# Nanosecond Pulsed Discharge Dynamics During Passage of a Transient Laminar
Flame

Colin A. Pavan and Carmen Guerra-Garcia  Massachusetts Institute of
Technology, 77 Massachusetts Ave, Cambridge, MA, USA<EMAIL_ADDRESS>
###### Abstract

This work presents an experimental study of a nanosecond repetitively pulsed
dielectric barrier discharge interacting with a transient laminar flame
propagating in a channel of height near the quenching distance of the flame.
The discharge and the flame are of comparable size, and the discharge is
favoured at a location where it is coupled with the reaction zone and burnt
gas. The primary goal is to determine how the discharge evolves on the time
scale of the flame passage, with the evolution driven by the changing gas
state produced by the moving flame front. This work complements the large body
of work investigating the effect of plasma to modify flame dynamics, by
considering the other side of the interaction (how the discharge is modified
by the flame). The hot gas produced by the combustion had a strong effect on
the discharge, with the discharge preferentially forming in the region of hot
combustion products. The per-pulse energy deposited by the discharge was
measured and found to increase with the size of the discharge region and
applied voltage. The pulse repetition frequency did not have a direct impact
on the per-pulse energy, but did have an effect on the morphology and size of
the discharge region. Two distinct discharge regimes were observed: uniform
and filamentary (microdischarges). Higher pulse repetition frequencies and
faster-cooling combustion products were more likely to transition to the
filamentary regime, while lower frequencies and slower-cooling combustion
products maintained a uniform regime for the entirety of the time the
discharge was active. This regime transition was influenced by the ratio of
the time scale of fluid motion to the pulse repetition rate (with no
noticeable impact caused by the reduced electric field), with the filamentary
regime preferentially observed in situations where this ratio was small. This
work demonstrates the importance of considering how the discharge properties
will change due to combustion processes in applications utilizing plasma
assistance for transient combustion systems.

Keywords: Plasma Assisted Combustion, Dielectric Barrier Discharge, Transient
Flame, Nanosecond Repetitively Pulsed Plasma, Discharge Regime Transitions

Submitted to Plasma Sources Sci. Technol.

##  1 Introduction and Motivation

Recent years have seen considerable interest in plasma assisted combustion
(PAC) technologies. The plasma interacts with the combustion processes in
multiple ways, and has shown particular promise in enabling or stabilizing
combustion in difficult regimes near flammability limits. These include lean
flammability limit extension and stabilization of lean flames [1, 2, 3, 4, 5]
and low residence time high-speed or supersonic combustion [6, 7, 8, 9]. In
the related field of plasma assisted ignition (PAI), the application of plasma
has been shown to reduce ignition delay times [10, 11, 12, 13] and enhance
low-temperature chemistry [14, 15]. The plasma acts in many different ways
depending on the way it is generated, the neutral gas composition and state,
and the magnitude of the reduced electric field, E/N. Lower reduced electric
fields, below around $10^{2}$Td in air plasmas [16], will put energy into
rotational and vibrational excitation modes, while higher fields, on the order
of $10^{2}$-$10^{3}$Td, can cause electronic excitation, electron-impact
dissociation and ionization [17, 18]. Experimentally determining the reduced
electric field can be challenging, but recent advances using the electric
field induced second harmonic generation (E-FISH) technique have allowed for
direct measurement of this quantity [19, 20].

These different mechanisms of interaction, and the modification of the
combustion process as influenced by the plasma, are also strongly dependent on
the positioning of the plasma with respect to the flame front. Several works
have focused on the influence of plasma generated upstream of the flame front,
in the reactant stream. Stange et al. [21] observed that an AC dielectric
barrier discharge (DBD) in the microdischarge mode applied to the fuel flow in
a coaxial air-propane nonpremixed flame configuration would cause a flame
speed increase, but the authors did not provide details into the potential
mechanisms enabling the interaction. More recent work by Elkholy et al. [22]
and Evans et al. [23] apply nanosecond pulsed plasma upstream of lean,
premixed methane air flames, with the former using a DBD configuration and the
latter a pin-to-pin configuration, and also observe flame speed increase.
Typical per-pulse energies used were of order $10^{-4}$J with pulse repetition
frequencies up to 10kHz for actuation on flames with power of order $10^{2}$W.
Other works have investigated plasma generated in-situ with the flame (e.g.
overlapping with the flame reaction front). This environment is considerably
more complex; not only because the plasma chemistry will more directly
interact with the combustion chemistry, but also because the presence of the
flame front will have an effect on the discharge properties through the
reduction in gas density and corresponding increase in E/N, and due to
changing ionization properties linked to combustion chemistry [24, 25, 20].
From the plasma perspective, there has been some work investigating these
environments; for example [26, 27] looked at how discharges would
preferentially form in regions of spatially inhomogeneous gas that favoured
breakdown. Other works [28, 29, 30] have looked at how discharge energy is
deposited primarily in regions heated by the gas combustion.

This work will focus on plasma generated using the nanosecond repetitively
pulsed dielectric barrier discharge (NRP DBD) method. This strategy is
attractive for plasma assisted combustion applications because it can generate
high reduced fields (several hundred Townsend) at a low average power [31, 24,
32, 33]. NRP DBD is also capable of producing a diffuse, uniform discharge at
atmospheric pressures, which is advantageous in many applications. However,
there are many factors, particularly in a combustion environment, that can
cause a transition to a filamentary microdischarge structure. One of these
factors is the strength of the electric field; Liu et al. give a threshold of
130kV/cm for room temperature air at atmospheric conditions [34, 35],
equivalent to 530 Td, for single nanosecond DBD pulses. The composition of the
mixture will also play a role in the uniformity, with the presence of
hydrocarbons and NO having been observed to more readily cause a uniform to
filamentary transition [24, 20]. A proposed mechanism for this transition is a
coupled plasma thermal-chemical instability [25, 20, 36]. Other contributing
effects include the gas residence time in the discharge region, with longer
residence times allowing the microdischarge instabilities to more readily
develop [37, 38]. This may be related to residual space and surface charge
from previous discharges creating a memory effect in the discharge region [39,
40, 41].

Changes in discharge structure driven by gas condition are particularly
relevant in situations involving transient combustion phenomena and moving
flame fronts where in-situ generated plasma for PAC may be of interest.
Examples include internal combustion engines, rotating detonation engines and
static or dynamic flame stability studies [42, 43, 44, 12, 45]. While there is
considerable work looking at how plasma affects combustion processes, the
other side of the interaction, namely how the discharge is affected by the
transient combustion environment, has been studied considerably less. There is
a significant reduction in the neutral gas number density as a flame front
passes, and this creates a corresponding increase in the reduced electric
field. This change in reduced field, on the order of 4-7 times (for a gas
starting at room temperature heated to 1200-2100K), will dramatically affect
the discharge behaviour. More subtly, the flame passage will also modify the
gas composition and flow field, which will also influence the discharge
behaviour. In cases where transient flame fronts are expected or situations
looking at flame stabilization where the flame location may change with time,
it is important to consider how the change in flame position and operating
conditions will modify the discharge behaviour to effectively design a control
scheme for plasma assistance that accounts for the evolving environment. This
work looks at the evolution of a coupled flame-discharge system as a flame
front propagates in between two dielectric-covered electrodes in a narrow
channel to investigate how the discharge is affected by the transient
environment created by the passing flame. Specific attention is given to how
the discharge evolves on the time scales of the flame propagation.

##  2 Setup and Methodology

###  2.1 Experimental Setup

The experimental setup is shown in figure 1. The main test section is a
tapered quartz channel with a width of 3cm and a height varying from 4.2mm to
1mm over 20cm. Premixed methane and air enters from the small end of the taper
and flows towards the large end. At the exit of the test section, a
conventional spark plug is used to ignite the mixture. The flame then flashes
back through the channel and is thermally quenched by the narrowing walls of
the chamber. This geometry was designed for experiments on quenching distance
modification using plasma assistance where the taper would allow the discharge
to be applied at heights varying continually from above, at or below the
quenching height of the flame, similar to the configuration of [46]. The
aspect ratio greater than ten is sufficient to minimize the effect of the
channel width on quenching height [47]. The height change is very gradual and
over short distances the opposite faces can be considered approximately
parallel (the height changes by less than 0.2mm over 1cm). Having the flame
propagate counter to the flow of gas slowed its propagation in the lab-frame
which was advantageous for the time-resolved measurements. The construction
out of quartz allows optical access and also acts as a dielectric barrier for
a DBD. The gas mixture flowing into the test section is controlled by two
Brooks GF40 mass flow controllers. High voltage is applied using a nanosecond
pulse generator (Transient Plasma Systems SSPG-20X-HP1). The high voltage
electrode is made of copper tape attached to the upper 2mm thick quartz plate.
The ground electrode is an aluminum plate that also acts as a support for the
test section. The size and position of the top electrode could be varied;
results in this work are reported for two cases. The _long electrode_ is 30mm
in length stretching over a region of the test section where the height
changes from 3.4 to 2.9mm, and the _short electrode_ is 5mm in length,
centered over the point where the channel height is 3.1mm. Note that thermal
quenching of the flame for a stoichiometric mixture happens at around 2.3 mm
of height for this experiment configuration. The free parameters in any set of
tests are the pulse repetition frequency (f), voltage (V), number of pulses
($N_{p}$), equivalence ratio ($\mathrm{\phi}$) and total flow rate (Q).

The system has three independent diagnostics systems, which are described in
detail in the following section. A high speed camera (Edgertronic SC2+) is
used for visual inspection of the flame and discharge. Electrical measurements
are taken using a high voltage probe (Lecroy PPE 20kV, 100MHz BW111BW is the
-3dB high frequency bandwidth reported in the equipment specifications.) and a
Rogowski-coil style current monitor (Pearson model 2877, 200MHz BW) placed on
the current return side of the circuit. Signals are collected at 100ps
resolution using a Teledyne Lecroy Waverunner 9254 oscilloscope (500MHz BW).
Dominant frequency components in current and waveform signals were
$\leq$50MHz. The discharge is also observed with an optical emission
spectroscopy system. The light collection is done using two planar-convex
lenses which focus the discharge emission from a 0.8mm long region of the
channel onto a fiber optic bundle that is attached to the spectrometer (Acton
SpecraPro 2750). The spectra are recorded using a Princeton Instruments PIMAX4
intensified CCD camera.

The experiment is operated by continuously flowing gas at the desired flow
rate and equivalence ratio. Since the flame propagates opposite the direction
of gas flow, the lab-frame flame speed is slower than the laminar flame speed.
The gentle taper of the channel and low flow rates ($\leq$1000sccm222
_standard conditions_ are referenced to 21.1oC and 1013.25mbar per the flow
controller manufacturer.) ensure that the unburnt gas flow remains laminar and
slower than the laminar flame speed at all points along the taper so that the
flame will continue moving upstream. Signal synchronization is done using a
delay generator (Berkeley Nucleonics Corp. Model 577-4C). A test begins by
simultaneously triggering the spark plug to ignite the mixture and starting
the high speed video. A train of pulses (typically 1000-4000 long) is
initiated after some delay, with the ICCD gate synchronized to capture the
pulse. All pulse timing can then be referenced to the start time of the high
speed video.

Figure 1: Schematic of the experimental setup with image of test section shown
in inset.

###  2.2 Diagnostics

####  2.2.1 Visual Flame Tracking

Flame propagation and discharge structure are observed using high speed
videography. The frame rates used in this experiment can resolve events that
occur on the timescale of the flame propagation (milliseconds) and on the
timescale of pulse repetition (hundreds of microseconds). Fiducial markers
were placed on the test section and the camera was focused to give a spatial
resolution of approximately 0.2mm$\times$0.2mm per pixel. With a lab-frame
flame speed of $\leq 40$cm/s, a camera frame rate of 2000 frames per second
(2kfps) ensures that the flame will travel at most one pixel per frame, and
resolution in flame front location is limited by the camera pixel density and
not the frame rate. To visually resolve individual pulses, a higher frame rate
is needed. The pulse repetition frequency used in this work was between
2-10kHz. . By operating the camera at a frame rate twice the pulse repetition
rate (referred to in this work as the high frame rate, or HFR, mode),
individual pulses could be observed with alternating frames with and without a
discharge. The camera exposure is approximately the inverse of the frame rate
(orders of magnitude longer than the pulse duration), so the images with a
discharge are the integrated emission of the discharge over the entire event.
Operating in this mode was useful for separating the flame from the discharge
in the monochrome images, and for discerning how sequential pulses differ (for
example, in the location of filamentary microchannels). Figure 2 shows the
analyses performed using the high speed video. The intensity gain of the
camera was adjusted based on the frame rate and additional brightening was
performed in post-processing to ensure the relevant structures were clearly
visible; all videos taken at 2kfps used an ISO gain of 25000 with an addition
1.5x artificial brightening in post processing. This includes all flame images
shown in this work, with the exception of figure 2.

(a) Flame Tracking Algorithm. (i) Raw video, (ii) Flame identified in channel,
flame front marked in red, (iii) Flame front mapped to channel dimensions.

(b) Two sequential frames from a video with a frame rate twice the pulse
repetition rate showing alternate no discharge (top) and discharge (bottom)
frames.

Figure 2: Analyses performed using the high speed video.

####  2.2.2 Energy Measurements

The energy was calculated by the product of the voltage and current integrated
over the initial pulse and all subsequent reflections according to equation 1.

$W_{\mathrm{p}}=\int_{t_{0}}^{t}V(t^{\prime})I(t^{\prime}-t_{\mathrm{lag}})\mathrm{d}t^{\prime}$
(1)

$t_{lag}$ accounts for the signal travel time differences, which must be
considered to get accurate energy measurements on the nanosecond timescale,
and $t_{0}$ is chosen just before the initial voltage rise. The reflections
are caused by a mismatched impedance; the pulser is rated for a 50$\Omega$
load but the variable impedance of the discharge gap (due to the changing gas
condition and presence or lack of residual electrons from the previous pulse)
mean this condition is rarely met. These reflections are common, with works
such as [48] displaying a similar behaviour. Technical details on how the
energy measurements were performed, including the method for determining
$t_{lag}$, are given in A.

####  2.2.3 Optical Emission Spectroscopy

Optical emission spectroscopy (OES) is performed to probe the temperature of
the coupled flame and discharge. Temperature measurements are used for two
main purposes in this work. The first is to determine if any gas heating is
caused by the discharge and the second is to estimate the reduced electric
field within the discharge gap, which is a key parameter governing the
discharge.

Due to fast relaxation times (of order 1ns [49]), and comparatively long
shutter gate width ($1\mu s$), rotational-translation (RT) equilibrium is
assumed. The RT temperature is determined from the N2 C-B (0,2) transition
around 380nm [50]. An 1800 groove/mm grating blazed at 500nm and centered on
376.5nm was used for these measurements.

The ICCD camera shutter is opened for 1$\mathrm{\mu}$s for each pulse and
several sequential spectra are acquired on the sensor before readout. Light
was collected from a window 0.8mm wide on the test section centered about the
point where the channel height was 3.1mm, and the number of accumulations was
set to integrate the light from pulses occurring within a 6ms time period. At
this location, the flame was moving at approximately 20cm/s in the lab frame,
which meant it would travel 1.2mm during the time it took to accumulate the
light for one spectra. Therefore, measured relative to the position of the
flame front, the collected spectra are the average for a region 2mm wide in
the flame propagation direction (window size + distance flame moves). Spectra
were acquired every 12.5ms, which was the fastest rate possible with the
equipment used (6ms acquisition, 6.5ms readout).The finite dimension of the
channel and Taylor-Saffman instabilities [51, 52] meant that the flame was
also not a perfect planar front, so there is some additional non-uniformity
caused by instabilities in the camera line of sight direction. In addition,
the flame has a crescent shape, as seen in figure 2(a) (ii), which is typical
of flames in laminar flow in small dimension channels [53, 54]. The
temperature rise caused by a single discharge pulse in a NRP DBD should be on
the order of a few Kelvin at most [55, 20], so the $1\mu s$ gate window and
integration over several pulses introduces minimal error to RT temperature
measurement.

Spectral fitting is done using SPECAIR [50]. Each test generates 20 spectra
and dozens of tests are performed; this creates thousands of spectra. To
automate the OES RT temperature measurements, synthetic spectra were generated
at 50K increments, which is approximately the resolution in RT measurement
that can be obtained by this method [50]. The synthetic spectra were generated
for wavelengths in the range 376.5-380.7nm, a region insensitive to
vibrational temperature. Each of these synthetic spectra were then compared
with the experimental spectra and the best fit was chosen based on finding the
temperature that maximized the cross correlation between the (normalized)
signals. An example of the procedure for determining RT temperature by
spectral fitting is shown in figure 3. Assuming that the fitting procedure
determines the best fit to within 50K and an additional 50K uncertainty
inherent in determining the R-T temperature from fitting the C-B (0,2)
transition using SPECAIR [50], the R-T temperature can be taken as accurate to
within $\pm 100$K..

Figure 3: Example of spectral fitting procedure. Experimental spectrum is
compared to each of the synthetic spectra (gray lines) generated at 50K
increments using SPECAIR.

##  3 Dynamic Evolution of a NRP DBD During Flame Passage

The experiments in this section were obtained using the _long electrode_.

###  3.1 Evolution of the Discharge Appearance

Using the long electrode, it is possible to observe the discharge evolution in
both space and time as the flame passes underneath it. This is shown in figure
4 for an example case. When the high voltage pulses begin, there is no visible
discharge in the inter-electrode gap and the only luminosity comes from the
flame (frame at 1ms). By 25ms, the flame has entered the discharge region and
the shape begins to change. The initially curved leading edge flattens out.
Using the HFR visualization, it was observed that the flame itself maintains
its curved structure and the apparent flattening of the front is caused by the
superimposed discharge. As time progresses, the size of the luminous region
increases. This is caused by a uniform discharge in the burnt gas behind the
flame. The uniform discharge extends from the flame front itself to the edge
of the electrode, see the 50ms frame. By 75ms, the trailing end of the
discharge has started to separate from the edge of the electrode. There are
two reasons for this. First, the taper of the channel means that the gap at
the left end of the electrode is slightly larger than at the right end, so the
electric field will be slightly lower. Second, the further behind the flame
front, the more the gas will have cooled against the walls of the chamber.
This cooling causes the density to increase and the reduced electric field to
decrease. These combined effects cause the discharge to form preferentially
closer to the flame front. This trend continues for the next 100ms, with the
discharge appearing stronger closer to the flame front. Note that the region
of increased luminosity at the very right end of the discharge region is the
flame itself (this was confirmed by the HFR videos). Up until this point, the
entire discharge remains uniform.

Between 175ms and 219ms the flame reaches its furthest most position and is
quenched before it exits the inter-electrode region. This quenching occurs
when the channel height is greater than the typical quenching distance of a
flame at the given equivalence ratio, and is caused by the discharge itself.
Further discussion of this premature quenching caused by the discharge in this
setup can be found in [56]. As soon as quenching occurs, the heat release
quickly drops and the gas rapidly cools. A regime transition is observed,
transitioning from the uniform discharge that was present throughout the burnt
gas to filamentary structures. At 219ms, filaments are beginning to form but
there is still a relatively uniform regime near where the flame front was. By
244ms, the uniform regime has almost completely disappeared and the discharge
gap is dominated by the filamentary regime. At this point, the discharge is no
longer preferentially forming at the right end of the electrode, where the
flame front had previously been and where the height is lowest. Instead, it
occurs closer to the center of the electrode. The cause of this regime
transition is discussed further in section LABEL:sec:regime_transition.

Figure 4: Evolution of the discharge appearance during flame passage. Video
was taken using long electrode at a frame rate of 2kfps. Flame moves left to
right and channel height decreases from left to right. Time 0ms corresponds to
the application of the first voltage pulse and pulsed voltage is applied
during all frames. Conditions: f=8kHz, V=13.8kV, $\mathrm{\phi}$=1,
Q=1000sccm, $N_{p}$=2000. Legend in colorbar is in arbitrary units.

###  3.2 Evolution of the Per-Pulse Energy

The energy deposited by the discharge as a function of pulse number is shown
in figure 5, for an example case. Each coloured line corresponds to an
individual test in a sequence of 20 independent tests. The flame enters the
inter-electrode region around pulse 400. There is an initial region of rapid
energy increase caused by the discharge growing in size. Around pulse 700
there is an inflection point. This corresponds to the discharge continuing to
grow as the flame front penetrates further into the inter-electrode region,
but also beginning to shrink at the wider end of the taper as the cooling gas
and larger gap makes it more difficult to sustain the discharge (see
discussion in preceding section). The net effect is still an overall increase
in the size of the discharge, but at a slower rate than previously.
Eventually, a peak is reached. In the case shown in figure 5, this corresponds
to the flame quenching. After this, as there is no longer a continuous heating
source and the gas cools, there is a rapid decay in the energy per-pulse and
the discharge transitions first to a filamentary structure and ultimately
disappears because the applied voltage is insufficient to maintain a discharge
in cold gas. When a lower-power discharge was applied, the flame would pass
through the electrode, although would be weakened by the discharge. In these
cases, there would be a decrease in per-pulse energy as the flame weakened
followed by an increase in per-pulse energy once the flame had passed through
the electrode. A more thorough discussion of this phenomenon is given in B.

Figure 5: Evolution of the per-pulse energy using the long electrode. f=10kHz,
V=12.9kV, $\mathrm{\phi}$=1, Q=1000sccm, Np = 4000.

##  4 Local Measurements to Reveal Plasma-Flame Interaction

The experiments in this section were obtained using the _short electrode_.

###  4.1 Advantages of Short Electrode

The preceding experiments looked at the evolution of the discharge throughout
the burnt gas region and with the discharge remaining coupled to the flame
front. Those experiments show how the coupled flame-discharge system evolve
simultaneously in both space and time. The length scale of spatial variation
is much smaller than the length of the electrode, so it is possible to observe
how the discharge behaves differently in front of, on top of, and at various
distances behind the flame at the same time. This configuration was well-
suited to performing visual analysis of the discharge evolution using the high
speed video. However, the spatial inhomogeneity makes it difficult to perform
more quantitative diagnostics like discharge energy measurements. In
particular, when there are both filamentary and uniform regimes present or
when the uniform region has a noticeable gradient in luminosity, it is unknown
how the energy deposition varies along the electrode and with the changes in
discharge structure. To better understand the spatial variations in the
system, the short electrode was used. With the 5mm electrode, the discharge
evolution is primarily governed by the changing gas condition underneath the
electrode as the hot gas behind the flame cools. Except for the time when the
flame front passes through this region, the short electrode length ensures
quite uniform background gas, which meant that when discharge energy and
temperature measurements were taken they could be directly associated with a
specific visual appearance and location. The other significant effect here is
that the flame would pass under the electrode and continue down the tapered
channel in all cases, and the high voltage would shut off before the flame had
quenched. This was opposite what happened with the long electrode; in that
case the application of the high voltage often caused the flame to quench
prematurely, before it exited the electrode. This results in a different gas
cooling behaviour. In the case of the long electrode, the hot gas cooled
against the walls of the chamber and at the same time the thermal source (the
flame reaction front) disappeared, causing rapid cooling. In the case of the
short electrode, only the former effect is present so the gas cooling occurs
more slowly.

###  4.2 Visual Evolution of Discharge

An example set of images taken during a test with the short electrode is shown
in figure 6.

Figure 6: Evolution of the discharge when using the short electrode, video
taken at 2kfps. Flame moves from left to right and channel height decreases
from left to right. Time 0ms corresponds to the application of the first
voltage pulse and pulsed voltage is applied during all frames. Conditions:
f=10kHz,V=14.3kV, $\mathrm{\phi}$=1, Q=1000sccm, $N_{p}$=3300. Legend in
colorbar is in arbitrary units.

The evolution with the short electrode shows some similarities with the long
electrode. The initial behaviour is the same, with a period of increasing
discharge size (70ms, 119ms frames). Unlike with the long electrode, there is
a point (160ms in figure 6) where the discharge and flame become separated.
The discharge persists in the burnt gas directly underneath the electrode, but
there is a region of hot gas that is not acted on by the high voltage between
the discharge and the flame that emits no (visible) light. The long term
behaviour is also different. With the long electrode, there was a transition
to a filamentary regime immediately after the flame was quenched. With the
short electrode, the discharge remains mostly uniform as it decreases in size
and amount of energy deposited. In certain conditions, typically with higher
frequency (8-10kHz) such as shown in figure 6, there was an observed
transition to a filamentary structure (see frame at 301ms). In other cases,
typically with lower frequency, the discharge would extinguish directly from a
contracted uniform region without first passing through the filamentary
regime.

###  4.3 Normalization by Flame Position

The flame speed in the lab frame is controlled by two parameters: the
equivalence ratio and the flow rate of gas. Changing the equivalence ratio
effects the quenching behaviour and burnt gas temperature; for the results
presented in this work it was fixed at $\phi$=1.0. Changing $\phi$ affected
the location of quenching, but not the overall behaviour; details can be found
in the authors’ previous work [56]. The flow rate was observed to have a minor
effect on the quenching distance; this is because the speed of the flame in
the lab frame will influence how it couples to the walls of the chamber and
the amount of stretch experienced in the boundary layers [54]. Since this
effect was minor, and the discharge is being applied before the channel has
narrowed to the quenching distance of the flame, the general procedure used
was to choose an equivalence ratio and then vary the flow rate to achieve the
desired flame speed. Since the electrodes are fixed in the lab frame, changing
the lab frame flame speed would modify the rate of discharge evolution. This
is shown in figure 7; both cases shown in the top two graphs were run with the
same equivalence ratio, frequency and voltage but different flow rates. When
the cases are plotted on the same graph, with the pulse number axis replaced
by the flame position, the two cases perfectly overlap. This indicates that
the evolution of the discharge is controlled entirely by the position of the
electrode relative to flame front. This way of looking at the data also helps
correct for small variations between tests that caused the flame to reach the
electrode at slightly different times. The zero-reference for the flame front
position axis is the edge of the electrode first encountered by the flame (the
left side of the electrode in figure 6).

Plotting against flame position also compensates for the changing flame speed
(in lab frame) as it propagates through the channel. The flame slows as it
passes through the tapered channel, both due to higher heat losses and because
of a higher velocity of the unburnt gas. This causes the energy curves plotted
on the pulse number axis in figure 7 to appear somewhat asymmetric, with the
per-pulse energy rise rate faster than the decay rate. When plotted against
the flame position, this asymmetry becomes much less pronounced because the
effect of changing lab-frame flame speed has been removed.

Figure 7: Normalization of the energy measurements by the flame position.
f=8kHz, V=13.5kV, $\phi=1$ and Np=2000 in both cases.

###  4.4 Energy Dependence on Frequency and Voltage

The energy measurements were repeated for the short electrode. The equivalence
ratio and flow rate were fixed and the frequency and voltage were varied. The
results are shown in figures 8(a) and 8(b). The solid line represents the
average of $\geq$20 tests and the dashed lines of the same colour are located
at $\pm 2\sigma/\sqrt{N}$ where $\sigma$ is the standard deviation and $N$ is
the number of tests. This is roughly a 95% confidence bound on the mean
energy. From figure 7, the pulse to pulse energy varies but the mean energy
across many tests is very consistent which explains the narrow bounds in
figure 8.

From figure 8(a), the frequency has almost no effect on the amount of energy
deposited per-pulse. This implies that the gas condition at the start of each
pulse is approximately the same regardless of pulse repetition frequency;
quantities like the electron density reach a steady state in between pulses
after relatively few pulses. When the flame is well past the electrode, there
starts to be some deviation in the curves with the higher frequency depositing
slightly more energy per-pulse than the lower frequency. The high speed videos
indicated that this difference was caused by a slightly larger discharge
region at the higher frequency. The effect of discharge region size is
investigated more in the next section. The slight dip in energy when the flame
is approximately 10mm past the start of the electrode for the 10kHz case
corresponds to the flame being disrupted before continuing on, similar to the
re-ignition phenomenon seen with the long electrode (see B). A larger
frequency having a greater disruptive effect was previously observed for this
setup [56]. The nanosecond pulser used in this work was limited to a maximum
PRF of 10kHz, so higher frequencies could not be investigated, but the
possible impacts are discussed further in section 5.3.

The voltage on the other hand has a very significant effect on the energy
deposited per-pulse, with the peak energy deposition doubling between 11.8kV
and 14.3kV and almost doubling again when the voltage was increased to 17.0kV.
The discharge also persisted for much longer after the flame had passed the
electrode when the higher voltage was applied. At 11.8kV, even though pulses
were still being applied, the discharge was no longer present after the flame
front had moved 20mm beyond the end of the electrode, while at higher voltages
a discharge was still visible and depositing energy well after the flame had
passed and until the high voltage pulses were stopped. The peak per-pulse
energy increases with the square of the applied voltage, as would be expected
for energy stored in a capacitor.

(a) Frequency variation at fixed voltage V=14.3kV.

(b) Voltage variation at fixed frequency f=6kHz.

Figure 8: Effect of varying the voltage and the frequency on the energy using
the short electrode.

###  4.5 Energy Dependence on Discharge Region Length

The high-voltage electrode had a length in the flame propagation direction of
5mm, but the ground electrode extended over the full length of the test
section. As a result, the discharge would often extend beyond the edges of the
electrode due to electrode edge effects. This can be seen at 70ms and 119ms in
figure 6. After the flame passed, the discharge region would become localized
underneath the electrode and then start to shrink in size as the gas cooled
further. To account for the effect of discharge region size, figure 8 can be
remade with the energy divided by the length of the discharge region. This
will indicate whether the differences in energy deposition are due to changes
in energy deposition density or changes in the size of the discharge region.
Figure 9(a) shows the evolution of discharge length along the electrodes and
per-pulse energy as the flame passes under the electrode (average of 20 tests
at f=6kHz, V=15.6kV). To account for the discharge extending beyond the extent
of the electrode, the region used for discharge detection extended 5mm beyond
the edge of the electrode in both upstream and downstream directions. This
region was chosen because it was observed from the energy measurements that
energy could begin to be deposited when the flame front was as early as 5mm
upstream of the electrode. The presence of a discharge was determined visually
from the high speed videos. The discharge and flame could be difficult to
distinguish in the monochrome videos, but it was assumed that the discharge
was always present on top of the flame when the flame was in the region below
the electrode $\pm$ 5mm. This is a reasonable assumption, since after the
flame had passed and the discharge and flame could be separated, the discharge
persisted in this region. When the flame is present, the temperature and
reduced electric field are higher, so the discharge is even more likely. The
discharge also caused a distortion in the flame front shape that could further
be used to confirm that it was overlapping with the luminous region of the
flame. This detection method is valid for uniform discharge mode only, which
was the situation in all cases shown in figure 9(b) except beyond 25mm. From
figure 9(a), it is observed that the per-pulse energy tracks with the
discharge region size, suggesting that the energy density of the discharge is
approximately constant in space. This is supported by figure 9(b) which shows
the per-pulse energy deposited per unit length of discharge ($W_{p}/L$). This
value stays relatively constant for most of the discharge duration. As with
the total per-pulse energy deposition, $W_{p}/L$ also increases with voltage.
This indicates that the energy deposition with voltage increase seen in figure
8(b) is indeed caused by the higher voltage, and not by a larger discharge
region.

(a) Size of discharge region and corresponding per-pulse energy.

(b) Per-pulse energy per unit length in flame propagation direction for the
cases of figure 8(b).

(c) Quadratic fit of pulse energy per unit length for figure 9(b) between 0
and 20mm.

Figure 9: Effect of discharge size on energy deposition.

The relatively constant value of the per-pulse energy deposition per unit
length at a given voltage indicates that the gas temperature has a minimal
effect on the energy density deposited in any region of gas. This is
significant, since the energy deposition pathways are highly dependent on the
value of the reduced electric field [16]. This seems to indicate that by
keeping the voltage constant, the same energy density would be deposited for
different temperature gas, yet it would activate different kinetic mechanisms
(determined by E/N). Nevertheless, the gas temperature does affect the size of
the discharge region, since discharge can only be sustained in regions with
elevated gas temperature. In particular, the discharge region size shrinks as
the gas cools. The value of $W_{p}/L$ as a function of peak voltage can be fit
reasonably well using a second order polynomial as shown in figure 9(c). The
data points and error bars in this figure are the mean value and twice the
standard deviation of the corresponding $W_{p}/L$ curves shown in figure 9(b)
sampled every 1mm between 0 and 20mm, the region where $W_{p}/L$ is
approximately constant.

###  4.6 Temperature Measurements using OES

The RT temperature was measured simultaneously with the energy for all cases
shown in figure 8 using the procedure described in section 2.2.3. In any
individual test, the low number of exposures accumulated on the ICCD sensor
resulted in a weak signal due to low amounts of light. To improve this, the
spectra recorded for all 20 tests were combined, giving a much cleaner signal.
To do this, the recorded spectra were placed into bins corresponding to a 1mm
range of flame front locations. Recall that the size of the optical window and
motion of the flame already cause some averaging of the spectra, so this
binning procedure does not reduce spatial resolution. All the spectra in a bin
were then summed, which greatly increased the signal strength. The temperature
fitting was only performed on bins with at least 5 spectra, since those with
less had too low a light level for a reliable temperature measurement.

The measured temperature profile for the various tests is shown by the thin
lines in figure 10. All cases yielded a very similar temperature once the
flame front had reached about 10mm past the beginning of the electrode. This
suggests that any difference in heating caused by the discharge was small.
This is compared to the maximum temperature rise that could be obtained if all
the electrical energy was deposited as heat, calculated by equation 2.

$\Delta T_{\mathrm{max}}=\frac{W_{p}f}{\dot{m}c_{p}}$ (2)

Where the $\dot{m}$ is the mass flow rate and $c_{p}$ is the constant pressure
heat capacity of the gas. Taking a value of $W_{p}=200\mu$J/pulse as typical
energy deposition, f=6kHz, $\dot{m}=18.7\mu g/s$ and $c_{p}=1.1kJ/kgK$, the
temperature rise caused by the discharge energy going entirely into heating is
58K, which is comparable to the RT reliable measurement precision. The
undetectable temperature change between cases also allows for the average of
the cases to be calculated, which is also shown in figure 10.

The optical window was centered at 3.5mm from the left end of the electrode.
When the flame front is near this region, there is a larger variation in the
measured temperature. This can be attributed to two factors. First, the steep
gradients in the temperature profile created by the flame meant that the
discharge would not be occurring at a uniform gas temperature everywhere in
the optical window and the coarse spatial resolution attainable in this setup
could not capture the steep gradient. Second the flame is not a perfect planar
front and non-uniformity across the cross-section, which varied between tests,
also contributed to a wider variation. When the flame front had passed the
optical window and the discharge was developing in the more uniform burnt gas
region, the variation in the measurements was reduced.

These factors that cause greater variation in the measured temperature when
the flame front is near the optical window also cause the measured maximal
temperature to be somewhat lower than what would be expected for a flame
propagating in a rectangular section of comparable size. This is supported by
modelling of the temperature profile in this configuration. A simple 1D model
based on the free-flame solver in Cantera [57] is used, with the finite size
channel accounted for by adding a heat loss term corresponding to laminar flow
in a rectangular cross-section pipe with constant wall temperature using a
Nusselt number of 5.60 [58]. Details of the model can be found in reference
[59]. The modelled temperature and experiment agree well beyond 10mm, where
the gas condition in the experiment is uniform and gradients are lower. Before
this point, the experimentally measured temperature is considerably lower than
what would be expected based on a purely planar flame front. It is therefore
concluded that the temperature measured when the flame front is beyond 10mm
from the electrode start are reasonably accurate, while the temperature
measured when the flame front is before this point corresponds to a spatial
average in the non-uniform environment of the observed section.

Figure 10: Measured rotational-translational temperature of the discharge for
various cases. Modelled temperature of purely planar flame shown in red for
reference.

##  5 Discussion: Discharge Dynamics and Regime Transitions

###  5.1 Role of E/N

Discharge mechanics are largely controlled by the reduced electric field, and
therefore it is useful to relate the observed discharge evolution to this
parameter. For the setup in this work, the voltage across the air gap
($V_{g}$) is related to the applied voltage ($V$) by equation 3, which is the
equation for three sequential parallel plate capacitors, dielectric-gas-
dielectric, with the first and last identical.

$\frac{V_{g}}{V}=\left[1+2\frac{\epsilon_{g}d_{q}}{\epsilon_{q}d_{g}}\right]^{-1}$
(3)

Where $\epsilon$ is the (relative) dielectric constant (1 for the gas, 3.8 for
quartz), $d$ is the thickness of the layer and the subscripts $q$ and $g$
refer to the quartz and the gas respectively. For the location of the short
electrode, this gives a ratio of the gas gap voltage to the applied voltage of
approximately 0.75. Prior to any ionization, the electric field in the gap is
given by $V_{g}/d_{g}$, however as the discharge evolves charge will build up
on the surfaces of the dielectrics and sheaths will form in the discharge.
Modelling of the reduced field as a function of position in a NRP DBD
discharge has been performed by [24, 16] and recently developed experimental
techniques have begun to allow it to be probed directly [19, 20].

Useful information can be obtained just based on the reduced electric field
calculated from the applied voltage (and not including the effect of space
charge, but including the presence of the dielectrics); for example the work
by Liu et al. on regime transitions identifies a threshold of 130kV/cm at
atmospheric pressure measured based on the applied voltage, corresponding to
530Td [34, 35, 60]. It should be noted that these works look at a single
nanosecond pulse, and not a NRP system, which will have additional effects
caused by the accumulated pulses. .The reduced field based on applied voltage
and the average measured temperature from figure 10 is shown in figure 11(a)
as a function of position and time. In this figure, position=0mm corresponds
to the left edge of the long electrode (gap height 3.4mm) and position=30mm
corresponds to the right edge (gap height=2.9mm) and a flame speed of 15cm/s
(lab frame) has been assumed. Contours corresponding to two critical
thresholds are highlighted. The first is a breakdown threshold based on the
Meek integral condition for streamer formation, equation 4, which is a metric
characterizing the increase in the number of electrons in an electron
avalanche [61, 62].

$K=\int_{0}^{\ell}\alpha\left(E/N\right)dx$ (4)

x is the coordinate along the electron avalanche direction, $\ell$ is the
length of the avalanche and $\alpha$ is the Townsend ionization coefficient
which is a function of the reduced electric field. A value of K=18-20
typically corresponds to the formation of a streamer [61, 62].

The other is metric is the threshold for uniform discharge proposed by Liu et
al. for a single pulse.. In figure 11(b), several images of the flame are
shown in a test with the same conditions as figure 11(a) (V=13kV, 30mm
electrode, flame speed $\approx$15cm/s in lab frame). By comparing figure
11(b) to figure 11(a), it is clear that the discharge is present in a region
approximately corresponding to where the Meek integral is $\geq$18.

(a) Reduced field (Td) as a function of space and time. Contours corresponding
to Meek breakdown criteria and E/N=530Td highlighted. Model is for the long
(30mm) electrode with a peak voltage of 13kV.

(b) Image of flame evolution under long electrode with f=8khz, V=13kV,
$\phi$=1.0, Q=1000sccm, Np=2000. Legend for colorbar is in arbitrary units.

Figure 11: Comparison of reduced electric field to flame appearance.

The Meek integral criterion, usually used as a threshold for streamer
formation but applicable here as a metric of the electron avalanche producing
a sufficient quantity of electrons to sustain the discharge, does a pretty
good job of estimating when the discharge will occur. Note the region at the
top left of figure 11(a) where the criterion is not met and compare this to
the dark region at the top left of figure 11(b) when the discharge has
disconnected from the left edge of the electrode. However, the criterion for
uniform to filamentary transition proposed by Liu et al. does not seem to be
applicable here, as the discharge appears uniform for a wide range of E/N. The
most likely explanation is that the accumulated affects of many pulses change
the dynamics of the transition, which is discussed further in the following
section.

###  5.2 Role of Gas Properties and the Thermal-Chemical Instability

Another factor affecting the discharge uniformity are the ionization
properties of the gas [24, 63, 20]. Certain combustion products, such as NO
[24], have a lower ionization energy than O2 and N2 and have been proposed to
promote instability. At the same time, the presence of water vapour in the
burnt gas may have a stabilizing effect due to its high attachment rate [25],
which could explain the long-lived uniform discharge observed in this work.
The existing population of electrons (as a result of previous pulses) will
also have an effect, with higher residual electrons being linked to earlier
breakdown [39] and an electron energy distribution function closer to
Maxwellian [40]. Another possibility, discussed in [25, 20] is heat release or
absorption by chemical reactions in the flow triggering the transition between
discharge regimes. Filaments are often seen after the flame has quenched. It
is possible that partial oxidation of fuel that has diffused into the hot
gasses is occurring and that this is contributing to the regime transition.
The diagnostics used in this work do not allow to quantify the impact of
chemistry on the regime transitions, but a number of recent papers have
focused on this contribution [25, 20].

###  5.3 Role of Fluid Residence Time and Pulsation Frequency

For a flowing gas, it may not be the pulse repetition rate on its own but
rather how this timescale compares to the gas residence time that matters [37,
38]. Zhang et al. [38], observed that a uniform discharge could be sustained
with a reduced field of 150Td (neglecting space charge) and a flow residence
time to pulse repetition period ratio
($\tau_{\mathrm{res}}/\tau_{\mathrm{PR}}$) oforder 10, while a filamentary
structure was more likely with ratios of 100 or greater at the same electrical
conditions. Khomich et al. [37] proposed a condition based on the fluid motion
of the gas between pulses transporting the gas far enough that residual
inhomogeneities from the previous pulse would not be amplified, estimating a
critical distance of 0.9mm.

The work of Zhang et al. used the electrode size as the characteristic
dimension for calculating residence time, while the work of Khomich et al.
used the discharge filament size. The latter metric will be used in the
following analysis, since it will be electrode area-independent. The geometry
of the discharge cell means that the mass flow rate will be constant. There is
a slight increase in the height of the channel h, and the channel has a
constant span, w. Thus, the flow rate can be calculated by equation 5

$u(x)=\frac{\dot{m}}{w}\frac{1}{\rho(x)h(x)}=\frac{\dot{m}R}{wP}\frac{T(x)}{h(x)}$
(5)

where u is the average flow velocity, x is the coordinate along the flame
propagation direction, $\dot{m}$ is the mass flow rate and $\rho$ is the
density, R is the gas constant, P is the pressure and T is the temperature.
Since the gas composition is dominated by nitrogen, R changes very little
between the burnt and unburnt mixture. Equation 6 is used to get a non-
dimensional ratio, $\tau$ of flow timescale to discharge timescale

$\tau=\frac{f_{p}d}{u(x)}=\frac{wPf_{p}d}{\dot{m}R}\frac{h(x)}{T(x)}$ (6)

where $f_{p}$ is the pulse repetition frequency and d is the characteristic
length of the instability leading to filament formation. Taking a fixed
observation location where the channel height is 3.1mm and assuming a
characteristic filament length scale of d=1mm, the contours of figure 12 are
constructed. In the experiment, transition to filamentary regime in the short
electrode was only observed at high frequency once the flame was well past the
electrode (see figure 6), which corresponds to the top right region of the
contour plot. At lower frequencies and voltages, the discharge would
extinguish before a regime transition was observed. Observations from the
experiment are shown on the contour plot as $\circ$ for uniform regime and
$\blacklozenge$ for filamentary regime. The transition observed in the
experiment is reasonably well explained by the residence time to pulse
repetition period ratio taking 30 as an approximate value for the transition.
The transition to filamentary discharge with the long electrode can also be
explained in this way, since with the long electrode the flame would generally
quench before exiting the electrode. In that case, the flow rate would drop
down to the oncoming gas flow rate (the far left side of the contour plot,
ahead of the flame front) regardless of temperature because the force
generated by the expanding gas no long exists. This may explain why the long
electrode almost always displayed transition to filamentary regime when the
flame extinguished. In cases where the flame nearly quenched and then
reignited the discharge (see B), the discharge always went back to a uniform
mode. As noted in section 4.4, the maximum frequency that could be used was
10kHz due to equipment limitations. It is expected that a higher PRF would
cause earlier transition to a filamentary regime, but further experiments are
necessary to confirm this.

Figure 12: Contours of $\tau$ (non-dimensional) for short electrode with
modelled flame temperature and density shown for reference. The thick red line
indicates a ratio of 30. Experimental data is shown as $\circ$ (uniform
regime) and $\blacklozenge$ (filamentary regime).

##  6 Conclusions

This work investigated the evolution of a NRP DBD discharge as it interacted
with a transient laminar flame on the time scales of the flame propagation.
Two configurations were tested; a long electrode was used to look at the
evolution of the discharge over a large region of space as the flame passed
under it, and a small electrode was used to look at the discharge evolution in
a local region as the gas condition changed. The discharge was strongly
affected by the presence of the flame, and the main observations can be
summarized as:

  * •

Discharge only occurred in the hot gases produced by the passage of the flame
(including possibly the reaction zone).

  * •

The per-pulse energy deposited by the discharge was directly related to the
size of the discharge region and the applied voltage, and was unaffected by
the frequency or flow rate of gas.

  * •

Transition from uniform to filamentary discharge mode was only observed in the
combustion products after the flame had quenched or travelled a significant
distance away from the discharge region. The relative time scales of gas
motion and pulse repetition rate can be used to explain this transition. The
reduced electric field alone can not explain the observations.

The coupled evolution of an NRP DBD discharge with a transient flame has
important implications for PAC systems. This work demonstrates that systems
involving transient combustion processes must consider the changing gas
environment and the effect it will have on the discharge behaviour, energy
deposition and energy pathways.

## Acknowledgments

This work has been supported by Lockheed Martin Corporation (LMC) as part of
its University Research portfolio, within the program managed by Dr. John
Rhoads and Mr. Jeffrey A. Mockelman acting as technical monitor. The authors
also acknowledge support from the Office of Naval Research (ONR) through the
Young Investigator Program, under Award Number N00014-21-1-2571, within the
program managed by Dr. Steven Martens.

##  Appendix A Energy Calculation

The energy calculation is performed by integrating the product of current and
voltage by equation 1. The mismatched impedance in the system caused many
pulse reflections of decreasing amplitude after the initial pulse. These
reflections continued for approximately 800ns before a switch in the pulser
damped them rapidly. Most energy deposition to the discharge occurred during
the initial pulse, but to get a proper energy measurement the signals had to
be integrated until about 700ns to account for the charging and discharging of
the capacitor. A typical trace of voltage, current and the integrated product
(energy) for a case with and without visible discharge is shown in figure
13(a) with the current shown in detail in figure 13(b). Three cases are shown
with no visible discharge, a low energy discharge and a high energy discharge.
In these cases, the energy deposited by the pulse is recorded as $4\mu J$,
$146\mu J$ and $326\mu J$ respectively. All three sets of waveforms were
generated during the same test, with the different behaviour caused by the
flame passage. The current waveforms show more reflections than the voltage
waveform because it was located on the low-voltage side of the discharge gap,
a necessity to avoid sparks to the Pearson coil. Downstream of the discharge
gap, there are superimposed waves coming from the the current flowing through
the discharge gap from the high voltage side (capacitive and conductive
components) and current reflected from the system ground connection, which is
why the frequency of the current oscillations is greater than the voltage
oscillations (which travel between the discharge gap and the pulser), however
these reflected pulses still represent a real current flowing towards the
discharge gap.

(a) Voltage and Current waveforms and energy measurement for three discharge
cases.

(b) Zoomed in view of current waveforms during the initial pulse and first two
reflected voltage pulses.

Figure 13: Example waveforms and energy measurements. Conditions: f=8khz,
V=13.0kV, $\phi$=1.0, Q=1000sccm, Np=3200. No discharge, low energy discharge
and high energy discharge are pulse 50, 450 and 1200 respectively.

The nanosecond timescale of the voltage rise means that signal travel time
delays are significant and must be accounted for to get accurate energy
measurements. The voltage probe was located on the high voltage side of the
test section, and the current probe on the current return side since
undesirable coronas formed between the probe and the wire when it was placed
on the high voltage side. This spatial separation, combined with different
length cables from the probes to the oscilloscope, led to a lag between the
current and voltage signals. To account for this lag, the high voltage was
pulsed at a low enough voltage that no discharge would form; this resulted in
the circuit behaving as a pure capacitor. In this case, no energy should be
dissipated in the load, so the lag was set so that the total energy dissipated
in the load is 0. This lag was on the order of a few nanoseconds, and could
vary based on small changes to the setup, but has a significant impact on the
energy calculation. A demonstration of this effect is shown in figure 14 for a
case without breakdown in the discharge gap. The red curve shows the energy
calculation (integrated product of voltage and current) without accounting for
the signal lag, the blue curve shows the energy calculation after accounting
for this lag. The remaining lines in gray, spaced at 200ps increments of
$t_{lag}$ indicate the sensitivity to this parameter. Before each sequence of
tests, several hundred pure capacitive pulses were measured and an average lag
was determined. The standard deviation of the lag calculated in 1000
calibration cases was 117ps. Since the scope resolution is discretized to
100ps, this indicates that the calculated signal lag was repeatable to within
about $\pm$2 discrete shifts. Based on figure 14 the error in energy
measurement caused by variation in the lag measurement is less than $\pm 20\mu
J$. This is expected to be the dominant source of systematic error and is on
the order of the random error in energy measurement seen between tests (see,
for example, figure 7). Overall, the energy measurements are expected to have
an uncertainty on the order of tens of $\mu J$.

Figure 14: Sensitivity of energy calculation to synchronization of signals.
Blue curve is without considering lag, red curve is after calibration, gray
curves demonstrate the sensitivity in the calculation to 200ps shifts in lag.
Conditions: f=4kHz, V=8kV, no flow, Np=1600.

An alternate method of determining the lag was also considered. For this
method, the current and time derivative of the voltage signals in a pure
capacitive case were Fourier decomposed and reconstructed using only the
common dominant frequencies. The cross-correlation of the reconstructed
signals was then used to determine the lag. This method allowed removing the
effect of high frequency reflections in the current signal, which represent a
real current across the load and must be considered when calculating the
energy, but make synchronization of the signals based on a pure-capacitor
assumption difficult. The lag calculated by this method was similar to that
calculated by assuming the pure capacitor dissipated zero energy, but was more
computationally intensive and found to have greater variance than the other
method, hence why the zero-energy condition was used.

##  Appendix B Discharge Re-ignition Driven by Invigoration of Flame

It was noted in the authors’ previous work [56] that application of the NRP
DBD discharge could cause premature flame quenching, with the effect more
significant at higher pulse repetition frequencies. The electrode position for
this experiment resulted in an interesting behaviour when the pulse repetition
frequency was set such that the flame would almost quench at the edge of the
electrode. In these cases, as the flame travelled along the electrodes, the
flame would become weaker and weaker and, consequently, the discharge would
almost completely disappear due to a lack of the hot gases needed to increase
the reduced electric field to the point that a discharge would occur. When
this weak flame did exit the electrode, it would then become stronger again
and resume production of large amounts of hot gas. As this gas is convected
along the channel behind the flame, it re-ignites a uniform discharge under
the electrode. This hot gas is propagating in the same direction of the
unburnt gas flow, and so in these cases the uniform discharge will grow from
the narrow to the wide end of the channel. This is shown visually in figure
15, and agrees with the energy measurements shown in figure 16. At 274ms, the
flame has almost extinguished and at 280-284ms there is not enough hot gas
being produced to sustain the discharge. By 287ms, as the flame exits the
electrodes, the flame front has begun to revive (as evidenced by the higher
luminosity) and the hot reaction products begin to reignite the uniform
discharge at the narrow end of the inter-electrode region. The speed at which
this uniform discharge region grows is approximately 1m/s; faster than the
laminar flame speed, but comparable with the speed at which the burnt gas
behind the flame front will be moving. This indicates that the growing
discharge is caused by the renewed production of hot combustion products by
the flame front propagating to the right.

Figure 15: Re-ignition of the uniform discharge caused by a revival of the
flame post electrode. Case of f=6kHz, V=11.7kV,$\mathrm{\phi}$=1, Q=1000sccm,
Np=2400. Legend in colorbar is in arbitrary units. Figure 16: Evolution of the
per-pulse energy in a case with partial flame quenching. The arrow indicates
the energy decrease when the flame is almost extinguished. f=6kHz,
V=11.7kV,$\mathrm{\phi}$=1, Q=1000sccm, Np=2400.

## References

## References

  * [1] Lacoste D A, Xu D A, Moeck J P and Laux C O 2013 Proceedings of the Combustion Institute 34 3259–3266
  * [2] Blanchard V P, Minesi N, Stepanyan S, Stancu G D and Laux C O 2021 Dynamics of a Lean Flame Stabilized by Nanosecond Discharges AIAA Scitech 2021 Forum January (Reston, Virginia: American Institute of Aeronautics and Astronautics) pp 1–10 ISBN 978-1-62410-609-5 URL https://arc.aiaa.org/doi/10.2514/6.2021-1700
  * [3] Starikovskaia S M 2006 Journal of Physics D: Applied Physics 39 R265–R299
  * [4] Pilla G, Galley D, Lacoste D A, Lacas F, Veynante D and Laux C O 2006 IEEE Transactions on Plasma Science 34 2471–2477
  * [5] Barbosa S, Pilla G, Lacoste D A, Scouflaire P, Ducruix S, Laux C O and Veynante D 2015 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373
  * [6] Ju Y and Sun W 2015 Progress in Energy and Combustion Science 48 21–83
  * [7] Starikovskiy A and Aleksandrov N 2013 Progress in Energy and Combustion Science 39 61–110
  * [8] Do H, Cappelli M A and Mungal M G 2010 Combustion and Flame 157 1783–1794
  * [9] Do H, Im S K, Cappelli M A and Mungal M G 2010 Combustion and Flame 157 2298–2305
  * [10] Aleksandrov N L, Kindysheva S V, Kosarev I N, Starikovskaia S M and Starikovskii A Y 2009 Proceedings of the Combustion Institute 32 205–212
  * [11] Castela M, Fiorina B, Coussement A, Gicquel O, Darabiha N and Laux C O 2016 Combustion and Flame 166 133–147
  * [12] Cathey C D, Tang T, Shiraishi T, Urushihara U, Kuthi A and Gundersen M A 2007 IEEE Transactions on Plasma Science 35 1664–1668
  * [13] Kosarev I N, Aleksandrov N L, Kindysheva S V, Starikovskaia S M and Starikovskii A Y 2008 Combustion and Flame 154 569–586
  * [14] Sun W, Won S H and Ju Y 2014 Combustion and Flame 161 2054–2063
  * [15] Yang S, Nagaraja S, Sun W and Yang V 2017 Journal of Physics D: Applied Physics 50 433001
  * [16] Nagaraja S, Yang V and Adamovich I 2013 Journal of Physics D: Applied Physics 46
  * [17] Popov N A 2016 Plasma Sources Science and Technology 25 043002
  * [18] Starikovskiy A 2015 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373 20150074
  * [19] Orr K, Yang X, Gulko I and Adamovich I V 2020 Plasma Sources Science and Technology 29 125022
  * [20] Rousso A C, Goldberg B M, Chen T Y, Wu S, Dogariu A, Miles R B, Kolemen E and Ju Y 2020 Plasma Sources Science and Technology 29 105012
  * [21] Stange S, Kim Y, Ferreri V, Rosocha L A and Coates D M 2005 IEEE Transactions on Plasma Science 33 316–317
  * [22] Elkholy A, Shoshyn Y, Nijdam S, van Oijen J A, van Veldhuizen E M, Ebert U and de Goey L P 2018 Experimental Thermal and Fluid Science 95 18–26
  * [23] Evans M D, Bergthorson J M and Coulombe S 2017 Journal of Applied Physics 122
  * [24] Adamovich I V, Nishihara M, Choi I, Uddi M and Lempert W R 2009 Physics of Plasmas 16 113505
  * [25] Zhong H, Shneider M N, Mokrov M S and Ju Y 2019 Journal of Physics D: Applied Physics 52 484001
  * [26] Guerra-Garcia C and Martinez-Sanchez M 2013 Journal of Physics D: Applied Physics 46 345204
  * [27] Guerra-Garcia C, Martinez-Sanchez M, Miles R B and Starikovskiy A 2015 Plasma Sources Science and Technology 24 055010
  * [28] Li T, Adamovich I V and Sutton J A 2013 Combustion Science and Technology 185 990–998
  * [29] Nagaraja S, Li T, Sutton J A, Adamovich I V and Yang V 2015 Proceedings of the Combustion Institute 35 3471–3478
  * [30] Massa L and Freund J B 2017 Combustion and Flame 184 208–232
  * [31] Adamovich I V, Choi I, Jiang N, Kim J H, Keshav S, Lempert W R, Mintusov E, Nishihara M, Samimy M and Uddi M 2009 Plasma Sources Science and Technology 18 034018
  * [32] Nikandrov D S, Tsendin L D, Kolobov V I and Arslanbekov R R 2008 IEEE Transactions on Plasma Science 36 131–139
  * [33] Lou G, Bao A, Nishihara M, Keshav S, Utkin Y G, Rich J W, Lempert W R and Adamovich I V 2007 Proceedings of the Combustion Institute 31 3327–3334
  * [34] Liu C, Fridman A and Dobrynin D 2018 Plasma Research Express 1 015007
  * [35] Liu C, Fridman A and Dobrynin D 2019 Journal of Physics D: Applied Physics 52
  * [36] Zhong H, Shneider M N and Ju Y 2021 Stability Analysis of Thermal-Chemical Instability in a Weakly Ionized Plasma AIAA Scitech 2021 Forum January (Reston, Virginia: American Institute of Aeronautics and Astronautics) pp 1–7 ISBN 978-1-62410-609-5 URL https://arc.aiaa.org/doi/10.2514/6.2021-1702
  * [37] Khomich V Y, Malanichev V E, Malashin M V and Moshkunov S I 2016 IEEE Transactions on Plasma Science 44 1349–1352
  * [38] Zhang Y and Guerra-Garcia C 2021 Discharge Regimes and Transitions in Pulsed Nanosecond Dielectric Barrier Discharges at Atmospheric Pressure AIAA AVIATION 2021 FORUM (Reston, Virginia: American Institute of Aeronautics and Astronautics) ISBN 978-1-62410-610-1 URL https://arc.aiaa.org/doi/10.2514/6.2021-3118
  * [39] Huang B D, Takashima K, Zhu X M and Pu Y K 2014 Journal of Physics D: Applied Physics 47
  * [40] Huang B D, Carbone E, Takashima K, Zhu X M, Czarnetzki U and Pu Y K 2018 Journal of Physics D: Applied Physics 51
  * [41] Huang B, Zhang C, Adamovich I, Akishev Y and Shao T 2020 Plasma Sources Science and Technology 29
  * [42] Moeck J, Lacoste D, Laux C and Paschereit C 2013 Control of combustion dynamics in a swirl-stabilized combustor with nanosecond repetitively pulsed discharges 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January (Reston, Virigina: American Institute of Aeronautics and Astronautics) pp 1–11 ISBN 978-1-62410-181-6 URL https://arc.aiaa.org/doi/10.2514/6.2013-565
  * [43] Kim W, Snyder J and Cohen J 2015 Proceedings of the Combustion Institute 35 3479–3486
  * [44] Zhu Y, Anand V, Jodele J, Knight E, Gutmark E J and Burnette D 2017 Plasma-assisted rotating detonation combustor operation 53rd AIAA/SAE/ASEE Joint Propulsion Conference p 4742
  * [45] Shanbhogue S J, Weibel D E, del Campo F G, Guerra-Garcia C and Ghoniem A F 2022 Active Control of Large Amplitude Combustion Oscillations using Nanosecond Repetitively Pulsed Plasmas AIAA Scitech 2022 Forum (San Diego, CA)
  * [46] Murphy D C, Sánchez-Sanz M and Fernandez-Pello C 2014 Journal of Physics: Conference Series 557 012076
  * [47] Berlad A L and Potter A E J 1957 Effect of Channel Geometry on the Quenching of Laminar Flames Tech. Rep. April National Advisory Committee for Aeronautics (NACA)
  * [48] Pancheshnyi S V, Lacoste D A, Bourdon A and Laux C O 2006 IEEE Transactions on Plasma Science 34 2478–2487
  * [49] Rusterholtz D L, Lacoste D A, Stancu G D, Pai D Z and Laux C O 2013 Journal of Physics D: Applied Physics 46
  * [50] Laux C O, Spence T G, Kruger C H and Zare R N 2003 Plasma Sources Science and Technology 12 125–138
  * [51] Saffman P G and Taylor G I 1958 Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 245 312–329
  * [52] Murphy D C 2015 The Measurement and Application of Electric Effects in Combustion Ph.D. thesis UC Berkeley URL https://escholarship.org/uc/item/47s6w7dm
  * [53] Lewis B and von Elbe G 1987 Combustion, Flames and Explosions of Gases 3rd ed (Orlando, FL: Academic Press, Inc.) ISBN 0-12-446751-2
  * [54] Ju Y and Xu B 2006 Combustion Science and Technology 178 1723–1753
  * [55] Lefkowitz J K, Uddi M, Windom B C, Lou G and Ju Y 2015 Proceedings of the Combustion Institute 35 3505–3512
  * [56] Pavan C A and Guerra-Garcia C 2021 Plasma Actuation of Mesoscale Flames AIAA AVIATION 2021 FORUM (Reston, Virginia: American Institute of Aeronautics and Astronautics) ISBN 978-1-62410-610-1 URL https://arc.aiaa.org/doi/10.2514/6.2021-3105
  * [57] Goodwin D G, Speth R L, Moffat H K and Weber B W 2018 Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes URL https://cantera.org/
  * [58] Bergman T, Lavine A, Incropera F and Dewitt D 2011 Fundamentals of Heat and Mass Transfer seventh ed (John Wiley & Sons, Inc) ISBN 978-0470-50197-9
  * [59] Pavan C A and Guerra-Garcia C 2022 Modelling the impact of a repetitively pulsed nanosecond dbd plasma on a mesoscale flame AIAA SCITECH 2022 Forum p 0975
  * [60] Liu C, Dobrynin D and Fridman A 2014 Journal of Physics D: Applied Physics 47
  * [61] Fridman A A and Kennedy L A 2011 Plasma physics and engineering 2nd ed (Boca Raton, FL: Taylor & Francis Group) ISBN 978-1-4398-1228-0
  * [62] Raizer Y P 1991 Gas Discharge Physics (Berlin: Springer-Verlag Berlin Heidelberg) ISBN 0-387-19462-2
  * [63] Gadkari S and Gu S 2017 Physics of Plasmas 24