used to characterize
the effect of cavity volume and orifice diameter on SparkJet cavity
pressure rise and the pressure sensor used to acquire cavity pressure
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Photograph of the SparkJet setup showing the Pearson current moni-
tor, differential voltage probe, installed pressure sensor, the SparkJet
actuator, SparkJet power supply, and a 12 inch ruler for scaling. . . 84
4.6 Circuit diagram of the 600 V external trigger SparkJet power supply. 85
4.7 Photograph of the 600ET SparkJet power supply circuit box with
dimensions of 110 mm x 79 mm x 38 mm. . . . . . . . . . . . . . . . 86
4.8 Circuit diagram of the 600 V pseudo-series trigger SparkJet power
supply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.9 Photograph of the 600PST SparkJet power supply circuit box with
dimensions of 133 mm x 133 mm x 55 mm. . . . . . . . . . . . . . . . 88
4.10 Photograph of the TENMA-72-6860 Pulse Generator used to trigger
the 600ET SparkJet circuit. . . . . . . . . . . . . . . . . . . . . . . . 94
4.11 Photograph of the TENMA-72-7245 Dual Channel Bench DC Power
Supply used to power the 600ET SparkJet circuit. . . . . . . . . . . . 94
4.12 Photograph of the Agilent Function Generator used to initialize the
SparkJet cycle for the 600PST circuit. . . . . . . . . . . . . . . . . . 95
4.13 Photograph of the Lambda GEN600-2.6 DC Power Supply and the
custom electronics box containing the Acopian power supply that are
used to power the 600PST SparkJet circuit. . . . . . . . . . . . . . . 95
4.14 Photograph of the uninstalled PCB 105C12 dynamic pressure sensor. 98
4.15 Drawing of the metal insert used to mate the SparkJet Macor housing
to the PCB pressure transducer. . . . . . . . . . . . . . . . . . . . . . 99
4.16 Cross section view of the SparkJet cavity, electrodes (configured for
an external trigger), installed pressure sensor, metal insert, brass ring,
and six layers of electrical tape. . . . . . . . . . . . . . . . . . . . . . 100
4.17 Comparison plots corresponding to the tape and no-tape test con-
figurations including the a) ensemble averaged pressure over 500 µs,
b) ensemble averaged pressure over 55 µs, and c) the FFTs of the
ensemble averaged pressure signals. . . . . . . . . . . . . . . . . . . 103
4.18 Pressure and FFT data demonstrating the effect of low-pass filtering
on the SparkJet internal cavity pressure measurements. . . . . . . . 108
x
5.1 Example of ensemble-averaged data acquisition output for the 600ET
setup and SparkJet cavity volume of 84.8 mm3, orifice diameter of
1.0 mm, and capacitance of 4.28 µF a) over 500 µs and b) over 22 µs.
(f < 1 Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Example of data acquisition output for the 600PST setup and Spark-
Jet cavity volume of 84.8 mm3, orifice diameter of 1.0 mm, electrode
gap of 1.0 mm, and capacitance of 3.72 µF a) over 500 µs and b) over
22 µs. (f < 1 Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3 Variation in the filtered pressure signal as a function of capacitance
for a cavity volume of 84.8 mm3, orifice diameter of 1.0 mm, and
electrode gap of 1.75 mm using the 600ET circuit. (f < 1 Hz) . . . . 117
5.4 Efficiency of the SparkJet actuator as a function of QC/E for an ori-
fice diameter of 1 mm, electrode gap of 1.75 mm, and cavity volumes
of 42.4 mm3, 84.8 mm3, and 169.6 mm3 using the 600ET circuit. . . . 118
5.5 Variation in the filtered pressure signal as a function of cavity volume
for a QC/E value near 12.5, orifice diameter of 1.0 mm, and electrode
gap of 1.75 mm using the 600ET circuit. (f < 1 Hz) . . . . . . . . . . 119
5.6 Variation in the filtered pressure signal as a function of orifice diame-
ter for a QC/E value of 28, cavity volume of 84.8 mm3, and electrode
gap of 1.75 mm using the 600ET circuit. (f < 1 Hz) . . . . . . . . . . 120
5.7 Comparison of pressure vs. time for three electrode gaps of 1.0, 2.0,
and 3.0 mm given the same cavity volume (84.4 mm3), orifice diameter
(1.0 mm) and QC/E of approximately 28. (f < 1 Hz) . . . . . . . . . 122
5.8 Comparison of efficiency vs. Q/E for three electrode gaps of 1.0, 2.0,
and 3.0 mm given the same cavity volume (84.8 mm3) and orifice
diameter (1 mm). (f < 1 Hz) . . . . . . . . . . . . . . . . . . . . . . 123
5.9 Comparison between experimental and modeled cavity pressure as a
function of time as a) input energy, b) cavity volume, and c) orifice
diameter are varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.10 Comparison between experimental and modeled cavity pressure as a
function of time as cavity volume is varied and the convective heat
transfer coefficient is increased to hin = 6000 W/m2K. . . . . . . . . . 127
5.11 Comparison between the modeled normalized, steady-state momen-
tum as a function of actuation frequency for convective heat trans-
fer coefficients of 125 W/m2 and 6000 W/m2. (QC = 0.45 J, v =
84.8 mm3, do = 1 mm) . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.1 Voltage, current, and power waveform curves in distinct groups corre-
sponding to each capacitance change for a cavity volume of 84.8 mm3
and orifice diameter of 1 mm. (f < 1 Hz) . . . . . . . . . . . . . . . . 134
6.2 Time-dependent variation in arc resistance for a capacitance of 4.28 µF
and electrode gap of 1.75 mm. (f < 1 Hz) . . . . . . . . . . . . . . . 135
6.3 Time-dependent variation in arc power and qualitative light gener-
ation by the arc for a capacitance of 0.95 µF and electrode gap of
1.75 mm. (f < 1 Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xi
6.4 Efficiency of the conversion of stored capacitor energy to the arc dis-
charge, calculated cavity temperature, and measured pressure rise as
a function of QC/E. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5 Comparison of the time dependent temperature and specific heat co-
efficient (Cv) assuming a calorically and thermally perfect gas as a
function of arc discharge current and voltage for (a) a single value of
QC/E = 27.8 and (b) the final temperature difference over a range
of QC/E values. (f < 1 Hz and gap of 1.75 mm) . . . . . . . . . . . . 139
6.6 Power drawn by the arc for an electrode gap of 1 mm, 2 mm and
3 mm. (f < 1 Hz, QC = 0.61 J) . . . . . . . . . . . . . . . . . . . . . 140
xii
Nomenclature
α thermal diffusivity (m2/s)
β phase constant between the vortex shedding and the aoustic wave response in
the cavity (unitless)
γ ratio of specific heats (unitless)
η efficiency (unitless)
ρ density (kg/m3)
ω frequency (rad/s)
A area (m2)
Bi Biot number (unitless)
C capacitance (F)
Cv specific heat for a constant volume (J/kgK)
Cp specific heat for a constant volume (J/kgK)
d diameter (m)
D depth (m)
e internal energy per unit mass (J/kg)
E internal energy (J)
f frequency (Hz)
h convective heat transfer coefficient (W/m2K)
H height (m)
I electric current (A)
k thermal conductivity (W/mK)
K ratio of vortex convection speed to freestream flow speed (unitless)
L length (m)
m mass (kg)
M Mach number
n mode number (unitless)
P pressure (Pa)
P0 total pressure (Pa)
q input power (J/s)
Q input energy (J)
r radius (m)
R specific gas constant (J/kgK)
t time (s)
T temperature (K)
U velocity (m/s)
v cavity volume (m3)
v orifice volume (m3)
V voltage (V)
xiii
Subscripts
∞ freestream condition
1 Conditions at the beginning of Stage 1
2 Conditions at the beginning of Stage 2
A arc
c cutoff
C capacitance
e exit
i spatial discretization
j temporal discretization
m measured
M Macor
n notional
o orifice
p calorically perfect
P pressure
T thermally perfect
W tungsten
Acronymns
600ET 600 V electrode voltage initiated with an external trigger
600PST 600 V electrode voltage initiated with a pseudo-series trigger
AFC Active Flow Control
AFOSR Air Force Office of Scientific Research
CFD Computational Fluid Dynamics
COMPACT Combustion Powered Actuator
DBD Dielectric Barrier Discharge
DST Digital Speckle Tomography
EMI Electromagnetic Interference
FFT Fast-Fourier Transform
FSU/AAPL Florida State University Advanced Aero-Propulsion Laboratory
JHU/APL Johns Hopkins University Applied Physics Laboratory
LAFPA Localized Arc Filament Plasma Actuator
MHD Magnetohydrodynamic
NASA National Aeronautics and Space Administration
ONERA Office National d’Etudes et de Recherches Arospatiales
PIV Particle Image Velocimetry
REM Resonance Enhanced Microjets
URANS Unsteady Reynolds-Averaged Navier-Stokes
V/STOL Vertical and/or Short Takeoff and Landing
ZNMF Zero-net-mass-flux
xiv
Chapter 1
Introduction
The ability to ensure safe separation of a weapon from the weapons bay of a
supersonic jet or reduce damaging noise from a jet engine exhaust depends on our
ability to properly control the high-speed flow environment. Controlling a large-
scale, high-speed flow environment requires a flow control device or array of flow
control devices which can influence a large area and have the requisite control au-
thority to manipulate high-momentum flow. The motivation for this work and any
flow control work is to develop, characterize, and understand an actuator capable
of improving otherwise adverse flow conditions. Specifically, this work attempts to
develop a simplified, one-dimensional model of a flow control actuator, the Spark-
Jet, capable of controlling high-speed flows. This model would enable simple and
fast design optimization of an array of devices specifically tailored to an application
such as controlling a jet exhaust or the shear layer over an open weapons bay. To
provide an understanding of the flow control applications where a high-momentum
flow control device would be required, the following sections describe a few of these
applications. Following a description of the applications, the devices and methods
of flow control for the previously mentioned applications are described.
1
1.1 High-Speed Flow Control Applications
The flow control challenges unique to high speed applications are primarily
related to the momentum and frequency requirements. These high-speed flow appli-
cations possess high-momentum, high-frequency instabilities within the boundary
layer. Before considering the appropriate flow control device and technique to effi-
ciently control flow, the details of each flow condition need to be understood. The
following sections describe some of the primary applications of interest for high-speed
flow control and the details useful for applying the ideal flow control technique.
1.1.1 Flow Separation
A common flow phenomenon of concern is flow separation which adversely af-
fects aircraft performance at low and high speeds. Over an airfoil, for example, flow
is considered attached when the flow near the wall, the boundary layer, is moving in
the same direction as the freestream. Flow separation occurs when an adverse pres-
sure gradient exists along a surfaces and causes the flow to lose momentum in the
boundary layer and eventually reverse direction. The adverse pressure gradient can
exist for a variety of reasons including a surface contour away from the freestream,
a normal shock on a transonic airfoil (Figure 1.1) or shock wave boundary layer in-
teraction between a reflected oblique shock and a boundary layer. Flow separation
is generally associated with increased drag or a loss of lift in all flows or a reattach-
ment zone that can lead to increased heat transfer to the surface in supersonic and
hypersonic flows.
2
-Flow Direction
Figure 1.1: Schlieren image of shock-induced flow separation on the upper surface of an
airfoil in transonic flow [1].
1.1.2 Boundary Layer Transition
Boundary layer transition is the process describing the transformation of a
boundary layer from laminar to turbulent. Boundary layer transition is caused by
the growth of particular instabilities in the flow near the surface. The instabilities
can exist due to surface roughness, surface features, or simply the natural growth
of the boundary layer. Figure 1.2 is a sketch from Schlichting [2] describing the
series of instabilities that transform laminar flow to fully turbulent flow. The differ-
ences between laminar and turbulent boundary layer flow make each type desirable
or undesirable. For example, the skin friction drag associated with laminar flow
is lower than with turbulent flow. However, laminar flow is more susceptible to
flow separation than turbulent flow because it can be influenced more easily by an
adverse pressure gradient or surface features. Therefore, both inducing or delaying
boundary layer transition are potential goals of flow control.
More advanced studies of boundary layer transition reveal a unique differ-
3
Figure 1.2: Sketch of flow over a flat plate through transition from a laminar to turbulent
boundary layer. The regions of flow from 1-6 are 1) stable laminar flow, 2)
unstable Tollmien-Schlichting waves, 3) three-dimensional waves and vortex
formation (Λ-structures), 4) vortex decay, 5) formation of turbulent spots,
and 6) fully turbulent flow, respectively [2].
4
ence between boundary layer development over two-dimensional, flat surfaces and
three-dimensional, curved surfaces. The surface curvature allows centrifugal forces
to create vortical motion within the boundary layer that introduce so-called Taylor-
Go¨rtler instabilities. These instabilities are considered primary and also include
secondary instabilities each of which occur at separate frequencies. When applying
flow control techniques, all modes of instability can prompt transition. Therefore,
the ideal flow control technique would prevent one type of instability without per-
turbing another type of instability. The instability responsible for transition is also a
function of velocity. For example, for Mach less than approximately four, Tollmein-
Schlichting instability growth rate dominates transition. However, for Mach greater
than four, the instability growth rate of higher frequency Mack modes dominate
transition [3]. Such dependence on velocity requires an actuator capable of tuning
the actuation frequency to the immediate flow conditions.
1.1.3 Supersonic Open Cavity Flow
Since the beginning of this dissertation research, the primary flow control
application of interest has been control of supersonic open cavity flow. An example
of when this type of flow condition can occur includes an aircraft flying at supersonic
velocities with open bomb bay doors. Figure 1.3 shows an F-22 Raptor with open
bomb bay doors and reveals the internal geometry the flow encounters when opened.
Analysis of this flow field is based on a simplified representation of the open cavity
with a characteristic length to depth ratio, (L/D). The primary feature of the open
5
Figure 1.3: F-22 Raptor flying with open bomb bay doors [5].
cavity flow is a self-sustaining, unsteady flow inside the cavity with a characteristic
frequency or frequencies based on L/D and Mach. The unsteady nature of the flow
can prevent reliable store separation or accelerate aircraft structural fatigue and,
thus, methods of reducing the unsteadiness are desired. As a result of wind tunnel
testing, the frequency, or tone, associated with cavity flow was first identified by
Rossiter [4] and is, subsequently, known as a Rossiter tone.
The targeted cavity geometry is characterized by an L/D of 5.6 in a Mach 1.5
flow. Following Rossiter’s work, Heller [6] determined that the cavity speed of
sound is actually equal to the freestream stagnation speed of sound. Therefore, the
frequency of the cavity tones is predicted by
f =
U
L
(n− β)
(
1
K +
M√
1+ γ−12 M
2
) (1.1)
and is commonly referred to in recent open cavity research. Using Equation 1.1
with K = 0.57 and β = 0.25, the 1st, 2nd and 3rd modes for the targeted cavity
6
geometry are 0.84, 1.69 and 2.53 kHz, respectively. Here, both passive and active
flow control (AFC) techniques have been applied and their ability to suppress these
tones is summarized in Section 1.2.
1.1.4 Jet Impingement / Jet Noise
Jet noise is the audible pressure unsteadiness that develops in a free or im-
pinging jet. Subsonic jet noise is characterized by aperiodic bursts of noise that
are hypothesized to be linked to large scale structures developing in the jet itself.
A noise event is identified when the instantaneous pressure exceeds 1.5 times the
root mean square (RMS) pressure. However, generation of these noise events is
not well understood and attempts to understand them are primarily experimental
[7, 8, 9]. As a result, without adequate understanding of the flow field, flow control
techniques have not yet been applied. In supersonic jets, however, free and imping-
ing jets exhibit both broadband and highly periodic noise. Both types of jet noise
are undesirable. Far-field noise suppression is required near humans and animals.
Near-field noise suppression is required to reduce the potential for structural fatigue
associated with jet impingement. The mechanism behind the noise generation is sim-
ilar for both broadband and tonal noise. A feedback loop consisting of disturbances
emanating from the nozzle lip which interfere with the shock-cell structure create
acoustic waves that propagate upstream. Once the acoustic waves interact with the
receptive disturbances at the nozzle lip, the feedback loop is complete. Screech tones
occur when the feedback loop frequency matches the fundamental screech frequency
7
[10]. Another noise generation mechanism is interaction of convective turbulence
structures passing through a standing shock structure in a supersonic jet [11].
An impinging jet is similar to a free jet except that the jet interacts with
a solid surface that is situated relatively close to the nozzle exit. This scenario
is commonly encountered on V/STOL (vertical and/or short take-off and landing)
aircraft during takeoff or landing. The noise associated with a supersonic impinging
jet is also broadband with a tonal component but at a higher noise level [12]. The
broadband noise is generated by eddies within the shear layer moving at supersonic
velocities. The tones associated with an impinging jet are called impingement tones
and the frequency of such tones are depending on the ratio of nozzle diameter to
distance from the solid surface. Similar to the free jet case, the tones emerge from
an acoustic feedback loop involving disturbances within the shear layer which create
acoustic waves that travel upstream toward the receptive area of the shear layer near
the nozzle lip. However, the proximity of a solid wall causes these disturbances to
create acoustic waves at a higher intensity than a free jet. Flow control techniques
have been applied to both free jet and impinging jet flows primarily targeting the
receptive shear layer. The devices and exact techniques are described in Section 1.2.
The variety of adverse flow phenomenon described in this section does not en-
compass all areas where flow control can be used but represent the majority of the
flow phenomenon targeted by high-speed flow control studies. The following section
describes the flow control devices used to affect these flows and how they have been
applied to the flow phenomenon described in this section.
8
1.2 High-Momentum Flow Control Actuators
In the broadest sense, all flow control devices can be categorized as either
passive or active devices. Passive devices include fences, ridges, bumps, thermal
bumps, or other built-in surface modifications that are designed to benefit an ad-
verse flow feature at specific, on-design flow conditions. These devices are useful
in volatile flow environments where flow enhancement is desired, but moving parts
would likely fail, and weight or power requirements would not allow an active device.
However, these devices can contribute to a performance penalty at off-design flow
conditions. To eliminate the drag penalty associated with passive devices, a wide
variety of active flow control devices have also been developed. For high-speed flow,
these tend to involve, high-momentum jets or rapid energy deposition using arcs,
plasma and magnetic fields. Devices that use rapid energy deposition include the
combustion powered actuator (COMPACT), magneto-hydrodynamic (MHD) actu-
ators, nanosecond dielectric barrier discharge (DBD) actuators, and localized arc
filament plasma actuators (LAFPAs). High-momentum fluidic devices include syn-
thetic jet actuators, microjet actuators, and resonance enhanced microjet (REM)
actuators. This section focuses on the flow control devices (including references)
and techniques applied to high-speed flows.
1.2.1 Passive Flow Control
Probably the most commonly used, tested, and modeled passive flow control
device is the vortex generator. A vortex generator is any device that creates a
9
streamwise vortex near a flow surface with the primary intention of delaying or pre-
venting flow separation. Most people have seen passive vortex generators on the
upper surface of commercial aircraft wings in the form of metal rectangular tabs.
This kind of vortex generator can be parallel to the flow or at a slight angle to
the flow and an array of vortex generators are employed to affect a larger area of
flow such as the entire span of a wing. The resulting vortices introduce high kinetic
energy air into a lower kinetic energy boundary layer that usually results in laminar
to turbulent flow transition and delayed or eliminated flow separation. In fact, con-
trolling flow separation is the primary use of vortex generators. Vortex generator
placement is just upstream of a potential flow separation location to ensure attached
flow throughout the entire flow field. As previously mentioned, vortex generators
are typically used on wings, and they are the most beneficial when the wings are
at a high angle of attack when flow separation is prevalent. Vortex generators are
also used in inlets where the incoming flow interacts with a highly curved transition
area from the inlet to the compressor.
Figure 1.4 demonstrates the beneficial effect of using vortex generators to pre-
vent flow separation in a supersonic inlet. Figure 1.4(a) shows a picture of the vortex
generators; in this case, they are designed to be used in pairs and there is an array of
three pairs that spans the wind tunnel test section. Figure 1.4(b) illustrates a large
separated flow region that forms over the ramp using oil flow visualization. The flow
separation is eliminated with the use of vortex generators as shown in Figure 1.4(c).
For controlling boundary layer transition, surface modifications are typically
used. The surface modifications are tailored to the natural boundary layer instabil-
10
ities that lead to transition as shown in Figure 1.2. Research conducted by Rasheed
et al. [14] evaluated the ability of surface porosity to affect transition in Mach 5
nitrogen and carbon dioxide flows. These holes, shown in Figure 1.5(a), cover a
5.06◦ sharp cone in a Mach 5 flow where the Mack mode dominates transition. This
flow control scheme directly targets the Mack mode instability via controlled spac-
ing between the holes. Through a comparison of schlieren images (Figure 1.5(b))
and heat transfer measurements over a smooth surface and a porous surface, the
porosity clearly delayed transition.
Another type of passive flow control that is used against high-speed flow
applications is fences, ridges or any other physical modification to the leading edge
of an open cavity flow. Passive devices are commonly used on subsonic cavity flow
but a collection of such devices were tested against supersonic open cavity flow by
Schmit et al. [15]. Four types of passive flow control devices (spoiler, 3D backward
step, ridges and a rod) were installed just upstream of the open cavity. Results
showed that the rod was most effective at reducing flow unsteadiness in the cavity
because the natural shedding from the rod disturbs the formation of large vortical
structures. The authors concluded that while these devices generated flow improve-
ments, the physical design should be tuned to the specific cavity flow conditions.
Contouring of jet nozzles is another passive flow control technique used to re-
duce jet noise. As described earlier, jet noise is caused by feedback from acoustic
waves to the shear layer near the nozzle exit. Therefore, contouring the nozzle exit
can be effective in reducing the feedback. This phenomenon was experimentally
demonstrated in work by Andre et al. [10] by reducing the amplitude of screech
11
tones. Figure 1.6(b) clearly shows that using a notched nozzle exit for Mach 1.35
reduces, if not eliminates, the screech tones where the black line shows the sound
pressure level (SPL) associated with the clean nozzle and the grey line shows the
SPL associated with the notched nozzle.
1.2.2 Active Flow Control
Active flow control devices are another technique designed to improve other-
wise adverse flow phenomenon without physically modifying the aerodynamic sur-
face. In fact, the use of active flow control is sometimes referred to as virtual shaping
[16, 17, 18, 19]. The most basic forms of active flow control involve steady blowing,
steady suction, or unsteady blowing and suction. These techniques are typically
used to control the thickness of the boundary layer and delay flow separation. A
disadvantage to operating steady suction or blowing devices is the requirement of
an external air supply or a vacuum chamber. Unsteady devices that draw on the
external flow environment and eliminate the need for an external air supply are
called zero-net-mass-flux (ZNMF) devices.
Applied specifically to high-speed flows, an actuator called a microjet has been
utilized for both supersonic open cavity flow and impinging jet noise. Microjets, de-
veloped by Florida State University, are steady jets supplied with high pressure air.
When arranged in an array, a single supply pressurizes a chamber and forces air
through multiple orifices. Microjets have been applied to two high-speed applica-
tions: open cavity flow and jet noise. To control noise and the unsteady shear layer
12
motion associated with a supersonic open cavity, an array of 12 microjets with a
0.4 mm orifice diameter were placed at the leading edge of the cavity [20]. The
aspect ratio of the cavity is L/D = 5.16 and L/W = 5.92 in a Mach 2.0 crossflow.
Using shadowgraph, particle image velocimetry (PIV), and unsteady acoustic pres-
sure measurements, the effect of microjets over a range of nozzle pressure ratios,
results demonstrated the ability of the microjets to reduce broadband noise up to
9 dB and tonal noise up to 20 dB.
While reducing noise levels is beneficial, another purpose of controlling su-
personic cavity unsteadiness is to ensure safe store separation. To that end, wind
tunnel tests were conducted to determine the efficacy of microjets creating a flow
environment suitable for store separation. As part of the High Frequency Excita-
tion (HIFEX) program funded by the Defense Advanced Research Projects Agency
(DARPA), wind tunnel testing was conducted to evaluate the effectiveness of mi-
crojets on a cavity flow field in Mach 2.5 flow [21]. The microjet array consisted
of two rows of 150 supersonic (M = 2.2) jets. The upstream array operated at a
momentum coefficient, Cµ, of 0.26 and the downstream array was operated at Cµ
of 0.18. The results of this study showed that the microjet arrays in conjunction
with a jet screen upstream of the microjet array provided the best control using the
least mass flow. The combination of flow control techniques is considered a tandem
array and was used to control store separation on a full-scale test at the high-speed
track at Hollomon Air Force Base [22]. The full-scale test confirmed that safe store
separation could be achieved at Mach 2 with the use of active flow control.
Combining both suction and blowing, the synthetic jet actuator is another
13
heavily used active flow control device also intended to control flow separation. The
synthetic jet actuator is probably the earliest, most heavily tested, modeled, and
characterized unsteady flow control device. A review of synthetic jets published
in 2002 [23] summarizes numerous studies toward characterizing and applying syn-
thetic jets to a wide variety of non-ideal flow conditions. The synthetic jets described
are created by physical changes to the cavity volume to force air in and out of an
orifice.
Synthetic jets, also known as vortex generator jets, are unsteady jets oscillat-
ing between blowing and suction at a prescribed frequency depending on the flow
environment instabilities. With no external air supply, these devices are considered
a ZNMF device. During the blowing phase, fluid forced through an orifice creates
a jet with a vortex pair at the jet front. This vortex pair causes high momentum
fluid in the freestream to interact with a lower momentum boundary layer that en-
ergizes the boundary layer leading to flow separation control. During the suction
phase, the local low-momentum boundary layer fluid is removed from the surface
above the synthetic jet orifice that also helps control separation and transition. Be-
cause these synthetic jets are mechanically created, they are limited in the ability to
produce simultaneous high-frequency and high-momentum throughput. Therefore,
other methods of creating an unsteady jet for high-speed applications have been
developed.
REMs were derived from microjets but operate with an inherent unsteadiness
to target natural unsteadiness in a flowfield. The REM design consists of a high
pressure air supply issuing through a 1 mm diameter nozzle into a cavity with ad-
14
justable height and diameter to produce a desired frequency. Opposite the 1 mm
nozzle, the fluid ejected into the cavity is passed through an array of orifices typically
0.4 mm in diameter. Figure 1.7(a) shows a sketch of the basic REM design with
the critical sizing parameters indicated. Development of REMs is fairly recent but
application oriented experimental tests have demonstrated the ability of the REMs
to produce an unsteady influence on Mach 1.5 flow over a flat plate [24] and reduce
overall and impingement tone noise levels [25, 26].
Another type of high-momentum flow control device is called a COMPACT,
developed by Georgia Institute of Technology, which uses a small combustion pro-
cess inside a chamber which ejects high-speed air through an orifice. This device
ignites a mixture of air and hydrogen with a spark to increase cavity pressure (see
Figure 1.8) [28]. Using a passive valve, the fuel and oxidizer flow into the cavity
once the cavity pressure drops below the supply pressure. Peak cavity pressure is
achieved between 1 and 3 ms after combustion is initiated and a cycle completes in
approximately 4-10 ms depending on the design parameters.
The COMPACT has been used successfully to delay flow separation over a
2-D airfoil [29] and a 3-D rotorcraft fuselage [30]. To control flow separation over
an airfoil, the COMPACTs were placed at 20% chord length from the leading edge.
The orifice shape was rectangular with the largest dimension being spanwise and
the resulting jets were normal to the airfoil surface. The study demonstrated the
transient effect of a single pulse from the array of COMPACTs where the flow be-
came momentarily attached for a duration of 8-10 convective time scales after the
COMPACT was initiated. For flow separation control on the generic, ROBIN (Ro-
15
tor Body Interaction) fuselage, several arrays of COMPACTs were arranged on the
lower surface of the transition section between the fuselage cabin and the tail boom.
Again, the orifices were rectangular and oriented in the spanwise direction but the
jets were configured to eject either normal or tangential to the flow surface. The
COMPACT arrays successfully reduced flow separation such that the drag coeffi-
cient was reduced between 12% and 17%.
The active flow control devices described so far have involved a fluidic jet
produced by a pressurized chamber of some sort. There are several techniques that
involve installing a device directly on the surface of a body or depositing energy
very near the surface of the body. A different type of actuation technique is the
use of electromagnetic fields to produce fluid motion. For very low-speeds, a DBD
actuator has been tested for multiple applications. A review of the device and the
multitude of experimental and computational studies are summarized in a review
by Corke et al. [31]. Since classic DBD actuators introduce only a small amount of
momentum and are limited to low-speed applications, they are not discussed here
in great detail. However, by actuating the DBD actuators with a nanosecond wide
pulse, the actuator produces a blast wave and can affect high-momentum flows.
Specifically, the nanosecond DBDs have delayed prevented flow separation over air-
foils in transonic flows as demonstrated in Figure 1.9 [32].
A LAFPA involves a sudden energy deposition in the form of an arc discharge
between two electrodes. The arc discharge produces significant heat via Joule heat-
ing that results in a blast wave and local heat addition to the flow [33]. This device
is similar to the SparkJet except that the LAFPA arc is not enclosed in a cavity
16
but rather open to the flow and recessed in a groove to shield the arcs from the
high-speed flow [34]. The electronics that support LAFPA actuation use a ramped
voltage up to 10 kV which creates an arc that is sustained by up to 0.25 A when an
array of eight LAFPAs are in use. LAFPAs have been primarily applied to jet noise
mitigation [35] and high-speed jet control [36] but also to shock wave boundary layer
interaction control for supersonic inlets [37].
Another unique type of flow control is through the use of MHD flow control.
This type of actuation relies on the presence of a volume of ionized gas on which a
magnetic field can apply a force. There are many challenges associated with MHD
control including producing an ionized gas with sufficient volume and charge density
while producing a strong enough magnetic field to achieve the needed Lorenz force
to change the momentum of a moving fluid. Therefore, there are several computa-
tional MHD studies, but experimental studies are lacking in number and variety. An
excellent summary of the challenges associated with MHD testing has been compiled
by Braun [38]. Braun also conducted experimental studies using an MHD actuator
array but applied to a low speed (6 - 25.6 m/s) boundary layer flow over a flat plate.
The MHD actuator array did indeed affect the boundary layer profile beneficially
but the energy consumed and mass of the power supplies would make this technique
prohibitive on an aerospace vehicle. A recent experimental test studied the ability
to control boundary layer transition in a Mach 3 flow [39].
The SparkJet actuator shares some commonalities with the above mentioned
active flow control devices. Specifically, the SparkJet uses an arc discharge similar
to the LAFPA actuator, it produces a fluidic jet similar to the microjets, and it pro-
17
duces an unsteady jet similar to a synthetic jet. However, no single device shares
the same characteristics of the SparkJet such that the SparkJet represents a unique
device with the ability to provide a unique input to high-speed flow applications.
1.3 Dissertation Goals
This chapter presented the need and requirements for high-speed flow control
and the devices attempting to fulfill those needs. This dissertation introduces an ac-
tuator, the SparkJet, capable of fulfilling several high-speed flow control needs with
the unique ability to produce simultaneous high-frequency and high-momentum jet
flow. The primary goal of this dissertation is to contribute a powerful, yet simple,
numerical model of the SparkJet actuator supported by experimental measurement
and prior existing CFD modeling. Previous work toward understanding the Spark-
Jet experimentally and numerically are described in Chapter 2. The modeling efforts
described in Chapter 3 are used to understand how various design parameters affect
actuator performance during a single actuation cycle and while operating at high fre-
quency. The modeling results are supported through experimental testing described
in Chapters 4 through 6 where current and voltage measurements are used to deter-
mine power draw from the SparkJet. Internal cavity pressure measurements were
collected to determine the amount of useful energy produced by the arc discharge.
Analysis of these results are used to determine any inefficiencies associated with
the SparkJet actuation process and support adjustments to the numerical model to
capture these inefficiencies.
18
(a) Picture of vortex generators.
(b) Oil flow visualization of separated supersonic flow without flow control.
(c) Oil flow visualization of attached supersonic flow with vortex generator control.
Figure 1.4: Demonstration of flow separation control in supersonic flow using passive
vortex generators [13].
19
(a) Surface porosity
(b) Schlieren images of delayed transition.
Figure 1.5: Picture of surface porosity demonstrating the hole size and distribution for
delaying boundary layer transition in hypersonic flow [14].
(a) Clean vs. Notched Nozzle Exit. (b) Sound pressure levels associ-
ated with a clean nozzle and a
notched nozzle.
Figure 1.6: Effect of a notched nozzle exit on the sound pressure levels versus Strouhal
number (Sr) associated with screech tones [10].
20
(a) Sketch of REM design sizing
parameters [27].
(b) Phase-averaged cavity pressure and cor-
responding effect on supersonic crossflow
[24].
Figure 1.7: Design requirements and output for unsteady REM actuation.
Figure 1.8: Sketch of the COMPACT [28].
21
Figure 1.9: Effect of nanosecond discharge on the coefficient of pressure (Cp) distribution
versus the percentage of the chord length (x/b, %) over an airfoil in transonic
flow [32].
22
Chapter 2
SparkJet Actuator
This chapter describes the basic operating principles of the SparkJet actuator
based on a three stage cycle. In addition, work conducted prior to, and outside
of, this dissertation is discussed to illustrate the state of the art. Through this
discussion, the need to further develop a simplified model and acquire additional
experimental data to support 1-D modeling is highlighted.
2.1 Device Description
The SparkJet is a solid-state device containing no moving parts and has no
external air supply making it a ZNMF device. It consists of a small cavity with em-
bedded electrodes and an orifice through which air can pass freely. The operation
of the SparkJet is illustrated in Figure 2.1 as a series of stages. Stage 1 is char-
acterized by a brief arc discharge within the cavity to produce hot, high-pressure
plasma and air. Stage 2 involves venting the high-pressure plasma and air through
the orifice, which converts the thermal energy of the discharge into kinetic energy.
Stage 3 involves a cooling/refresh phase (Stage 3) prior to the beginning of the next
arc discharge (Stage 1).
The basic SparkJet cycle is dependent on a multitude of parameters including
the magnitude of input energy; actuation frequency; orifice area and shape; cavity
23
Figure 2.1: Basic schematic of the three stages of SparkJet operation.
volume and shape; electrode spacing, diameter, and shape; and cavity wall thermal
properties. When considering flow applications, an array of SparkJets interacting
with external flow conditions will be dependent on external flow conditions (pres-
sure, temperature, velocity, inherent instability frequency). In addition, the number
of and spacing between collocated SparkJets would affect spatial and temporal in-
teractions between devices. The physics and other details pertaining to each stage
are described in the following sections.
2.1.1 Stage 1 - Arc Discharge
The primary goal of Stage 1 is to raise the cavity air temperature and pressure
quickly and with maximum efficiency. The method of accomplishing this goal is by
the use of a brief, high-current, arc discharge sustained by a charged capacitance
parallel to the anode (high-voltage) and cathode (ground). The maximum amount
of input energy, QC , is predicted by
QC =
1
2
CV 2, (2.1)
where C is the capacitance and V is the voltage. Another parameter important to
understanding Stage 1 is the internal cavity energy, E, just before the arc discharge
24
given by
E = mCvT. (2.2)
At this point, a valuable parameter QC/E can also be defined as
QC
E
=
1/2CV 2
mCvT
, (2.3)
which provides a sense of the amount of energy stored in the SparkJet cavity prior
to Stage 2. This parameter is simply the ratio of input energy to the internal cavity
energy.
Currently, the method of initiating the arc discharge is through a high-voltage,
low-current trigger spark. In the very first SparkJet device [40], Stage 1 was initi-
ated by increasing the voltage across the anode and cathode until the breakdown
voltage between the electrode tips was exceeded and the trigger spark was initiated.
However, this technique posed problems when attempting to acquire characteri-
zation data because the time between the voltage increase and the arc discharge
would vary, making data difficult to capture experimentally. Therefore, the initia-
tion technique changed to the use of a trigger spark between a trigger electrode and
the cathode that was achieved using an external trigger circuit. Further triggering
details are discussed in Chapter 4 including an improved triggering technique.
Regardless of the exact trigger setup, once the trigger mechanism induces the
capacitive arc breakdown, the rest of Stage 1 involves the conversion of capacitor en-
ergy into Joule heat causing the cavity pressure to rise. The energy which increases
the temperature and, therefore, pressure in the SparkJet cavity is of most interest to
SparkJet performance. However, experimental results presented in Chapter 5 expose
25
some inefficiencies based on comparison between experimental data and modeling
solutions. Analysis of the arc power in Chapter 6 helps identify the cause of the
energy losses. Once the arc discharge is complete and the temperature and pressure
of the air in the cavity have reached their maximum values, Stage 1 is complete.
2.1.2 Stage 2 - Jet Flow
The SparkJet jet formation and flow is a very unsteady process initiated by
the sudden pressure differential across the orifice face generated by high-temperature
and high-pressure cavity air from Stage 1. As the cavity air is forced through the
orifice, the cavity pressure decreases due primarily to the decrease in cavity density
and secondarily to the thermal losses through forced and free convection. Collab-
orative efforts outside this dissertation work with Florida State University (FSU)
have enabled the acquisition of microschlieren imagery of the early jet development.
Figures 2.2(a) and 2.2(b) show microschlieren images of the SparkJet flow 12 µs and
100 µs, respectively, after Stage 1 was initiated. These images show that the first
evidence of the SparkJet formation is a blast wave (called a precursor shock by other
researchers [41]) which appears only 6 µs after the Stage 1 initiates. Immediately
following the blast wave is the jet front which, over the course of 100 µs develops
into a fully turbulent jet. By phase-averaging multiple images, evidence of locally
supersonic flow is apparent [42].
The duration of Stage 2 is primarily controlled by the cavity volume, orifice
area, and heat transfer. In general, decreasing orifice area and cavity volume in-
26
(a) 12 µs (b) 100 µs
Figure 2.2: Microschlieren images at two time delays after Stage 1 initiation showing the
a) blast wave and b) turbulent jet formation in the early portion of Stage 2
[42].
crease the duration of Stage 2. The jet flow can also be affected by contouring the
orifice throat. The studies presented in this dissertation only involve a constant area
throat but other studies have investigated a converging orifice [41] and a converging-
diverging orifice [43] to increase jet Mach number. At the very end of Stage 2, the
cavity pressure decreases to ambient pressure but the momentum of the flow pass-
ing through the orifice further decreases the cavity pressure below ambient pressure
extending Stage 2 until the adverse pressure gradient decreases the jet momentum
to zero.
27
2.1.3 Stage 3 - Refresh
Finally, once the fluid momentum is zero, the decreased pressure in the cavity
forces the flow to reverse and increase the pressure and density of the cavity air. In
addition, the ingested relatively cool external flow mixes with the high-temperature
cavity air thereby reducing the overall cavity temperature. Further reduction in
cavity temperature occurs due to free convection of the air to the cooling cavity
walls and electrodes. The primary parameters that affect the duration of Stage 3
are again cavity volume, orifice diameter, and thermal conductivity of the walls and
electrodes.
Stage 3 typically lasts long enough to raise concerns about SparkJet frequency
limitation. For example, if the time from Stage 1 to the end of Stage 3 exceeds 1 ms
and an actuation frequency of 1 kHz is desired, the arc discharge will occur when
the cavity density is lower and the temperature is higher than the first actuation
event. While this leads to a higher QC/E value due to a lower E value, the lack
of cavity density and mass leads to a lower momentum output during Stage 2 and
a decreased ability to affect the external flow. This dissertation uses the modeling
results to learn the design parameters that control high-frequency performance.
2.2 Previous Work
Prior to the work presented in this dissertation, previous studies had taken
place at JHU/APL to study the SparkJet and its performance. In addition, several
other organizations have begun conducting independent evaluations of SparkJet de-
28
vice (or similar) performance. These previous studies are outlined in the following
paragraphs which describe analytical modeling, numerical modeling, and experi-
mental testing efforts by JHU/APL and other organizations.
2.2.1 Analytical Modeling
The desire to simplify the complexities of the SparkJet process is evident by
the efforts outlined in this section to produce an analytical model. The development
of the SparkJet originated at JHU/APL and, therefore, the first experimental stud-
ies and modeling of the device was also performed by JHU/APL. Initial SparkJet
analysis debuted with a simplified, first-order, one-dimensional analytical model, a
limited collection of schlieren images and CFD simulations of a SparkJet issuing into
quiescent flow [40]. The analytical model was based on the conservation of energy
equation,
d
dt
(ρev) = Q˙− m˙ht − Q˙′ASURFACE, (2.4)
which equates the time dependent change in energy to the summation of three terms
that are separately independent of time but represent the three stages of SparkJet
operation. The primary assumptions in this model include an instantaneous energy
deposition during Stage 1 with negligible heat loss and the cavity air is calorically
perfect.
The first term on the right-hand side of Equation 2.4 represents the energy
gain during Stage 1 due to the arc discharge. The second term represents the
energy loss during Stage 2 due to enthalpy loss as the high-temperature and high-
29
Figure 2.3: SparkJet flow velocity at the orifice based on a simplified, one-dimensional,
analytical model developed by Grossman, et al. [40].
pressure cavity air exits the control volume through the orifice. The third term
represents the final energy loss through heat transfer to the surface while the control
volume temperature, pressure, and density return to equilibrium. Calculation of the
three terms led to a physics-based understanding of the SparkJet operation and the
influence of various parameters on SparkJet performance. Figure 2.3 shows the
predicted maximum exit velocity of the SparkJet during Stage 2 which increases
with Q/E, the ratio of input energy to internal cavity energy. This quantity, Q/E
was first identified in this paper as an important non-dimensional parameter for
understanding SparkJet performance and is continuously used throughout follow-on
papers. In this dissertation, several energy sources are identified; therefore, QC in
this dissertation is equivalent to Q in previous papers. No further modifications
to this first analytical model were made; however, other simplified modeling efforts
have emerged from other organizations.
In 2012, Anderson and Knight at Rutgers University documented a one-
dimensional, analytical model which included an in-depth dimensional analysis [44].
Anderson’s model and the early JHU/APL modeling efforts are similar in that they
30
simplify the SparkJet to a 1-D representation, assume calorically perfect gas, inviscid
flow, adiabatic walls and prescribed, instantaneous energy input. Based on these
assumptions, the models seek to quantify several performance parameters based
on fluidics. Both of these studies consider a single cycle with limited discussion
of Stage 3 operation. Therefore, these models are not capable of capturing high-
frequency actuation.
2.2.2 Computational Modeling
Higher-fidelity simulations of the SparkJet actuator involve CFD analysis.
Several organizations have conducted CFD studies of the SparkJet actuator includ-
ing JHU/APL, Rutgers University and the Office National d’Etudes et de Recherches
Arospatiales (ONERA). For analysis of a single SparkJet with a single orifice, the
simulations are two-dimensional taking advantage of the axisymmetric properties
of the basic SparkJet design. Initial CFD studies began at JHU/APL simulating a
SparkJet firing into quiescent flow using unsteady Reynolds-Averaged Navier-Stokes
(URANS) simulations. Computational studies using CFD++ were used understand
the SparkJet influence of a quiescent flow and then to determine of the SparkJet
concept was capable of influencing the boundary layer of a Mach 3 flat plate cross-
flow [45]. Computational results showed that not only did the SparkJet influence
the supersonic boundary layer but also penetrated into the freestream crossflow.
Follow-on computational efforts were made to allow for analysis of a SparkJet actu-
ator array on a large-scale application such as a missile body [46].
31
Figure 2.4: First CFD simulation of SparkJet interaction with a Mach 3 crossflow over
a flat plate [45].
At Rutgers University, Anderson and Knight used a finite volume code called
GASPex (export version of GASP) to simulate the SparkJet flow [47]. The simula-
tion efforts considered a single SparkJet (called plasma jet in the paper) interacting
with both a quiescent flow and a Mach 3 turbulent boundary layer over a flat plate.
The quiescent flow cases considered the SparkJet performance for a range of input
energy levels, QC , while the flat plate simulations considered the effect of orifice
diameter, do, on the SparkJet impulse. Comparisons were made between the com-
putational simulations and the Rutgers-developed analytical model. Results showed
that the analytical model and computational simulation agree very well for inter-
actions with a quiescent flow. The Mach 3 flat plate cases show that the impulse
generated by the SparkJet is significantly affected by the flow environment. For
example, the impulse generated in quiescent flow was compared to the impulse gen-
erated in the flat plate flow and results showed that the impulse is significantly lower
for the quiescent flow than the flat plate flow case. While the simulations showed
that the jet velocity at the orifice is lower for the flat plate flow case than the qui-
32
escent case, the difference in impulse is primarily due to the longer jet duration in
the flat plate flow case.
ONERA has also conducted a two-dimensional, axisymmetric, URANS simu-
lation of the SparkJet (called a plasma synthetic jet) interaction with a quiescent
flow and a crossflow assuming a prescribed energy input using their in-house CFD
code called CEDRE CFD [43]. Follow-on work also included simulation of the
arc discharge [48]. The fluidic simulations of the quiescent flow interactions track
the cavity pressure and temperature as well as the orifice velocity and mass flow
rate. The CFD analysis was used to study the effect of shaping the SparkJet ori-
fice and cavity shape for a fixed volume. Results showed, not surprisingly, that a
converging-diverging nozzle provides the best SparkJet performance based on exit
Mach number. The parametric cavity shape simulations showed that the height to
diameter ratio of the cavity primarily affect the influence of viscous effects. For ex-
ample, a tall, narrow cavity results in a relatively longer jet duration but lower total
mass ejection than a short, wide cavity. This is due to the fact that viscous losses
on the cavity walls are more dominant in the tall, narrow cavity. However, viscous
effects are prevalent on the walls of the orifice for the short, wide cavity shape be-
cause the streamlines are highly curved as the fluid leaves the cavity through the
orifice leading to a thicker boundary layer. Finally, the simulations also show that
walls with high conductive heat transfer lead to shorter refresh durations. Three-
dimensional, URANS simulations of the SparkJet interacting with a relatively low
speed (21.5 m/s) crossflow and preliminary comparisons to experimental data veri-
fied that the simulation captures the pair of vortices surrounding the jet orifice and
33
the jet penetration of the boundary layer.
2.2.3 Experimental Characterization
While simulations and modeling have estimated the potential strength and
usefulness of the SparkJet actuator, experimental tests were conducted to confirm
the modeling results. At JHU/APL, the first experimental results were acquired
using schlieren with a continuous light source and a camera featuring a fast shutter
speed [40]. The qualitative schlieren images provided visualization of the SparkJet
jet flow and a sense of the operation. Quantitative data acquired from the schlieren
imagery was limited to jet duration and an initial estimation of the jet front velocity.
To acquire more quantitative data, particle image velocimetry (PIV) was used
to analyze the local velocity within the core of the SparkJet jet flow [49]. At this
time, verification of supersonic flow features in the SparkJet jet flow was of interest.
However, the seed particles used for the PIV studies were too large to follow the
highest velocity flow in the core of the SparkJet jet flow and the maximum velocity
measured was only 100 m/s and 50 m/s in a separate PIV measurement attempt
[50]. The flow velocities captured by the PIV were of the entrained flow rather than
the jet flow.
In a collaborative effort between JHU/APL and JHU, improved experimental
results were found using a completely non-intrusive, optical technique called digital
speckle tomography (DST) [50]. This technique relies on density or temperature
gradients in the SparkJet jet flow altering a speckled background image. The opti-
34
cal movement of the speckles is calibrated such that a density or temperature can
be quantified along a line that intersects the jet flow. This study captured the tem-
perature profile across the SparkJet 1.8 mm above the orifice as a function of time.
These results showed that for an input energy of 0.03 J, the maximum temperature
measured was 1600 K which aligned well with computational results.
Another attempt to quantify SparkJet performance was using a custom-built
thrust stand to measure impulse [49]. The thrust stand results showed that the
SparkJet impulse varied linearly with the amount of energy deposition. Specifically,
the impulse varied from 1.5 µN-s at 0.03 J and 2.7 µN-s at 0.9 J.
As seen with the analytical and computational modeling efforts, several other
organizations have experimentally tested the SparkJet including the University of
Texas at Austin, ONERA, the University of Illinois, the National Aeronautics and
Space Administration (NASA) Langley Research Center, and the University of
Florida. At the University of Texas at Austin, the actuator (called a pulsed plasma
jet (PSJ)) design was similar to the SparkJet in that an arc within the cavity ini-
tiated the jet flow. However, the PSJ actuator was only evaluated at very low
pressures (45 torr). The actuator design consisted of a 5 mm electrode gap and the
arc current was controlled such that it was maintained for 20 − 50 µs at 1.1-3.9 A
[51]. Experimental testing was primarily based on schlieren imagery in addition to
planar laser scattering (PLS) of flow interactions with quiescent and a Mach 3 flow
over a 30◦ corner. While significant differences in design and operating conditions
exist between the SparkJet and the PSJ actuator, the experimental results demon-
strated important flow interactions. To summarize the highlight of the results, the
35
flow responsiveness was directly related to the frequency of the actuator and the
actuator had the most effect on the separation bubble at the compression corner
when placed upstream of the separation bubble. These results demonstrated the
ability of such an actuator to control the frequency of fluid structures at Mach 3
flow conditions.
At the University of Illinois, both quiescent and Mach 3 crossflow experiments
were conducted. The actuator design used for these studies was very similar to the
SparkJet but the researchers refer to the actuator as a pulsed plasma jet actuator
(PPJA). For the quiescent flow studies [41], schlieren, and PIV images were used
to characterize the PPJA flow over a wide range of input energy values controlled
by the capacitance values of 0.25 µF, 2 µF, 25 µF and 68 µF. The results show
that the PPJA flow starts with a blast wave referred to as a precursor shock with
a velocity independent of the energy deposition level. The contact surface (i.e. jet
front) velocity varies with input energy as seen in Figure 2.5.
The PIV images obtained from the quiescent experiments at University of
Illinois are very revealing of the PPJA flow and their ability to seed the ambient flow
is a clear improvement over previous PIV efforts by JHU/APL. Figure 2.6 shows the
velocity field contours of the PPJA flow at 30 µs, 50 µs, 70 µs and 90 µs delay times
for a capacitance of 25 µF. The maximum flow velocity easily exceeds the speed of
sound based on ambient temperature; however, shock cells are not visible because
the local temperature in the jet is higher than the ambient temperature such that
the local Mach number is below unity. Further data processing of the PIV images
show that beyond 90 µs, the jet velocity gradually decreases. The magnitude of
36
Figure 2.5: Variation in velocities determined from schlieren images for the pulsed
plasma jet for various capacitive energy depositions [41].
the velocity is proportional to the magnitude of input energy. The axial velocity
profiles also show that the velocity across the jet is fairly uniform even at an early
time delay of 30 µs. Only the low energy deposition case shows non-uniformity but
that is likely due to the strong presence of vortices. The transverse velocity profiles
show that the jet has an influence on the flow three orifice diameters wide.
The Mach 3 crossflow studies conducted by Illinois used PIV and schlieren
to investigate the interaction between a single SparkJet in a Mach 3 crossflow [52].
The results showed that the SparkJet, using 2 µF across the anode and cathode
weakly affected the crossflow. The PIV results showed that the maximum velocity
perpendicular to the crossflow from the SparkJet was 60 m/s. The primary conclu-
sion from this work was that further investigation is needed to understand the weak
influence.
Extensive experimental work related to the SparkJet (called a plasma synthetic
jet (PSJ)) has occurred at ONERA [43, 53]. First experimental tests characterized
37
Figure 2.6: Average velocity field contours and vectors for the 25 µF discharge at 30 µs,
50 µs, 70 µs and 90 µs delay times [41].
the PSJ design as a function of cavity wall thermal conductivity, orifice diameter,
frequency, and input energy. The PSJ design was very similar to the first SparkJet
design by JHU/APL that contained a center anode and used a metal lid as the
cathode. The experiments examined the effect of the orifice diameter, lid mate-
rial, and frequency on the pressure rise in the cavity. The results showed that as
orifice diameter decreases and actuation frequency increases, the pressure rise in
the chamber decreases. They also show that as actuation frequency increases and
lid thermal conductivity decreases, the pressure rise decreases. These results sug-
gest that maintenance of a prescribed pressure rise in the cavity at high actuation
frequencies requires an increase in thermal heat transfer through the actuator mate-
rials. ONERA also experimentally tested a single PSJ in a wind tunnel to examine
interactions between a pitched and skewed PSJ and a 40 m/s crossflow. Results
showed that the PSJ influences the flow much like a synthetic jet or vortex gen-
38
erator. Experimental testing was also conducted to study the interaction between
a PSJ and a high subsonic jet (Mach 0.6 and 0.9). Based on schlieren imagery,
the PSJ clearly affects the jet shear layer and demonstrates potential to control jet
noise.
Only very recently, NASA Langley Research Center (in collaboration with
Rutgers University) has begun testing the SparkJet actuator in quiescent flow [54].
The purpose of the experiments was to measure the impulse (and efficiency) provided
by the SparkJet by fixing the SparkJet to the end of a pendulum and measuring
the displacement due to SparkJet operation. The measured displacement was con-
verted to an impulse and then compared to an analytical model created by Rutgers
University. The specific dimensions of the SparkJet actuator used in these studies
include a cavity volume of 214.9 mm3, an orifice diameter of 1 mm, a capacitance
of 3-40 µF , a charging voltage of 550 V, and an electrode gap of 1 mm. The results
showed that the actuator design used for this study provided an angular deflection
up to 0.015◦ which corresponded to an efficiency of only 8%. This low efficiency
value is most likely due to the small electrode gap coupled with voltage potential
drops in the long cables connecting the actuator to the power supply.
The prior work presented in this chapter illustrates the need for further ex-
ploration of SparkJet. For example, modeling of Stage 3 of the SparkJet cycle is
needed for high-frequency actuation modeling. In addition, several organizations
have studied the resulting jet flow during Stage 2 but the internal cavity conditions
are still largely unknown experimentally. Chapter 3 of this dissertation presents
a simplified 1-D model which includes Stage 3 modeling and, therefore, the effect
39
of design parameters on high-frequency actuation performance is explored. Addi-
tionally, Chapters 5 and 6 present experimentally acquired cavity pressure and arc
power data. These experimental results answer questions pertaining to actuator
efficiency and the sources of inefficiency in actuator design.
40
Chapter 3
1-D Analytical Model
The goal of this chapter is to present and demonstrate a one-dimensional
numerical model to describe the SparkJet operation. Through variation of several
design and operating parameters, methods to optimize the design for high-frequency
operation are understood. In addition, a comparative study is included between a
SparkJet CFD simulation previously conducted [55] and the 1-D model presented
in this chapter to gain confidence in the 1-D model.
3.1 Description of the Problem
The development of a 1-D model of the SparkJet actuator cycle began in
2003 [40]. The work presented in 2003 presented a three-stage model for the initial
energy deposition, isentropic choked jet flow followed by unchoked jet flow, and
refresh stage. A similar approach is used in this chapter by modeling the SparkJet
according to the three stages of actuation. Each stage of the SparkJet cycle (shown
in Figure 2.1) is analyzed separately. To one-dimensionalize the problem, pressure,
temperature, and density are assumed averaged over the entire cavity volume and
the jet velocity is assumed significant only at the throat of the orifice and zero in
the cavity. Figure 3.1 shows a sketch of the control volume used to analyze the
SparkJet; the larger volume represents the cavity and smaller volume represents the
41
Figure 3.1: Sketch of the assumed control volume where the larger volume represents
the cavity volume, the smaller volume represents the orifice volume.
throat. The dashed lines surrounding the control volume represent the walls of the
SparkJet actuator. The orifice, at the top of the throat volume, is the only surface
area through which air is able to pass.
3.2 Governing Equations
The governing equations used to model the SparkJet cycle involve basic ther-
modynamics and fluid dynamics. The plasma chemistry involved in the arc forma-
tion and resulting heat generation is modeled using simple thermodynamics. The
fluid dynamics are represented by a 1-D derivation of the Euler equations combined
with a thermal model of heat loss to the internal SparkJet actuator surfaces.
3.2.1 Stage 1
To initiate Stage 1 of the SparkJet cycle, an instantaneous energy input, QC ,
of an assumed value from the capacitive arc discharge is used to determine the
temperature rise from T1, the temperature prior to the arc discharge, to T2, the
42
temperature after the arc discharge. The generic thermodynamic equation used to
determine a temperature rise due to an energy input is given by
∆T =
q
mCv
. (3.1)
Defining the temperature rise ∆T = T2 − T2, q = QC , and recalling Equation 2.2,
T2 = T1
(
1 +
QC
E
)
(3.2)
is used to calculate the peak temperature at the beginning of Stage 2. Using the
ideal gas law and under the assumption that no mass has exited the cavity during
this instantaneous energy deposition, the peak pressure, P2, is determined by
P2 = ρRT2 = ρRT1
(
1 +
QC
E
)
. (3.3)
Several assumptions were made to simplify modeling Stage 1. The real arc
discharge process for the SparkJet actuator takes place over hundreds of nanosec-
onds. The temperatures within the arc column vary from 5000 to 30000 K during a
single discharge. During the discharge, a thermal and pressure wave emanate from
the arc column at a speed near 500 m/s [42]. When considering a real arc discharge,
one realizes that the process is highly time and space dependent. However, a goal
of this dissertation is to simplify such complex processes using various assumptions
and also to determine the validity of such assumptions for evaluating the SparkJet
cycle. In addition to the complexities of the arc discharge itself, transferring the
capacitive discharge energy, QC , to thermal energy is known to be an inefficient
process [56]. These inefficiencies are explored in Chapters 5 and 6.
43
3.2.2 Stage 2
To initiate Stage 2, T2 and P2 from Stage 1 are used as initial conditions and
no more energy is added to the system. The Euler equations, separately identified
as the conservation of mass, momentum, and energy equations, are used to describe
the change in cavity conditions for Stage 2 as a function of time. The complete,
integral form of the conservation equations are given by
∂
∂t
∫∫∫
v
ρdv +
∫∫
S
ρU · dS = 0, (3.4)
∂
∂t
∫∫∫
v
ρUdv +
∫∫
S
(ρU · dS)U = −
∫∫
S
pdS +
∫∫∫
v
ρfdv + Fviscous, (3.5)
and
(3.6)
∂
∂t
∫∫∫
v
ρ
(
e+
U2
2
)
dv +
∫∫
S
ρ
(
e+
U2
2
)
U · dS
=
∫∫∫
v
q˙ρdv + Q˙viscous −
∫∫
s
pU · ds +
∫∫∫
v
ρ (f ·U) dv + W˙viscous.
The assumptions used to simplify Equations 3.4-3.6 are consolidated below followed
by the derivations corresponding to these assumptions.
1. Body forces are negligible
2. Inviscid flow
3. Velocity is negligible in the cavity volume and significant through the throat
volume
4. Velocity, pressure, temperature, and density across the orifice are uniform and
represented by scalar values
44
5. Pressure, temperature, and density are uniform inside the cavity volume (1-D)
6. Stages 2 and 3 are calorically perfect
The first assumption used to simplify these equations is assuming that viscous
effects and body forces, f , are negligible. Body forces include forces due to gravity,
electromagnetism and other forces which “act at a distance” [57] and are small
compared to pressure forces. Viscous effects are considered negligible due primarily
to the small scale of the cavity and the small throat length to diameter ratio. If
the throat were long and narrow, viscous forces would need to be considered. The
resulting inviscid, integral form of the conservation equations are given by
∂
∂t
∫∫∫
v
ρdv +
∫∫
S
ρU · dS = 0, (3.7)
∂
∂t
∫∫∫
v
ρUdv +
∫∫
S
(ρU · dS)U = −
∫∫
S
pdS, (3.8)
and
∂
∂t
∫∫∫
v
ρ
(
e+
U2
2
)
dv +
∫∫
S
(
ρ
(
e+
U2
2
)
U · dS
)
=
∫∫∫
v
q˙ρdv −
∫∫
s
pU · ds.
(3.9)
To simplify the surface integrals, the control volume shown in Figure 3.1 is con-
sidered. The only surface area through which fluid can pass or a pressure gradient
can act upon the fluid is at the interface between the orifice and the external flow,
Ao. Furthermore, the pressure, temperature, density, and velocity across the orifice
interface are assumed constant. The velocity through the orifice is assumed exactly
parallel to the surface normal vector and is converted to a scalar rather than a
vector. The pressure term, p, at the orifice interface is equivalent to the difference
45
between the cavity pressure, P , and the exit pressure, Pe, such that p = P − Pe.
The equations showing the effect of applying these assumptions are given as
∂
∂t
∫∫∫
v
ρdv + ρUAo = 0, (3.10)
∂
∂t
∫∫∫
v
ρUdv + ρU2Ao = − (P − Pe)Ao, (3.11)
and
∂
∂t
∫∫∫
v
(ρe) dv+
∂
∂t
∫∫∫
v
ρ
U2
2
dv+ ρ
(
e+
U2
2
)
UAo =
∫∫∫
v
q˙ρdv− (P −Pe)UAo.
(3.12)
In preparation for simplifying the volume integrals, the unsteady terms are also
expanded in the energy equation.
The next assumption to simplify the equations is that the density, pressure,
and temperature inside the cavity are uniform and that the only significant velocity
values are uniform in the orifice volume only. Therefore, any volume integrals with
a velocity term are integrated over the orifice volume whereas volume integrals
without a velocity term are integrated over the cavity volume. Acknowledging that
the cavity and orifice volumes are independent of time, the equations resulting from
these simplifications are
v
∂ρ
∂t
+ ρUAo = 0, (3.13)
vo
∂ (ρU)
∂t
+ ρU2Ao = − (P − Pe)Ao, (3.14)
and
(3.15)v
∂ (ρe)
∂t
+ vo
∂
(
ρU
2
2
)
∂t
+ ρ
(
e+
U2
2
)
UAo = q˙ρv − (P − Pe)UAo.
46
All terms in Equations 3.13-3.15 not included in the time derivative are moved to
the right-hand side. The conservation of mass equation then becomes
v
∂ρ
∂t
= −ρUAo. (3.16)
Additionally, using the product rule for derivatives, the time derivatives in the
momentum (Equation 3.14) and energy (Equation 3.15) equations are expanded
into
vo
(
ρ
∂U
∂t
+ U
∂ρ
∂t
)
= − (P − Pe)Ao − ρU
2Ao (3.17)
and
(3.18)v
∂ (ρe)
∂t
+ vo
(
2ρU
∂U
∂t
+ U2
∂ρ
∂t
)
= q˙ρv − (P − Pe)UAo − ρ
(
e+
U2
2
)
UAo,
respectively.
Assuming that the equations are solved in sequence starting with the con-
tinuity equation, ∂ρ/∂t and ∂U/∂t are considered knowns in Equations 3.17 and
3.18 and are also moved to the right-hand side. At this point, the derivation of the
conservation of mass,
∂ρ
∂t
= −
(
ρUAo
v
)
, (3.19)
and momentum,
∂U
∂t
=
1
ρ
(
− (P − Pe)Ao − ρU2Ao
vo
− U
∂ρ
∂t
)
, (3.20)
equations is complete. However, for the energy equation,
(3.21)v
∂ (ρe)
∂t
= q˙ρv− (P − Pe)UAo − ρ
(
e+
U2
2
)
UAo − vo
(
2ρU
∂U
∂t
+U2
∂ρ
∂t
)
,
we need to assume a calorically perfect gas and that Stage 2 is a constant volume
process, such that the internal energy can be defined as e = CvT . Applying this
47
definition of internal energy to Equation 3.21 allows the energy equation to be given
as
v
∂ (ρCvT )
∂t
= q˙ρv − (P − Pe)UAo − ρ
(
CvT +
U2
2
)
UAo − vo
(
2ρU
∂U
∂t
+ U2
∂ρ
∂t
)
.
(3.22)
Applying the ideal gas law to the left-hand side of Equation 3.22, the temperature
derivative can be converted to a pressure derivative as shown by the equality
(3.23)v
∂ (ρCvT )
∂t
= v
∂
(
ρCv PρR
)
∂t
.
By removing the constants Cv and R from the time derivative,
(3.24)
vCv
R
∂P
∂t
is the resulting term on the left hand side of Equation 3.22. Finally, the energy
equation is given by
dP
dt
=
(
q˙ρv − (P − Pe)UAo − ρ
(
CvT + 12U
2
)
UAo − vo
(
2ρU dUdt + U
2 dρ
dt
))
(
vCv
R
) (3.25)
and represented as the time derivative of pressure. While it is very likely that
the temperatures reached in the cavity exceed that which allows us to make the
assumption that Cv is constant, this assumption is made during this first model
development iteration.
During Stage 2, the pressure ratio between the cavity and outside leads to
the need for a choked and unchoked flow condition. The isentropic pressure ratio
equation at Mach 1,
P
P∞
=
(
1 +
γ − 1
2
M2
) γ
γ−1
=
(
1 +
γ − 1
2
) γ
γ−1
= 1.893, (3.26)
48
which corresponds to the onset of choked flow, is used to determine when each
condition exists. Therefore, Stage 2 is segmented into a choked flow condition
followed by an unchoked flow condition that depends on the pressure ratio between
the cavity pressure and the external pressure. Assuming a specific heat ratio for
air, γ = 1.4, the solution at Mach 1 shows that if the pressure inside the cavity is
above 1.893P∞, the flow is considered choked and the pressure boundary condition
at the orifice is set to Pe = P/1.893. Otherwise, the pressure at the orifice is equal
to the ambient pressure for unchoked flow (Pe = P∞). Stage 2 ends with zero orifice
velocity and high temperature, low density and slightly below atmospheric pressure
air inside the cavity.
3.2.3 Stage 3
During Stage 3, the cavity is refreshed with relatively cool, high-density, am-
bient air due to the slight pressure gradient across the orifice which is maintained
by the continued heat transfer to the internal cavity surfaces. As heat transfers to
the Macor walls and electrodes, the air loses heat and is cooled which maintains a
slight pressure gradient and continues to draw in air to bring the density up to am-
bient. Since the flow velocity is low throughout Stage 3, no choked flow conditions
are experienced. The derivation and equations described for Stage 2 are identical
to those used to simulate Stage 3 except where the equations describe the flow
passing through the orifice. Therefore, the surface integrals in the original conser-
vation equations involve ambient conditions rather than cavity conditions. Stage 3
49
modeling is accomplished using
dρ
dt
= −
(
ρ∞AoU
v
)
, (3.27)
dU
dt
=
1
ρ∞
(
(P − P∞)Ao − ρ∞U2Ao
vo
− U
dρ
dt
)
, (3.28)
and
dP
dt
=
(
q˙ρv − (P − P∞)UAo − ρ∞
(
CvT∞ + 12U
2
)
UAo − vo
(
2ρ∞U dUdt + U
2 dρ
dt
))
(
vCv
R
)
(3.29)
to solve for the conservation of mass, momentum, and energy, respectively.
3.2.4 Thermal Modeling
As mentioned in Section 3.2.3, thermal heat transfer drives Stage 3, and there-
fore, cannot be ignored. Heat transfer is also present in Stages 1 and 2 such that a
wide variety of heat transfer scenarios are considered here. This section considers
the significance of each heat transfer mechanism (radiation, convection, and conduc-
tion) during a single SparkJet cycle and, as a result, considers two thermal modeling
techniques: a combined lumped capacitance and thermal resistance model; and a
model based on finite-difference discretization of the heat equation. These models
are applied to identical thermal scenarios to determine which method accomplishes
a suitable balance of simplicity and accuracy.
Based on arc discharge literature, the typical plasma temperatures reached for
an arc in local thermal equilibrium at 1 atm are on the order of 5000-30000 K.
50
At such high temperatures, radiative heat effects require attention. The time
constant associated with radiative heat transfer corresponds to the speed of light
(299,792 mm/µs). Therefore, if radiative heat transfer is significant, the cavity air
and surrounding Macor walls would increase temperature almost immediately.
According to Section 10.9.5 in Raizer [58], in air at a pressure of 1 atm, “ra-
diative losses make up from one to several percent of the power input.” The primary
reason for the relatively low radiative energy output is because most of the radiative
energy generated by the highest temperature plasma is absorbed by the surround-
ing plasma such that only a small amount of radiative energy is emitted by the arc
column. With the intention of investigating an arc discharge such as that utilized
during Stage 1, both ONERA [48] and UT Austin [59] have also provided numerical
results based on a 2-D axisymmetric grid supporting the low losses due to radiative
heat transfer. Therefore, radiative heat transfer in air at 1 atm is considered negli-
gible for this thermal modeling. If operated at high pressure (≈ 10 atm) radiative
losses should be included in the thermal model.
Assuming the radiative losses are not significant, the remaining heat trans-
fer mechanisms are conduction and convection. During the initial arc discharge, the
primary mechanism is thermal conduction to the surrounding air and the electrodes.
As the air is rapidly heated, an expanding, cylindrical blast wave emanates from the
nearly cylindrical arc column. In the case of the SparkJet, the thermally conducting
blast wave expands until it reaches the Macor walls. Based on the experimental
cavity pressure measurements presented in Chapter 5, the blast wave takes approx-
imately 5 µs to reach the walls. This relatively long delay until a pressure rise is
51
measured also supports the assumption of negligible radiative heat transfer. If ra-
diative heat transfer was significant, a pressure rise would have begun immediately
after the arc discharge initiated. Since the SparkJet cavity is shaped such that the
cavity height and diameter are equivalent, the blast wave reaches all Macor walls
at approximately the same instant neglecting the added time required to reach the
cavity corners.
The subsequent pressure waves interact with the Macor walls resulting in an
unsteady heat transfer to the walls. The frequency of this heat transfer primarily
depends on the cavity dimensions. Also, because the cavity geometry (cylindrical)
does not match the blast wave geometry (spherical), areas of the walls impacted by
the blast wave at an angle will also experience convective heat transfer as the fluid
“slips” along the surface of the walls. As heat is convectively transferred to the
walls, conductive heat transfer transports the thermal energy through the SparkJet
walls. Therefore, the thermal heat transfer during the SparkJet cycle includes con-
current conductive and convective heat transfer.
To properly model the thermal heat transfer process, the thermal energy trans-
ferred to the walls needs to be calculated from the beginning of Stage 2. There are
a variety of thermal models available depending on the assumptions that can be
made. Balancing simplicity and unsteady thermal effects, the lumped capacitance
model is appealing. This model first uses the Biot number,
Bi =
hL
k
(3.30)
52
to determine the validity of using this method where h is the convective heat trans-
fer coefficient, L is the characteristic length, and k is the conductive heat transfer
coefficient. If the Biot number is less than 0.1, the lumped capacitance method is
considered valid. Physically, a low Biot number means that the spatial temperature
distribution across the wall is constant such that conductive heat transfer is far more
significant than convective heat transfer. Since the SparkJet walls offer two thermal
heat transfer paths corresponding to each material, Macor walls and tungsten elec-
trodes, two values for the Biot number are determined and both should be less than
0.1 to have confidence in the lumped capacitance method alone. The Biot number is
approximately 0.34 for Macor and 0.003 for tungsten assuming h is 125 W/m2; the
wall thickness, L, is 4 mm; and thermal conductivity of Macor, kM , and tungsten,
kW , are 1.46 W/mK and 173 W/mK, respectively. The low thermal conductivity
of Macor, 1.46 W/mK, leads to a more substantial Biot number. Despite the Biot
number of Macor exceeding 0.1, this method will be evaluated in a hybrid approach
explained later. Additionally, the hybrid approach, which includes several simpli-
fying assumptions, will be compared against higher-fidelity method that includes
fewer assumptions.
Another commonly used thermal modeling method is based on thermal resis-
tance. This method assumes that the system has reached a thermal steady-state
condition and that the spatial thermal distribution is linear through the walls. While
the SparkJet thermal condition is not steady-state, the modeling solution can be
solved for every time step such that within each time step, the system is considered
in a steady-state condition. The wall temperature values from the previous time
53
step can be carried over to the next step to determine new heat transfer values.
Both the lumped capacitance and thermal resistance methods include features
of interest for modeling SparkJet heat transfer. Therefore, these techniques are
combined and a representative thermal circuit is used to model the thermal heat
transfer and monitor the wall temperature. The general sequence of heat transfer
begins with convection from the cavity air to the interior cavity surfaces. Convective
heat transfer is then followed by parallel thermal conduction through the cavity ma-
terials and completed by thermal convection from the exterior SparkJet surfaces to
the ambient air temperature. The thermal energy source is represented electrically
by an ideal voltage source.
The cavity air temperature is represented by the equivalent voltage, TA. This
value will be used to link the thermal model to the fluid dynamic model described
in the previous sections. The only means of heat transfer from the air to the Ma-
cor and tungsten is through thermal convection, which is represented by a resistor,
Rh,in. The voltage labeled as Tw,in represents the interior wall temperature, assum-
ing the wall temperature of the Macor and tungsten are equivalent. The Macor
has a thermal capacitance represented as CM . The thermal conduction through the
Macor and tungsten is represented by two parallel thermal resistances, RM and RW ,
respectively. The voltage labeled as Tw,out represents the exterior wall temperature.
With the exterior wall temperature and ambient air temperature known, the final
thermal heat transfer mechanism, thermal convection, is represented as a resistor,
Rh,out. All thermal heat transfer mechanisms ultimately reach electrical ground or,
T∞, the ambient air temperature.
54
Figure 3.2: Electrical representation of the thermal heat transfer process for the Spark-
Jet actuator.
Once the thermal circuit has been defined, the equivalent thermal resistance
and capacitance of the Macor and tungsten need to be determined. The thermal
resistance of Macor and tungsten are given by
RM =
LM
kMAM
(3.31)
and
RW =
LW
kWAW
, (3.32)
respectively. The thermal resistances associated with convective heat transfer on
the inside and outside of the SparkJet device are given by
Rh,in =
1
hinAin
(3.33)
and
Rh,out =
1
houtAout
, (3.34)
respectively. The ability of a material to store thermal energy is represented by an
equivalent capacitance. Here, only the thermal capacitance of the Macor, given by
CM = ρMvMCp, (3.35)
55
is considered as the thermal capacitance of the tungsten electrodes is small due to
their small volume. With all of the circuit components defined,
TA − Tw,in
Rh,in
= CM
dTw,in
dt
+
Tw,in − T∞
Rh,out + 11
RM
+ 1RW
(3.36)
is used to solve for the wall temperature.
Another separate method for estimating SparkJet heat transfer is represent-
ing the differential form of the heat equation with finite difference equations. This
method provides the most spatially and temporally accurate representation of the
thermal effects involving the SparkJet cycle. However, this method can be more
computationally time consuming. To maintain some level of simplicity the exact
SparkJet Macor shape is not modeled but a single line through a side wall is ana-
lyzed. Due to the cylindrical shape of the SparkJet cavity and surrounding walls,
the heat equation is analyzed in cylindrical coordinates as given by
∂T
∂t
= α
∂2T
∂x2
=
α
r
∂
∂r
(
r
∂T
∂r
)
= α
∂2T
∂r2
+
α
r
∂T
∂r
, (3.37)
where
α =
kM
ρMCp,M
(3.38)
represents the thermal diffusivity.
For this work, discretization of the partial differential heat equation is ac-
complished by using explicit first-order, forward-difference discretization in time and
second-order, central-difference discretization in space. The discretized time rate of
change in temperature is given by
∂T
∂t
=
T j+1i − T
j
i
∆t
(3.39)
56
Figure 3.3: Sketch of the thermal heat transfer modeled using finite difference equations
and the discretized radial rate of change in temperature is given by
α
∂2T
∂r2
+
α
r
∂T
∂r
≈ α
T ji+1 − 2T
j
i + T
j
i−1
∆r2
+
α
ri
T ji+1 − T
j
i
∆r
. (3.40)
This discretization solves for the conductive heat transfer through the interior of
the SparkJet walls at time step j and mesh point i. Separate equations are used to
approximate the heat equation at the boundaries which involve the convective heat
transfer as well.
To solve for the internal mesh points, the above equations are combined into
T j+1i − T
j
i
∆t
= α
T ji+1 − 2T
j
i + T
j
i−1
∆r2
+
α
ri
T ji+1 − T
j
i
∆r
, (3.41)
which is the discretized form of the cylindrical heat equation. At the boundaries,
however, the effects of convection are taken into account. The internal boundary
condition is given by
hA
(
T jA − T
j
i=1
)
+ kA
T ji=2 − T
j
i=1
∆r
= ρAC
∆r
2
T j+1i=1 − T
j
i=1
∆t
(3.42)
57
and the external boundary condition is given by
hA
(
T j∞ − T
j
i=M
)
+ kA
T ji=M − T
j
i=M−1
∆r
= ρAC
∆r
2
T j+1i=M − T
j
M
∆t
. (3.43)
The highly unsteady nature of the SparkJet thermal heat transfer requires very
small time steps in this simulation. In general, small time steps require small grid
spacing to maintain numerical stability, especially when using an explicit method.
To perform the stability analysis, all terms in Equation 3.41 at time step j and mesh
point i are collected and set to be greater than or equal to zero, which results in the
inequality
1− 2α
∆t
∆r2
− α
∆t
∆r2
∆r
ri
≥ 0. (3.44)
Solving for ∆t,
∆t ≤
1
α∆r
(
2
∆r +
1
ri
) (3.45)
gives the maximum allowable time step for numerical stability. The mesh spacing,
∆r, is given while the radius of each mesh point, ri, is chosen to minimize ∆t which
corresponds to the radius of the SparkJet cavity.
To compare the two thermal modeling methods for the SparkJet cycle, a no-
tional temperature profile, Tn, was applied to the internal wall. A negative expo-
nential function of a magnitude of 2000 K,
Tn = (2000− T∞) e
−2000t + T∞, (3.46)
mimics a typical SparkJet cavity air temperature profile. For comparison, the inter-
nal wall temperature, Tw,in, was monitored using both methods. Figure 3.4 shows
the notional temperature profile as a function of time. Also for comparative pur-
58
0 200 400 600 800 10000
500
1000
1500
2000
Time (µs)
Tem
pera
ture 
(K)
Figure 3.4: Temperature profile used to compare the combined thermal resistance and
capacitance method and the discretization method
poses, both thermal modeling methods model heat transfer though an infinitely long
hollow Macor tube.
Figure 3.5 shows the wall temperature response to the notional temperature
profile shown in Figure 3.4 over 1000 µs. Initially, the wall temperature, Tw,in, rises
quickly and then reaches an asymptotic value of 288.01 K. Surprisingly, the peak
wall temperature is not significantly above the initial wall temperature of 288 K.
Based on these results, an isothermal wall condition at 288 K may be sufficient for
modeling the thermal heat transfer during a single SparkJet cycle. However, when
considering multiple SparkJet cycles, recall that the wall temperature does not equal
the ambient wall temperature at 1000 µs. When modeling high-frequency actua-
tion, the wall temperature rise can become more significant. Figure 3.6 shows the
wall temperature response over 50 cycles at 1 kHz. The wall temperature steadily
increases with each cycle but the total temperature rise is still less than 1 K such
that an isothermal wall condition may still be sufficient.
Figure 3.7 shows the spatial and temporal response of the cylindrical Macor
59
0 500 1000 1500 2000 2500 3000 3500 4000288
288.002
288.004
288.006
288.008
288.01
Time (µs)
T w,in 
(K)
Figure 3.5: Wall temperature response to the notional temperature profile using the
combined thermal capacitive and resistive model.
    0 10000 20000 30000 40000 50000288
288.05288.1
288.15288.2
288.25288.3
288.35288.4
Time (µs)
T w,in 
(K)
Figure 3.6: Wall temperature response to the notional temperature profile simulating
1 kHz actuation over 50 cycles.
60
walls to the notional temperature profile using the finite-difference equations. Dur-
ing a single cycle, the temperature at the surface rises quickly and then more slowly
decreases toward room temperature. Note, however, that the wall temperature only
rises to a couple degrees above room temperature which compares well with the
simplified thermal model. As time increases, the wall temperature decays but the
interior temperature distribution rises as the thermal energy diffuses into the Ma-
cor. Also note that the thermal diffusion does not penetrate far into the Macor. In
fact, the outer wall temperature rise is negligible and external thermal convection
is irrelevant.
When considering the thermal effects over several SparkJet cycles, the internal
wall temperature remains above the atmospheric temperature. Therefore, subse-
quent cycles will gradually raise the internal wall temperature such that prolonged
cycles will result in a significant temperature rise as seen in Figure 3.8. Specifically,
Figure 3.8(a) shows the wall temperature as a function of time and Figure 3.8(b)
shows the internal Macor temperature at several instances. Together, these plots
show that the Macor at the wall and internally increase with each SparkJet cycle.
However, the total temperature rise is only 14 K. Based on these results, a single
SparkJet cycle can be modeled with an isothermal wall boundary condition with-
out sacrificing modeling fidelity. However, for high-frequency actuation, detailed
thermal modeling is recommended.
Regardless of the method used to model heat transfer, defining the convective
heat transfer coefficient remains a challenge. Inside the SparkJet cavity, unsteady
pressure waves generate a forced convection condition while free convection exists
61
0 1000 2000
30002.4 2.6 2.8
288
289
290
291
Time(µs)Radius (mm)
Wall
 Tem
pera
ture 
(K)
Figure 3.7: Time and space dependent thermal response of the Macor material to the
notional temperature profile applied to the internal wall of the Macor during
a single cycle.
outside the SparkJet walls. The convective heat transfer coefficient can be estimated
based on Reynolds number, Prandtl number, and Nusselt number. However, these
estimations describe convective heat transfer in relatively steady flows, which is not
the case inside the SparkJet cavity where unsteady pressure oscillations exist. As a
result, a convective heat transfer coefficient of 125 W/m2 is used based on compar-
ison to the CFD simulation discussed in the next section [55].
Considering the original thermal modeling goal of balancing fidelity with sim-
plicity, the thermal model selected to be used with the fluid dynamic portion of the
SparkJet 1-D model is the combined capacitive and resistive model. Both models
show a wall temperature increases of less than 10 K even over several cycles, which
is a small temperature rise compared to the expected cavity temperature rise. The
following sections will show that the SparkJet momentum throughput reaches a
steady-state value over 3-30 cycles. The previous plots showed that over 50 cycles,
62
    0  5000 10000 15000 20000 25000 30000 35000 40000 45000 50000288
290
292
294
296
298
300
302
304
Time (µs)
Wall T
empe
rature
 (K)
(a) Wall temperature versus time.
2.4 2.5 2.6 2.7 2.8 2.9 3288
290
292
294
296
298
300
302
Radius (mm)
Intern
al Ma
cor Te
mpera
ture (K)
 
 t = 4.9 µst = 44.9 µst = 84.8 µst = 124.8 µst = 164.8 µst = 204.7 µst = 244.7 µst = 284.6 µst = 324.6 µst = 364.6 µst = 404.5 µst = 444.5 µst = 484.5 µs
(b) Sample internal temperature distribution versus radius at various time steps.
Figure 3.8: Time and space dependent thermal response of the Macor material to the
notional cavity air temperature profile applied at a frequency of 1 kHz over
50 cycles.
63
the wall temperature rises to only a few degrees above 288 K. While the finite differ-
ence modeling reveals some interesting thermal trends and phenomena, the simpler
method best satisfies the modeling goal of this dissertation.
3.3 Single Cycle Simulation
The system of equations described in the previous sections were solved using
the initial value problem solver in MATLAB called ode45. The MATLAB code used
to run the simulations is presented in the Appendix. The ode45 solver is based on
the explicit Runge-Kutta numerical approximation method designed to solve a set
of variables as a function of a single variable, time in this case. The simulation
requires multiple input parameters to properly represent an actual SparkJet cycle
and its physical design. These parameters include ambient pressure, temperature,
and density; cavity volume (excluding orifice volume); orifice diameter and height;
stored capacitor energy; specific heat capacity for a fixed volume of air; and the gas
constant of air. If considering the conductive thermal losses, thermal properties of
the SparkJet cavity materials and electrodes are also required. To demonstrate this
1-D analytical model, Table 1 provides values for the required inputs. These values
were specifically chosen for comparison with pre-existing CFD simulations of the
SparkJet actuator interacting with quiescent flow [60]. The CFD simulation only
modeled convective heat transfer to an isothermal wall so the physical properties of
Macor and tungsten are not included in Table 3.1.
Figure 3.9 shows the time history of the cavity pressure, temperature, and den-
64
sity as well as the orifice velocity during Stages 1 and 2 and the beginning of Stage 3.
The cavity pressure and temperature is instantly increased as a result of Stage 1.
During Stage 2, the pressure and temperature start to decrease toward ambient
conditions and the cavity density decreases as mass exits the orifice. The velocity
of the flow through the orifice remains fairly constant when the flow is choked. The
slight decrease in choked flow velocity is due to the decreasing temperature and,
therefore, speed of sound. The slope of the velocity curve changes near 200 µs when
the flow becomes unchoked. As cavity pressure nears ambient pressure, the velocity
decreases toward zero and the cavity density reaches a minimum. Stage 3 starts
when the velocity reaches zero. During Stage 3, the flow velocity through the orifice
is negative indicating the external flow is being drawn into the cavity. As a result,
the cavity density starts to increases and the cavity temperature decreases more
rapidly reaches ambient temperature. Figure 3.10 shows the full cycle ending when
all cavity conditions have returned to ambient conditions.
To support the modeling assumptions related to viscosity, convective heat
transfer, choked jet flow, and one-dimensionality of pressure, temperature, density
and velocity, the numerical modeling results are compared to the CFD simulation
results previously shown in Figure 3.12. For the specific case of a SparkJet under
the conditions given in Table 3.1. The details of the CFD simulation are described
in Reference [55] and are, therefore, only summarized here.
The CFD++, URANS simulation was run axisymmetric and 2nd order ac-
curate in time and space using a thermally perfect single air species. A cubic k-
turbulence model was used with an isothermal 288 K solve-to-the-wall boundary
65
Table 3.1: Design parameters for numerical model used for comparison to CFD results.
Parameter Value Units
P∞ 101325 Pa
T∞ 288 K
R 287.15 J/kg-K
Cv 716.85 J/kg-K
v 42.4 mm3
do 1.0 mm
ho 0.5 mm
Q 0.089 J
hin 125 W/m2K
66
0 500 1000 1500 2000 2500 30000
500
1000
1500
2000
2500
3000
3500
Time (µs)
See 
Lege
nd
 
 
1−D Density (g/m3)1−D Velocity (m/s)1−D Pressure (kPa)1−D Temperature (K)
Figure 3.9: Simulation of the time-dependent variation in the cavity pressure, cavity
temperature, cavity density, and the velocity through the orifice during early
portion of the SparkJet cycle.
    0  5000 10000 15000 20000 25000 30000 35000 40000 450000
500
1000
1500
2000
2500
3000
3500
Time (µs)
See 
Lege
nd
 
 
1−D Density (g/m3)1−D Velocity (m/s)1−D Pressure (kPa)1−D Temperature (K)
Figure 3.10: Simulation of the time-dependent variation in cavity pressure, cavity tem-
perature, cavity density, and the velocity through the orifice for the entire
SparkJet cycle. Note that cavity temperature, pressure, and density have
returned to original ambient conditions as given in Table 3.1.
67
Figure 3.11: Structured, axi-symmetric grid representation of the SparkJet internal ge-
ometry with grid axis units in millimeters. The red section represents the
grid cells raised to an elevated temperature and pressure to represent the
energy deposition in Stage 1.
condition. Figure 3.11 shows a zoomed view of the grid in the SparkJet cavity,
throat and external flow.
Relatively short duration (≈ 500 µs) simulations, focused on the initial pres-
sure rise, were conducted over a range of energy deposition efficiencies and ended
near the start of the refresh cycle. The red section of the grid in Figure 3.11 was
initialized separately from the rest of the flow-field and it is about 65% of the cavity
volume. The flow was initialized at rest, a temperature of 288 K and a pressure of
101325 Pa. The red section of the grid was kept at the same density and at rest
but was increased in pressure and temperature to model Stage 1 of the SparkJet
actuation cycle. Once the initial conditions were defined, the solution was run time-
accurate. A separate boundary condition family was created at the base of the cavity
that was the width of the experimental dynamic pressure sensor to be described in
68
Chapter 4. Comparisons to the experimental data are discussed in Chapter 5.
Comparison to the 1-D model is also achieved by calculating the volume-
averaged pressure, temperature, and density in the cavity as well as the area-
averaged velocity in the throat. Figure 3.12 shows these time-dependent values
as a result of the CFD simulation. After the initial pressure and temperature rise,
the orifice velocity increases, which leads to the decrease in cavity density. As the
jet flow continues, the pressure and temperature also decrease. The slope change in
the velocity around 120 µs, is indicative of the transition from choked to unchoked
flow through the orifice. Once the velocity reaches zero, the refresh stage starts
as is visible by the gradual increase in cavity density. As the density increases,
the relatively cool air is drawn into the cavity, which quickly decreases the cavity
temperature. The cavity pressure remains just below ambient pressure which forces
the refresh stage to continue. The pressure gradient is primarily maintained by the
continued convective heat transfer at the walls.
The 1-D model and CFD results are compared in Figure 3.13. This figure
shows that the overall comparison between the pressure, temperature, density, and
velocity curves are quite good. The largest disparities occur at the beginning of
the SparkJet cycle and are likely due to the highly unsteady flow present in the
cavity and orifice as the jet formation begins. However, as the cycle continues, the
two modeling solutions merge. This comparison does not necessarily prove the as-
sumptions but certainly supports them. With this level of support in the initial 1-D
model development, the model is expanded to high-frequency actuation.
69
0 100 200 300 400 500 600 700 800 900 10000
500
1000
1500
2000
2500
3000
3500
Time (µs)
See 
Lege
nd
 
 CFD Density (g/m3)CFD Temperature (K)CFD Velocity (m/s)CFD Pressure (kPa)
Figure 3.12: Time-dependent CFD simulation of the volume-averaged cavity pressure,
temperature, and density and area-averaged jet velocity through the orifice.
0 100 200 300 400 500 600 700 800 900 10000
500
1000
1500
2000
2500
3000
3500
Time (µs)
See 
Lege
nd
 
 1−D Density (g/m3)1−D Temperature (K)1−D Velocity (m/s)1−D Pressure (kPa)CFD Density (g/m3)CFD Temperature (K)CFD Velocity (m/s)CFD Pressure (kPa)
Figure 3.13: Comparison of the cavity pressure, temperature, and density and orifice
velocity versus time based on the CFD simulations and the simplified nu-
merical model.
70
    0  5000 10000 15000 20000 25000 300000
500
1000
1500
2000
2500
3000
Time (µs)
See L
egen
d
 
 1−D Density (g/m3)1−D Velocity (m/s)1−D Pressure (kPa)1−D Temperature (K)
Figure 3.14: Simulation of the time dependent variation in cavity pressure, temperature,
and density and the velocity through the orifice for three SparkJet cycles
at 100 Hz.
3.4 Simulating High-Frequency Actuation
As discussed in Chapter 1, most AFC techniques involve actuation at frequen-
cies matching, or at a harmonic of, a natural instability in the flow. This model was
modified to study the effect of high-frequency actuation on SparkJet performance.
Based on a prescribed actuation frequency, the ODE solver is exited at the time step
corresponding to the beginning of the next actuation cycle. The cavity pressure,
temperature, density, and orifice velocity become the initial conditions for the next
cycle and the ODE solver is restarted. Each cycle is initiated with the same value
of QC . Figure 3.14 shows the variation in cavity pressure, temperature, and density
and the velocity through the orifice when the SparkJet is actuated under the same
conditions listed in Table 1 at 100 Hz. Figure 3.14 demonstrates the effect of high-
frequency actuation on SparkJet performance. Even at 100 Hz, the model shows
that the second cycle encroaches on the end of Stage 3 of the first cycle such that
71
the peak temperature for the second cycle is slightly higher than that of the first
cycle. This increase in peak temperature is indicative of the effect of a very slow,
unassisted refresh cycle. Because momentum is an important performance param-
eter for AFC devices, Figure 3.15(a) shows that the effect on momentum is visible
but not significant for an actuation frequency of 100 Hz. However, as frequency
is increased, this effect is even more evident as shown in Figure 3.15(b) when the
SparkJet actuation frequency is modeled at 1 kHz and 5 kHz. When the simulation
is run over a range of frequencies from 1 Hz to 10 kHz, the steady-state momentum,
normalized by the peak momentum of the first cycle, can be plotted as a function of
frequency as shown in Figure 3.16. The normalized steady-state momentum output
remains constant until approximately 100 Hz and decreases as frequency increases
above 100 Hz.
The dependence of the normalized, steady-state momentum due to design pa-
rameters such as orifice diameter, cavity volume, and QC are analyzed using the
model. Figure 3.16 shows that increasing the orifice diameter allows for higher
momentum throughput as frequency increases. However, all three diameters are
subject to a decrease in momentum throughput around 100 Hz. Figure 3.17 shows
that increasing the input energy does not affect the high-frequency, normalized,
momentum-based performance. As QC increases, the pressure gradient across the
orifice face during Stage 2 increases, which drives the jet velocity up. Additionally,
the increased cavity temperature allows for a higher jet velocity associated with
choked flow. During the transition from Stage 2 to Stage 3, the increased jet veloc-
ity results in a larger decrease in the cavity pressure as jet momentum over-exhausts
72
    0  5000 10000 15000 20000 25000 30000−0.05
00.05
0.10.15
0.20.25
Time (µs)
Mom
entum
 (µN−s
)
(a) 100 Hz.
0 1000 2000 3000 4000 5000 6000−0.05
00.05
0.10.15
0.20.25
Time (µs)
Mom
entum
 (µN−s
)
(b) 1000 Hz.
0 500 1000 1500 2000 2500−0.05
00.05
0.10.15
0.20.25
Time (µs)
Mom
entum
 (µN−s
)
(c) 5000 Hz.
Figure 3.15: Simulation of the momentum through the SparkJet orifice as a function of
time when operated at 100, 1000 and 5000 Hz.
73
the cavity. Again, a relatively larger pressure gradient across the orifice at the be-
ginning of Stage 3 leads to a higher intake velocity. As a result of these phenomena,
the durations of Stage 2 and Stage 3 are not significantly influenced by variation in
QC . Only the magnitudes of the cavity pressure, temperature, density, jet velocity
and jet momentum are affected. Figure 3.18 shows that increasing the cavity volume
decreases the high-frequency momentum-based performance of the actuator. Taking
each of these trends into consideration, the high-frequency performance is improved
by design parameter changes that essentially shorten the duration of the complete
SparkJet cycle. Unfortunately, the high-frequency results are not validated against
CFD or experimental results and future work toward validation is desired. There-
fore, the quantitative rollover frequencies shown in Figures 3.16-3.18 require further
support.
3.5 Summary
This chapter defined the control volume of interest for SparkJet analysis and
presented the governing equations utilized to develop the 1-D model of the SparkJet
cycle. Several assumptions were made in the development of the model and the va-
lidity of most of the assumptions is considered. The model first simulated a complete
single SparkJet cycle and was compared to CFD analysis under the same conditions
to support several modeling assumptions. The comparison to CFD supported the
assumptions that the flow through the orifice can be assumed inviscid, the 2-D ax-
isymmetric flow can be simplified by volume averaging, the process is non-adiabatic
74
100 101 102 103 104 10510−2
10−1
100
Frequency (Hz)
Mome
ntum 
(µN−s)
 
 do = 0.4 mmdo = 1.0 mmdo = 2.0 mm
Figure 3.16: Magnitude plot of the normalized, steady-state momentum throughput
from the SparkJet actuator operated from 1 Hz to 10 kHz for orifice diam-
eters of 0.4, 1.0 and 2.0 mm, QC = 0.45 J, and v = 84.8 mm3 as predicted
by the 1-D model.
75
100 101 102 103 104 10510−2
10−1
100
Frequency (Hz)
Mome
ntum 
(µN−s)
 
 QC = 0.24 JQC = 0.45 JQC = 0.90 J
Figure 3.17: Magnitude plot of the normalized, steady-state momentum throughput
from the SparkJet actuator operated from 1 Hz to 10 kHz for energy de-
position values of 0.24, 0.45 and 0.90 J, do = 1 mm, and v = 84.8 mm3 as
predicted by the 1-D model.
76
100 101 102 103 104 10510−2
10−1
100
Frequency (Hz)
Mome
ntum 
(µN−s)
 
 V = 42.4 mm3V = 84.8 mm3V = 169.6 mm3
Figure 3.18: Magnitude plot of the normalized, steady-state momentum throughput
from the SparkJet actuator operated from 1 Hz to 10 kHz for cavity volumes
of 42.4, 84.8 and 169.6 mm3, QC = 0.45 J, and do = 1 mm as predicted by
the 1-D model.
77
and the choked flow condition can be defined with isentropic relations. This model
is unique from previous efforts primarily due to the addition of heat transfer which
enabled modeling of Stage 3 and high-frequency actuation.
With modest confidence in the SparkJet model, the model was exercised to
explore performance at high-frequency actuation. The results illustrate well that, as
frequency increases, performance generally suffers when quantified using jet momen-
tum. As the high-frequency actuation demands increase, investigation into improv-
ing performance at high-frequency is required. For the SparkJet actuator, this is
primarily controlled by reducing the duration of Stages 2 and 3. This can be achieved
by reducing cavity volume, or increasing orifice diameter. However, while peak mass
flow rate and momentum may be increased, the duration of high-momentum flow is
decreased. Another technique is to increase heat transfer from the cavity by using
wall materials of a higher thermal conductivity such that the duration of Stages 2
and 3 are decreased.
The following chapters focus on experimental data to determine the validity
of the remaining assumptions primarily related to Stage 1. The peak cavity pres-
sure and power drawn by the arc during Stage 1 are used to determine an energy
transfer efficiency. Further analysis considers failures in the assumptions to explain
the energy losses and changes in efficiency as certain design parameters change.
78
Chapter 4
Experimental Details and Data Acquisition
While comparison to CFD supports assumptions related to fluid dynamics,
the assumptions for energy transfer modeling during Stage 1 require experimental
support. A description of the equipment used to power the SparkJet and to acquire
data are described in this chapter and an overview of the experimental setup is
shown in Figure 4.1. Further details regarding this setup are discussed throughout
this chapter. Experimental measurements were primarily acquired to understand
the plasma-physics, support Stage 1 modeling, and to examine Stage 2. The design
parameters varied to understand the effects on SparkJet performance include cavity
volume, orifice diameter, capacitance, and electrode gap. As these parameters were
varied, the SparkJet actuator was characterized with simultaneous internal cavity
pressure measurements and arc power measurements to quantify SparkJet operation
and efficiency.
Prior to this work, experimental efforts were limited and did not provide infor-
mation on internal SparkJet cavity conditions necessary to support Stage 1 model-
ing. Therefore, attention was turned to measuring the pressure inside the SparkJet
cavity in addition to arc power measurements. The pressure measurements were
completed using a high-frequency response, dynamic pressure transducer. The arc
power was calculated using the product of current and voltage measurements. These
79
Figure 4.1: Diagram showing the basic electrical connections and equipment used to
acquire experimental data and power the SparkJet power supplies.
experiments led to a demonstration of SparkJet performance and estimation of the
operating efficiency. This new knowledge led to modifications to Stage 1 modeling
and exposed a disparity related to Stage 2 modeling.
4.1 SparkJet Actuator Device
The SparkJet actuator design used to support this dissertation included the
use of a Macor housing, tungsten anode and cathode, and a copper trigger electrode.
The SparkJet actuator was assembled from two Macor parts identified as a lid and a
base. Figures 4.2 and 4.3 show dimensional drawings of the lid and base for a cavity
volume of 84.8 mm3 and orifice diameter of 1 mm. The lids were designed with
three slots into which the electrodes could be placed. To characterize the SparkJet
operation, several design parameters were varied to analyze the effect on the peak
pressure rise in the cavity. These parameters include the orifice diameter, cavity
volume, capacitance across the electrodes, and electrode gap.
Figure 4.4 shows a photograph of the variety of Macor SparkJet bases and lids
80
Figure 4.2: Drawing of the SparkJet Macor lid with a 1 mm orifice diameter and
84.8 mm3 cavity volume.
corresponding to variations in cavity volume (42.4, 84.8, and 169.6 mm3) and orifice
diameter (0.4, 1.0, and 2.0 mm). The top row of white components are the bases
and the white components below the bases are the lids compatible with each cavity
volume. For a cavity volume of 84.8 mm3, all three orifice diameters were evaluated.
For cavity volumes of 42.4 mm3 and 169.6 mm3, an orifice diameter of 1 mm was
evaluated. Also, at the top of this image, the metal housing that was inserted into
the bottom of the SparkJet base with the pressure sensor installed (small circle in
the center of the metal face) are visible. Further details on the sensor and installa-
tion are discussed in Section 4.5 in this chapter.
The remaining design parameters were functions of the electrodes or the
electronics. The electrode spacing was controlled during the assembly process and
measured using a digital micrometer. The electrodes are placed in the pre-machined
slots shown in Figure 4.2 and adhered to the Macor using 5-minute epoxy such that
81
Figure 4.3: Drawing of the base component of the SparkJet for a cavity volume of
84.8 mm3.
Figure 4.4: Photograph of the SparkJet cavities and lids used to characterize the effect
of cavity volume and orifice diameter on SparkJet cavity pressure rise and
the pressure sensor used to acquire cavity pressure data.
82
the anode and cathode tips were approximately equidistant from the center of the
SparkJet cavity. This electrode gap was carefully selected to be small enough to en-
sure SparkJet reliability but large enough to maximize efficiency. The voltage across
the electrodes was controlled by an external power supply, but was typically set
near 600 V for maximum output energy. The capacitance across the electrodes was
controlled by adding or subtracting individual ceramic capacitors (Vishay 820 nF,
630 V) from a bank of capacitors as seen in Figure 4.5. Also note in this photo-
graph that the wires from the capacitor bank to the SparkJet are approximately
0.2 m long. The wires were kept relatively short to help minimize power losses that
would otherwise exist in unnecessarily long wires. While the resistance of wires is
low, the high current flowing through during the arc discharge results in a volt-
age drop that reduces the actual voltage potential across the electrode tips and,
therefore, the power available to the arc.
4.2 SparkJet Power Supplies
As the SparkJet operation became better understood over the duration of this
work, the SparkJet electronics evolved to meet high-frequency demands and im-
prove SparkJet performance. Two circuit designs were used to operate the SparkJet
actuator involving an external trigger and pseudo-series trigger. The generic circuit
used to generate the majority of the experimental results is shown in Figure 4.6.
This circuit is based on an externally triggered arc discharge concept that is rated
to 600 V. For the remainder of this dissertation, this circuit is referred to as 600ET.
83
Figure 4.5: Photograph of the SparkJet setup showing the Pearson current monitor,
differential voltage probe, installed pressure sensor, the SparkJet actuator,
SparkJet power supply, and a 12 inch ruler for scaling.
84
Figure 4.6: Circuit diagram of the 600 V external trigger SparkJet power supply.
The maximum voltage rating of 600 V for the 600ET circuit was chosen for practical-
ity because electrical components rated to 600 V are less expensive and are readily
available. A photograph of this power supply is shown in Figure 4.7 to demonstrate
the small size of this circuit which is beneficial for aircraft installation.
Based on further understanding of SparkJet performance, the generic circuit
shown in Figure 4.8 corresponds to the pseudo-series trigger SparkJet power supply
also with a maximum capacitor voltage of 600 V. For the remainder of this disser-
tation, this circuit is identified as 600PST. Transition to this circuit was motivated
by reliability issues. With the intention of improving SparkJet reliability, results
85
Figure 4.7: Photograph of the 600ET SparkJet power supply circuit box with dimensions
of 110 mm x 79 mm x 38 mm.
86
Figure 4.8: Circuit diagram of the 600 V pseudo-series trigger SparkJet power supply.
discussed in Chapter 5 will show that the 600PST circuit had additional efficiency
benefits. A photograph of this power supply is shown in Figure 4.9 to demonstrate
the small size of this circuit as well.
Basic operation of both of the power supplies is very similar. The SparkJet
operates on a triggered capacitive arc discharge; therefore, each SparkJet power
supply includes a means of triggering the discharge (trigger circuit) and a bank of
capacitors parallel to the SparkJet anode and cathode, which sustains the trigger
arc (sustain circuit). The sustain and trigger circuits are identified in Figures 4.6
and 4.8 by a red, dash-dot rectangle surrounding the sustain circuit and a blue, dash
87
Figure 4.9: Photograph of the 600PST SparkJet power supply circuit box with dimen-
sions of 133 mm x 133 mm x 55 mm.
88
rectangle surrounding the trigger circuit. Separate DC power sources are typically
used to provide power to each section of the SparkJet power supply. An external DC
power supply is used to charge the capacitor bank to a set voltage (typically near
600 V). The output current limit of the power supply determines the time required
to charge the capacitor bank. Therefore, one determining factor in the frequency
limit of the SparkJet is due to the DC power supply current limit. Additional, non-
essential electrical components are also placed in parallel with the capacitors such
as a resistor for operator safety, a surge suppressor to encourage a “clean” discharge,
and diodes immediately before the trigger electrode to prevent an arc discharge from
the anode to the trigger electrode.
Separate from the sustain circuit is the trigger circuit. A capacitor is charged
by an additional DC power supply and the charging rate is regulated by a resistor.
When the SparkJet is triggered, a low-voltage square pulse activates a transistor
such that current can flow across it. Once the transistor gate is open, the trigger
circuit capacitor begins to discharge across a high-voltage transformer that quickly
(≈ 0.5 µs) raises the voltage potential at the trigger electrode tip. If the breakdown
voltage between the trigger electrode and cathode is less than the peak trigger
voltage (near 10 kV), a trigger spark will form between the trigger electrode and
cathode. Subsequently, if the resulting trigger spark locally reduces the breakdown
voltage below the electrode voltage, Stage 1 begins as the main capacitor bank dis-
charges across the anode and cathode in the form of an arc. In the case of the
pseudo-series circuit, the high-voltage transformer output is connected to the anode
but high-voltage, high-current, blocking diodes prevent the high voltage from inter-
89
acting with the capacitor bank portion of the circuit. Further detail explaining the
difference between the external and pseudo-series triggers is provided in the follow-
ing paragraphs.
The purpose of the trigger spark is to momentarily reduce the local breakdown
voltage between the anode and cathode. While using the external trigger circuit,
maximizing the triggering reliability resulted in the ideal trigger electrode tip place-
ment as close to the anode as possible without the trigger spark occurring between
the trigger electrode and anode. Therefore, the trigger electrode tip was placed
approximately half way between the anode and cathode. In addition, the trigger
spark only ionizes a portion of the electrode gap. As the gap between the anode and
cathode increases, placement of the trigger electrode becomes a delicate and time-
consuming process. Because the exact placement varied from SparkJet to SparkJet,
the trigger electrode had to be a flexible metal such that copper was typically used.
Because copper does not survive the arc environment well, the trigger electrode tip
would frequently need to be cleaned and repositioned which further added to the
time consumption associated with using an external trigger circuit. In addition, the
maximum achievable gap using the external trigger mechanism while maintaining
moderately reliable arc breakdown was approximately 1.75 mm. The problems as-
sociated with reliability prompted an effort to change the trigger mechanism.
The trigger circuit design change was based on online documentation by Am-
glo [61] and Perkin Elmer [62] describing methods of causing an arc breakdown for
arc lamps. Trigger circuits can be categorized into three types: external, series and
pseudo-series. All of these circuit types contain a triggering circuit and a sustain
90
circuit but differ in the portions of the circuits which overlap. The external trigger
circuit, which is used in the 600ET circuit, only overlaps at the electrode gap by
sharing a common cathode. The sustain and trigger circuits each have a separate
anode. The electrical components in the trigger circuit are rated to tolerate the
high-voltage associated with the trigger spark but not the high-current associated
with the sustain circuit and vice versa.
In a series trigger circuit, several portions of the circuit overlap such that many
of the electrical components in the trigger and sustain circuits experience both high-
voltage and high-current including the transformer in the trigger circuit. The most
costly consequence of the series trigger circuit is due to the transformer requirement
for low gauge wiring making the transformer heavy and bulky. Also, using com-
ponents tolerant of high-voltage and high-current are more difficult to procure and
are generally more costly. Finally, the pseudo-series trigger circuit is a compromise
between the external and series trigger circuits such that the sustain and trigger
circuits only overlap at the anode and cathode. Sharing the anode requires that
high-voltage blocking diodes force the trigger voltage to pass between the anode
and cathode rather than into the sustain circuit.
Utilizing the pseudo-series trigger circuit offers both electrical and physical
performance benefits over the external trigger circuit. With the pseudo-series trig-
ger circuit, the electrode configuration reduces to two electrodes while maintaining
the ability to synchronize with data acquisition equipment. In this design, the trigger
spark occurs between the anode and cathode thus reducing the breakdown voltage
of the entire gap rather than a portion of it as with the external trigger circuit.
91
This design feature also increases reliability in producing an arc breakdown and,
thus, operating the SparkJet actuator. In addition, removing the dedicated trigger
electrode eliminates the time consumption associated with placement. The pseudo-
series trigger circuit also allows for flexibility in selecting the electrode gap without
sacrificing reliability. In Chapter 5, efficiency benefits as a function of electrode gap
are made apparent when measuring the cavity pressure.
4.3 External Supplies
Figure 4.1 shows the basic experimental setup including the equipment used to
operate the SparkJet and instruments used to acquire pressure, voltage, and current
data. This section describes the external power or voltage supplies in detail. For
this work, operating the SparkJet requires a low voltage (0-10 V) square pulse with
a pulse width of approximately 40 µs. For the work utilizing the 600ET circuit, a
TENMA-72-6860 pulse generator was used to provide this pulse. The pulse generator
can operate at a set frequency or using a manual trigger. Because only single pulse
data acquisition was acquired from these power supplies, the manual push-button
trigger function was used. When triggered, the low voltage pulse is applied to the
custom trigger circuit as shown in Figure 4.6. For improved control of high-frequency
actuation, the low-voltage pulse controlling the circuit in Figure 4.8 was generated
by an Agilent 33521A Function/Arbitrary Waveform Generator. Transition to the
function generator resulted from the ability of the function generator to generate a
burst of pulses at a desired frequency, which was helpful for high-frequency actuation
92
on the benchtop. This equipment was used with the 600PST circuit with the purpose
of performing high-frequency actuation tests. However, for the tests described in
this dissertation, only single pulses were applied to the SparkJet circuits.
Each SparkJet circuit requires DC power sources for charging the capacitors
parallel to the SparkJet electrodes and for the high-voltage trigger circuit. The
600ET circuit used a TENMA-72-7245 Dual Channel Bench DC Power Supply to
provide power to the trigger and sustain circuits. The TENMA power supply is
capable of outputting up to 30 VDC and 3 A per channel. The 600PST SparkJet
circuit is powered by a TDK Lambda GEN600-2.6 DC power supply to charge the
capacitors and an Acopian U275Y20M Power Supply Module to power the trigger
circuit. The TDK power supply is capable of outputting up to 600 VDC and 3 A
and the Acopian power supply is capable of outputting up to 275 V and 0.2 A. The
high voltage and current output of the TDK power supply was used for the ability
to support high-frequency actuation. The output from the Lambda power supply is
controlled with two, high-wattage, parallel resistors that also isolate the capacitance
within the Lambda power supply from the SparkJet power supply.
4.4 Data Acquisition
The primary instrument used to collect data was an Agilent AT-DSO5014A
- 100MHz 4CH portable oscilloscope which monitored the voltage across the elec-
trodes, the pressure sensor output, and the current through the arc at a sampling
frequency of 250 MHz. Depending on other required diagnostic testing through the
93
Figure 4.10: Photograph of the TENMA-72-6860 Pulse Generator used to trigger the
600ET SparkJet circuit.
Figure 4.11: Photograph of the TENMA-72-7245 Dual Channel Bench DC Power Supply
used to power the 600ET SparkJet circuit.
94
Figure 4.12: Photograph of the Agilent Function Generator used to initialize the Spark-
Jet cycle for the 600PST circuit.
Figure 4.13: Photograph of the Lambda GEN600-2.6 DC Power Supply and the custom
electronics box containing the Acopian power supply that are used to power
the 600PST SparkJet circuit.
95
course of this dissertation work, the fourth channel was used for the low-voltage
pulse, a photodetector, trigger voltage measurements, or a single-ended voltage
probe.
To evaluate SparkJet operation and efficiency, the pressure sensor, current,
voltage, and capacitance measurements were of primary interest. The pressure sen-
sor output illustrated in Figure 4.1 passes through a PCB Model 482C Signal Con-
ditioner designed for PCB pressure sensors before reaching the oscilloscope. Further
details on the pressure sensor itself are provided in the next section. A CIC Research
Model DP02-10K high-voltage differential probe was used to monitor the voltage
difference across the anode and cathode. Current measurements were made using
a Pearson Model 110 monitor and a Pearson Model A10 x10 attenuator to capture
peak currents above 500 A. The capacitance parallel to the sustain electrodes was
measured before each test using a Fluke Model 179 True RMS multimeter.
A variety of diagnostic measurements were also acquired. Monitoring the low-
voltage pulse was through a BNC cable from the pulse generator to the oscilloscope.
A BK Precision Model PR2000 200 MHz Oscilloscope High-Voltage, Single-Ended
Probe with x100 attenuation and rated to 2 kV was used to measure the volt-
age across the electrodes on the 600ET circuits. To monitor the trigger voltage
which typically exceeds 5 kV, a Tektronics P6015A 1000x high voltage passive test
probe rated to 20 kV DC was used. Due to the design of the 600PST circuit,
this probe also measured the voltage across the capacitors on the 600PST circuit.
With simultaneous current and voltage measurements, the power drawn by the arc
was directly calculated using Ohm’s Law. The photodetector, an Electro Optics
96
Technology Biased Silicon Detector ET-2040, was used to diagnose the voltage and
current waveforms. These waveforms tend to oscillate during the arc discharge and
questions about how the oscillations correspond to the arc discharge prompted the
use of the photodetector. The results of this diagnostic technique are discussed later
in Chapter 6.
4.5 Cavity Pressure Sensor
The high-frequency pressure data was obtained using a PCB 105C12 dynamic
pressure sensor (Figure 4.14) installed in the bottom of the SparkJet cavity, opposite
the orifice. The sensor design includes threads for installation and a brass ring to
provide a pressure seal. The force from the sensor threads due to the recommended
torque (1.69 Nm) exceeds the strength of the Macor. Therefore, a stainless steel
component was inserted into the Macor in order to support the sensor installation.
The stainless steel component was secured to the Macor using 5-min epoxy. A de-
tailed drawing of the metal insert is shown in Figure 4.15. The Macor and stainless
steel are dimensioned such that the face of the pressure sensor is recessed from the
bottom of the cavity. The recessed depth was chosen to allow for a stack of six 10 mil
thick layers of electrical tape discs sized to match the diameter of the pressure sensor.
A cross-sectional view of the assembled SparkJet actuator, metal insert, electrodes,
pressure sensor, and tape is shown in Figure 4.16. The PCB 105C12 pressure sensor
was chosen for its small size (2.5 mm sensing diameter), fast response time (< 2 µs),
and high flash temperature tolerance (1922 K). An additional benefit of using this
97
Figure 4.14: Photograph of the uninstalled PCB 105C12 dynamic pressure sensor.
sensor is the low sensitivity to electromagnetic interference (EMI) because the sensor
is connected to the PCB Model 482C signal conditioner via a shielded BNC cable.
The PCB sensing technology includes a preloaded quartz crystal surrounded
by a stainless steel housing. The loading on the quartz determines the output signal.
Ideally, only pressure changes at the sensor face affect the output signal. However,
thermal loads can also affect the output signal due to the small size of the sensor
and the proximity to the hot arc. To reduce the effects of the thermal shock associ-
ated with the initial blast wave, the initial test configurations involved coating the
exposed face of the sensor with a layer of black RTV (Room Temperature Vulcan-
ized) sealant. However, the long-term effects of the high-temperature air inside of
the chamber expands the stainless steel housing and reduces the output signal such
that the apparent pressure signal is more negative than the predicted chamber pres-
sure. This thermal expansion hypothesis has been supported based on discussions
with PCB technical engineers and a positive adjustment in the long-term pressure
signal when more thermal insulation covers the sensor face. In addition, fast-Fourier
transform (FFT) analysis of the pressure signal output indicated the sensor hous-
ing resonant frequency was being excited. As a result, the test configuration was
98
Figure 4.15: Drawing of the metal insert used to mate the SparkJet Macor housing to
the PCB pressure transducer.
altered.
Based on conversations with PCB engineers, resonating the sensor housing
can lead to nonlinear sensor output, signal attenuation, and significant signal un-
certainty. PCB engineers suggested covering the sensor face with layers of electrical
tape to dampen the initial shock wave effects exciting the resonance and protect the
sensor from subsequent large thermal loads. This method, of course, raised concerns
about the effect on sensor output. Therefore, tests were conducted to specifically
understand the impact of using tape on the sensor face.
The goal of this comparison testing (tape vs. no-tape) was to show that cov-
ering the sensor with tape did not affect the ultimate efficiency measurement (peak
filtered pressure). The expected differences in these testing configurations include
99
Figure 4.16: Cross section view of the SparkJet cavity, electrodes (configured for an
external trigger), installed pressure sensor, metal insert, brass ring, and six
layers of electrical tape.
100
a delayed initial pressure rise (time-of-arrival data), a reduction in thermal expan-
sion of the sensor housing, and the elimination of excitation of the sensor housing
resonance. The latter two effects are desired outcomes of using tape while the first
effect is not desired. However, a goal of these comparative tests was to show that a
delayed pressure response does not affect the peak pressure measurements required
for efficiency estimation. The test conditions under which this comparative study
was conducted were also carefully considered.
The SparkJet design used for this test included the largest of the three vol-
umes (169.6 mm3) in the test matrix and the 1 mm orifice diameter. The largest
volume was chosen because a low QC/E was desired to avoid the thermal effects
on the sensor housing. Therefore, E was maximized and, correspondingly, QC was
minimized until the ultimate efficiency measurements were within a few percent of
each other. At such a low QC/E, thermal effects become insignificant and the long
term thermal effects were also insignificant. These conditions were met at a capac-
itance of 0.33 µF charged to approximately 300 V for value of QC = 14.9 mJ and
QC/E = 0.35.
Figure 4.17 shows a comparison between tape vs. no-tape output over a long
time frame (4.17(a), output over a short time frame (4.17(b)), and the FFT (4.17(c))
of the pressure signal. The pressure comparison over a large time frame shows that
using the tape to protect the pressure sensor does not affect the magnitude of the
output significantly, and the overall shape of the pressure oscillation bounds are sim-
ilar. In addition, at the end of the pressure signal, both test configurations settle
near 101 kPa which indicates that long term thermal effects are also not significant.
101
When focusing on the initial pressure sensor output in Figure 4.17(b), the first
pressure rise occurs sooner for the no-tape configuration than for the tape configu-
ration. This effect was expected as the force from the initial blast wave must travel
through the six layers of 10 mil thick tape to impact the pressure sensor face. From
this figure, it appears the time delay is approximately 1 µs.
Finally, the comparison between the FFT of the two test configurations shows
that the frequency content is similar in the low frequency range (below 100 kHz).
At 310 kHz, however, there is a large peak. According to the PCB 105C12 data
specifications, the sensor resonance frequency occurs at or above 250 kHz. The
peak at 310 kHz is the resonant frequency for this sensor which is evident in the
no-tape testing configuration. However, the resonant peak is not visible in the tape
configuration demonstrating the successful reduction of sensor housing resonance
due to the pressure waves. The magnitude of the frequency content, however, is
generally lower for the no-tape configuration than for the tape configuration. Based
on conversations with PCB engineers, excitation of the sensor housing resonant fre-
quency can cause signal attenuation across all frequencies. The ultimate goal of this
comparison of the test configurations was to verify that the use of the tape does not
affect the efficiency estimation value. Since the estimation involves filtering the raw
pressure data, a discussion of the pressure signal post-processing is important.
102
50 100 150 200 250 300 350 400 450 5000
50
100
150
200
Time (µs)
Press
ure (kP
a)
 
 TapeNo Tape
(a) Pressure output over large time frame
50 55 60 65 70 75 80 85 90 95 1000
50
100
150
200
Time (µs)
Press
ure (kP
a)
 
 TapeNo Tape
(b) Pressure output over small time frame
0 50 100 150 200 250 300 350 4000
5
10
15
Frequency (kHz)
|P(f)−P
∞| (kPa
)
 
 TapeNo Tape
(c) FFT comparison
Figure 4.17: Comparison plots corresponding to the tape and no-tape test configura-
tions including the a) ensemble averaged pressure over 500 µs, b) ensemble
averaged pressure over 55 µs, and c) the FFTs of the ensemble averaged
pressure signals.
103
4.5.1 Signal Post-Processing
For each SparkJet design, the voltage, current and pressure were acquired
five times such that each signal could be ensemble averaged to reduce measure-
ment uncertainty. The uncertainty for the differential voltage, current, and pressure
measurements are ±10 V, ±10 A and ±50 kPa, respectively, based on the steady
state oscillations before each arc discharge. Through ensemble averaging, these un-
certainties were reduced to ±5 V ±3 A, and ±25 kPa, respectively. Uncertainty
associated with the capacitance measurement before each discharge was 0.01 µF.
For the voltage and current signals, this level of post-processing was sufficient to
obtain a reliable signal with an improved signal-to-noise ratio. The unsteadiness
observed in the pressure data is very repeatable; however, for efficiency analysis,
unsteadiness obscures the volume-averaged peak pressure measurement. Therefore,
the pressure output was low-pass filtered.
The cavity pressure data acquired is unsteady beginning with the very high
pressure associated with the initial blast wave and subsequent reflected waves within
the cavity. Following each pressure wave, the pressure signal drops to a very low
value and can even be negative at high QC/E. Part of the oscillations are due to
mechanical resonance of the sensor at 310 kHz as illustrated in the previous section.
Natural pressure wave reflections occur at lower frequencies (≈100-250 kHz) depend-
ing primarily on cavity dimensions. Because the purpose of this efficiency analysis
is to improve modeling accuracy in a 1-D model utilizing a volume-averaged cavity
pressure, there is a need to apply a low-pass filter to the high-frequency oscillations.
104
The PCB output is linear up to one fifth the resonance frequency (≈ 310 kHz) based
on an FFT analysis of the pressure signals. Therefore, each pressure signal was de-
composed into frequency components. Only components less than 60 kHz were used
to provide a volume-averaged peak pressure estimate. The pressure signal in the
time domain is converted to the frequency domains using the equations
a0 =
1
T
T∑
t=0
P (t)dt, (4.1)
an =
2
T
T∑
t=0
P (t) cos(nω0t)dt, (4.2)
and
bn =
2
T
T∑
t=0
P (t) sin(nω0t)dt, (4.3)
where ω0 = 2pifsampling (fsampling =250 MHz). Selecting the frequency components
up to 60 kHz,
P (t) = a0 +
nmax∑
n=1
an cos(nω0t) + bn sin(nω0t) (4.4)
is used to reconstruct the filtered pressure in the time domain. The maximum
value of the resulting filtered pressure signal is used to determine the pressure-based
efficiency. The mode corresponding to the cutoff frequency is given by
nmax = 2pi
ω0
ωc
. (4.5)
Figure 4.18 shows several comparative plots demonstrating the effect of filter-
ing on the pressure signals. Specifically, Figure 4.18(a) shows the effect of filtering
the ensemble averaged pressure signal. For both tape and no-tape test configura-
tions, the filtered pressure signal contains far fewer oscillations than the unfiltered
105
pressure signal. Also noteworthy is that the maximum values of the filtered pres-
sure signals are very similar. The maximum value is used to quantify pressure-based
efficiency as discussed in the following section. The equation used to calculate the
energy required to raise the cavity pressure to the maximum value (which will be
derived in Section 4.6) is compared to the stored capacitor energy to provide an
estimation of the pressure-based efficiency. The no-tape test configuration provides
an estimate of 47.0% efficiency and the tape configuration provides an estimate of
50.4% efficiency. This small variation in efficiency estimate represents the uncer-
tainty associated with using the tape to protect the pressure sensor face of 3.4%.
Figure 4.18(b) demonstrates the effect of applying the filtering technique de-
scribed above. The filtered and unfiltered FFT results track almost exactly up to
the cutoff frequency at 60 kHz. Beyond the cutoff frequency, the FFT of the filtered
signal is nearly zero. This plot comparison clearly demonstrates the effectiveness of
the low-pass filter.
After presenting the effect of filtering the low QC/E case, one may wonder if
the filtering presents non-physical oscillations that would affect the efficiency pre-
diction. To show that the oscillations actually do not significantly contribute to
the peak pressure measurement, a high QC/E case (QC/E = 18.0) is presented in
Figure 4.18(c) where the ensemble averaged pressure signal is filtered. Here the fil-
tered signal clearly represents an averaged pressure signal. Again, both the tape and
no-tape signals are presented to show that the tape still does not affect the peak pres-
sure measurement at the beginning of the pressure rise. However, beyond the initial
pressure rise, there is a slight difference between the tape and no-tape configuration
106
that lasts the remainder of the pressure signal. This difference is indicative of the
long term temperature effects on the pressure sensor due to the high-temperature
SparkJet cavity conditions. Also demonstrated in this figure is the visible reduction
in pressure oscillations in the tape configuration due to the successful suppression
of the pressure sensor resonance. Based on these test configuration comparisons, all
testing was performed with tape covering the pressure sensor.
4.6 Efficiency
The arc discharge and resultant Joule heating are complex processes that are
dependent on circuit design and localization of the arc discharge within the larger
cavity. The purpose of this efficiency analysis is to provide an understanding of the
SparkJet efficiency, the source of inefficiencies, and methods to improve efficiency
starting with a calorically and electrically ideal assumptions. In addition, the results
of this efficiency analysis will support on-going Stage 1 modeling to be included in
a simplified model of the entire SparkJet cycle.
Efficiency is evaluated by determining the energy associated with the multiple
processes involved in Stage 1. To begin, the maximum possible energy is defined as
the stored capacitor energy, QC , as defined in Equation 2.1. Assuming no losses, this
energy deposition can be used to estimate the cavity pressure and temperature rise
as was shown in Chapter 3. However, there are several electrical and physical effects
which reduce the energy output by the SparkJet actuator. During the conversion of
stored capacitor energy to the arc, there are energy losses outside those described by
107
0 50 100 150 200 250 300 350 400 450 500
0
50
100
150
200
250
Time (µs)
Pres
sure
 (kPa
)
 
 Unfiltered, No TapeUnflltered, TapeFiltered, No TapeFiltered, Tape
(a) Comparison of the time-dependent unfiltered and filtered pressure at QC/E = 0.35.
0 50 100 150 200 250 300 350 4000
5
10
15
Frequency (kHz)
|P(f)−P
∞| (kPa
)
 
 Unfiltered, TapeFiltered, Tape
(b) FFT comparison between the unfiltered and filtered pressure.
0 50 100 150 200 250 300 350 400 450 500−2000
−1000
0
1000
2000
3000
Time (µs)
Pres
sure
 (kPa
)
 
 Unfiltered, No TapeUnfiltered, TapeFiltered, No TapeFiltered, Tape
(c) Comparison of the time-dependent unfiltered and filtered pressure at QC/E =
Figure 4.18: Pressure and FFT data demonstrating the effect of low-pass filtering on
the SparkJet internal cavity pressure measurements.
108
Narayanaswamy [63] which include resistance in the wires leading to the electrode
tips, an increasing Cv as the cavity temperature increases, and localized, rather than
distributed, energy deposition and heating. These phenomena result in an efficiency,
η, less than 1.
The total energy released due to the power drawn by the arc is estimated by
QA =
∫
V (t)I(t)dt, (4.6)
where V and I are directly measured. The efficiency related to converting stored
capacitor energy to arc power is given by
ηA = QA/QC . (4.7)
Efficiency losses here are related to parasitic resistance and inductance in wires and
other circuit components that depend on circuit design.
The heat produced transfers from the arc column to the surrounding air or
materials. Some heat is inevitably lost to the electrodes because the heat is released
at the electrode tips [56]. The remaining heat, however, is transferred to the cavity
air. The energy deposition calculation assumes the energy is added to the entire
cavity volume to raise the cavity temperature. This assumption does not take into
account the highly spatial effects associated with the arc discharge but provides
some insight into the efficiency losses.
In an attempt to incorporate the effects of the high cavity temperatures, Cv is
estimated as a function of cavity air temperature resulting in a temperature depen-
dent Cv and a thermally perfect gas assumption. Once the new cavity temperature
109
is determined, the value of Cv is adjusted according to the simple harmonic oscillator
model,
Cv(T ) = Cv,p
(
1 + (γp − 1)
[(
θ
T
)2 eθ/T
(eθ/T − 1)2
])
, (4.8)
and assuming a thermally perfect gas [64]. Here, γp and Cv,p correspond to a calor-
ically perfect gas values and θ is the molecular vibrational energy constant equal to
3055.6 K. At the conclusion of the energy deposition,
QT = mCv,p(T2 − T1) (4.9)
uses the final cavity temperature to determine the energy required to raise the cavity
temperature assuming the idealized calorically perfect conditions. The efficiency
associated with the conversion of stored capacitor energy to thermal energy, ηT , is
defined as
ηT = QT/QC . (4.10)
The additional loss due to a calorically imperfect gas helps explain the efficiency loss
between the arc discharge energy and the measured peak cavity pressure presented
in Chapter 5.
Finally, the peak pressure measured by the pressure transducer is used to
determine the efficiency based on pressure measurements. The energy required to
raise the cavity pressure to the measured peak pressure is determined by
QP =
(
Pm
ρRT1
− 1
)
E (4.11)
110
assuming the original calorically perfect assumptions. The ratio of QP to QC is used
to calculate the pressure-based efficiency, ηP , defined as
ηP = QP/QC . (4.12)
The following chapters show the resulting pressure data, arc power measure-
ments and the corresponding efficiency analysis. A discussion of these results also
leads to a modification to the Stage 1 modeling.
111
Chapter 5
Pressure-Based Results and Analysis
Using the experimental setups described in Chapter 4, the SparkJet perfor-
mance based on cavity pressure, electrode voltage, and arc current is studied. This
chapter focuses on the pressure-based results with analysis and discussion. First, a
general understanding of the SparkJet cavity pressure as a function of several de-
sign parameters is considered. Second, the filtered peak pressure is converted to an
efficiency metric as described at the end of Chapter 4. Finally, the time-dependent,
filtered pressure results are compared to the 1-D model results.
5.1 Basic Operation
Before presenting the bulk of the experimental results, a look at a single Spark-
Jet data acquisition cycle is considered here for the 600ET and 600PST circuits
operated at low frequency (< 1 Hz). Figure 5.1(a) shows the typical output from
the 600ET circuit. The dashed line shows the raw pressure transducer output and
the solid black line shows the filtered pressure output to filter the transducer reso-
nant frequency above 60 kHz. To observe the arc voltage and current, Figure 5.1(b)
shows these signals over a 22 µs time frame. The blue, dash-dot line shows the volt-
age across the electrodes and the orange, dashed line shows the arc current. Note
that the voltage across the actuators is approximately 600 V prior to initiating the
112
SparkJet. This value is not set to exactly 600 V and, therefore, analysis of each
data set involves averaging the voltage over the first 4 µs (1000 samples) to deter-
mine the applied voltage across the actuators and, therefore, QC . The raw pressure
transducer output is initially affected by EMI from the trigger spark as indicated
by the negative spike at 50 µs but the signal conditioner prevents the output signal
from continuing to be contaminated. While the EMI spike is not related to pres-
sure, it indicates when the trigger spark was initiated. After the EMI spike, the
first significant pressure rise provides the time of arrival for the first blast wave from
the arc, typically 3-6 µs after the EMI spike. Beyond the first pressure rise, the
signal represents the pressure due to the arc acting on the tape layers covering the
face of the PCB sensor. The filtered pressure transducer output is the source of the
efficiency and Stage 2 analysis in the subsequent sections.
Figure 5.2(a) shows the typical output for the 600PST circuit which is sim-
ilar plot to Figure 5.1(a) except for one difference. The initial voltage across the
electrodes is approximately 600 V, but there is a large rise in the voltage near 50 µs
which represents the trigger voltage since both the trigger and capacitor voltages
are applied to the anode. In this particular plot, the maximum value of the trigger
voltage is not captured (saturated output) in order to maintain measurement res-
olution of the voltage during Stage 1. Figure 5.2(b) shows a zoomed-in section of
Figure 5.2(a) focusing on the output signals during Stage 1. The arc current shown
in Figure 5.2(b) represents the typical waveform seen for the 600PST circuit. The
arc current data are presented in Chapter 6 with power-based analysis.
113
0 50 100 150 200 250 300 350 400 450 500−1500
−1000
−500
0
500
1000
1500
2000
2500
Time (µs)
See 
Lege
nd
 
 
Arc Current (A)Arc Voltage (V)Cavity Pressure (kPa)Filtered Cavity Pressure (kPa)
(a)
48 50 52 54 56 58 60 62 64 66 68 70−1000
−500
0
500
1000
1500
2000
2500
Time (µs)
See 
Lege
nd
 
 
Arc Current (A)Arc Voltage (V)Cavity Pressure (kPa)Filtered Cavity Pressure (kPa)
(b)
Figure 5.1: Example of ensemble-averaged data acquisition output for the 600ET setup
and SparkJet cavity volume of 84.8 mm3, orifice diameter of 1.0 mm, and
capacitance of 4.28 µF a) over 500 µs and b) over 22 µs. (f < 1 Hz)
114
0 50 100 150 200 250 300 350 400 450 500−3000
−2000
−1000
0
1000
2000
3000
Time (µs)
See 
Lege
nd
 
 
Arc Current (A)Arc Voltage (V)Cavity Pressure (kPa)Filtered Cavity Pressure (kPa)
(a)
48 50 52 54 56 58 60 62 64 66 68 70−3000
−2000
−1000
0
1000
2000
3000
Time (µs)
See L
egen
d
 
 
Arc Current (A)Arc Voltage (V)Cavity Pressure (kPa)Filtered Cavity Pressure (kPa)
(b)
Figure 5.2: Example of data acquisition output for the 600PST setup and SparkJet
cavity volume of 84.8 mm3, orifice diameter of 1.0 mm, electrode gap of
1.0 mm, and capacitance of 3.72 µF a) over 500 µs and b) over 22 µs.
(f < 1 Hz)
115
5.2 Results
As mentioned in Chapter 4, the SparkJet performance was evaluated over a
range of cavity volumes, orifice diameters, electrode spacings, and input energy.
This section will discuss and demonstrate in detail the effects of varying each of
these parameters on the filtered cavity pressure. All results will be based on the
600ET circuit except for the electrode spacing results which are based on the 600PST
circuit.
5.2.1 Variation with Input Energy
Using the 600ET circuit, the effect of varying the input energy has been stud-
ied. For these tests, the voltage remained near 600 V and the capacitance across
the electrodes was varied by adding or removing capacitors. Five capacitors were
used that summed to values of C = 0.95, 1.79, 2.57, 3.42, and 4.28 µF. Figure 5.3
shows the filtered cavity pressure output as the capacitance increased from 0.95 µF
to 4.28 µF for a fixed volume of 84.8 mm3 and orifice diameter of 1 mm. As capaci-
tance increases, the energy input into the cavity increases and, correspondingly, the
cavity pressure response is increased. Each pressure curve shape is very similar to
each other and show that the duration of Stage 2 is relatively unaffected by increas-
ing energy input. This trend is consistent with the results presented in Figure 3.17,
demonstrating that both modeling and experimental results show increasing input
energy does not affect the duration of Stages 2.
Another observation to note in Figure 5.3 is that the long-term pressure
116
0 50 100 150 200 250 300 350 400 450 500
0
200
400
600
800
1000
Time (µs)
Pres
sure
 (kPa)
 
 C=0.95 µFC=1.79 µFC=2.57 µFC=3.42 µFC=4.28 µF
Increasing Energy Input
Figure 5.3: Variation in the filtered pressure signal as a function of capacitance for a
cavity volume of 84.8 mm3, orifice diameter of 1.0 mm, and electrode gap of
1.75 mm using the 600ET circuit. (f < 1 Hz)
signals vary slightly. The variation is due to the long-term thermal effects on the
pressure sensor that are move prevalent as the input energy and, therefore, peak
temperature increases. For the lowest capacitance value, the long-term thermal
effects are minimal; however, the effects are present for large capacitance values
despite the layers of insulating tape on the pressure sensor. While the tape does not
completely eliminate the effects of arc heat, the effects are reduced.
As mentioned in Chapter 4, the peak filtered pressure is used to determine the
efficiency, ηp, of each arc discharge. Figure 5.4 shows how the efficiency varies as the
non-dimensional parameter QC/E increases. This figure also shows how efficiency
varies for all three cavity volumes tested at a fixed orifice diameter of 1 mm and
electrode gap of 1.75 mm. Over a very wide range of QC/E values, the efficiency
values for all three volumes coalesce to one curve which decreases slowly from 42%
to 21% as QC/E increases.
117
Figure 5.4: Efficiency of the SparkJet actuator as a function of QC/E for an orifice
diameter of 1 mm, electrode gap of 1.75 mm, and cavity volumes of 42.4 mm3,
84.8 mm3, and 169.6 mm3 using the 600ET circuit.
5.2.2 Variation with Cavity Volume
Using the 600ET circuit, the effect of varying cavity volume on the filtered
cavity pressure has been evaluated. For these tests, the orifice diameter remained
constant at 1 mm and the input energy was varied over the same range as in the
previous section. The electrode gap was also kept constant at 1.75 mm. Figure 5.5
shows the filtered cavity pressure as a function of cavity volume. Since changes to
the cavity volume alone affect the energy ratio QC/E, test cases of similar QC/E
values were chosen and fall near 12.5. This figure shows that cavity volume does not
have a significant effect on the peak pressure but does have a significant effect on
the shape of the pressure signal. In general, increasing cavity volume increases the
duration of Stage 2. Again, this trend is consistent with the results shown in the
high-frequency modeling section demonstrating that increasing the cavity volume
increases the duration of Stage 2 as is evident in Figure 3.18.
118
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
Time (µs)
Pres
sure
 (kPa
)
 
 V=42.4 mm3V=84.8 mm3V=169.6 mm3
Increasing Cavity Volume
Figure 5.5: Variation in the filtered pressure signal as a function of cavity volume for
a QC/E value near 12.5, orifice diameter of 1.0 mm, and electrode gap of
1.75 mm using the 600ET circuit. (f < 1 Hz)
5.2.3 Variation with Orifice Diameter
Using the 600ET circuit, the effect of varying orifice diameter on the filtered
cavity pressure has been evaluated. For these tests, the cavity volume remained
constant at 84.8 mm3 and the input energy was varied over the same range as in the
previous sections. The electrode gap was again kept constant at 1.75 mm. Figure 5.6
shows the filtered cavity pressure as a function of orifice diameter for do =0.4, 1.0,
and 2.0 mm. Similar to the cavity volume results, the orifice diameter primarily
affects the shape of the cavity pressure output. At an orifice diameter of 2.0 mm,
the cavity pressure decreases quickly to 101 kPa after less than 100 µs. At an orifice
diameter of 1.0 mm, the cavity pressure decreases more slowly and reaches 101 kPa
after 200 µs. Finally, at an orifice diameter of 0.4 mm, the cavity pressure decreases
over the longest duration of the three orifice diameters and reaches 101 kPa after
119
0 50 100 150 200 250 300 350 400 450 500
0
200
400
600
800
1000
Time (µs)
Pres
sure
 (kPa
)
 
 do=2.0 mmdo=1.0 mmdo=0.4 mm
Decreasing Orifice Diameter
Figure 5.6: Variation in the filtered pressure signal as a function of orifice diameter for a
QC/E value of 28, cavity volume of 84.8 mm3, and electrode gap of 1.75 mm
using the 600ET circuit. (f < 1 Hz)
250 µs. These results are consistent with the thought that the pressurized cavity
air takes a longer duration to escape through a small diameter orifice than through
a large diameter orifice.
Yet again, these experimental results are consistent with the trends discussed
in the 1-D modeling section. Figure 3.16 shows that for a small orifice diameter, the
high-frequency performance degrades at a lower frequency than that of a large orifice
diameter. This result implies, and the experimental data confirms, that decreasing
orifice diameter increases the duration of Stage 2. Variation in orifice diameter does
not, however, affect SparkJet efficiency as the orifice diameter has no effect on the
peak pressure rise.
120
5.2.4 Variation with Electrode Gap
While the main goal of introducing the pseudo-series trigger circuit was to
increase reliability, an unexpected benefit was the increase to the maximum electrode
gap. This benefit was explored experimentally by varying the electrode gap and
quantifying the corresponding peak cavity pressure. The SparkJet actuator used to
quantify the effect of changing the trigger mechanism on efficiency is identical to
that described in Chapter 4 except without a trigger electrode. The SparkJet had a
cavity volume of 84.8 mm3, an orifice diameter of 1 mm and the electrode gap was
evaluated at 1.0, 2.0, and 3.0 mm.
Testing demonstrates that increasing the electrode gap also increases the peak
cavity pressure during Stage 1. Figure 5.7 shows the internal cavity pressure as a
function of time for each electrode gap tested. This plot shows that increasing the
electrode gap increases the energy transferred from the arc to the surrounding air
during Stage 1. Figure 5.8 summarizes the curves shown in Figure 5.7 over a range
of QC/E values. This plot shows that efficiency as a function of QC/E for various
gap sizes follows similar exponentially decreasing trends as shown in Figure 5.4. For
the 1 mm gap case, however, trends cannot be determined because all results are
within the 3% uncertainty associated with the efficiency measurements. The largest
change to the efficiency is seen for QC/E near 8 where the efficiency rises from 9%
to 30% for a tip distance change from 1.0 mm to 3.0 mm, respectively. The surface
area of the arc column between the electrodes is now much larger and interacts
with a larger fraction of the surrounding air. With increased efficiency, the same
121
50 100 150 200 250 300 350 400 450 500
0
100
200
300
400
500
600
700
800
Time (µs)
Press
ure (kP
a)
 
 
Gap = 1.0 mmGap = 2.0 mmGap = 3.0 mm
Figure 5.7: Comparison of pressure vs. time for three electrode gaps of 1.0, 2.0, and
3.0 mm given the same cavity volume (84.4 mm3), orifice diameter (1.0 mm)
and QC/E of approximately 28. (f < 1 Hz)
momentum throughput can be achieved for a lower energy input, QC/E.
5.3 Analysis
The previous section presented experimental data which showed the effect of
varying cavity volume, orifice diameter, input energy, and electrode gap on the
SparkJet cavity pressure. The trends showed that the duration of Stage 2 is de-
pendent on cavity volume and orifice diameter, and independent of input energy
and electrode gap. These trends associated with cavity volume, orifice diameter,
and input energy were explored in Chapter 3 and the experimental results are con-
sistent with the modeling results. However, the quantitative pressure results are
not consistent with the modeling results. For example, as Figure 5.4 showed, the
transfer of stored capacitor energy, QC , to raising the cavity pressure is inefficient.
The efficiency is primarily a function of QC/E and the electrode gap. The modeling
122
0 10 20 30 400
20
40
60
80
100
Energy Ratio (QC/E)
Effic
iency
 (%)
Figure 5.8: Comparison of efficiency vs. Q/E for three electrode gaps of 1.0, 2.0, and
3.0 mm given the same cavity volume (84.8 mm3) and orifice diameter
(1 mm). (f < 1 Hz)
results presented in Chapter 3 assumed all stored capacitor energy resulted in in-
creasing the cavity pressure; however, it is clear from the experimental results that
inefficiencies need to be incorporated into the model. In addition, the source of the
inefficiencies need to be explored.
The experimental results in this chapter show that the method of transferring
stored capacitor energy to energy involved in raising the cavity pressure is inefficient.
There are several possible sources of inefficiency. First, the assumption that all of
the capacitor energy is deposited in the arc ignores losses in circuit wiring and the
possibility that the capacitor bank does not fully discharge during Stage 1. If the
losses between the capacitor bank and arc are significant, the total energy applied
to the cavity air is already lower than estimated. The measured power drawn by
the arc is explored further in Chapter 6.
Another source of inefficiency could lie in the method of energy deposition.
123
The 1-D model assumes that the energy is deposited uniformly over the entire cav-
ity volume. However, the arc discharge, the source of the thermal energy, is a
concentrated filament between the electrode tips, which must conduct heat away
from the arc into the cavity air. This process is highly spatially and temporally de-
pendent. In addition, the extremely high arc temperature mentioned in the thermal
modeling section of Chapter 3 far exceeds the temperature range associated with a
calorically perfect gas. Therefore, at locally high cavity air temperatures, a unit of
energy input does not raise the cavity temperature as much as that same unit of
energy input into a locally low-temperature region of cavity air.
Finally, the protruding tungsten and copper electrodes may cause thermal heat
transfer out of the cavity air before the cavity pressure reaches the pressure sensor.
The electrode tips are, by nature, closest to the arc discharge and are an immedi-
ate avenue for heat transfer. The high thermal conductivity of both tungsten and
copper also lead to very rapid heat transfer.
Incorporating the source of the inefficiency into the 1-D model would involve
modeling the circuit or modeling the three-dimensionality of the arc energy deposi-
tion process. The first method of modeling the inefficiency would be simple except
for the varying resistance of the arc throughout the discharge process. The second
method is not practical for this 1-D modeling task as the arc energy deposition pro-
cess is highly three-dimensional. Therefore, the inefficiency is incorporated into the
1-D model based on the experimental results as a coefficient, η, which ranges from
124
0 to 1.0. Including the pressure based efficiency,
P2 = ρ2RT1
(
1 + ηP
QC
E
)
(5.1)
gives the model peak cavity pressure.
5.4 Comparison to 1-D Model
Direct comparison between the experimental results and the 1-D model are
useful to assess the validity of the 1-D model. Comparisons as a function of cavity
volume, orifice diameter, and input energy can be made, but cannot be made as
a function of electrode gap since electrode gap is not modeled. These comparisons
also incorporate the experimentally estimated efficiency.
Figures 5.9(a), 5.9(b) and 5.9(c) all confirm the trends related to the duration
of Stage 2 and the dependence on cavity volume, orifice diameter and input energy.
However, all of these figures also show that the model significantly over-predicts the
duration of Stage 2. Based on the results presented in this chapter, factors such as
cavity volume and orifice diameter can affect the duration of Stage 2. Increasing
orifice diameter would lead to a shorter Stage 2; however, an effectively larger orifice
diameter does not make physical sense. Decreasing the cavity volume would also
lead to a shorter Stage 2; however, the amount of decreased effective volume is too
significant to be physically possible.
Another possible reason for the over-predicted duration of Stage 2 is an under-
prediction of heat transfer. Since the convective heat transfer rate was determined
from CFD analysis, it is possible that CFD simulations and, therefore, the 1-D model
125
50 100 150 200 250 300 350 400 4500
100200
300400
500600
700800
900
Time (µs)
Press
ure (kPa
)
 
 Experiment (C = 0.95 µF)Experiment (C = 1.79 µF)Experiment (C = 2.57 µF)Experiment (C = 3.42 µF)Experiment (C = 4.28 µF)Model (C = 0.95 µF)Model (C = 1.79 µF)Model (C = 2.57 µF)Model (C = 3.42 µF)Model (C = 4.28 µF)
Increasing Energy Input
(a)
0 50 100 150 200 250 300 350 400 450 5000
100200
300400
500600
700800
9001000
Time (µs)
Press
ure (kPa
)
 
 Experiment (V = 42.4 mm3)Experiment (V = 84.8 mm3)Experiment (V = 169.6 mm3)Model (V = 42.4 mm3)Model (V = 84.8 mm3)Model (V = 169.6 mm3)
Increasing Volume
(b)
50 100 150 200 250 300 350 400 4500
100200
300400
500600
700800
9001000
Time (µs)
Press
ure (kPa
)
 
 
Experiment (do = 0.4 mm)Experiment (do = 1.0 mmExperiment (do = 2.0 mmModel (do = 0.4 mm)Model (do = 1.0 mm)Model (do = 2.0 mm)
Decreasing Orifice Diameter
(c)
Figure 5.9: Comparison between experimental and modeled cavity pressure as a function
of time as a) input energy, b) cavity volume, and c) orifice diameter are
varied.
126
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
Time (µs)
Pres
sure
 (kPa)
 
 Experiment (V = 42.4 mm3)Experiment (V = 84.8 mm3)Experiment (V = 169.6 mm3)Model (V = 42.4 mm3)Model (V = 84.8 mm3)Model (V = 169.6 mm3)
Increasing Volume
Figure 5.10: Comparison between experimental and modeled cavity pressure as a func-
tion of time as cavity volume is varied and the convective heat transfer
coefficient is increased to hin = 6000 W/m2K.
under-predict the actual convective heat transfer rate. Arbitrary adjustments to the
convective heat transfer rate within the 1-D model show that a value of 6000 W/m2K
provides improved comparison between the experimental data and the 1-D model.
Figure 5.10 shows an example of the effect of increasing hin to 6000 W/m2K. The
model and experimental values are in better agreement; however, this figure also
highlights a potential issue with the efficiency estimation method. Rather than
using the peak pressure to estimate efficiency, a method for averaging the initial
peaks would be a better representation of the volume-averaged pressure.
This large change to the convective heat transfer coefficient has prompted
some further exploration into the source of the differences. In particular, analytical
methods for estimating the convective heat transfer related to entrance flows, such
as the flow from the cavity to the orifice, have been investigated. To determine the
127
analytical equation to use for this estimation, the approximate Reynolds number,
ReD, associated with the flow through the orifice was estimated to be near 13,000
assuming air density is 1.225 kg/m3, U = 600 m/s, D = do = 1 mm, and µ ≈
6×10−6 kg/ms at elevated temperatures. Therefore, the flow through the orifice is
considered laminar. The average Nusselt number, NuD, for laminar flow in a duct
of length L can be estimated using
NuD = 3.66 +
0.065 (D/L)ReDPr
1 + 0.04 [(D/L)ReDPr]
2/3
, (5.2)
where Pr is the Prandtl number, which is approximately 0.7 for air [65]. Using
Equation 5.2, the average Nusselt number is 66.45. Nusselt number is defined as
Nu =
hD
k
(5.3)
where h is the convective heat transfer and k is the convective heat transfer coeffi-
cient of air. Rearranging Equation 5.3 to solve for h and assuming the conductivity
of air is approximately 0.05 W/m2, the approximated convective heat transfer co-
efficient is 3322 W/m2. This value is much larger than the originally estimated
125 W/m2 but not quite as large as 6000 W/m2. Regardless, this analytical check
provides some support for increasing the convective heat transfer coefficient used in
the model. At the same time, further investigation into the disparity between the
model and experimental results is still required.
The over-prediction of the duration of Stage 2 brings the rollover frequencies
presented in Chapter 3 and Figures 3.16-3.18 into question. A reduced duration of
Stage 2 would lead to an improved high-frequency performance. To demonstrate
128
100 101 102 103 104 10510−2
10−1
100
Frequency (Hz)
Mom
entum
 (µN−s
)
 
 
h = 125 W/m2h = 6000 W/m2
Figure 5.11: Comparison between the modeled normalized, steady-state momentum as
a function of actuation frequency for convective heat transfer coefficients
of 125 W/m2 and 6000 W/m2. (QC = 0.45 J, v = 84.8 mm3, do = 1 mm)
the effect of the increased heat transfer rate, Figure 5.11 shows the comparison be-
tween the high-frequency performances corresponding to hin = 125 W/m2 (replotted
from Figure 3.16) and hin = 6000 W/m2. All other parameters are identical. The
comparison plot shows that the rollover frequency increases significantly from ap-
proximately 100 Hz to 1000 Hz thereby demonstrating the improved high-frequency
performance with an increased convective heat transfer coefficient.
The increased convective heat transfer coefficient is significantly larger than
the original value of 125 W/m2K as determined through comparisons to CFD sim-
ulations. At the same time, the high-temperature waves that impact the SparkJet
cavity walls can easily be significantly higher than the CFD and 1-D model pre-
dictions. A larger temperature difference between the air and the wall would lead
to larger convective heat transfer, which is essentially the effect of increasing the
convective heat transfer rate. Another possible cause is the inability to capture heat
loss to the electrodes. The exact cause for the difference in the duration of Stage 2
129
is unknown; however, incorrect assumptions related to heat transfer (convective or
conductive to electrodes) is the most plausible explanation.
5.5 Summary
In this chapter, the SparkJet cavity pressure response to an arc discharge has
been experimentally explored. A look at the unfiltered pressure data and the corre-
sponding arc current and voltage show that the initial pressure rise occurs between
3 and 6 µs after the arc discharge begins. The resulting unfiltered pressure signal is
highly unsteady due to excitation of the pressure sensor resonant frequency. There-
fore, the pressure signal is filtered and the maximum value of the filtered pressure
signal is used to estimate efficiency. The filtered pressure signal is also used to un-
derstand the effect of cavity volume, orifice diameter, input energy, and electrode
gap on the duration of Stage 2.
Stage 2 is affected by cavity volume and orifice diameter but is independent
of input energy and electrode gap. The duration of Stage 2 directly affects the
ability to maintain high-momentum throughput during Stage 2 when operated at
high-frequency. A long Stage 2 results in poor high-frequency performance while a
short Stage 2 duration results in good high-frequency performance. The cavity is
able to refill with external air during Stage 3 sooner if Stage 2 is short. Therefore,
for high-frequency applications, a small cavity volume and large orifice diameter are
ideal. The input energy and electrode gap allow the SparkJet designer to control the
peak cavity pressure and efficiency of delivering energy to the cavity. However, as
130
input energy increases, the SparkJet efficiency decreases. Also, as the electrode gap
increases, efficiency increases but, based on user experience, reliability also tends to
decrease.
Quantitative comparisons between the 1-D model and experimental results
show significant differences in the prediction of the duration of Stage 2. The model
results predict a much longer Stage 2 duration than is shown in the experimental
results. The most plausible explanation for these differences is in the mis-prediction
of heat transfer related to non-1-D effects.
The experimental results presented in this chapter showed that pressure-based
efficiency was as low as 20%, whereas the model assumes 100% efficiency. The differ-
ences between experiment and modeling are considered part of an overall SparkJet
inefficiency. Several possible reasons are provided to explain the source of the inef-
ficiency. One of these reasons is the inefficient energy transfer from the capacitors
to the arc. The following chapter uses experimental measurement of the arc current
and voltage to explore this hypothesis.
131
Chapter 6
Power-Based Results and Analysis
The results presented in Chapter 5 demonstrated that the experimentally mea-
sured pressure-based efficiency was as low as 20% whereas the model assumes 100%
efficiency. Initial analysis discussed the role of the power drawn by the arc to explain
one of the contributing factors toward the loss in SparkJet efficiency. This chapter
looks into measuring and quantifying the power and energy drawn by the arc based
on voltage and arc current measurements.
6.1 Arc Power Results
In order to investigate the source of the low pressure-based efficiencies shown
in the previous chapter, the power drawn by the arc has been evaluated. Because arc
power is independent of parameters such as cavity volume and orifice diameter, these
parameters are kept constant for this analysis. Therefore, the arc discharge power
was evaluated for a SparkJet with a cavity volume of 84.8 mm3, orifice diameter of
1 mm, capacitor voltage of approximately 600 V, and five capacitance values ranging
from 0.95 µF to 4.28 µF. Corresponding to each capacitance value, Figure 6.1 shows
five current, voltage and power waveforms for a constant electrode gap of 1.75 mm.
Figure 6.1(a) shows that as capacitance increases, both the current magnitude and
period increase. Correspondingly, Figure 6.1(b) shows that as capacitance increases,
132
the duration of the voltage drop increases such that time derivative of the voltage
drop during the arc discharge decreases in magnitude. Evaluating the product of
the current and voltage, the power drawn by the arc is shown as a function of time
in Figure 6.1(c).
Additionally, the resistance can also be calculated when the voltage and
current across the arc are known using Ohm’s Law. Figure 6.2 shows the calcu-
lated resistance as a function of time for a capacitance of 4.28 µF. Referring to
Figure 6.1(c), the power curve for this capacitance level experiences three surges.
Also note that Figure 6.2 shows three regions of varying resistance that correspond
to the power curve. For each power surge, the resistance exponentially decreases
to a small value. Each subsequent region, however, decreases to a slightly higher
resistance than the previous region. The increasing resistance indicates that the arc
column is decreasing in temperature since arc column resistance is inversely propor-
tional to arc temperature [66]. Referring also to Figure 6.1(b), the spikes present in
Figure 6.2 correspond to when the voltage crosses zero.
Figures 6.1(a) and 6.1(b) also show that inductance in the circuit drives the
voltage and current to continue to oscillate. To determine if these oscillations corre-
spond to the arc discharge or are artifacts of circuit inductance, a photodetector was
placed near the SparkJet. Assuming light production corresponds to the presence of
the arc, an increasing trend in the photodetector output during the entire duration
of the power waveform indicates the corresponding presence of the arc. To aid in
the qualitative comparison, the arc power drawn is correlated to the light output in
Figure 6.3. This plot shows that the light output increases significantly during the
133
48 50 52 54 56 58 60 62 64 66 68
−500
0
500
1000
Time (µs)
Curre
nt (A)
 
 C=0.95 µFC=1.79 µFC=2.75 µFC=3.42 µFC=4.28 µF
Increasing Capacitance
(a) Current Waveforms.
48 50 52 54 56 58 60 62 64 66 68−100
0100
200300
400500
600
Time (µs)
Voltag
e (V)
 
 
C=0.95 µFC=1.79 µFC=2.57 µFC=3.42 µFC=4.28 µF
Increasing Capacitance
(b) Voltage Waveforms.
48 50 52 54 56 58 60 62 64 66 680
20
40
60
80
100
120
Time (µs)
Powe
r (kW)
 
 
C=0.95 µFC=1.79 µFC=2.57 µFC=3.42 µFC=4.28 µF
Increasing Capacitance
(c) Power Waveforms.
Figure 6.1: Voltage, current, and power waveform curves in distinct groups correspond-
ing to each capacitance change for a cavity volume of 84.8 mm3 and orifice
diameter of 1 mm. (f < 1 Hz)
134
50 52 54 56 58 60 62 64 66 68−15
−10
−5
0
5
10
15
Time (µs)
Resis
tance
 (Ω)
Figure 6.2: Time-dependent variation in arc resistance for a capacitance of 4.28 µF and
electrode gap of 1.75 mm. (f < 1 Hz)
48 50 52 54 56 58 60 62 64 66 680
10
20
30
40
50
60
Time (µs)
See 
Lege
nd
 
 
Arc Power (kW)Light Signal (mV)
Figure 6.3: Time-dependent variation in arc power and qualitative light generation by
the arc for a capacitance of 0.95 µF and electrode gap of 1.75 mm. (f < 1 Hz)
first power surge. Beyond the first rise in the light output, the light output does not
rise as significantly as the initial increase but it does increase slightly. Only after
the power drawn reaches zero does the light output start to continuously decrease.
Based on these results, the power drawn after the initial power spike does contribute
to the overall arc power.
With an understanding of the voltage, current, and power measurements
during the arc discharge process, the energy drawn by the arc is calculated by in-
135
tegrating the power curve between the initial voltage drop and when the current
reaches zero according to Equation 4.6. The energy drawn is divided by the stored
capacitor energy, QC , to calculate the capacitor to arc power efficiency and the re-
sults are shown by the triangles in Figure 6.4. These results show that approximately
80% of the energy stored in the charged capacitor bank is converted to power drawn
by the arc. These losses are likely due to the parasitic resistance and inductance
corresponding to the 0.46 m of 22 AWG wires leading from the capacitors to the
arc. Based on wire resistance tables, this length of wire and wire gauge corresponds
to 0.024 Ω. Assuming the arc current also passes through these wires, there is ap-
proximately an 8% power loss due to the wires alone. The remaining power loss is
likely due to contact resistance of the connectors between the wires and electrodes,
incomplete energy transfer from the capacitors, or other sources of parasitic circuit
resistance.
For comparison, the pressure-based efficiency for these tests is shown in Fig-
ure 6.4 by the squares using the same analysis as described in Chapter 5. These
pressure-based efficiencies shows that the conversion of energy stored in the capaci-
tors to the energy required to raise the cavity pressure (assuming calorically perfect
gas) is 30-50% efficient. Therefore, the majority of the efficiency loss stems from
the efficiency loss from the arc to the cavity air. To explain some of the additional
efficiency losses and estimate temperature rise in the cavity, the instantaneous arc
discharge and calorically perfect gas assumptions are removed using Equation 4.8.
The third set of points in Figure 6.4 correspond to the estimated efficiency
based on the non-instantaneous, thermally perfect energy addition where QC = 0.6 J
136
Figure 6.4: Efficiency of the conversion of stored capacitor energy to the arc discharge,
calculated cavity temperature, and measured pressure rise as a function of
QC/E.
(C = 4.28 µF). The effect of assuming a thermally perfect gas according to Equa-
tion 4.8 is shown in Figure 6.5. This figure shows the power drawn by the arc
discharge that results in the energy deposition into the cavity. Assuming a non-
instantaneous energy deposition into a calorically perfect gas where Cv is constant,
the dash-dot curve shows the corresponding temperature variation. For comparison,
the non-instantaneous energy deposition into a thermally perfect gas where Cv is a
function of cavity temperature is shown with the dotted curve. The corresponding
variation in the specific heat constant is also shown for reference with the dashed
curve. This comparison shows that incorporating a thermally perfect assumption
reduces the final cavity temperature, T2 up to 1500 K. The temperature difference
assuming a thermally perfect gas increases as QC increases as shown in Figure 6.5(b).
The efficiency based on these calculations is shown in Figure 6.4 with the circles.
Based on this analysis, the removal of the calorically perfect assumption accounts
137
for approximately 10-25% of the pressure-based efficiency losses.
The remaining pressure-based efficiency losses are most likely due to heat
loss to the electrodes. Roth et al. [56] measured heat loss to electrodes due to arc
discharges in a variety of gases and for a range of electrode spacings and electrode
diameters. While none of Roth’s tests were in air, the thermal diffusivity of Argon,
one of the gases tested, is very similar to that of air. Similar to the SparkJet experi-
ments, Roth also tested heat loss for electrodes of a diameter of 1 mm and electrode
gap of 2 mm in Argon pressurized at 101 kPa. The results showed that approxi-
mately 50% of the heat generated by the arc discharge was lost to the electrodes.
This electrode heat loss is quite significant but, based on the results presented by
Roth, can be significantly reduced by increasing the electrode gap. The following
section explores this method of reducing heat loss.
6.2 Pseudo-Series Trigger Results
To explore the effect of varying the electrode gap on power drawn by the arc,
the 600PST circuit was used with a SparkJet actuator of a cavity volume of 84.8 mm3
and orifice diameter of 1 mm. The effect of the electrode gap was evaluated at 1 mm,
2 mm and 3 mm. In the previous chapter, the pressure based results showed that the
peak pressure increases with increased electrode gap. In this section, the increase in
peak pressure is supported by arc power measurements. Figure 6.6 shows the power
drawn by the arc for QC = 0.61 J. While the differences in the power curves are not
significant, the area under the power curve for the 1 mm electrode gap configuration
138
50 52 54 56 58 60 62 64 0
1000
2000
3000
4000
5000
6000
7000
8000
Time (µs)
Temp
eratur
e (K)
 
 
−20
0
20
40
60
80
100
120
Powe
r (kW) Cv(T)Cv=constant
(a)
0 5 10 15 20 25 300
200
400
600
800
1000
1200
1400
1600
QC/E
Tem
pera
ture 
Diffe
renc
e (K)
(b)
Figure 6.5: Comparison of the time dependent temperature and specific heat coefficient
(Cv) assuming a calorically and thermally perfect gas as a function of arc
discharge current and voltage for (a) a single value of QC/E = 27.8 and (b)
the final temperature difference over a range of QC/E values. (f < 1 Hz
and gap of 1.75 mm)
139
48 50 52 54 56 58 600
50
100
150
200
250
300
Time (µs)
Powe
r (kW)
 
 Gap = 1.0 mmGap = 2.0 mmGap = 3.0 mm
Figure 6.6: Power drawn by the arc for an electrode gap of 1 mm, 2 mm and 3 mm.
(f < 1 Hz, QC = 0.61 J)
is visibly less than the area under the curve for 3 mm. Therefore, the reduced arc
energy is likely the source of the decrease in efficiency as the electrode gap decreases.
6.3 Summary
Through this experimental analysis of the arc power, the source of efficiency
losses during Stage 1 of the SparkJet cycle are better understood. The results pre-
sented in this chapter show that arc power increases as a function of capacitance
and electrode gap. These results also show that removing the 1-D modeling assump-
tions of an instantaneous energy deposition and calorically perfect gas accounts for
some of these efficiency losses. However, to capture these effects, time-dependent
current and voltage information are required. This information can be acquired
experimentally; however, in the interest of building a self-contained 1-D model, the
entire SparkJet circuit could be modeled. The primary challenge associated with
modeling the circuit, however, is modeling the time-dependent variation in arc re-
140
sistance. As shown in Figure 6.2, this resistance is complex and challenging to
simulate. Therefore, it is recommended that the power drawn be experimentally
obtained and applied to the model for varying input energy and electrode gap.
With an understanding of the losses, the SparkJet design can be improved
to minimize losses. For example, the wires leading from the capacitor bank to the
electrode gap should be shortened and of a larger gauge. Also, the circuit should be
designed to promote larger electrode gaps to, not only reduce heat loss to the elec-
trodes but, enlarge the arc channel to disperse heat from the arc to the surrounding
air more quickly. Another advantage of increasing the electrode gap is the reduced
electric field between the electrode tips such that the arc resistance relative to the
parasitic wire resistance is increased; therefore, the relative power drawn by the arc
is higher than the power drawn by the wires.
141
Chapter 7
Conclusions and Future Work
This final chapter summarizes the wide variety of results and conclusions pre-
sented in this dissertation. The original goal of this work was to address the hy-
pothesis that the complexities of the SparkJet actuator can be modeled using 1-D
analysis. The primary benefit of developing such a model is the ability to analyze
several SparkJet designs faster than generating CFD simulations. Several aspects
of this research have demonstrated that the SparkJet can, indeed, be simplified by
a 1-D model. However, the model has difficulty capturing aspects related to the arc
discharge and heat transfer beyond empirically-based coefficients. Defining these
aspects is an area considered for future work.
7.1 Summary of Results
This dissertation has provided a significant contribution to the state of the art
pertaining to 1-D model development and experimental results to understand design
factors that affect SparkJet performance. The first contribution from this disserta-
tion was the development of the 1-D SparkJet model to rapidly generate SparkJet
performance information for a variety of design variations. Using the Navier-Stokes
equations, this model was developed under several assumptions. These assumptions
include inviscid flow, time-dependent 1-D representation of the cavity pressure, tem-
142
perature, density, and jet velocity, a calorically perfect gas, and instantaneous energy
deposition. Additionally, a thermal model, based on a lumped circuit-element rep-
resentation, has also been developed, which incorporates convective and conductive
heat transfer.
Through comparison with a CFD simulation, the assumptions of inviscid flow
and 1-D cavity pressure, temperature, density and jet velocity were well supported.
The CFD simulation was also used to estimate the convective heat transfer coef-
ficient. During the initial 200 µs, the CFD results captured unsteadiness within
the cavity whereas the 1-D model did not. However, beyond the first 200 µs, the
1-D model and CFD results align quite well. Agreement in the later portion of the
SparkJet cycle is important for modeling high-frequency actuation.
Supported by the comparison to CFD results, the model was exercised to
explore SparkJet high-frequency performance and evaluate the effect of increasing
actuation frequency on SparkJet momentum throughput as a function of several
design parameters. The high-frequency evaluations showed that high-frequency per-
formance is primarily affected by cavity volume and orifice diameter. For example,
the model shows that a typical SparkJet design (1 mm orifice diameter, 84.8 mm3
cavity volume and 0.5 J energy input) operated over a range of frequencies from
1 Hz to 10 kHz shows a decrease in peak momentum corresponding to an actuation
cutoff frequency of 800 Hz. By halving the orifice diameter to 0.5 mm, the cutoff fre-
quency was near 300 Hz and by doubling the orifice diameter to 2.0 mm, the cutoff
frequency is near 2.5 kHz. As cavity volume increases, high-frequency performance
decreases; as orifice diameter increases, high-frequency performance increases. The
143
effect of input energy on high-frequency performance was also considered but model-
ing results showed that input energy does not have an effect on the cutoff frequency.
To support aspects of the 1-D model related to Stage 1 (arc discharge), experi-
mental measurements of cavity pressure and arc power were acquired. These results
not only provided insight into Stage 1, but also Stage 2 (jet flow). Regarding Stage 1,
results showed that the experimentally measured peak pressure is significantly (60-
80%) lower than assumed in the 1-D model. This discrepancy signifies inefficiencies
associated with transferring capacitor energy to the cavity air. The efficiency is a
function of both input energy and electrode gap, which decreases slightly as input
energy increases and increases moderately as electrode gap increases.
Regarding Stage 2, the experimentally observed trends pertaining to the du-
ration of Stage 2 and, therefore, high-frequency performance are the same as was
seen in the modeling results. The modeled dependence on orifice diameter, cav-
ity volume, and input energy were confirmed by experimental results. However,
the quantitative duration of Stage 2 is not well modeled such that the model over-
predicts the duration of Stage 2. Consideration of several aspects of the 1-D model
lead to the thought that inaccurate representation of heat transfer stemming from
the convective heat transfer coefficient itself, underestimation of the cavity temper-
ature, or heat loss to the electrodes is the source of the difference in the duration of
Stage 2.
To follow up with Stage 1 inefficiencies observed via cavity pressure measure-
ments, the arc power was also analyzed as a function of input energy and electrode
gap. These results show that 20% of the stored capacitor energy is lost to parasitic
144
resistance in circuit components and wires connecting the capacitor bank to the
SparkJet. To further analyze efficiency losses, the initial Stage 1 modeling assump-
tions of an instantaneous and calorically perfect energy deposition were removed.
Time-dependent calculations of the corresponding cavity temperature rise showed
that an additional 10-25% of the energy losses can be attributed to the extremely
high temperatures reached in the cavity. The remaining energy loss is likely due to
heat lost to the electrodes. Efficiency as a function of electrode gap was also ex-
plored using the pseudo-series triggered circuit which was also designed for improved
reliability. Using a maximum trigger voltage of 12 kV, an electrode gap increase
from 1.0 mm to 3.0 mm increases the efficiency from 9% to 30%. In summary, the
efficiency losses are due primarily to circuit losses, extremely high cavity air tem-
peratures, and heat lost to electrodes.
This dissertation work has presented the development of a 1-D model to sim-
ulate the complexities associated with the SparkJet actuator. Through comparison
to high-fidelity CFD and experimental results, this 1-D model is well-supported.
However, these comparisons have also highlighted some deficiencies in the model
which should be addressed in future work.
7.2 Future Work
The area of characterizing the SparkJet actuator and demonstrating the actu-
ator performance in a high-speed flow environment has been explored only over the
last 10 years. Due to the high-voltage and high-current nature of the actuator, this
145
exploration has been slow and there are several avenues for future work.
Related to this dissertation work, shortcomings in the 1-D model related to
modeling the arc discharge and heat transfer are of immediate interest for future
work. Once these modeling aspects are addressed, the 1-D model can more accu-
rately predict high-frequency performance rather than only modeling trends. Several
other areas of future work have been conceived based on the results presented in
this dissertation. For example, confidence in the high-frequency modeling, above
100 Hz, would be better supported with experimental high-frequency tests or CFD
designed to model sequential SparkJet cycles. Also, the 1-D model comparison
exploited differences between the estimation and measurement of the duration of
Stage 2. However, future work could be performed to link the duration of Stage 2 to
the rollover frequency since Stage 2 measurements are easier to obtain than Stage 3
measurements. Additionally, exploring the possibilities of designing the SparkJet
device to take advantage of hydraulic amplification would also be an interesting area
for future work. Similarly, exploring the effect of the ratio of the orifice and cavity
cross-sectional areas on the jet momentum with attention to orifice edge effects and
jet vortex generation. Finally, the high-temperature gas dynamics discussed may
also indicate the possibility of non-equilibrium gas effects such that a significant
portion of the arc energy is converted to vibrational molecular energy rendering it
unusable by the cavity air. Further exploration into the effects of ionized gas phe-
nomena could be beneficial for Stage 1 modeling and estimating inefficiencies.
In the interest of evaluating the SparkJet interaction with a flowfield, expand-
ing the 1-D model to represent actuator performance in the presence of an external
146
flow would be a valuable tool for design the SparkJet for a particular flow applica-
tion. To support this model expansion, experimental data to measure the SparkJet
cavity pressure and arc power measurements while interacting with an external flow
would also be beneficial. Applying the SparkJet actuator to a realistic installation
would likely involve a large array of SparkJet actuators. Further expansion of the
model to include effects of coupling with nearby SparkJet plumes would help illus-
trate the effect of actuator orifice spacing and phasing regions of the actuator array
to beneficially manipulate an external flowfield.
Finally, the flight conditions of a vehicle poised to benefit from an array of
SparkJet actuators would likely operate at high-altitude where low pressures and
temperatures exist. Therefore, an evaluation of the actuator performance under tac-
tical conditions would be beneficial. This evaluation could be accomplished using
the 1-D model. However, to capture the effects of low-density air on SparkJet relia-
bility and circuit requirements would require experimental evaluation. Also related
to testing under tactical conditions is exploration of the effects of environmental
challenges such as the presence of dirt, ice, or water on actuator performance and
reliability.
147
Appendix
This appendix contains the MATLAB code used to define and run the 1-D
model of the SparkJet cycle. The primary routine is called run sparkjet thermal.m
and uses the ode45 solver to solve the equations derived in Chapter 3, which are
contained in sparkjet thermal.m. The design parameters for the SparkJet actuator
are defined in parameters.m.
%****************************************************************
%This is the main function the one-dimensional analytical model,
%which utilizes one of MATLAB’s built in ODE solvers (ode45)
%based on explicit Runge-Kutta. See MATLAB %help on ode45 for
%more information. The ODE solver finds the time history of
%cavity pressure, density, and temperature; the velocity of the
%air moving through the orifice of a SparkJet; and the wall
%internal wall temperature.
%
%INPUTS:
%This program only requires access to parameters.m as an input
%and sparkjet_thermal.m for the ode45 solver.
%
%OUTPUTS:
%This program outputs the time-dependent cavity pressure,
%temperature, and density and orifice velocity and momentum.
148
%%Written by Sarah Haack Popkin
%Original version: 11 Dec 2010
%*****************************************************************
clear
clc
close all
format compact
global Ao V Cv Patm vo R Tatm RM Re Rair rhoatm Q eff CM Rout
% Load SparkJet design parameters
[Ao, V, Cv, Patm, vo, R, Tatm, RM, Re, Rair, rhoatm, Q, eff,...
CM, Rout]=parameters;
%Actuation frequency (Hz)
f=1000;
%Initialize steady state power vector
E_ss=zeros(length(f),1);
M_ss=zeros(length(f),1);
%Define Initial Conditions
for k=1:length(f)
149
%Define Initial Conditions
rho0=rhoatm; %Initial density
m=rho0*V; %Calculate mass inside cavity
U0=1e-5; %Initial Velocity
T0=Tatm+eff*Q/(m*Cv); %Initial Temperature
P0=rho0*R*T0; %Initial Pressure
Tw=Tatm; %Initial Wall Temperature
freq=f(k);
events=ceil(freq/100); %Number of times Stage 1 occurs
T=[]; %Initialize T vector
Y=[]; %Initialize Y vector
Emax=zeros(events,1); %Initialize peak power vector
Mmax=zeros(events,1); %Initialize peak momentum vector
for i=1:events
%Run ODE solver with (redefined) initial conditions
options = odeset(’RelTol’,1e-6,’AbsTol’,...
[1e-6 1e-6 1e-6 1e-6]);
[t,y]=ode45(@sparkjet_thermal,...
[(i-1)*(1/freq+1e-9) i*1/freq],[rho0 U0 P0 Tw],options);
%Build time and variable vectors
Y(length(Y)+1:length(Y)+length(t),1:4)=y;
T(length(T)+1:length(T)+length(t),1)=t;
150
%Redefine initial conditions for Stage 1 as the final
%conditions in Stage 3
rho0=y(end,1);
T0=y(end,3)/(rho0*R)+eff*Q/(rho0*V*Cv);
P0=rho0*R*T0;
U0=y(end,2);
Tw=y(end,4);
%Find the maximum output power (E/s) during a cycle
Emax(i)=max(y(:,1).*y(:,2)*Ao);
Mmax(i)=max(y(:,1)*vo.*y(:,2));
%Save the max power from the first shot as the reference
%power for a frequency of 1 Hz
if i==1
Emax1=Emax(i);
Mmax1=Mmax(i);
elseif abs(Emax(i)-Emax(i-1))<1e-8
%Save max power once steady state conditions are
%reached as the state power
E_ss(k)=(Emax(i)+Emax(i-1))/2;
M_ss(k)=(Mmax(i)+Mmax(i-1))/2;
break %Stop if steady state conditions have been achieved
else
151
%Save max power once steady state conditions are
%reached as the state power
E_ss(k)=(Emax(i)+Emax(i-1))/2;
M_ss(k)=(Mmax(i)+Mmax(i-1))/2;
end
end
end
P=Y(:,3); %Pressure
rho=Y(:,1); %Density
U=Y(:,2); %Velocity
T=T*1e6; %Temperature
Mom=rho*vo.*U; %Momentum
%Append power value for f = 1 Hz actuation
E_ss=[Emax1; E_ss];
M_ss=[Mmax1; M_ss]/Mmax1;
f=[1;f];
%Plot normalized steady-state momentum as a function of frequency
figure
loglog(f,M_ss,’LineWidth’,2)
xlabel(’Frequency (Hz)’,’FontSize’,24)
152
ylabel(’Momentum (\muN-s)’,’FontSize’,24)
set(gca,’FontSize’,24)
% Plot time history of density, orifice velocity, pressure (kPa)
figure
plot(T,Y(:,1)*1000,’--g’,’LineWidth’,3)
hold on
plot(T,Y(:,2),’--m’,’LineWidth’,3)
plot(T,Y(:,3)/1000,’--b’,’LineWidth’,2)
plot(T,Y(:,3)./(Y(:,1)*R),’--r’,’LineWidth’,3)
xlabel(’Time (\mus)’,’FontSize’,24)
ylabel(’See Legend’,’FontSize’,24)
set(gca,’FontSize’,24)
legend(’1-D Density (g/m^3)’,’1-D Velocity (m/s)’,...
’1-D Pressure (kPa)’,’1-D Temperature (K)’)
function dy = sparkjet_thermal(t,y)
%*****************************************************************
%This function defines system of ODEs as given in Chapter 3.
%This is also where the temperature and heat transfer are defined.
The change in boundary conditions are also determined here.
%
%INPUTS
153
% t - time (sec)
% y - 4x1 vector containing the values of density,
% velocity, pressure, and wall temperature,
% respectively, with respect to time.
%
%OUTPUTS
% dy - 4x1 vector containing the changes in density,
% velocity, pressure, and wall temperature,
% respectively, with respect to time.
%
% Written by Sarah Haack Popkin
% Original date 11 Dec 2010
%******************************************************************
global Ao V Cv Patm vo R Tatm RM Re Rair rhoatm CM Rout
dy=zeros(4,1);
%Algebraic Equations
T = y(3)/(y(1)*R); %Ideal Gas Law
%Simple harmonic oscillator model for thermally perfect gas
Cvt=Cv*(1+(1.4-1)*((5500/1.8/T)^2*exp(5500/1.8/T)/...
(exp(5500/1.8/T)-1)^2));
154
%Define pressure boundary condition at orifice between choked and
%unchoked flow
if y(3)>1.893*Patm
Pe=y(3)/1.893;
elseif y(3)<=1.893*Patm
Pe=Patm;
end
% Convective heat transfer equation for air only
qdot=(1/Rair*(y(4)-T))/(y(1)*V); %J/(s*kg)
%Differential Equations
if y(2)>0 %Stage 2 Discharge
%Mass
dy(1) = -y(1)*y(2)*Ao/V;
%Momentum
dy(2) = 1/y(1)*(((y(3)-Pe)*Ao-y(1)*abs(y(2))*y(2)*Ao)/vo-...
y(2)*dy(1));
%Energy
dy(3) = (qdot*y(1)*V-((y(3)-Pe)+y(1)*(Cvt*T+(y(2)^2)/2))*...
y(2)*Ao-vo* (2*y(1)*y(2)*dy(2) + y(2)^2*dy(1)) ) /...
(Cvt*V/R);
155
elseif y(2)<0 %Stage 3 Refresh
%Mass
dy(1) = -rhoatm*y(2)*Ao/V;
%Momentum
dy(2) = 1/rhoatm*(((y(3)-Patm)*Ao-rhoatm*abs(y(2))*y(2)*Ao)...
/vo-y(2)*dy(1));
%Energy
dy(3) = (qdot*y(1)*V-((y(3)-Patm)+rhoatm*(Cv*Tatm+(y(2)^2)/2))*...
y(2)*Ao-vo*(2*rhoatm*y(2)*dy(2)+y(2)^2*dy(1)) )/(Cvt*V/R);
end
%Wall Temperature
dy(4) = 1/CM*((y(4)-Tatm)/(Rout+1/(1/RM+1/Re))-(T-y(4))/Rair);
function [Ao, V, Cv, Patm, vo, R, Tatm, RM, Re, Rair, rhoatm, Q,...
eff, CM, Rout]=parameters
%******************************************************************
%This function defines the design parameters for the SparkJet.
%
%INPUTS (NA)
%
%OUTPUTS
% Ao - Orifice Area (m^2)
% V - Cavity volume (m^3)
156
% Cv - Specific heat of air
% Patm - Atmospheric Pressure (Pa)
% vo - Orifice Volume (m^3)
% R - Specific gas constant for air (J/kg*K)
% Tatm - Atmospheric Temperature (K)
% RM - Thermal Resistance of Macor
% Re - Thermal Resistance of electrodes
% Rair - Thermal Resistance of air inside SparkJet
% cavity
% rhoatm - Atmospheric Density (kg/m^3)
% Q - Capacitor Energy (J)
% eff - Heat transfer efficiency
% CM - Macor Capacitance
% Rout - Thermal Resistance of air outside SparkJet
% cavity
%
% Written by Sarah Haack Popkin
% Original date 11 Dec 2010
%*****************************************************************
R=287.05; %Specific gas constant for air in J/kg*K
Cp=1004; %Specific heat of air for constant pressure
Cv=Cp-R; %Specific heat of air for constant volume
157
V=8.484e-8; %Cavity volume
Tatm=288; %External flow temperature
Patm=101325; %External flow pressure
rhoatm=Patm/(Tatm*R); %External flow density using ideal gas law
Vc=600; %Capacitor voltage
C=0.9e-6; %Capacitance
eff=1; %Efficiency
%Stage 1 Parameters for Energy Deposition
Q=1/2*C*Vc^2; %Energy discharged by spark
%Material properties and geometry
%Electrodes
Dw=0.001; %Electrode diameter
rw=Dw/2; %Electrode radius
Swc=pi*rw^2; %Electrode cross-sectional area
%Orifice
no=1; %Number of Orifices
ro=0.001/2; %Orifice radius (m)
Ao=no*pi*ro^2; %Orifice area
ho=0.0005; %Orifice height
158
vo=Ao*ho; %Throat volume
%Cavity
hc=(4*V/pi)^(1/3); %Cavity height
Acc=pi*(hc/2)^2; %Cavity cross-sectional area
dc=hc; %Cavity Diameter
Am=2*Acc+pi*dc*hc-Ao; %Cavity internal area
%Outer SparkJet Dimensions
Hout=0.00855; %Outer height (m)
Dout=0.01275; %Outer diameter (m)
VM=pi*(Dout/2)^2*Hout-V; %Volume of Macor (m^3)
Aout=2*pi*(Dout/2)^2+2*pi*Dout*Hout;%Outer surface area of Macor (m^3)
%Resistance
%Cavity Air
hin=125; %Convective heat transfer coefficient(W/m^2*K)
Rair=1/(hin*Am); %Thermal resistance of air
hout=25; %Convective heat transfer coefficient(W/m^2*K)
Rout=1/(hout*Aout); %Convective thermal resistance on outside of
%SparkJet walls
%Macor
159
k_M=1.46; %Thermal conductivity of Macor
Rw=1/(2*pi*hc*k_M)*log(Dout/dc); %Thermal resistance of Macor Wall
Rb=(Hout-hc)/2/(Acc*k_M); %Thermal resistance of Macor bottom
Rt=(Hout-hc)/2/((Acc-Ao)*k_M); %Thermal resistance of Macor top
RM=1/(1/Rw+1/Rb+1/Rt); %Thermal resistance of Macor
rhoM=2520; %Density of Macor (kg/m^3)
C_m=794.95; %Specific heat of Macor (J/kg*K)
CM=VM*rhoM*C_m; %Capacitance of Macor
%Electrodes
k_e=173; %Thermal conductivity of Tungsten
Re=(Hout-hc)/(k_e*Swc); %Thermal resistance of electrodes
160
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