# The Backward Problem in Plasma-Assisted Combustion: Experiments of
Nanosecond Pulsed Discharges Driven by Flames

Carmen Guerra-Garcia  Colin A. Pavan

###### Abstract

Plasma technologies are a promising way of addressing a number of challenges
in combustion, ranging from stabilization and flame-holding for hypersonic
vehicles, to enabling low-emissions propulsion and power. Whereas most works
in the literature have centered on the forward problem: the impact of the
plasma on different combustion metrics, including static and dynamic flame
stabilization; fewer studies have been devoted to the backward problem: or the
significance of having a strongly inhomogeneous environment, often unsteady in
time, on the discharge characteristics. In this perspectives article, the
backward problem is addressed in a hierarchical manner. First, the
implications of the discharge proceeding in an inhomogeneous background gas,
with gradients in density and composition, and nonuniform flows, are
discussed. Next, these foundation experiments are used as elementary building
blocks to explain experiments of nanosecond pulsed discharges in dynamic
combustion environments, including a transient laminar flame and a combustor
undergoing dynamics. Finally, the strong coupling between the forward and
backward problem is illustrated with an imaging study, using transparent
electrodes, of a mesoscale premixed flame and a nanosecond pulsed dielectric
barrier discharge. The experiments demonstrate the importance of considering
the pulse-to-pulse evolution due to changes in the background environment when
using plasma assistance in unsteady combustion processes, including active
control of combustion dynamics.

###### keywords:

plasma-assisted combustion , forward problem , backward problem , two-way
coupling , nanosecond pulsed discharges , flame static stability , combustion
dynamics , non-thermal plasmas , dielectric barrier discharge , inhomogeneous
media , transparent electrodes

††journal:  Applications in Energy and Combustion Science

[inst1]organization=Department of Aeronautics and Astronautics,addressline=77
Massachusetts Avenue, city=Cambridge, postcode=02139, state=MA, country=USA

##  1 Introduction

###  1.1 Challenges in Combustion that can Benefit from Plasma-Assistance

Plasma technologies offer an extra knob to enhance and control combustion
regimes that are inherently challenging. For aerospace applications, these
challenging conditions appear as a result of extreme environments intrinsic to
the field itself, e.g., supersonic and hypersonic flight; or driven by the
need to address environmental concerns.

For hypersonic flight, reliable thrust and high propulsion efficiency over a
wide range of flight Mach numbers, typically 5-25, and altitudes, 20-55km, is
needed [1]. This demands effective strategies for combustion stabilization and
flameholding in an extremely hostile environment, with comparable timescales
of the fluid flow and the chemical kinetics. Viable methods to achieve
stabilization include (i) mixing enhancement of the fuel and air streams, by
introducing turbulence and accelerating molecular diffusion; and (ii)
chemistry enhancement, by accelerating the chain reaction process through
either a local temperature increase or by generating a pool of radicals and
excited species. For different conditions, pursuing either chemistry or mixing
enhancement can be more effective. E.g., whereas pure mixing methods, such as
vortex generators, can be efficient in the lower Mach range; incorporating
chemical enhancement, such as heat recirculation using cavity flame holders,
is beneficial at higher Mach [2]. In the case of plasma, all three effects can
be accessed: heat, reactive radicals and excited species, as well as turbulent
kinetic energy and vorticity generation [3, 4].

From the environmental perspective, transportation contributes to climate
change and global warming through a complex set of processes including CO2
emissions, NO${}_{\text{x}}$ emissions, water vapor, soot, and sulfate
aerosols. In particular, air transportation contributes over 2% of all man-
made carbon dioxide, but its impact on the effective radiative forcing is
about three times higher due to contributions from nitrogen oxides and
contrail cirrus [5]. The general approach to reduce NO${}_{\text{x}}$
emissions is to reduce the process temperature, by burning lean and ultra-lean
mixtures [6], but these mixtures are prone to combustion dynamics, that can
cause wear and damage to combustor components and in extreme cases downstream
turbine elements [7]. Hence, novel technologies are needed to extend the
static stability (lean blowoff) limits of flames, as well as provide a means
for suppressing combustion dynamics. To address the carbon emissions,
sustainable fuels will be introduced. In the near term, these will be drop-in
fuels with similar physical and chemical properties to conventional fuels but
about 80% lower CO2 emissions on a lifecycle basis, and therefore similar
combustion challenges to be addressed in addition to fuel-flexibility
constraints. In the longer term, the propulsion system will need to be
significantly modified to safely burn hydrogen, or reliably and stably burn
its carrier ammonia [8, 9]. Plasma technologies for ignition and flame
stabilization may mitigate many of the challenges associated with using these
novel fuels in propulsion.

###  1.2 The Forward Problem: Plasma on Flame

A broad classification of the literature of plasma-assisted combustion can be
made in terms of demonstration works, that testify to the ability of plasma to
promote ignition and flame stability; and fundamental works, that clarify the
energy pathways activated by the plasma that enhance combustion.

From the practical perspective, numerous demonstrations for hydrocarbon fuels
and hydrogen are reported in the literature including: reductions in ignition
delay times and minimum ignition temperatures, flame speed enhancement,
extensions of flammability limits, and mitigation of combustion dynamics [10,
11, 12, 13, 14, 15, 16, 17]. Related to the emissions concern, there is
compelling evidence that plasma can extend the lean blowout limit [18, 19,
20], with Barbosa et al. reporting a three-fold extension in the equivalence
ratio for extinction [21]. It is still unclear however whether this will have
significant impact on the NO${}_{\text{x}}$ emissions, as the plasma activates
new pathways of NO${}_{\text{x}}$ production [22]. Similar strategies have
also been successful in mitigating dynamic instabilities, including
suppression of pressure oscillations [18, 19, 20, 21, 23], with recent work by
Shanbhogue et al. [24] demonstrating significant attenuation of large (a few
percent of the operating condition) amplitude pressure oscillations in a
swirl-stabilized burner. Current efforts have also addressed plasma-
enhancement of ammonia flames [25, 26, 27], particularly exciting are the
results by Choe et al. [28] showing 35% lean limit extension, in this case
accompanied by significant NO${}_{\text{x}}$ emissions reduction.

At a fundamental level, the pathways responsible for these combustion
enhancements include: kinetic, thermal, and transport effects [10, 11].
Kinetic effects refer to ionization, excitation, and dissociation triggered by
electron-impact reactions. Thermal effects refer to local heating of the gas
that in turn increases the Arrhenius-type reaction rates, and can follow
kinetic effects through slow vibrational relaxation processes, in microsecond-
timescales, and fast electronic relaxation processes, in nanosecond-timescales
[29]. Transport effects include ion-driven winds [30], modified diffusivities,
and pressure wave generation [31]. Specific to plasma-strategies are the
kinetic effects, namely electron-impact reactions whose reaction rates are
dependent on the electron temperature rather than the significantly lower
temperature of the gas. How the electrical energy is expended can therefore be
controlled by the reduced electric field (electric field divided by neutral
gas number density, or $E/N$111The reduced electric field is measured in
townsend units: 1 townsend, or 1 Td, is equivalent to 250 V applied across a 1
cm gap at standard pressure and temperature.) as the electrons gain energy
from the electric field and spend that energy in collisions [32, 33, 34]. At
reduced fields beyond about 150 Td (easily achieved using pulsed discharges of
tens of nanoseconds duration), a significant energy deposition pathway becomes
electronic excitation of nitrogen and oxygen, figure 1. The quenching of these
excited species leads to: (1) radical seeding, including oxygen radicals,
providing a useful method of bypassing chain initiation reactions [35, 36,
37], and (2) fast gas heating [35, 29], that occurs at nanosecond time scales.
A recent review by Popov and Starikovskaia [29] provides an overview of the
kinetics of fast gas heating, and considers the dependency of the fraction of
energy spent on this process as a function of $E/N$.

Figure 1: Fractional energy deposition into different internal energy modes
for a stoichiometric methane-air mixture, as a function of the reduced
electric field. Calculated using BOLSIG+[38] using cross sections from [39,
40, 41]. The area of each region at a given value of E/N represents the
relative contribution of that loss mechanism; for example, at 300Td about 5%
of the total input energy is consumed to produce N2(A).

The absolute concentration of active species produced by the plasma can be
further determined by the energy density deposited in the gas, or equivalently
the electron number density. E.g., the concentration of atomic oxygen produced
through electron impact dissociation of molecular oxygen,
($e^{-}+\text{O}_{2}\rightarrow e^{-}+\text{O}+\text{O}$), can be estimated by
an energy balance as:

$[\text{O}]=\frac{\varepsilon_{tot}\cdot f_{\text{O}_{2}(dis)}\cdot
N_{A}}{\varepsilon_{\text{O}=\text{O}}/2},$ (1)

where $\varepsilon_{tot}$ is the total energy density of the discharge in
$J/m^{3}$, $f_{\text{O}_{2}(dis)}$ is the fraction of energy that goes to
dissociation of molecular oxygen and depends on $E/N$ (figure 1),
$\varepsilon_{\text{O}=\text{O}}$ is the $\text{O}=\text{O}$ bond energy in
$J/mol$, and $N_{A}$ is the Avogadro number. With the cross-sections used to
produce figure 1, the electron impact dissociation of oxygen is a two-step
process caused by rapid quenching of electronically excited O2 produced by the
pathway labelled “O2 Electronic
($\varepsilon>\varepsilon_{\text{O}=\text{O}}$)”. The fraction of energy used
for direct dissociation of O2, $f_{\text{O}_{2}(dis)}$, is approximately given
by this value.222When the electronically excited molecule of O2 undergoes
dissociative quenching, additional energy beyond the threshold
$\varepsilon_{\text{O}=\text{O}}$ is converted to kinetic or internal energy
in the product species/third body quencher and so the energy actually expended
on O2 dissociation will be less than the energy used to electronically excite
O2 to an internal energy state with energy $>\varepsilon_{\text{O}=\text{O}}$.
Similarly, the fraction of energy going into methane dissociation could be
estimated using the pathway labelled “CH4 Electronic (7.9eV)” and assuming
these electronically excited states rapidly underwent dissociative quenching.
This is for direct dissociation only; additional dissociation of O2 and CH4
will be caused when electronically excited states of nitrogen quench by
collision with these molecules. At a fundamental level then, $E/N$ and
$\varepsilon_{tot}$, or equivalently electron temperature and electron
density, determine the plasma-activated chemistry.

###  1.3 The Backward Problem: Flame on Plasma

Based on the above description, it is tempting to envision being able to
externally control the two relevant fundamental parameters, $E/N$ and
$\varepsilon_{tot}$, to favor certain reaction pathways and deliver specific
amounts of plasma-activated species to affect combustion. Although some level
of control is indeed possible by adequately choosing the plasma source,
electrical parameters, positioning and timing, these parameters tend to self-
adjust as the plasma is generated, e.g., for applied fields above the
breakdown threshold ($E/N\sim$ 120Td in air) the electron concentration
rapidly grows effectively shielding the field. Moreover, a flame is an unusual
environment for a gas discharge to proceed. Whereas electrical breakdown is
typically studied in homogeneous and uniform gas conditions (constant
composition, pressure, and temperature), and the main source of inhomogeneity
comes from the background electric field (e.g., for needle electrodes) [42,
43, 44]; when using plasmas for combustion applications, these assumptions no
longer hold. Combustion processes inherently constitute inhomogeneous
environments (flame fronts, hot spots, concentration gradients, etc.), and
this background gas will guide the accessible $E/N$ and $\varepsilon_{tot}$
values, the plasma regimes encountered, the coupling of the electrical energy,
and even feed back into the combustion response that can be achieved. In
addition, most combustion processes of practical interest present turbulent
conditions, and/or transient behaviour, requiring a consideration of the
dynamic environment. A solid understanding of how the combustion background
affects the plasma, in static and dynamic conditions, is therefore crucial to
ensure the control of the plasma source and enable the design of predictable
and robust flame modification and actuation strategies.

###  1.4 Scope of this Work

As seen in the previous sections, plasma can be used to modify and control
flames, and this is the practical application the community is interested in.
However, the combustion environment equally affects the plasma dynamics and
kinetics, limiting our ability to control the energy deposition process. Of
course in reality the two-way problem of plasma-assisted combustion is fully
coupled and the effects of plasma-on-combustion and combustion-on-plasma need
to be simultaneously considered. Whereas the majority of the literature
focuses on the forward problem, Lacoste [45] identified in her recent review
that the effects of pressure, mixture composition and flow on the plasma, at
relevant engine conditions, are relatively poorly documented. This article
aims to be a first step at filling that literature gap. The scope of this work
is to present the authors’ perspectives on the backward problem. The focus is
on the plasma dynamics of Nanosecond Pulsed Discharges (NPD) in subsonic
flows, gaseous blends, laminar and turbulent flames, in premixed and
nonpremixed conditions.

This article is organized as follows. Section 2 describes, from a gas
discharge physics perspective, the relevant combustion processes that will
serve as the baseline environments for the studies in the next sections.
Sections 3 and 4 summarize prior work by the authors and collaborators
documenting the development of NPD plasmas in steady, but nonuniform, flame
environments and their elementary surrogates; as well as dynamic combustion
environments, both laminar and turbulent. Section 5 extends these works
through an imaging study of the response of a nanosecond pulsed dielectric
barrier discharge to the passage of a transient laminar premixed flame. The
images present a visualization of the discharge-flame interaction not observed
before, enabled by the use of transparent electrodes. Section 6 summarizes the
implications of the results presented on the forward problem. Finally, section
7 presents the conclusions of this work.

##  2 Combustion Environments

The applicability of plasmas to enhance combustion processes has been
demonstrated across various combustion modes including supersonic combustion,
homogeneous ignition, premixed flames, and nonpremixed flames. In all of these
environments, the gas state (composition, temperature, and pressure) and
velocity field will be varying, either as a function of time (for homogeneous
processes), space (for the case of localized combustion zones or flames), or
both (for unsteady flames). These inhomogeneous, unsteady, backgrounds allow
for two different scenarios that need to be accounted for, when designing a
plasma-actuation strategy.

The first scenario happens when the discharge gap is comparable to, or greater
than, the characteristic length scale for appreciable changes in temperature,
pressure, composition or velocity. Examples are given in section 2.1 and
figure 2. In this situation, the discharge will develop in a highly
inhomogeneous environment, leading to inhomogeneous plasma generation and
energy deposition, and possible reinforcement of the plasma in regions where
ionization is favored (e.g., hot spots).

The second scenario happens when the discharge gap is smaller than the
characteristic length scale for appreciable changes in temperature, pressure,
composition or velocity. In this case, the plasma source can be contained in a
locally uniform region, and examples are given in section 2.2 and figure 3. In
this case, if the combustion process is in steady state, the question is where
to locate the plasma to maximize the benefits of plasma-assistance,
considering also that the plasma characteristics will depend on this choice.
If the combustion process is unsteady, e.g., a flame undergoing thermoacoustic
oscillations, it is to be expected that the plasma state will be evolving at
the timescale of combustion dynamics, with implications on the response that
can be achieved. In what follows, some relevant combustion environments will
be described from the electrical breakdown perspective.

###  2.1 Laminar Premixed and Nonpremixed Flames

The first classification of flames can be made in terms of whether the
reactants are premixed or not. Premixed flames require that the fuel and
oxidizer form a solution, and are characterized by the self-sustaining
propagation of a localized combustion zone or discrete combustion wave, figure
2(a). Nonpremixed flames form when the fuel and oxidizer are originally
separated, they have no characteristic speed associated to them, and typically
burn more slowly than premixed flames, figure 2(b). In both cases, the
characteristic length scale for changes in the state of the gas is of the
order of millimeters (at atmospheric pressure), e.g., the temperature varies
by a factor of 4-7 within the thin thermal mixing layer of the laminar
nonpremixed flame depicted in figure 2(b).

(a) Laminar premixed flame.

(b) Laminar nonpremixed flame.

Figure 2: Inhomogeneous conditions for 1D premixed and nonpremixed laminar
flames. These simple environments serve as building blocks for more complex
combustion processes, e.g., turbulent flames can be pictured as ensembles of
laminar flame elements (or flamelets) embedded in a turbulent flow field.

In the plasma-assisted combustion literature, academic premixed and
nonpremixed burners have been used to isolate the effect of different plasma
species [17]; to explore different actuation strategies, including pre-
conditioning of the reactants (by placing the plasma ahead of the flame) [46]
and in-situ plasma deposition (by overlapping plasma and flame) [47]; in
addition to exploring the impact of plasma on different global combustion
parameters, including the flame speed [48].

###  2.2 Bluff Body- and Swirl-Stabilized Flames

Since the propagation speed of flames is much lower than local flow speeds in
any practical system, practical burners typically require some flame
stabilization scheme to anchor the flame. These can be realized using
geometric alterations, including rapid expansions or bluff bodies, as well as
swirlers, to locally slow down the flow as well as recirculate hot products,
figure 3. These configurations are more representative of industrial reactors,
including gas turbines for power generation and aeropropulsion (swirl-flame
stabilized methods), and afterburners (bluff body-flame stabilized methods).
In both cases, the environment presents large gradients in composition and
temperature, as well as is characterized by complex velocity fields. At the
burner scale, different regions corresponding to the cold reactant gases, the
flame, and the hot post-combustion products can be identified, with
characteristic length scale that of the geometrical elements of the burner
(order 1-10 cm). Note that, at the local level, the turbulent flames typically
expected under these circumstances can be pictured as ensembles of laminar
flame elements, or flamelets, embedded in a turbulent flow field, so that the
configurations presented in figure 2 might still be relevant here.

(a) Bluff body-stabilized premixed flame.

(b) Swirl-stabilized premixed flame.

Figure 3: Inhomogeneous gas background for practical flame stabilization
systems, including flow recirculation introduced by bluff bodies and swirlers.

In the plasma-assisted combustion literature both, bluff body- and swirl-
stabilized burners, have been studied to demonstrate flame static and dynamic
stability enhancement by plasma. For static stability studies, or extensions
of the blowoff conditions by plasma-assistance, the flame environment can be
mostly considered in steady state. For dynamic stability studies, or
suppression of combustion instabilities by plasma-assistance, the environment
is intrinsically unsteady, as it is undergoing oscillations in pressure and
flame location. The electrode arrangements used primarily depend on the flame
stabilization scheme: for bluff body-stabilized burners [49, 18, 19] and dump
combustors [50], pin-to-pin electrode configurations are typically selected;
whereas for swirl-stabilized burners [21, 23, 20, 51], a pin-to-ring/ cylinder
arrangement is typically preferred, so that the plasma is left to rotate
freely with the flow rather than being constrained in space.

##  3 Nanosecond Pulsed Discharges in Inhomogeneous Media

Whether static or dynamic, common to these environments is the presence of
sharp gradients in composition, temperature, pressure, as well as flowing gas
conditions. Other sources of inhomogeneity arise from the electric field; as
well as the structure of the discharge itself, often developing in the form of
filamentary structures. In particular, the geometry of the electrodes guides
the localization of the discharge, volume of plasma activation, and possible
localization in regions of high curvature, feeding into the ignition and
combustion problems [52, 53].

Under pulsed voltage, streamer breakdown is favored and several authors have
studied the implications of nonuniform media in streamer development.
Nonuniform environments have been considered in the context of atmospheric
pressure plasma jets and streamer development along helium-air mixing layers
[54, 55, 56, 57, 58, 59, 60, 61]; weakly ionized discharges in the upper
atmosphere (such as sprites) and density gradients along the propagation
length of the streamer [62, 63, 64, 65, 66, 67]; streamer development in two-
phase media [68, 69, 70]; plasma aerodynamic flow control and the development
of streamers with shock and rarefaction waves [71]; and plasma-assisted
combustion [72, 61, 73]. The impact of these nonuniformities on the breakdown
conditions of NPD, as relevant to flame environments, is discussed in what
follows.

###  3.1 Composition and Density Gradients

The first contributor to the medium inhomogeneity is the variation in the
ionization properties of the gas, as affected by the local composition and gas
number density. The impact of the gas composition and number density on the
ionization properties is captured by the first Townsend coefficient, $\alpha$
[32, 74]:

$\frac{\alpha}{N}=F\exp{\left(-\frac{G}{E/N}\right)},$ (2)

where $F$ and $G$ are composition-dependent. Figure 4 shows the Townsend
ionization coefficients for the major species found in methane flames, and
some popular dilutants, as calculated by BOLSIG+ [38]. The cross sections used
in the calculation are those reported in the Hayashi database [39] for CH4,
CO2 and H2O, the Phelps database [40] for O2, and the SIGLO database [41] for
N2, Ar and He. Table 1 summarizes the $F$ and $G$ coefficients for those same
species by fitting the data in figure 4, in the range 100-500Td, to equation
2.333These values are for the pure gases. When looking at mixtures, the
weighted molar average of the pure substances, known as the Wieland
approximation [75], can be misleading as the electron energy distribution
function (EEDF) can deviate from that of its constituents, particularly for
monoatomic/molecular mixtures [76]. For consistency, the effective ionization
coefficient has to be evaluated by solving the EEDF of the mixture and adding
the individual coefficients of each species.

Figure 4: Townsend ionization coeffcients for major species in flame and typical dilutants used, calculated using BOLSIG+. Cross sections used are detailed in text. Table 1: Townsend ionization constants for major species in flame and typical dilutants used, calculated by fitting the data in figure 4, in the range 100-500Td, with equation 2. | Gas | F [$10^{-21}m^{2}$] | G [$Td$]
---|---|---|---
Reactants | O2 | 28.8 | 631
| CH4 | 24.3 | 550
Dilutants | He | 8.0 | 174
| N2 | 24.2 | 867
| Ar | 19.9 | 323
Products | CO2 | 19.3 | 558
| H2O | 28.4 | 490

When looking at the elementary flame environments in figure 2, if a set of
electrodes is placed parallel to the flame front with gap distance greater
than the flame thickness, the medium can be represented by three gaseous
layers in series, of different electrical properties [72, 61, 73, 77]. In this
case, the main electric field is parallel to the dominant composition and
temperature gradients, and a reinforcement of the discharge in the regions
where the Townsend ionization coefficient is strengthened (higher temperature
or favorable composition) is to be expected. For extreme conditions, if the
disparity between the electrical properties of the different layers is large
enough, streamer development can be blocked by the more insulating layers [71]
and the discharge is confined to regions of favorable ionization.

Figure 5: Gas-confined Barrier Discharge (GBD) experiments with N2-He-N2
parallel jets and NPD discharges. Schematics of the experiment: (a) view of
three-parallel jets, (b) detail of AA’ section serving as discharge cell. (c)
photograph of discharge proceeding in He layer and N2 layers remaining non-
ionizing; blue region marks the discharge cell pictured in (b). Conditions:
6mm discharge gap (2 mm per layer), 15kV, 3kHz. The experiments are reported
in Ref. [72].

This Gas-confined Barrier Discharge (GBD) concept was demonstrated in Ref.
[72, 61] by applying NPD discharges across three parallel jets, with helium
injected in the central stream and nitrogen injected in the side jets, figure
5. Helium and nitrogen were chosen for their very disparate ionization
properties, as shown in figure 4 and table 1. For the experimental conditions
selected, the discharge proceeded in the helium stream alone and the nitrogen
layers acted like a barrier to the discharge. Although the nitrogen layers are
non-ionizing, they still allow for transport of charged species, including
convection, electric drift, and diffusion. Alternatively, selective ionization
when using NPD discharges can be realized by having a region of reduced local
density, N. This situation was explored in Ref. [73, 77] by applying NPD
pulses across two opposed jets of different temperature, figure 6. In the
experiment, the applied voltage amplitude was gradually increased to observe a
transition between: a discharge confined to the hot jet, figure 6 (b), and
breakdown of the full gap, figure 6 (c).

Figure 6: Counter-flow helium jets at disparate temperatures, and atmospheric
pressure. Two mesh electrodes are placed at the exit of the counter-facing
nozzles. NPD discharges at 3 kHz are applied with variable voltage amplitude.
(a) Schematic of the discharge cell, (b) case of selective excitation of the
high temperature (low density region) for 7kV, (c) case of full breakdown of
the discharge gap for 8.9kV. The experiments are reported in Ref. [73, 77].

Whether or not breakdown can be sustained is only one piece of the puzzle.
Much of the work in the field has centered on the role of composition on the
regime transitions observed, including uniform to filamentary modes. In
particular, certain combustion products can promote filamentarization of the
discharge (e.g., NO), and this has been attributed to their different
ionization properties [78, 79, 80]; alternatively, certain species can have a
stabilizing effect (e.g., water vapor), attributed to higher attachment rates
[81]. The role of heat release or absorption by chemical reactions in the
transition between uniform to filamentary regimes, the so-called thermo-
chemical instability mechanism, has also been a matter of study [81, 80].

###  3.2 Flow Fields

Aside from the local state of the gas (pressure, temperature, composition),
the flow field within the discharge volume can play a major role in the
discharge dynamics, controlling aspects such as the discharge regime
encountered or the spatial structure of the energy deposition.

Several studies by Leonov et al. [82, 83, 84, 85] have considered the
influence of airflow on so-called near-surface electric discharges, including
dielectric barrier discharges (DBDs), pulsed DC discharges, and quasi-DC
discharges, mostly in the context of aerodynamic flow control and supersonic
conditions. The two-way interaction between flow and discharge is well
exemplified by the case of quasi-DC discharges. Despite the DC voltage, the
discharge is pulsating and the transient nature of the filamentary discharges,
with temperatures of about 3000K moving with the flow downstream until they
split and reattach to a new location [83], is the result of a strong coupling
of the moving gas and the discharge dynamics [85].

Figure 7: (Top) Image of the experimental setup seen from the perspective of
the camera. The ITO electrode is transparent; its border is indicated by the
dashed red line. (Bottom) Top view showing the flow inlet location and open
sides of the test section. Flow is out of page as indicated by the $\odot$
symbol. Figure 8: Head-on visualization of the discharge as a function of
pulse repetition frequency, PRF, and a characteristic flow replenishment
distance, $d_{r}$ (ratio of the inlet speed-to-PRF). The different
replenishment distances are obtained by varying the mass flow rate in the
reactor at fixed PRF. Each image corresponds to the time-integrated intensity
of pulse 501 in a train with fixed applied voltage of 18 kV. Flow going up.
The experiments are reported in Ref. [86]. The view direction shown in these
images is parallel to the applied electric field and the discharge is observed
through the clear ITO anode. Each image shows the time-integrated visible-
spectrum light from a single pulse. Camera settings were kept fixed for each
PRF, but camera sensitivity and frame rate were varied as PRF was changed;
relative intensity can be compared between columns but not between rows.

When using NPD, bulk flow will affect the synergistic effect of consecutive
pulses, relating the initial conditions of a pulse (electron densities,
temperature, excited species densities, etc) to the preceding one. Filamentary
discharges have also been observed to be advected by the flow in this case.
The experiment reported in Ref. [86] using a dielectric barrier discharge,
DBD, reactor in flowing air at atmospheric pressure and ambient temperature
illustrates these points. The reactor used a transparent anode, consisting of
a 1 mm thick glass slide with a thin film of indium tin oxide (ITO) deposited
on the dry-side. The cathode was made of regular copper tape. The choice of
electrodes allows to take head-on images of the discharge. The ITO anode is
placed orthogonal to the camera’s line of sight, so that the view of the
discharge is through the ITO anode and co-linear to the dominant electric
fields and the propagation direction of the filamentary discharges that may
form. This setup is shown in figure 7. Another peculiarity of this rectangular
reactor was that gas was injected from one side and the three other sides were
left open to the atmosphere, resulting in a 2D non-uniform flow field with
diverging streamlines. The setup was used to explore the effects of flow
residence time, compared to the inter-pulse period, by varying the mass flow
rate (or inlet speed) and the pulse repetition frequency. The key finding of
the study is shown in figure 8. A characteristic flow replenishment distance,
$d_{r}$, defined as the ratio of the inlet speed to the pulse repetition
frequency, was identified as the parameter controlling the discharge dynamics,
rather than the inlet speed or the pulse repetition frequency alone. This
distance represents the characteristic distance travelled by a fluid parcel
between two consecutive pulses. For $d_{r}$ below some critical value order
$\sim 100\mu m$, microdischarges are favored; for larger $d_{r}$ the discharge
is no longer visibly observed or becomes uniform. The transition is gradual
and can be spatially resolved within the reactor due to the diverging flow
field structure. As $d_{r}$ increases, a decreasing region of micro-
discharges, downstream of the transition zone where the speed is highest, is
observed. The boundary of the transition zone moves downstream with $d_{r}$
(either increasing inlet speed or decreasing pulse repetition frequency).
Interestingly, the critical replenishment distance for the transition seems to
be comparable to typical streamer radii at these conditions [87], suggesting
that filamentarization is more likely when there is a stronger synergy between
pulses and any instability present will be more likely to get amplified.

###  3.3 Flame Front and Reaction Zone Environments

In the previous examples, the different contributors to the medium non-
uniformity were to some extent isolated to evaluate their separate impact on
the breakdown phenomena. Even the simplest flame environments depicted in
figure 2 present all these effects simultaneously, in addition to a pre-
ionization region contained within the high-temperature zone [88]. For
hydrocarbon flames, ions and electrons are generated in the reaction zone of
the flame in amounts much in excess of thermodynamic equilibrium values as a
result of chemi-ionization reactions, in which species undergo a chemical
rearrangement that is sufficiently exothermic to ionize one of the products
[89, 90, 91, 92]. The most important reaction contributing to the formation of
ions and electrons is:

$\text{CH}+\text{O}\longrightarrow\text{CHO}^{+}+e^{-},$ (3)

where the CH is either in ground state or electronically excited [93].
Although CHO+ is the precursor ion, the dominant ion is usually H3O+. The
region of the flame containing charged species is comparable in size to the
reaction zone thickness, e.g., for stoichiometric premixed methane-air flames
this thickness is $\delta_{i}\sim$0.6mm [94, 95]. Note that this thickness may
vary with the flame parameters, but will be of the order of a fraction of a
mm, and much less than the thickness of the high temperature region. The
ionization fraction within this region varies in the range $n_{i}/N\approx
10^{-8}$-$10^{-7}$, for methane-air flames, peaking close to stoichiometric
conditions (slightly fuel rich) [96]. The presence of these chemi-ionization
electrons and ions impacts the way the electrical energy is coupled to the gas
in a number of ways.

First, this region of pre-ionization is critical when using sub-breakdown
fields. For DC and AC waveforms, the chemi-ionization ions are responsible for
the generation of ion-driven winds, a phenomenon that has been studied since
the 1960s [97, 89, 30]. When using sub-critical microwave (MW) energy, the
energy can be directly coupled to the chemi-ionization electrons, which are
heated up selectively [95, 98, 99, 100].

In the case of NPD, the presence of the chemi-ionization electrons has several
implications. They contribute to the sustainability and repeatability of the
discharge: typically the NPD strategy relies on the use of kHz frequencies,
chosen to match the recombination times of electrons, in order to sustain a
minimum electron population at all times [101]. In the case of a flame front,
a baseline level for the electrons already exists, with magnitude comparable
to the minimum electron mole fractions measured between pulses in glow-like
NPD discharges [102]. Pre-ionization strategies can also be used to improve
the uniformity of the discharge [103, 104]: electron avalanches can overlap,
smoothing out local gradients, and favoring a homogeneous breakdown condition.
Huang et al. [105] discussed the role of pre-ionization in the breakdown
properties of the gas in the context of NPD discharges at different pulse
repetition frequencies. Their experiments of NPD discharges, at atmospheric
pressure using pin-to-pin electrodes in Ne and Ar mixtures and variable
frequency in the 5-667kHz range, showed a reduction in the breakdown voltage
for the highest frequencies tested, which they attributed to the high residual
charge density between pulses. Subsequent experiments with Fast Ionization
Waves by the same team [106], proposed that higher pulse repetition
frequencies also lowered the degree of non-equilibrium of the discharge. Note
that a lower local breakdown threshold at the flame front location would lead
to the coupling of the discharge to this region. Such a selective excitation
of the flame to the flame front region, by NPD plasma, is presented in Ref.
[73] using a counterflow nonpremixed methane/air burner with mesh electrodes
parallel to the flame, figure 9. The photographs shown in figure 9(b) were
taken by a PicoStar HR12 (LaVision GmbX) intensified charge-coupled device
(ICCD) camera using a gate of 0.5ns and correspond to the integrated emission
over a broad wavelength range. Within an applied voltage range, the luminosity
of the plasma was fully contained within the flame front region, with no
contact with the electrodes, suggesting localized energy deposition to the
flame.

Figure 9: Plasma emission coupled to flame front in a counterflow nonpremixed
burner. (a) Discharge cell schematic. (b) Timeline of discharge evolution
during voltage pulse application; time stamps correspond to time with respect
to NPD pulse onset, and white-dashed lines indicate location of the electrode
meshes located at the exit of the counterflow nozzles. Images correspond to
pulse number 1000 in a train and the exposure time is 0.5ns. The plasma
luminosity is contained within the flame and the edges of the powered
electrode. The experiments are reported in Ref. [73].

##  4 Nanosecond Pulsed Discharges in Dynamic Environments

Much of the work in the field exploits the repeatability of the pulses and
centers on the characterization of the discharge once a quasi-periodic
behavior has been reached [107]. This occurs after a few tens or hundreds of
pulses, if the medium is not significantly changing at the timescale of the
applied train of pulses. For operation at a few kilohertz, e.g. 10 kHz, and
trains of $\sim$100 pulses, this timescale is of the order of $\sim$10 ms.
Many combustion processes of interest cannot be considered steady within that
time-frame. E.g., plasma-assisted ignition is greatly influenced by inter-
pulse coupling considerations; a laminar methane-air premixed flame will
travel about 4 mm in 10ms, which is typically larger than the flame thickness;
and a combustor presenting an instability with a frequency of oscillation of
100 Hz will have undergone a full cycle of variation when pulse 100 is
applied. For these cases, and also when considering turbulent flows, a steady-
state pulsing behavior will not be encountered, and individual pulses will
need to be studied separately to have the complete picture of the interaction.
Three examples are reviewed in what follows.

###  4.1 Inter-Pulse Coupling and Cumulative Effects of Pulses

In otherwise uniform conditions, the application of a train of NPD discharges
introduces a modification to the medium as each consecutive pulse is applied.
In this situation, individual pulses are triggered in different background
conditions (in terms of temperature, composition, and pre-ionization), as
modified by the prior pulse. One of the first works reporting the cumulative
effects of pulses was Ref. [108], who measured the properties of individual
pulses in a train, in terms of energy deposited, temperature increase, and
optical spectra; both in atmospheric pressure air and in a plasma-assisted
ignition experiment with variable pressure and propane/air mixtures. The study
revealed that during a train of 10 pulses in air, the first few pulses had
negligible conduction current and energy limited to around a hundred
microjoules, whereas the last pulses in the train delivered about ten times
more energy (order mJ). The difference was attributed to the pool of electrons
and other active species (radicals, excited molecules) building up, as well as
the cumulative heating effects of the pulses, leading to more favourable
conditions for the discharge for the later pulses in the train. The number of
pulses for the transition depends on the state of the gas, as well as the
electrical conditions selected, and plays an important role in plasma-assisted
ignition studies.

The implications of the cumulative effects of pulses on the ignition
probability of a flowing mixture was studied by Lefkowitz and Ombrello in Ref.
[109]. The experiments varied a number of parameters including total energy
deposition, inter-pulse time, equivalence ratio, gap distance, and flow
velocity; and identified three regimes of inter-pulse coupling: fully-coupled,
partially-coupled, and decoupled. The fully-coupled regime occurs at the
highest pulse repetition frequencies (for given flow residence times), and in
this regime all the pulses are deposited in the same small volume, resulting
in high energy density, and a high probability of developing an ignition
kernel that can freely propagate into a flame. For the decoupled regime,
occurring at the lowest pulse repetition frequencies (for given flow residence
time), the ignition kernels do not interact until they have expanded
significantly, so that the total ignition probability is proportional to the
ignition probability by a single pulse. Refs. [109, 110] exemplify the
complexity of optimizing a plasma-assisted ignition and combustion problem,
given the interactions between plasma, flow, and combustion.

###  4.2 Transient Laminar Flame Passage

A simple platform to visualize the impact of a transient combustion
environment on the discharge behavior is provided in Ref. [111]. In this work,
the evolution of a DBD discharge under NPD voltage during passage of a laminar
premixed methane-air flame, in a narrow channel, is presented. For a discharge
developing in this background gas, all of the effects considered in section 3
will be present, and in addition every pulse applied will see a slightly
different environment.

(a) Setup for mesoscale flame experiments in tapered channel. Cathode is 20cm
long, extending the entire length of the test section; anode position and
length are variable. Test section is 3cm wide in the y-direction.

(b) Direct imaging of individual pulses/ flame interaction.

(c) Per-pulse energy measurements.

Figure 10: (a) Experimental setup and (b,c) sample results for experiments in
tapered channel with laminar flame. Conditions for results shown: f=8kHz,
V=14.7kV, $\mathrm{\phi}$=1.1, Q=200sccm, camera frame rate 2kfps. The
discharge transitions from a uniform regime to micro-discharges, with
consequential modifications to the spatial structure of the energy deposition,
as well as total energy delivery. The experiments are reported in Ref. [111].

Figure 10 shows a schematic of the experimental setup and presents an example
of the discharge evolution, in response to flame passage, for a case of pulses
at f=8kHz, V=14.7kV and flow conditions of $\mathrm{\phi}$=1.1, 200sccm. The
images shown time-integrate the visible spectrum light from 4 sequential
pulses and were taken using an Edgertonic SC2+ monochrome high speed camera.
To distinguish flame from discharge, the experiment was also performed with a
camera frame rate twice the discharge frequency; an example of this is shown
in the original work [111]. The first few pulses (pulse 20) are applied before
the flame enters the inter-electrode gap, and no visible discharge is
appreciated. As the flame enters the inter-electrode gap (pulse 270), a
uniform plasma is ignited in the hot burnt gas behind the flame, and that
uniform plasma region increases in size as the flame continues to propagate
(pulses 520, 770, 1020). During this time of plasma size increase, the energy
deposition increases close to linearly with pulse number (or discharge size
extension), figure 10(c). An inflection point is seen around pulse 750, when
the leftmost edge of the discharge detaches from the electrode (the cooling of
the gas at the downstream side of the flame cannot support breakdown in that
region anymore). Between pulse 1020 and pulse 1270, the flame reaches its
furthest most position and is quenched before it exits the inter-electrode
region, as a result of adverse interactions with the plasma and the tapered
walls of the test cell [111]. This point also coincides with the peak energy
deposition. At this point, the heat source keeping the gas hot is extinguished
and the discharge is seen to transition into a microdischarge regime (pulse
1270, 1520), which persists until the pulses are turned off (pulse 1770).
After flame extinction, the per-pulse energy rapidly decays and remains
approximately constant while the micro-discharges persist. The test section
was constructed with a shallow taper to study plasma interaction with flames
near the quenching distance, and the extinction of the flame is partially
caused by the narrowing walls of the channel. This decreasing gap size also
causes the filaments to form at the right side of the electrode (pulses 270,
520). Additionally, the plasma has been observed to cause premature quenching
of the flame, as reported in Ref. [112]. While the precise mechanism is not
currently known, a possible cause is discharge-produced flow disturbances
predicted by subsequent modelling work [113].

This example illustrates that the combustion environment is driving the volume
of gas activated by plasma, the energy delivery, as well as the plasma regime
encountered. The effects discussed for static environments are present, e.g.,
selective ionization of hot gas regions and regime transitions, with the added
complexity of the pulse-to-pulse variability of the discharge. This has
important implications when considering plasma actuation strategies. Any
control scheme will likely be based around a particular discharge mode and
expected energy addition to the gas, however, a changing gas environment
driven by combustion dynamics can lead to different behaviour. The bench-top
scale platform here is useful for probing the mechanisms of interaction in a
way not possible in complex environments of more practical relevance, one of
which will be discussed in the following section. The mesoscale burner will be
revisited in section 5.

###  4.3 Swirl-Stabilized Burner Undergoing Combustion Dynamics

Whereas the experiment discussed in the previous section represents an
academic platform, the high level observations should also translate to the
industrially-relevant environments discussed in section 2.2. In particular,
such a strong coupling of the discharge behavior to transient flame
environments could have important implications when using NPD plasmas to
stabilize lean flames and suppress thermoacoustic instabilities, section 1.
Such an environment has been studied in Refs. [24, 114] and is briefly
summarized in what follows. The studies consider a premixed swirl-stabilized
burner described in detail in Ref. [24], which presents combustion dynamics
with a dominant frequency around 120 Hz at the lean equivalence ratios tested.
The reactor operates at atmospheric temperature and pressure with lean
methane/air mixtures, and the operating flame power is around 14 kW. The flow
is in this case turbulent. The electrode system is provided by FGC Plasma
Solutions and consists on a center-body pin-anode inserted into the swirler
hub, to which NPD discharges are applied; the cathode is the annular body of
the injector section. The NPD plasma was shown to provide significant
attenuation of the pressure oscillations [24], and the discharge was seen to
be strongly affected by the combustion dynamics [114], as illustrated in
figure 11. In this plot, the pressure trace is used to track the combustion
dynamics, and should not be interpreted as the driver of the plasma-regime
modifications (the change in pressure is at most $\sim 4\%$ of the ambient
value). Rather, it is correlated to the heat release rate and the position of
the flame with respect to the central electrode, which are the more likely
drivers of the discharge evolution. Under these conditions, during the
instability cycle, three different regimes were identified, corresponding to
different energy depositions and visual appearances: (i) a high-energy ($\sim
20mJ$) ns-spark; (ii) a low-energy ($\sim 10mJ$) streamer corona444In the
streamer corona mode, most of the energy is not deposited in the corona itself
but rather upstream of the electrode via parasitic resistance and discharge,
see [114].; and (iii) a transitional regime.

Figure 11: Correspondence between energy deposition by NPD discharge and
pressure oscillation cycle during thermoacoustic instability. (Top) Sample
time-trace. (Bottom) Polar plot of the discharge energy, radial coordinate in
mJ, as a function of pressure phase angle. Blue markers are streamer corona,
red markers are nanosecond spark, orange markers are transitional. Conditions
$\phi=0.78$, PRF=$9kHz$, $V_{peak}$=12.4kV. Experiments are reported in Ref.
[114].

The high-energy regime appears within a specific region of the pressure phase
cycle, coinciding with conditions where the flame luminosity is at its maximum
(flame strengthened). The lower-regime appears at the opposite region of the
cycle, when the flame is weakest. The transition between both modes is
gradual, with pulses depositing intermediate energy levels in between. In this
situation, the dynamic combustion environment is controlling the discharge
energy deposition process, with possible implications on its ability to
suppress the instability, if not adequately accounted for.

##  5 Coupling of the Forward and Backward Problems: Imaging Study of a
Mesoscale Flame with Plasma

To further emphasize the two-way discharge-flame interaction, the experiments
reviewed in section 4.2 are extended in this perspectives article to include a
novel visualization enabled by the use of transparent ITO electrodes. By
applying the NPD to an ITO anode, the interaction between flame and plasma can
be visualized in 2-dimensions, revealing features that were previously hidden.
The experimental setup used is the same as that described in Ref. [111] and
shown in figure 10(a), and is here summarized for completeness, including the
new elements added. The reactor is a tapered quartz channel with a width of
3cm and a height varying from 4.2mm to 1mm over 20cm (the height changes by
less than 0.2mm over 1cm). Premixed methane-air, at an equivalence ratio of
$\phi=1.06$, enters from the small end of the taper and flows towards the
large end, the flow rate is set to 400sccm. At the exit of the reactor, a
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.
The 2mm quartz walls of the reactor are used as dielectric barriers for the
DBD. The main difference with the earlier work [111] is that the high voltage
electrode is substituted by a glass slide of 1mm thickness with a thin ITO
coating at the dry-side of the reactor, which is attached to the upper quartz
plate. This electrode is 36mm long and centered over the point where the
channel height is 3.4mm. High voltage is applied using a nanosecond pulse
generator (Transient Plasma Systems SSPG-20X-HP1) at 8kHz and variable
amplitude. The ground electrode is an aluminum plate. Electrical measurements
are acquired using a high voltage probe (Lecroy PPE 20kV) and a Rogowski-coil
style current monitor (Pearson model 6595) placed on the high voltage side of
the circuit, close to the high-voltage probe. Signals are collected at 100ps
resolution using a Teledyne Lecroy Waverunner 9254 oscilloscope. High speed
video is taken using a high speed camera (Edgertronic SC2+), looking directly
through the transparent high voltage electrode (along the z-axis in figure
10(a)). For the images presented, the frame rate of the camera was set to
2kfps, as a compromise between accurate flame tracking under the electrode and
isolation of individual pulses, and adequate resolution of the structures in
low light. All signal synchronization is done using a delay generator
(Berkeley Nucleonics Corp. Model 577-4C).

Figure 12: Interaction between premixed mesoscale flame propagating in narrow
channel and NPD DBD plasma ignited by flame passage: in-situ strategy. The
view presented is through the transparent ITO anode. Train of 2000 pulses
applied at 20.4kV and 8kHz. Images correspond to visible-spectrum light
accumulated in 4 pulses + flame emission (camera frame rate 2kfps). Electrode
edge marked by dashed white lines. Figure 13: Per-pulse energy measurements
corresponding to images shown in figure 12.

Figure 12 shows images of several pulses within a train of 2000 pulses, at
20.4kV, during passage of the laminar flame, in conditions phenomenologically
similar to those depicted in section 4.2. The evolution of the energy
deposition is pictured in figure 13. For the backward problem (flame-on-
plasma) the timeline is as follows:

  * 1.

No visible discharge is observed for pulses applied before the flame front
reaches the inter-electrode gap (pulse 10).

  * 2.

A uniform discharge ignites behind the flame as soon as its front reaches the
electrode (pulse 150), in the region of burnt gas contained underneath the
electrode (pulse 300).

  * 3.

The uniform discharge grows in size, as the flame propagates (pulse 440, 580,
720), until the flame reaches the end of the electrode (pulse 870). As the
discharge grows in size, the per-pulse energy deposited increases.

  * 4.

The uniform discharge remains contained to the inter-electrode space, and
detaches from the flame front as the flame continues its propagation beyond
the electrode (pulse 1010, 1150). During this time the per-pulse energy has
reached its maximum. This uniform region starts to shrink, in particular
detaching from the burner walls that represent the coldest region of gas.

  * 5.

The discharge begins to transition to a microdischarge regime as the flame
moves further away (pulse 1290, 1440) and eventually quenches (pulse 1580). As
the discharge transitions to the filamentary mode and the discharge volume
starts to shrink, the energy deposition starts to decrease. The filaments
cover more of the volume, until the uniform contribution fully disappears
(pulse 1720).

  * 6.

The microdischarges persist (pulse 1860, 2000) until the end of the train,
even though the flame is no longer present. During this time the energy
deposition decreases, likely due to the cooling gas conditions.

As the discharge and the flame are of comparable size, the two-way interaction
is amplified and the plasma greatly modifies the flame properties in a dynamic
manner. For the forward problem (plasma-on-flame) the timeline is as follows:

  * 1.

The flame propagates in the channel with close to constant speed, comparable
to that expected for the mixture at hand in a channel with heat loss. The
flame front presents a two-lobe structure as a consequence of the finite
dimension of the channel and the development of fluid instabilities. The shape
is typical of flames in confined channels, and the structure is sometimes
referred to as a tulip flame [115, 116].

  * 2.

The flame presents more wrinkling (three-lobe structure instead of two) and
speeds up (see below), as it enters the inter-electrode space (pulse 150, 300,
440, 580, 720).

  * 3.

The flame seems to come to a halt as it reaches the end of the electrode
(pulse 870, 1010).

  * 4.

The flame continues its propagation beyond the electrode, with a two-lobe
structure (pulse 1150, 1290).

  * 5.

As the flame detaches from the electrode and eventually quenches (pulse 1580),
the flowfield structure of the post-combustion gases can be inferred from the
visible microdischarges and uniform discharge patches.

Figure 14: Interaction between premixed mesoscale flame propagating in narrow
channel and NPD DBD plasma ignited by flame passage: ahead-of flame strategy.
The view presented is through the transparent ITO anode. Train of 3200 pulses
applied at 22.5kV and 8kHz. Images correspond to visible-spectrum light
accumulated in 4 pulses + flame emission (camera frame rate 2kfps). Electrode
edge marked by dashed white lines. Figure 15: Per-pulse energy measurements
corresponding to images shown in figure 14. Figure 16: Shape and position of
flame front at equally spaced time points. Dashed vertical lines mark the
edges of the electrode and labels indicate the time in ms. Train of 2000
pulses applied at 22.5kV and 8kHz. Pulse train begins at approximately 120ms.
Note that while the discharge conditions are the same as figure 14, it is a
different test, hence the slightly different flame structure.

A case with slightly higher voltage, e.g. 22.5kV, is presented in figures 14
and 15. The chronology of the interaction is similar to the previous example,
with the difference that in this case the applied voltage is high enough for
breakdown to occur ahead of the flame, in the narrowing section of the
channel, before the flame arrives to the electrodes. This strategy allows for
pre-activation of the reactants, ahead of the flame, as compared to the in-
situ strategy (fully coupled flame and plasma) described in the previous case.
A few additional observations follow. From the discharge perspective, the
discharge in the cold reactants (pulse 150, 300) proceeds in the
microdischarge regime. As soon as the flame enters the inter-electrode space
transition to uniform mode is observed in the burnt gas region (pulse 440,
580). As the flame quenches and the full inter-electrode gap sustains mostly
micro-discharges (pulse 1720, 1860, 2000) the per-pulse energy remains
constant and does not drop like in the previous case as the voltage is enough
to sustain the discharge even in the cold gas. From the flame perspective, the
enhanced flame wrinkling is observed before the flame reaches the electrode
(pulse 300). The flame speed-up is also more noticeable in this higher voltage
case. This flame speed increase, while the flame is underneath the electrodes,
can be appreciated in figure 16 by plotting the flame contours every 20ms: the
flame clearly moves a greater distance, within 20ms, while located underneath
the electrode as compared to other locations.

In this configuration, the discharge transitions between uniform and
filamentary regimes multiple times. If the applied voltage is high enough to
allow breakdown in the cold reactant gas, the discharge often starts uniform
for the first several pulses before transitioning to filaments. This is
thought to be caused in part by a plasma thermal-chemical instability [117,
80] and part due to slow gas flow since it was possible to achieve a uniform
discharge in pure air with high flow rates but not in methane-air mixtures.
Behind the flame front, the fuel oxidation reactions have largely been
completed, the gas has been accelerated, and the temperature is high. All
these aid in creating a uniform regime; since most oxidative reactions have
ceased, the plasma thermal-chemical instability is less important, the fast
moving gas prevents instability amplification (see section 3.2), and the
higher temperature aids with diffusion, a uniformizing process. As the gas
cools and slows, notably from the walls far behind the flame, the latter two
effects are lessened and the discharge transitions back to filamentary.

##  6 Implications for Enhancement and Control of Combustion by Plasma

While the forward problem represents the driver for research in plasma-
assisted combustion, the backward problem might well be the enabler for the
success of the strategy. Several studies have focused on NPD plasmas in pin-
to-pin configurations and steady environments, including air and post-
combustion gases [102, 108, 118, 101, 107, 31, 119], and have set the
foundation of our understanding of the plasma-chemical activation as well as
the energy share that goes to other processes, such as shock wave generation
or rapid gas heating. The implications of moving to higher pressure
environments, engine-relevant fuels, or fast fluid flows, is still to be
addressed. Whereas pin-to-pin discharges represent an ideal platform for
optical studies, due to the fixed location of the plasma in space, and are
already relevant for applications such as bluff-body stabilized burner
configurations (section 2.2), other electrode configurations also need to be
characterized. The diagnostics for situations where filamentary plasmas are
left to move freely with the flow is more difficult to realize. In addition,
and as motivated in previous sections, non-traditional studies of electrical
breakdown, that move beyond the uniform and static gas background conditions,
are also required to understand the development of NPD discharges in flames.

Since this nonuniform and unsteady environment will guide the plasma dynamics
and energy deposition process, it needs to be accounted for when designing an
actuation strategy. In particular, our ability to address the grand challenges
in combustion discussed in section 1 will require consideration of:

  * 1.

Plasma response to combustion environment, when designing actuation strategies
for control of combustion dynamics. The plasma can evolve at the timescale of
the instability, in the worst case resulting in a weakened discharge as the
flame detaches from the flame-holder element. Our flameholding ability relies
on accounting for this two-way coupling and controlling the plasma parameters
by other means, if necessary.

  * 2.

Reinforcement of plasma in hot spots, and flamelets, when considering
turbulent flames. Typically, the discharge volume is significantly smaller
than the flame volume, and the location of the plasma source affects the
degree of combustion enhancement, or disturbance. For situations where the
discharge characteristic size is comparable to the flame features,
reinforcement of the plasma in high temperature regions will be encountered.
This can have a beneficial effect, if plasma deposition occurs close to the
reaction front, but can also lead to wasted energy, if the energy deposition
happens in the post-combustion gases.

  * 3.

Realistic flow conditions, when characterizing plasma sources for practical
system. The implications of fast flowing gases are typically overlooked and
affect the plasma dynamics, the spatial structure of the energy deposition,
the synergy between pulses, and the plasma regime transitions.

##  7 Conclusions

The problem of plasma-assisted combustion has been presented in terms of a
two-way problem. The forward problem refers to the ability of impacting
ignition and flames using a plasma. This side of the interaction is the driver
of research in the field and is widely documented in the literature, both in
terms of fundamental efforts and practical demonstrations. The backward
problem refers to the influence of combustion on electrical breakdown and
discharge dynamics. This side of the interaction is the focus of this paper
and has been less documented in the literature, with most prior efforts
focusing on detailed kinetic characterization of the plasma in uniform
background gases. This article has presented a series of experiments that
exemplify the relevance of addressing the backward problem, in particular when
considering practical applications and dynamic combustion environments. It is
argued that, when using pulsed nanosecond discharges for control of dynamic
processes, or when the characteristic length scales of combustion are
comparable to the inter-electrode gap length, the coupled interaction can not
be overlooked. In these cases, the success of the strategy may well depend on
properly accounting for the backward problem.

## Acknowledgements

This work was partially funded by the Office of Naval Research (ONR) Young
Investigator Award _Transient Corona Discharges for Ignition and Flameholding
in Navy Relevant Combustion_ , N00014-21-1-2571. The authors would like to
acknowledge fruitful collaborations throughout the years with the co-authors
of some of the works reviewed in this paper: M. Martinez-Sanchez, A.
Starikovskiy, R. Miles, F. Gomez del Campo, D. Weibel, S. Shanbhogue, and A.
Ghoniem.

## References

  * [1] W. Heiser, D. Pratt, D. Daley, U. Mehta, Hypersonic Airbreathing Propulsion, American Institute of Aeronautics and Astronautics, Inc., Washington, DC, 1994\.  doi:10.2514/4.470356.
  * [2] Q. Liu, D. Baccarella, T. Lee, Review of combustion stabilization for hypersonic airbreathing propulsion, Progress in Aerospace Sciences 119 (2020) 100636.  doi:10.1016/j.paerosci.2020.100636.
  * [3] B. McGann, C. D. Carter, T. Ombrello, H. Do, Direct spectrum matching of laser-induced breakdown for concentration and gas density measurements in turbulent reacting flows, Combustion and Flame 162 (12) (2015) 4479–4485.  doi:10.1016/j.combustflame.2015.08.021.
  * [4] S. Leonov, D. Yarantsev, C. Carter, Experiments on Electrically Controlled Flameholding on a Plane Wall in a Supersonic Airflow, Journal of Propulsion and Power 25 (2) (2009) 289–294.  doi:10.2514/1.38002.
  * [5] D. S. Lee, D. W. Fahey, A. Skowron, M. R. Allen, U. Burkhardt, Q. Chen, S. J. Doherty, S. Freeman, P. M. Forster, J. Fuglestvedt, A. Gettelman, R. R. De León, L. L. Lim, M. T. Lund, R. J. Millar, B. Owen, J. E. Penner, G. Pitari, M. J. Prather, R. Sausen, L. J. Wilcox, The contribution of global aviation to anthropogenic climate forcing for 2000 to 2018, Atmospheric Environment 244 (2020).  doi:10.1016/j.atmosenv.2020.117834.
  * [6] S. R. Turns, An introduction to combustion, concepts and applications, 2nd Edition, McGraw-Hill Higher Education, 2000.
  * [7] T. C. Lieuwen, Unsteady combustor physics, Cambridge University Press, 2012.  doi:10.1017/9781108889001.
  * [8] A. R. Masri, Challenges for turbulent combustion, Proceedings of the Combustion Institute 38 (1) (2021) 121–155.  doi:10.1016/j.proci.2020.07.144.
  * [9] H. Kobayashi, A. Hayakawa, K. D. A. Somarathne, E. C. Okafor, Science and technology of ammonia combustion, Proceedings of the Combustion Institute 37 (1) (2019) 109–133.  doi:10.1016/j.proci.2018.09.029.
  * [10] Y. Ju, W. Sun, Plasma assisted combustion: Dynamics and chemistry, Progress in Energy and Combustion Science 48 (2015) 21–83.  doi:10.1016/j.pecs.2014.12.002.
  * [11] A. Starikovskiy, Physics and chemistry of plasma-assisted combustion, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373 (2015) 20150074.  doi:10.1098/rsta.2015.0074.
  * [12] I. V. Adamovich, W. R. Lempert, Challenges in understanding and predictive model development of plasma-assisted combustion, Plasma Physics and Controlled Fusion 57 (1) (2015).  doi:10.1088/0741-3335/57/1/014001.
  * [13] K. Shinohara, N. Takada, K. Sasaki, Enhancement of burning velocity in premixed burner flame by irradiating microwave power, Journal of Physics D: Applied Physics 42 (18) (2009) 182008.
  * [14] E. S. Stockman, S. H. Zaidi, R. B. Miles, C. D. Carter, M. D. Ryan, Measurements of combustion properties in a microwave enhanced flame, Combustion and Flame 156 (7) (2009) 1453–1461.
  * [15] X. Yu, J. Peng, P. Yang, R. Sun, Y. Yi, Y. Zhao, D. Chen, J. Yu, Enhancement of a laminar premixed methane/oxygen/nitrogen flame speed using femtosecond-laser-induced plasma, Applied Physics Letters 97 (1) (2010) 11503\.
  * [16] A. Elkholy, Y. Shoshyn, S. Nijdam, J. A. van Oijen, E. M. van Veldhuizen, U. Ebert, L. P. de Goey, Burning velocity measurement of lean methane-air flames in a new nanosecond DBD microplasma burner platform, Experimental Thermal and Fluid Science 95 (2018) 18–26.  doi:10.1016/j.expthermflusci.2018.01.011.
  * [17] T. Ombrello, S. H. Won, Y. Ju, S. Williams, Flame propagation enhancement by plasma excitation of oxygen. Part II: effects of O2 (a1$\Delta$g), Combustion and Flame 157 (10) (2010) 1906–1915.
  * [18] G. Pilla, D. Galley, D. A. Lacoste, F. Lacas, D. Veynante, C. O. Laux, Stabilization of a Turbulent premixed flame using a nanosecond repetitively pulsed discharge, IEEE Transactions on Plasma Science 34 (6) (2006) 2471–2477.
  * [19] Q. L. L. Pham, D. A. Lacoste, C. O. Laux, Stabilization of a premixed methane-air flame using nanosecond repetitively pulsed discharges, IEEE Transactions on Plasma Science 39 (11) (2011) 2264–2265.  doi:10.1109/TPS.2011.2163806.
  * [20] F. Di Sabatino, D. A. Lacoste, Enhancement of the lean stability and blow-off limits of methane-air swirl flames at elevated pressures by nanosecond repetitively pulsed discharges, Journal of Physics D: Applied Physics 53 (35) (2020) 355201.  doi:10.1088/1361-6463/ab8f54.
  * [21] S. Barbosa, G. Pilla, D. A. Lacoste, P. Scouflaire, S. Ducruix, C. O. Laux, D. Veynante, Influence of nanosecond repetitively pulsed discharges on the stability of a swirled propane/air burner representative of an aeronautical combustor, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373 (2048) (2015).  doi:10.1098/rsta.2014.0335.
  * [22] J. Choe, W. Sun, Blowoff hysteresis, flame morphology and the effect of plasma in a swirling flow, Journal of Physics D: Applied Physics 51 (36) (2018).  doi:10.1088/1361-6463/aad4dc.
  * [23] J. P. Moeck, D. A. Lacoste, C. O. Laux, C. O. Paschereit, Control of combustion dynamics in a swirl-stabilized combustor with nanosecond repetitively pulsed discharges, 51st AIAA Meeting, Grapevine, Texas (2013) 2013–0565.
  * [24] S. J. Shanbhogue, C. A. Pavan, D. E. Weibel, F. Gomez del Campo, C. Guerra-Garcia, A. F. Ghoniem, Control of Large-Amplitude Combustion Oscillations Using Nanosecond Repetitively Pulsed Plasmas, Journal of Propulsion and Power (2023) 1–13doi:10.2514/1.B38883.
  * [25] G. Faingold, J. K. Lefkowitz, A numerical investigation of NH3/O2/He ignition limits in a non-thermal plasma, Proceedings of the Combustion Institute 38 (4) (2021) 6661–6669.  doi:10.1016/j.proci.2020.08.033.
  * [26] A. Shioyoke, J. Hayashi, R. Murai, N. Nakatsuka, F. Akamatsu, Numerical Investigation on Effects of Nonequilibrium Plasma on Laminar Burning Velocity of Ammonia Flame, Energy and Fuels 32 (3) (2018) 3824–3832.  doi:10.1021/acs.energyfuels.7b02733.
  * [27] T. S. Taneja, S. Yang, Numerical modeling of plasma assisted pyrolysis and combustion of ammonia, in: AIAA Scitech 2021 Forum, no. January, 2021, pp. 2021–1972.  doi:10.2514/6.2021-1972.
  * [28] J. Choe, W. Sun, T. Ombrello, C. Carter, Plasma assisted ammonia combustion: Simultaneous NOx reduction and flame enhancement, Combustion and Flame 228 (2021) 430–432.  doi:10.1016/j.combustflame.2021.02.016.
  * [29] N. A. Popov, S. M. Starikovskaia, Relaxation of electronic excitation in nitrogen/oxygen and fuel/air mixtures: fast gas heating in plasma-assisted ignition and flame stabilization, Progress in Energy and Combustion Science 91 (2022) 100928.  doi:10.1016/j.pecs.2021.100928.
  * [30] C. Guerra-Garcia, M. Martinez-Sanchez, Counterflow nonpremixed flame DC displacement under AC electric field, Combustion and Flame 162 (11) (2015) 4254–4263.  doi:10.1016/j.combustflame.2015.07.038.
  * [31] D. A. Xu, D. A. Lacoste, D. L. Rusterholtz, P. Q. Elias, G. D. Stancu, C. O. Laux, Experimental study of the hydrodynamic expansion following a nanosecond repetitively pulsed discharge in air, Applied Physics Letters 99 (12) (2011) 121502.  doi:10.1063/1.3641413.
  * [32] Y. Raitzer, Gas Discharge Physics, Springer-Verlag, Berlin, 1991.
  * [33] A. Starikovskiy, N. Aleksandrov, Review: Plasma-assisted ignition and combustion, Progress in Energy and Combustion Science 39 (1) (2013) 61–110.  doi:10.1016/j.pecs.2012.05.003.
  * [34] S. Nagaraja, V. Yang, I. Adamovich, Multi-scale modelling of pulsed nanosecond dielectric barrier plasma discharges in plane-to-plane geometry, Journal of Physics D: Applied Physics 46 (15) (2013).  doi:10.1088/0022-3727/46/15/155205.
  * [35] N. A. Popov, The effect of nonequilibrium excitation on the ignition of hydrogen-oxygen mixtures, High Temperature 45 (2) (2007) 261–279.  doi:10.1134/S0018151X07020174.
  * [36] N. L. Aleksandrov, S. V. Kindysheva, M. M. Nudnova, A. Y. Starikovskiy, Mechanism of ultra-fast heating in a non-equilibrium weakly ionized air discharge plasma in high electric fields, Journal of Physics D: Applied Physics 43 (25) (2010) 255201.  doi:10.1088/0022-3727/43/25/255201.
  * [37] A. Starikovskiy, N. Aleksandrov, A. Rakitin, Plasma-assisted ignition and deflagration-to-detonation transition, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370 (1960) (2012) 740–773.  doi:10.1098/rsta.2011.0344.
  * [38] G. J. M. Hagelaar, L. C. Pitchford, Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models, Plasma Sources Science and Technology 14 (4) (2005) 722–733.  doi:10.1088/0963-0252/14/4/011.
  * [39] M. Hayashi, Hayashi database (retrieved on November 9, 2021).
URL www.lxcat.net/Hayashi

  * [40] A. Phelps, Phelps database (retrieved on November 9, 2021).
URL www.lxcat.net/Phelps

  * [41] A. Phelps, L. Pitchford, Siglo database (retrieved on November 9, 2021).
URL www.lxcat.net/SIGLO

  * [42] S. K. Dhali, P. F. Williams, Two-dimensional studies of streamers in gases, Journal of Applied Physics 62 (12) (1987) 4696–4707.
  * [43] G. J. M. Hagelaar, G. M. W. Kroesen, G. J. M. Hageelar, G. M. W. Kroesen, Speeding up fluid models for gas discharges by implicit treatment of the electron energy source term, Journal of Computational Physics 159 (1) (2000) 1–12.
  * [44] S. Pancheshnyi, P. Segur, J. Capeillere, A. Bourdon, Numerical simulation of filamentary discharges with parallel adaptive mesh refinement, Journal of Computational Physics 227 (13) (2008) 6574–6590.
  * [45] D. A. Lacoste, Flames with plasmas, Proceedings of the Combustion Institute (2022).  doi:10.1016/j.proci.2022.06.025.
  * [46] W. Sun, M. Uddi, T. Ombrello, S. H. Won, C. Carter, Y. Ju, Effects of non-equilibrium plasma discharge on counterflow diffusion flame extinction, Proceedings of the Combustion Institute 33 (2) (2011) 3211–3218.
  * [47] W. Sun, S. H. Won, T. Ombrello, C. Carter, Y. Ju, Direct ignition and S-curve transition by in situ nano-second pulsed discharge in methane/oxygen/helium counterflow flame, Proceedings of the Combustion Institute 34 (1) (2013) 847–855.
  * [48] S. Stange, Y. Kim, V. Ferreri, L.A.Rosocha, D. M. Coates, Flame Images Indicating Combustion Enhancement by Dielectric Barrier Discharges, IEEE Transactions on Plasma Science 33 (2) (2005).
  * [49] D. Pai, D. A. Lacoste, C. O. Laux, Transitions between corona, glow, and spark regimes of nanosecond repetitively pulsed discharges in air at atmospheric pressure, Journal of Applied Physics 107 (9) (2010) 93303.
  * [50] W. Kim, J. Cohen, Plasma-Assisted Combustor Dynamics Control at Realistic Gas Turbine Conditions, Combustion Science and Technology (2019) 1–20doi:10.1080/00102202.2019.1676743.
  * [51] S. Shanbhogue, D. Weibel, F. Gomez del Campo, C. Guerra-Garcia, A. Ghoniem, Active control of large amplitude combustion oscillations using nanosecond repetitively pulsed plasmas, in: AIAA Scitech 2022 Forum, 2022, pp. 2022–1450.  doi:10.2514/6.2022-1450.
  * [52] X. Mao, H. Zhong, T. Zhang, A. Starikovskiy, Y. Ju, Modeling of the effects of non-equilibrium excitation and electrode geometry on H2/air ignition in a nanosecond plasma discharge, Combustion and Flame 240 (2022) 112046.  doi:10.1016/j.combustflame.2022.112046.
  * [53] S. P. Bane, J. L. Ziegler, J. E. Shepherd, Investigation of the effect of electrode geometry on spark ignition, Combustion and Flame 162 (2) (2015) 462–469.  doi:10.1016/j.combustflame.2014.07.017.
  * [54] J. P. Boeuf, L. L. Yang, L. C. Pitchford, Dynamics of a guided streamer (plasma bullet) in a helium jet in air at atmospheric pressure, Journal of Physics D: Applied Physics 46 (1) (2013) 15201.
  * [55] D. Breden, K. Miki, L. L. Raja, Self-consistent two-dimensional modeling of cold atmospheric-pressure plasma jets/bullets, Plasma Sources Science and Technology 21 (3) (2012) 34011.
  * [56] G. V. Naidis, Modelling of streamer propagation in atmospheric-pressure helium plasma jets, Journal of Physics D: Applied Physics 43 (40) (2010) 402001.
  * [57] G. V. Naidis, Simulation of a Single Streamer Traveling along two counterpropagating Helium jets in ambient air, IEEE Transactions on Plasma Science 40 (11) (2012) 2866–2869.
  * [58] C. Douat, M. Fleury, M. Laroussi, V. Puech, Interactions Between Two Counterpropagating Plasma Bullets, IEEE Transactions on Plasma Science 39 (11) (2011) 2298–2299.
  * [59] G. V. Naidis, Modelling of plasma bullet propagation along a helium jet in ambient air, Journal of Physics D: Applied Physics 44 (21) (2011) 215203.
  * [60] D. Breden, K. Miki, L. L. Raja, Computational study of cold atmospheric nanosecond pulsed helium plasma jet in air, Applied Physics Letters 99 (11) (2011) 111501.
  * [61] C. Guerra-Garcia, M. Martinez-Sanchez, Can insulating gaseous layers provide a dielectric barrier discharge effect?, Applied Physics Letters 106 (4) (2015) 4–7.  doi:10.1063/1.4906417.
  * [62] A. Luque, U. Ebert, Sprites in varying air density: charge conservation, glowing negative trails and changing velocity, Geophysical Research Letters 37 (2010) L06806.
  * [63] A. Luque, U. Ebert, Density models for streamer discharges: beyond cylindrical symmetry and homogeneous media, Journal of Computational Physics 231 (3) (2012) 904–918.
  * [64] D. F. Opaits, M. N. Shneider, P. J. Howard, R. B. Miles, G. M. Milikh, Study of streamers in gradient density air: table top modeling of red sprites, Geophysical Research Letters 37 (2010) L14801.
  * [65] R. S. Sigmond, T. Sigmond, L. Rolfseng, A. F. Bohman, F. T. Stormo, L. Hvidsten, The aiming of the bolt: how a flashover finds the weak spot, IEEE Transactions on Plasma Science 32 (5) (2004) 1812–1818.
  * [66] N. Y. Babaeva, M. J. Kushner, Effect of inhomogeneities on streamer propagation: I. Intersection with isolated bubbles and particles, Plasma Sources Science and Technology 18 (3) (2009) 35009.
  * [67] N. Y. Babaeva, M. J. Kushner, Effect of inhomogeneities on streamer propagation: II. Streamer dynamics in high pressure humid air with bubbles, Plasma Sources Science and Technology 18 (3) (2009) 35010.
  * [68] N. Pillai, N. L. Sponsel, J. T. Mast, M. J. Kushner, I. A. Bolotnov, K. Stapelmann, Plasma breakdown in bubbles passing between two pin electrodes, Journal of Physics D: Applied Physics 55 (47) (2022) 475203.  doi:10.1088/1361-6463/ac9538.
  * [69] N. Y. Babaeva, M. J. Kushner, Structure of positive streamers inside gaseous bubbles immersed in liquids, Journal of Physics D: Applied Physics 42 (13) (2009) 132003.  doi:10.1088/0022-3727/42/13/132003.
  * [70] P. Bruggeman, C. Leys, Non-thermal plasmas in and in contact with liquids, Journal of Physics D: Applied Physics 42 (5) (2009) 53001.
  * [71] A. Y. Starikovskiy, N. L. Aleksandrov, Blocking streamer development by plane gaseous layers of various densities, Plasma Sources Science and Technology 29 (3) (2020) 034002.  doi:10.1088/1361-6595/ab5837.
  * [72] C. Guerra-Garcia, M. Martinez-Sanchez, Gas-confined barrier discharges: a simplified model for plasma dynamics in flame environments, Journal of Physics D: Applied Physics 46 (34) (2013) 345204.  doi:10.1088/0022-3727/46/34/345204.
  * [73] C. Guerra-Garcia, M. Martinez-Sanchez, R. B. Miles, A. Starikovskiy, Localized pulsed nanosecond discharges in a counterflow nonpremixed flame environment, Plasma Sources Science and Technology 24 (5) (2015).  doi:10.1088/0963-0252/24/5/055010.
  * [74] G. G. Raju, Gaseous electronics. Theory and practice, Taylor and Francis Group, 2006.
  * [75] P. J. Chantry, R. E. Wootton, A critique of methods for calculating the dielectric strength of gas mixtures, Journal of Applied Physics 52 (4) (1981) 2731–2739.
  * [76] D. Maric, M. Radmilovic-Radenovic, Z. L. Petrovic, On parametrization and mixture laws for electron ionization coefficients, European Physical Journal D 35 (2005) 313–321.
  * [77] C. Guerra-Garcia, Non-thermal plasmas in flames and other inhomogeneous environments, Ph.D. thesis, Massachusetts Institute of Technology, PhD thesis (2015).
  * [78] I. V. Adamovich, M. Nishihara, I. Choi, M. Uddi, W. R. Lempert, Energy coupling to the plasma in repetitive nanosecond pulse discharges, Physics of Plasmas 16 (11) (2009) 113505.  doi:10.1063/1.3264740.
  * [79] S. Gadkari, S. Gu, Numerical investigation of co-axial DBD: Influence of relative permittivity of the dielectric barrier, applied voltage amplitude, and frequency, Physics of Plasmas 24 (5) (2017) 053517.  doi:10.1063/1.4982657.
  * [80] A. C. Rousso, B. M. Goldberg, T. Y. Chen, S. Wu, A. Dogariu, R. B. Miles, E. Kolemen, Y. Ju, Time and space resolved diagnostics for plasma thermal-chemical instability of fuel oxidation in nanosecond plasma discharges, Plasma Sources Science and Technology 29 (10) (2020) 105012.  doi:10.1088/1361-6595/abb7be.
  * [81] H. Zhong, M. N. Shneider, M. S. Mokrov, Y. Ju, Thermal-chemical instability of weakly ionized plasma in a reactive flow, Journal of Physics D: Applied Physics 52 (48) (2019) 484001.  doi:10.1088/1361-6463/ab3d69.
  * [82] S. B. Leonov, I. V. Adamovich, V. R. Soloviev, Dynamics of near-surface electric discharges and mechanisms of their interaction with the airflow, Plasma Sources Science and Technology 25 (6) (2016) 063001.  doi:10.1088/0963-0252/25/6/063001.
  * [83] S. B. Leonov, D. A. Yarantsev, Near-Surface Electrical Discharge in Supersonic Airflow: Properties and Flow Control, Journal of Propulsion and Power 24 (6) (2008) 1168–1181.  doi:10.2514/1.24585.
  * [84] S. B. Leonov, V. Petrishchev, I. V. Adamovich, Dynamics of energy coupling and thermalization in barrier discharges over dielectric and weakly conducting surfaces on micro-s to ms time scales, Journal of Physics D: Applied Physics 47 (46) (2014) 465201.  doi:10.1088/0022-3727/47/46/465201.
  * [85] A. Houpt, B. Hedlund, S. Leonov, T. Ombrello, C. Carter, Quasi-DC electrical discharge characterization in a supersonic flow, Experiments in Fluids 58 (4) (2017) 25.  doi:10.1007/s00348-016-2295-5.
  * [86] C. Guerra-Garcia, C. A. Pavan, S. Rao, R. J. Dijoud, Influence of Airflow on Nanosecond Pulsed Discharges, in: AIAA SciTech 2023 Forum, American Institute of Aeronautics and Astronautics, Reston, Virginia, 2023, pp. 1–8.  doi:10.2514/6.2023-1866.
  * [87] C. Pavan, M. Martinez-Sanchez, C. Guerra-Garcia, Investigations of positive streamers as quasi-steady structures using reduced order models, Plasma Sources Science and Technology 29 (9) (2020) 95004.  doi:10.1088/1361-6595/aba863.
  * [88] J. Lawton, F. J. Weinberg, Maximum ion currents from flames and the maximum practical effects of applied electric fields, Proceedings of the Royal Society of London. Series A 277 (1371) (1964) 468–497.
  * [89] J. Lawton, F. J. Weinberg, Electrical aspects of combustion, Oxford University Press, 1969.
  * [90] E. S. Semenov, A. S. Sokolik, Thermal and chemical ionization in flames, Combustion, Explosion, and Shock Waves 6 (1) (1970) 37–48.
  * [91] T. Pedersen, R. C. Brown, Simulation of electric field effects in premixed methane flames, Combustion and Flame 94 (4) (1993) 433–448.
  * [92] A. M. Starik, N. S. Titova, Kinetics of ion formation in the volumetric reaction of methane with air, Combustion, Explosion and Shock waves 38 (3) (2002) 253–268.
  * [93] T. A. Cool, P. J. H. Tjossem, Direct observations of chemiionization in hydrocarbon flames enhanced by laser excited CH, Chemical Physics Letters 111 (1-2) (1984) 82–88.
  * [94] C. S. MacLatchy, Langmuir probe measurements of ion density in atmospheric-pressure air-propane flame, Combustion and Flame 36 (1979) 171–178.
  * [95] Y. Ju, S. O. Macheret, M. N. Shneider, R. B. Miles, Numerical study of the effect of microwave discharge on the premixed methane-air flame, 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit (2004) 2004–3707.
  * [96] H. F. Calcote, Ion production and recombination in flames, Symposium (International) on Combustion 8 (1) (1961) 184–199.
  * [97] J. Lawton, P. J. Mayo, F. J. Weinberg, Electrical control of gas flows in combustion processes, Proceedings of the Royal Society of London. Series A 303 (1474) (1968) 275–298.
  * [98] D. J. Sullivan, S. H. Zaidi, S. O. Macheret, Y. Ju, R. B. Miles, Microwave techniques for the combustion enhancement of laminar flames, 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit (2004) 2004–3713.
  * [99] E. M. Stockman, Microwave enhanced combustion of laminar hydrocarbon flame fronts, Ph.D. thesis, Princeton University (2009).
  * [100] J. B. Michael, Localized Microwave Pulsed Plasmas for Ignition and Flame Front Enhancement, Ph.D. thesis, Princeton University (2012).
  * [101] D. Pai, G. D. Stancu, D. A. Lacoste, C. O. Laux, Nanosecond repetitively pulsed discharges in air at atmospheric pressure - the glow regime, Plasma Sources Science and Technology 18 (4) (2009) 45030.
  * [102] C. H. Kruger, C. O. Laux, L. Yu, D. M. Packan, L. Pierrot, Nonequilibrium discharges in air and nitrogen plasmas at atmospheric pressure, Pure and Applied Chemistry 74 (3) (2002) 337–347.
  * [103] A. J. Palmer, A physical model on the initiation of atmospheric-pressure glow discharges, Applied Physics Letters 25 (3) (1974) 138–140.
  * [104] J. I. Levatter, S. Lin, Necessary conditions for the homogeneous formation of pulsed avalanche discharges at high gas pressures, Journal of Applied Physics 51 (1) (1980) 210–222.
  * [105] B.-D. Huang, K. Takashima, X.-M. Zhu, Y.-K. Pu, The influence of the repetition rate on the nanosecond pulsed pin-to-pin microdischarges, Journal of Physics D: Applied Physics 47 (42) (2014) 422003.  doi:10.1088/0022-3727/47/42/422003.
  * [106] B.-D. Huang, E. Carbone, K. Takashima, X.-M. Zhu, U. Czarnetzki, Y.-K. Pu, The effect of the pulse repetition rate on the fast ionization wave discharge, Journal of Physics D: Applied Physics 51 (22) (2018) 225202.  doi:10.1088/1361-6463/aabf2d.
  * [107] N. Q. Minesi, V. P. Blanchard, E. Pannier, G. D. Stancu, C. O. Laux, Plasma-assisted combustion with nanosecond discharges. I: Discharge effects characterization in the burnt gases of a lean flame, Plasma Sources Science and Technology 31 (4) (2022) 045029.  doi:10.1088/1361-6595/ac5cd4.
  * [108] S. V. Pancheshnyi, D. A. Lacoste, A. Bourdon, C. O. Laux, Ignition of propane-air mixtures by a repetitively pulsed nanosecond discharge, IEEE Transactions on Plasma Science 34 (6) (2006) 2478–2487.  doi:10.1109/TPS.2006.876421.
  * [109] J. K. Lefkowitz, T. Ombrello, An exploration of inter-pulse coupling in nanosecond pulsed high frequency discharge ignition, Combustion and Flame 180 (2017) 136–147.  doi:10.1016/j.combustflame.2017.02.032.
  * [110] X. Mao, H. Zhong, Z. Wang, T. Ombrello, Y. Ju, Effects of inter-pulse coupling on nanosecond pulsed high frequency discharge ignition in a flowing mixture, Proceedings of the Combustion Institute (2022) 1–8doi:10.1016/j.proci.2022.06.018.
  * [111] C. A. Pavan, C. Guerra-Garcia, Nanosecond pulsed discharge dynamics during passage of a transient laminar flame, Plasma Sources Science and Technology 31 (11) (2022) 115016.  doi:10.1088/1361-6595/aca0bc.
  * [112] C. Pavan, Y. Zhang, C. Guerra-Garcia, Plasma Actuation of Mesoscale Flames, 2021 AIAA Aviation Forum, Online (2021) 1–11doi:10.2514/6.2021-3105.
  * [113] C. A. Pavan, C. Guerra-Garcia, Modeling Flame Speed Modification by Nanosecond Pulsed Discharges to Inform Experimental Design, in: AIAA SCITECH 2023 Forum, American Institute of Aeronautics and Astronautics, Reston, Virginia, 2023, pp. 1–15.  doi:10.2514/6.2023-2056.
  * [114] C. A. Pavan, C. Guerra-Garcia, D. Weibel, M. Nishihara, F. del Campo, S. Shanbhogue, A. Ghoniem, Nanosecond Pulsed Discharge Dynamics in a Swirl-Stabilized Combustor with an Unstable Flame, in: AIAA Aviation 2022 Forum, American Institute of Aeronautics and Astronautics, Reston, Virginia, 2022, pp. 1–11.  doi:10.2514/6.2022-3574.
  * [115] C. Clanet, G. Searby, On the “tulip flame” phenomenon, Combustion and Flame 105 (1-2) (1996) 225–238.  doi:10.1016/0010-2180(95)00195-6.
  * [116] M. Gonzalez, R. Borghi, A. Saouab, Interaction of a flame front with its self-generated flow in an enclosure: The “tulip flame” phenomenon, Combustion and Flame 88 (2) (1992) 201–220.  doi:10.1016/0010-2180(92)90052-Q.
  * [117] H. Zhong, M. N. Shneider, X. Mao, Y. Ju, Dynamics and chemical mode analysis of plasma thermal-chemical instability, Plasma Sources Science and Technology 30 (3) (2021) 035002.  doi:10.1088/1361-6595/abde1c.
  * [118] D. L. Rusterholtz, D. A. Lacoste, G. D. Stancu, D. Z. Pai, C. O. Laux, Ultrafast heating and oxygen dissociation in atmospheric pressure air by nanosecond repetitively pulsed discharges, Journal of Physics D: Applied Physics 46 (46) (2013) 464010.  doi:10.1088/0022-3727/46/46/464010.
  * [119] D. A. Xu, M. N. Shneider, D. A. Lacoste, C. O. Laux, Thermal and hydrodynamic effects of nanosecond discharges in atmospheric pressure air, Journal of Physics D: Applied Physics 47 (23) (2014) 235202.  doi:10.1088/0022-3727/47/23/235202.