Combustion and Flame 184 (2017) 208–232

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Combustion and Flame

journal homepage: www.elsevier.com/locate/combustflame

Plasma-combustion coupling in a dielectric-barrier discharge actuated

fuel jet

Luca Massa

a , ∗, Jonathan B. Freund

b

a Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA,United States b Mechanical Science & Engineering and Aerospace Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, United States

a r t i c l e i n f o

Article history:

Received 26 November 2016

Revised 8 January 2017

Accepted 14 June 2017

Available online 1 July 2017

Keywords:

Ignition

Plasma-assisted combustion

a b s t r a c t

A plasma-combustion coupling mechanism is proposed and applied to the laser-induced atmospheric-

pressure ignition and combustion of a hydrogen jet as assisted by a dielectric-barrier discharge (DBD).

The specific configuration matches corresponding experiments, and the proposed coupling mechanism

leads to an improvement of the prediction for ignition probability and explains the observed electrical

power increase during burning conditions. To realize this, the model includes the key effects of the fast

DBD microflimentary plasma structure on combustion time scales, which would not be included in a sim-

pler quasi-steady approximation. It also explains observed plasma emission patterns and the dependence

of the DBD power absorbed on the cross-flow velocity. The main conclusion of the present computa-

tional analysis is that the interaction of plasma and combustion supports a two-way coupling rooted in

the electron and neutral energy equations. The coupling selectively amplifies the energy and radical con-

tributions by the discharge at the ignition hot spot. These contributions dominate the evolution of hot

spots interacting with the local electric field over dielectric surfaces and are a key ingredient of predictive

ignition models. Results are discussed in the context of the lower pressure, lower equivalence ratio and

lower dimensional (often premixed and quasi-one-dimensional) studies that provide insights for develop-

ing this integrated model while illuminating the important differences of the coupling in non-premixed

conditions at atmospheric pressure.

© 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

s

t

1. Introduction

It is well-understood that plasma can accelerate combustion

rates [1–6] . While most of this work has focused on how the

electrical discharg e can affect chemical kinetics, Savelkin et al.

[7] have recently shown a strong reverse coupling effect: combus-

tion and mixing can affect the discharge leading to an increase of

the electrical power absorbed and a broadening of the plasma’s ex-

tent. Yet the mechanisms of this two-way coupling are not fully

understood, and it remains uncertain what factors are important

in any particular configuration. We develop a model to analyze the

mechanisms that are active in the laser-induced ignition of a round

fuel jet in a turbulent cross-flow in the presence of a dielectric-

barrier discharge (DBD) plasma. Our goal is to identify and rep-

resent in a simulation how the plasma-combustion interaction af-

fects the ignition probability. Experimental observations for this

∗ Corresponding author.

E-mail addresses: lmassa@vt.edu (L. Massa), jbfreund@illinois.edu (J.B. Freund).

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http://dx.doi.org/10.1016/j.combustflame.2017.06.008

0010-2180/© 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved

ame configuration are recently reported [8,9] ; observations of par-

icular interest are summarized here:

• The DBD plasma enhances ignition above a threshold, although

it also slightly hinders is probability for low applied voltages.

A two-stage ignition and associated blow-off process has been

identified in the regime when the DBD supports ignition.

• Although the actuator is axisymmetric, the plasma light emis-

sion is not once the jet is burning. The absorbed electrical

power is significantly increased by the flame, and, more inter-

estingly, decreases when the flame is in a cross-flow. Both of

these observations indicate coupling between the flow, plasma,

and flame.

• Light emissions at 720 nm (near the water vapor infrared band)

are more intense and more spatially distributed with DBD-

actuation, pointing to additional coupling.

To explain these observations we develop a model for the in-

eraction of plasma and combustion on the time scales of turbu-

ent combustion ( ∼ 10 −2 s). These are fast relative to the flow,

et around 10 6 times slower than the characteristic time for

lectron transport and ionization in atmospheric air ( ∼ 10 −8 s ),

.

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 209

Fig. 1. DBD setup with the electrodes drawn in brown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this

article.)

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hich occurs for reduced electric field strength E / N � 100 Td.

herefore, we anticipate at the outset that a detailed resolu-

ion of the electron transport is unneeded and would likely be

omputationally infeasible. However, despite this time-scale dif-

erence, we can also anticipate that an only weak coupling of

uid and plasma dynamics [10] , whereby the fluid would be

pproximated as unchanging on the plasma time scale, would

iss essential interactions that are tightly coupled at atmospheric

ressure [11, ch. 9] . The primary two-way coupling can be an-

icipated to arise from a decrease in specific collisional loss

ith increasing temperature and Joule heating [ 11 , p. 223]. Be-

ause the ratio between electron drift is expected to far exceed

uid velocity for our conditions ( v d / U ∞

� 10 4 ), the avenue of

uch a coupling is anticipated to be through the relatively slow

hange of the target state that the electrons impact in the their

rift.

The quantitative description of the coupling mechanism we de-

elop is based on the microstructure of DBD plasma, particularly

hat it is supported by many quasi-periodic microstreamers, each

ccurring on a ∼ 10 −9 s time scale. Surface charging provides these

ith an effective memory, so that they recur in the same locations

t each change of the voltage polarity [12] . Our specific model

s motivated by the observed intensification in the presence of

he flame as well as the corresponding electric power absorption

hanges.

At low gas temperature and large electric frequencies, mi-

rostreamers are inefficient and Joule heating is negligible [10] .

owever recent measurements show a dramatic change when

here is a sustained flame, presumably due to the high tempera-

ures supported by turbulent combustion. In this case it is seen

hat the filament coverage of the dielectric surface increases, has

ow intermittency, and tracks the flame location [8] . We propose

hat this coupling is due to the linear decrease in three-body at-

achment to oxygen with increasing temperature. Our analysis re-

ults in a criterion for the formation of self-sustained plasma fila-

ents, which bypass the random avalanche phase [ 11 , p. 328] and

orm in the pre-ionized gas of previous microstreamers. This is de-

eloped to explain these observations and plasma–flame interac-

ions primarily in Section 4.4.3 .

Recent analysis of the effect of plasma and combustion in

ow-dimensional configurations has led to the improvement of ki-

etic models and the coupling between the Boltzmann equation

ith trusted models for neutral chemistry [13,14] . Yet studying

lasma-combustion coupling in one-dimension is difficult. Thermal

oupling between the discharge and the fluid enthalpy contracts

he plasma column [11] and selectively amplifies the energy and

adical production at distinct locations [8] . Experiments demon-

trate the difficulty in obtaining a one-dimensional ignition front,

ven at very low pressure [14] . Realistic, higher-pressure condi-

ions accentuate this [7,8] . Our current target is thus intrinsically

hree-dimensional, and mechanism are integrated into a model

f a genuinely three-dimensional configuration with correspond-

ng experimental observations. Because the time scales of the elec-

ron drift are small and thus neglected, this comparison depends

t

oremost upon the coupling model. Lower-dimensional simulations

ave been used to determine some model parameters, though in a

ay such that the three-dimensional results are true predictions.

The following Section 2 provides a description of the target

onfiguration and measurements. Then, Section 3 provides addi-

ional detailed motivation and reviews the specific assumptions

nvoked in crafting the integrated model. Section 4 describes

he ion-chemistry model, Section 5 the practical tabulation strat-

gy to include the plasma sources in the governing equations,

ection 6 the governing equations, Section 7 the simulation strat-

gy, and Section 8 the results. Conclusions are revisited with addi-

ional discussion in Section 9 .

. Experiments

.1. Apparatus

The experiments were conducted in a subsonic windtunnel

ith a test section of 0.4 m × 0.4 m cross section and 1.19 m

ong. A 40-grit sandpaper roughness strip of total height 1.64 and

0.8 mm width trips the boundary layer turbulent 333 mm up-

tream of the center of the fuel port. PIV measurements confirm

hat a fully-developed turbulent boundary layer was obtained [9] .

The hydrogen fuel enters vertically through the windtunnel

oor through a port with diameter D H = 4 . 83 mm at flow rate Q =7 . 83 cm

3 / s . The Reynolds number in the tube is Re D = 4 Q /νD

2 H

=4 . 5 , so the fuel flow is laminar. A laser-induced optical break-

own with measured power P int = 17 . 64 ± 6 . 12 mJ was used to ig-

ite the fuel.

The DBD actuator shown in Fig. 1 was operated with a 12 kV,

0 kHz sine-wave. The dielectric material is quartz; the exposed

lectrode is a coaxially aligned copper tube with wall thickness

.51 mm and is recessed 4.8 mm from the top surface of the

uartz, which in turn is flush with the windtunnel floor. The other

lectrode is buried 4.425 mm below the exposed quartz surface.

t is a ring of thickness 0.4 mm that extends radially from r i = . 375 mm r e = 19 . 05 mm.

.2. Measurements

A complete description the measurements used to support the

nalysis in the present study is reported elsewhere [8,9] . The prob-

bility of igniting the H 2 jet was measured as the position of a

aser spark varied. For each breakdown position, 50 independent

xperiments were conducted, in which the laser energy and flame

gnition status were determined. Ignition was determined based on

oth schlieren imaging and water emission spectra from the ro-

ovibrational bands at 717 nm [15,16] using a lens with a 720 nm

10 nm FWHM) bandpass filter. The discharges were analyzed with

ight emission and power measurements to ascertain their contri-

ution of streamers to the current and electric power coupled into

he fluid.

210 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 2. Flattening of standburner flame with increasing voltage �V filtered emis-

sions (water ∼ 720 nm) for �V = 0 , �V = 5 kV, �V = 7 kV, and �V = 8 . 8 kV.

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3. Model design: motivation and main assumptions

3.1. Motivation

As listed in Section 1 , even the visible-light emissions suggest

that a coupling occurs between the plasma, combustion, and flow.

The obvious emissions without cross-flow have a distinct cylindri-

cal symmetry ( Fig. 3 a); when the H 2 is burning in a 15 m/s cross-

flow, the corresponding emissions are skewed downstream, and

the region of brightest plasma glow overlaps the plume, as seen

in ( Fig. 3 b). A corresponding 10-fold increase in the electric power

is required to maintain the plasma [8] . An H 2 flame increases the

power at all voltages by a factor of approximately 1.5, indicating

additional coupling [8] . Retter et al. [8] also show that the pres-

ence of the flame, or even just the heating of the dielectric sur-

face, cause a transition between highly intermittent microstream-

ers to a diffuse, repetitive filamentary plasma. This transition is ac-

companied by an increase of the anodic current and a flattening

of the flame, anticipated to be caused by Coulombic body forces.

The flame flattening substantially intensifies emissions, which we

anticipate to be due to the overlapping of the flame and the elec-

tric field layer. An example of this flattening is shown in Fig. 2 .

Flame-on measurements also manifest an increase in frequency

and magnitude of the current pulses when the exposed electrode

acts as the anode. Therefore, this effect would be consistent with

the flame energizing the anode-directed microstreamers formed

near the dielectric wall, as seen in Figs. 7 and 8 of Retter et al. [8] .

Modeling plasma-combustion coupling in a turbulent field is

challenging because of the inherent three-dimensionality of tur-

bulence, the separation between chemical induction and electron

drift temporal and spatial scales, and the stochastic character of

the electric field supported by the dielectric barrier discharge. The

following observations are deemed to be particular important in

developing a predictive model [8] :

• The axial symmetry of the electrodes leads to a primarily ra-

dial inhomogeneity in electric field. Retter et al. [8] report that

changing the size of the ring electrode significantly changes the

distribution of the microdischarges in the flame-on measure-

ments. Thus, the radial geometry must be accounted for, which

is done by solving an axisymmetric Poisson equation for the

electrostatic potential.

• Plasma is localized near the electrodes and orifice, and its cou-

pling with the flow and combustion is only in the sheath layer

( λ ≈ 1 mm), which is thinner than the momentum boundary

layer ( δ∗ ≈ 2 mm) [17] .

• Plasma exists in a region where we anticipate there to be a

mixture of air and H 2 . These have significantly different at-

tachment cross sections (electro-positivity), so it is expected

that mixing affects the plasma formation, which will likely be

spatially dependent. This is supported by the observation that

flowing H 2 enlarges the region where microdischarges occur

[8] , consistent with the weaker electropositivity of H and thus

2

lower ionization breakdown potential. This effect is not due

to plasma advection because a comparable Q of air does not

change the plasma emissions [8] .

• These same measures show that an increase in temperature re-

duces the electric field necessary to sustain the plasma, with

greater sensitivity at lower gas temperature. A reduction of

collisional losses is expected to provide this phenomenology.

Molecular oxygen, in particular, has a strong three-body attach-

ment, which might be anticipated to lead to the linear reduc-

tion of attachment rate with the temperature and a consequent

reduction of the breakdown threshold. This translates in a re-

duction of the specific collisional energy loss per electron-ion

pair produced [ 18 , p. 81].

• The laser-induced breakdown is intense and forms a localized

overvoltage on the dielectric surface that induces the genera-

tion of microstreamers on a time-scale much faster than that of

ignition. Plasma-ignition coupling is therefore due to the low-

ering of the particle density N and corresponding increase in

reduced electric field E / N .

• At the atmospheric pressure, most ionization is expected to oc-

cur at peak electric-field strength because of the fast decrease

in electronic temperature after the micro discharge extinguish-

ment (see Section 1 ). Subsequent production of electrons, in

the afterglow of this nominal event, is expected to be bal-

anced by recombination in the microdischarge remnant [19] . A

quasi-steady approximation, which is deemed accurate in some

regimes [ 20 , p. 228], would therefore risk mis-representation of

key features of such a process.

Based on these observation and anticipate underlying effects,

e identify three main modeling needs. (1) The first is the quan-

itative representation of the pressure, temperature, composition

nd external electric-field conditions for microdischarge existence,

ince this will so affect the extend and intensity of the plasma.

iven the plasma character, it is also important to represent both

he (2) plasma-to-fluid energy transfer rates and (3) chemical rates

f radical production. The specific assumptions invoked to develop

hese models are discussed in the following subsection.

.2. Assumptions

To develop a model for the effect of the microdischarges [12] ,

e invoke the filamentary plasma structure as proposed by Frid-

an et al. [19] . This is predicated on the assumption that a slow

urface-charge relaxation leads to microdischarges reforming on

he same spot with each change of polarity. The electric field in

he microdischarges can be considered the superposition of three

ontributions: an external component supported by the electrodes,

n internal component due to the polarization space charge at the

treamer head, and a surface component due to charge deposi-

ion on the dielectric walls [21] . Because the fields nearly can-

el after a microdischarge, and assuming that the local field fol-

ows the external value (see justification below), we propose to

odel a filament with a duty cycle that reflects the local elec-

ric field. We note that this is an alternative approach to that pro-

osed by Goldberg et al. [22] , who determine the field by solving

f a decoupled one-dimensional discharge with the drift-diffusion

pproximation.

.2.1. External field

The external field is deemed that which is present in sheaths

lose to the boundaries, governed by a screened Poisson equation

· (ε∇ φ)

= F ( φ) ,

here F is assumed to be independent of the combustion and

ixing events and is evaluated as described at the beginning of

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 211

Fig. 3. Visible-light plasma emissions in the DBD actuator [8] at 12 kV.

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ection 8 . Thus, the local charge F depends only on the local po-

ential, which is an established approximation [23,24] . In essence,

t follows from three principal assumptions: (1) the region with

ignificant charge is localized in thin sheets along the boundary,

2) the plasma is in a Boltzmann equilibrium, and (3) the Poisson–

oltzmann equation can be linearized. This view is supported by

he analysis in Section 8 , where the body forces and sheath ex-

ension are deduced based on the measured velocity field over an

ctuated stand-burner without cross-flow.

.2.2. Space and surface charge fields

Within microstreamers, the (combined) electric field will in-

lude both the influence of the polarization space charge, which

esponds quickly to reduce the field after the ionization wave,

nd of the transfer of charge to the dielectric surface, which

loses the conductive channel, thus extinguishing the discharge.

his scenario is fundamentally multi-dimensional, whereby trans-

erse derivatives of the polarization space charge create a non-

lanar wave with a strong peak [25] . We assume a wave-

orm based on similarity, based on the lack of geometrical

ength scale, the slow time-dependence of the external field,

nd a negligible Peclet number dependence [26,27] . In this

imit, the reactive time and the drift velocity provide the rel-

vant scales [26,27] . Insensitivity to initial conditions has been

stablished [28] .

In many combustion conditions, the variation of fluid transport

oefficients with the mixture fraction is small compared to their

emperature dependence [29] . We assume this also for the drift-

iffusion equations, with relatively low sensitivity of transport to

eutral particle density N . Using μ0 and χ0 to denote respec-

ively the mobility and Townsend coefficient at the reference state,

ives time scale τ ≡ 1 μ0 χ0 E 0

and length scale ≡ 1 χ0

. Based on first-

rinciples arguments (e.g., Hagelaar and Pitchford [30] ), χ∝ N and

∝ N

−1 , so ∝ N

−1 , and τ∝ E 0 .

There are two possible limits that provide an E 0 : it can match

he field ahead of the wave E ∞

, and so be independent of N , or the

haracteristic field of the space charge E sc ≡ n 0 e /( χ0 ɛ 0 ), where n 0 s the plasma density and e the unit charge. If the wave propaga-

ion is driven by photo-ionization, n 0 ∝ N

2 [26] and the time scale

ased on the space charge is τsc ∝ N

−1 . There are three situations

hen E 0 ≈ E ∞

, and for our purposes we can neglect the quadratic

ependence on n 0 : (1) short-lived streamers, for which the space

harge is set by the initiation condition, giving E sc = E ∞

[31,32] ; (2)

treamers driven by pre-ionization, for which n 0 is constant and

atches the pre-breakdown density [33] ; and (3) streamers that

re well-described by a one-dimensional ionization waves [21] . We

ake E 0 ≈ E ∞

because conditions (1) and (2) are both satisfied.

he success of similitude based on E 0 describing the experiments

nd computations of Pancheshnyi et al. [25] provides additional

upport.

With E 0 ≈ E ∞

time scale is independent of N (and thus evolu-

ion of the combustion), we assume

ˆ ≡ | ∇ φ| , E(r, y, t)

=

ˆ E ( r, y )

⎡

⎣

tanh

(δc −2 t m

2 t f

)+ tanh

(δc +2 t m

2 t f

)2

⎤

⎦ | sin ( ω act t ) | ,

ith t m

≡ mod (t − δD + T act / 4 , T act / 2) − T act / 4 . The time-period is

act ≡ 1 / f = 2 π/ω act . The parameters δD , t f , δc are set to match the

ower in cold flow in Section 4.6 based on the same anticipated

emperature independence of the pulse form.

With the approximation ∝ T established, the fast ionization

ave thickness �a that can exceed 1 cm [34] for high temper-

tures and low densities. When the characteristic thickness ex-

eeds the geometrical scale of the actuator �g , quenching of sur-

ace charge is expected to introduce an dependence on �g / �a . To

ccount for such an effect, we limit the wave thickness in (3.2.2)

ith the cut-off wave-length

∗c = min ( δc , �g / ( μ0 E 0 θ ) )

n place of δc . The electron mobility μ0 is evaluated at the refer-

nce electric field E 0 , and θ is the ratio between wave velocity and

lectron mobility, which is taken to be θ = 10 based on fast-wave

xperiments [21] . The reference length of the actuator is the ra-

ial extent of the covered electrode, �g = 1 . 9 cm (cf. Section 2.1 ).

he use of (3.2.2) also limits Joule heating and radical generation

or large E / N . Overall, for plasma-combustion coupling, it reduces

he thermal instability of atmospheric plasmas induced by the mu-

ual reinforcement of Joule heating and temperature [ 11 , p. 222].

therwise, it does not affect the scaling at low temperature and,

herefore, parameters can be selected to match baseline cold-flow

ower data.

.2.3. Charged particle transport

Because plasma time scales so exceed those of the flow, elec-

ron and ion transport are not resolved. Here we confirm that in

he fast-wave limit θ � 1 their contribution to the overall ioniza-

ion budget will be small, especially when compared to the colli-

ional source terms. First, ion transport in the remnants is negli-

ible compared to ion production and destruction in the pulse be-

ause the Bohm velocity in the afterglow is expected to be small.

his assertion follows from δc / T act 1 and the fast electron tem-

erature relaxation time (as discussed in Section 1 ), so that the

lectron energy drops quickly after the pulse, the Bohm velocity

t the sheath edge is small, and plasma ion losses are small [ 18 ,

p. 376–380]. Second, in an ionization wave propagating at several

imes the electron drift velocity, electron transport is important

nly in determining the polarization space charge, while collisions

ominate particle production in the wave, as supported by planar

elf-similar theory [21, ch. 3] and considered in more detail in the

ollowing. Taking θ ≡ V / μE , α = μ| E | χ the ionization frequency,

0

212 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 4. Contributions to the electron density change across a planar ionization wave.

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(

i

e

χ ≡ A exp

(− ˆ E / ̃ E

)the Townsend coefficient, μ the constant elec-

tron mobility, ˜ E ≡ | E | /E 0 the scaled electric field, then the change

in electron density ( n e ≡ N e − ) across an anode-directed wave is

n e, −∞

− n e, ∞

=

∫ ∞

−∞

χ ˜ E − ∂ ̃ E ∂ξ

θ − ˜ E n e d ξ .

Because the two members in the numerator of the integrand do

not change sign, the two terms

T 1 ≡∫ ∞

−∞

χ ˜ E

θ− ˜ E n e d ξ

n e, −∞

− n e, ∞

T 2 ≡∫ ∞

−∞

∂ ̃ E ∂ξ

θ− ˜ E n e d ξ

n e, −∞

− n e, ∞

(1)

represent the contributions of collisional process and drift relative

to the particle production. They are plotted against θ in Fig. 4 for

three values of the ionization parameter ˆ E . We confirm that for

realistic values θ ≈ 10 [21] , the contribution of drift is negligible

in the determination of the particle production. We will use this

result to remove the electron transport terms and obtain the cou-

pling terms through an ODE solution illustrated in Section 4.4 .

4. Ion chemistry and plasma model

4.1. Overview and parameterization

The objective of the ion chemistry model is to represent

plasma-combustion of our weakly ionized H 2 /O 2 /N 2 /H 2 O mixtures

for temperatures T = 300 to 30 0 0 K and pressures near atmo-

spheric. Because the ionization is weak, collisions between elec-

trons and radicals are neglected during the pulse. Likewise, with

only one exception (reaction 6 in (2) ) the bimolecular reactions

between ions and radicals are assumed not to affect the radical

production and energy transfers supported by the plasma.

Parameters for the ion chemistry scheme for the plasma are

based on six main sources: (1) Tawara et al. [35] for the cross sec-

tions of the collisions between hydrogen molecules and ions with

electrons, (2) Itikawa [36] for the cross section of oxygen molecules

and ions, (3) Itikawa and Mason [37] for the cross sections of wa-

ter molecules, (4) Buckman and Phelps [38] for the cross sections

of nitrogen molecules, (5) Aleksandrov et al. [39] for the ther-

mal rates of plasma decay in the afterglow (remnant) period of

the microdischarge, and (6) the UMIST database [40] (and refer-

ences therein) for the other reactions induced by collisions be-

tween heavy particles. Additional data [41] have also been used to

extend viable temperature ranges of some of the bimolecular rates

to the high temperatures in the flame and igniter.

All collisions leading to excitation, ionization and attachment

are included, and all associated energy transfers affect electron

energy distribution function and the energy coupled in the gas

hase (see Section 8.4 ). However, not all excited states affect com-

ustion through non-equilibrium initiation of the chain branching.

adical generation from from vibrationally excited nitrogen was

hown to be unimportant at low pressure (50 Torr) [42] ; we ex-

ect their effect to be smaller in the current conditions. Radical

roduction by electronically excited (triplet) nitrogen was shown

o be comparable to the electron impact dissociation of H 2 by

agaraja et al. [43] at low pressure (25 Torr) and low equiva-

ence ratios (premixed mixture with φ ≈ 0.1). Their relative impor-

ance is smaller at atmospheric pressure because (1) the quench-

ng decay-time is ∝ 1/ P [44] , while the microdischarge interval

25] and the electric (actuation) period do not scale with the pres-

ure; (2) strong non-linear effects [45] result in the peak concen-

ration of N

∗2 behind a fast ionization wave to be substantially

igher at lower pressure, while the electron number density and

nergy are weakly affected. The equivalence ratio plays also a role

n determining relevant radical production paths: the direct path

s dominant in non-premixed combustion because the majority of

adicals are produced close or above stoichiometric conditions, as

iscussed in more details in Appendix A. Thus, triplet nitrogen

s assumed in a quasi-steady state with equilibrium statistics for

ollisions.

Electronic excitations of oxygen states include the transitions

o O 2 (a 1 �g ), O 2 ( b

1 �+ g ) , O 2 (

1, 3 �g ), O 2 ( B

3 �−u

) , O 2 ( 1 �u ). The first

wo states are also discussed in Appendix A, while the other

hree lead via pre-dissociation to corresponding excited states

f atomic oxygen [46] . Atomic oxygen can also be assumed in

hemical quasi-equilibrium [47] because (1) the time scales of

are fast compared to other processes in the reaction layer

f H 2 / Airflames , and (2) the chain-branching initiation of the

2 /O 2 system depends, to the leading order in X H 2 /X O 2

, on

he concentration of H only [47] . Corrections of higher order in

H 2 /X O 2

are implemented using the scaling described by Boivin

t al. [47] . We also remark that in our conditions the fuel en-

rgy density ρH 2 �HHV is much larger than that coupled by the

lasma,

H 2 Q �HHV ≈ 250 W �

∫ ˙ ω E d V ≈ 25 W,

hus the radical production by the flame pool is much stronger

han that of the discharge. This is different from the low-pressure

xperiments of Nagaraja et al. [43] , where the power coupled by

he plasma accounts for ≈ 50% of the heat of combustion. In our

onditions, the DBD-produced radicals are relevant only during the

hain-initiation. For these reasons, we focus on the plasma contri-

utions to H production.

The rate constants and branching ratios of all dissociative re-

ombination reactions were obtained using the cross section data

isted below and the electron density distribution function from a

oltzmann equation solution, as described in Section 4.3 . The to-

al cross section for dissociative excitation of H 2 is evaluated by

umming the contributions of predissociation and dissociation via

xcitation to Lyman, Werner and metastable states [35] . The pre-

issociation cross section is in turn the sum of the excitation to

he lowest repulsive state H 2 ( b 3 �u

+) , and to H 2 ( a 3 �g +) that de-

ays to H 2 ( b 3 �u

+) via fast radiative-collisional exchange [4 8,4 9] .

he dissociative recombination cross section for H

+ 2 (reaction 4 in

ection 4.2 ) is that of Tawara et al. [35] , for O

+ 2 (reaction 10) is

iven by Peverall et al. [50] , the cross section for N

+ 2 (reaction 17)

s from Sheehan and St-Maurice [51] , the cross section for H 2 O

+

reaction 21) is from Rosén et al. [52] , that for H 3 O

+ (reaction 25)

s from Neau et al. [53] , that for N 2 H

+ (reaction 26) is from Vigren

t al. [54] .

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 213

4

4

.

t

w

y

n

s

d

f

4

.

T

e

o

t

o

e

h

a

t

t

t

a

a

a

t

r

p

o

G

c

i

4

T

s

w

t

n

4

w

g

h

[

l

e

.

4

B

.2. Reactions

.2.1. Hydrogen reactions

The represented reactions primarily involving hydrogen are

1. Ionization: e − + H 2 − > H

+ 2 + 2 e −

2. Dissociative attachment: e − + H 2 − > H

− + H

3. Ion–ion recombination: H

− + H

+ 2 − > H 2 + H

4. Dissociative recombination: H

+ 2 + e −− > 2 H

5. Electron collision detachment: H

− + e −− > H + 2 e −

6. Associative detachment: H

− + H − > H 2 + e −

7. Dissociation: H 2 + e −− > 2 H + e −

(2)

Associative detachment has a limited but consistent effect on

he model. The H concentration in reaction 6 is assumed constant

ithin the ion chemistry solution based on the time-scale anal-

sis reported below in Section 4.4.1 , thus its value is set by the

eutral chemistry evolution with reported rate parameters [55] . A

ensitivity analysis of the other reaction paths for the associative

etachment from H

− [40] suggest at most secondary importance

or to present configuration.

.2.2. Oxygen reactions

The represented reaching primarily involving oxygen are

8. Ionization: e − + O 2 − > O

+ 2 + 2 e −

9a. 3B attachment: e − + O 2 + O 2 − > O

−2 + O 2

9b. 3B attachment: e − + O 2 + N 2 − > O

−2 + N 2

9c. 3B attachment: e − + O 2 + H 2 O − > O

−2 + H 2 O

9d. 2B attachment: e − + O 2 − > O + O

−

10. Dissociative recombination: O

+ 2 + e −− > O + O

11a. Detachment: O

−2 + O 2 − > e − + 2 O 2

11b. Detachment: O

−2 + N 2 − > e − + O 2 + N 2

11c. Hydration: O

−2 + H 2 O + M − > H 2 O

−3 + M

12a. 3B ion–ion: O

−2 + O

+ 2 + M − > 2 O 2 + M

12b. 2B ion–ion: O

−2 + O

+ 2 − > 2 O 2

13. Ion–ion: O

− + H

+ 2 − > O + H 2

14. Detachment: H 2 + O

−− > H 2 O + e −

15. Charge transfer: O 2 + O

−− > O

−2 + O

(3)

he first six reactions are evaluated by integrating the electron en-

rgy distribution function (eedf; see Section 4.3 ). Two reported sets

f cross sections have been used: The electron impact cross sec-

ions of Itikawa and Mason [37] yield essential the same results as

thers that are available in other sources [38,56] . Following Skalny

t al. [57] , the rates for processes 9a, 9b and 9c are assumed to

ave identical third-body efficiency. Pack and Phelps [58] report

significantly higher efficiency for the attachment rate when wa-

er is the third body and a significantly lower efficiency when ni-

rogen acts as the third body, but the corresponding changes in

he cross sections are unknown. The formation of cluster ions O

−4

nd N

−4 having higher dissociative recombination rates than the di-

tomic analogs is estimated to be unimportant at the high temper-

ture of hydrogen/air ignition [39] . The ions are assumed to be in

hermal equilibrium, so the corresponding temperature dependent

ates are taken directly from the UMIST data [40] with reported

arameter [39] . The hydration reaction rate of the anion super-

xide radical and water (reaction 11.c above) is that suggested by

ravendeel and De Hoog [59] . The temperature dependence is un-

ertain, and we consider its effect on plasma-combustion coupling

n Section 8.4 .

.2.3. Nitrogen and water reactions

The reactions primarily involving nitrogen and water are

16. Ionization: e − + N 2 − > N

+ 2 + 2 e −

17. Dissociative recombination: N

+ 2 + e −− > 2 N

18. Charge transfer: N

+ 2 + O 2 − > O

+ 2 + N 2

19. Ionization: e −+ H 2 O − > H 2 O

+ +2 e −

20. Charge transfer: H 2 O

+ + O 2 − > O

+ 2 + H 2 O

OH + H γ1

21. Dissociative recombination: H 2 O

+ + e −− > O + H 2 γ2

O + 2 H γ3

22. Dissociative attachment: e − + H 2 O − > H

−+ H + O

23a. Charge transfer: H

+ 2 + H 2 O − > H 2 O

+ + H 2

23b. Ion-neutral: H

+ 2 + H 2 O − > H 3 O

+ + H

24. Charge transfer: N

+ 2 + H 2 O − > H 2 O

+ + N 2

(4)

he cross section and the normalized branching ratios of the dis-

ociative recombination of H 2 O

+ are those of Rosén et al. [52] ,

ho suggests γ1 = 0 . 20 , γ2 = 0 . 09 , γ3 = 0 . 71 . The parameters of

he charge transfer rate are those suggested by Rakshit and War-

eck [60] .

.2.4. Ion clusters

Ion clustering with water are known to alter the discharge in

et air [61,62] . To account for such clusters and generally investi-

ate the effect of hydration on plasmas in hydrogen–air flames we

ave included the relevant reactions for H 3 O

+ H 3 O

+ [53] and N 2 H

+

54] . Rates of ion–ion recombination in local thermodynamic equi-

ibrium (LTE) are from the UMIST data [40] . The reactions consid-

red are

25. Dissociative recombination: H 3 O

+ + e −− > OH + 2 H

26. Dissociative recombination: N 2 H

+ + e −− > N 2 + H

27. Ion-neutral: H 2 O + H 2 O

+ − > H 3 O

+ + OH

28. Ion-neutral: H 2 + H 2 O

+ − > H 3 O

+ + H

29. Ion–ion: H

− + H 3 O

+ − > 2 H + H 2 O

30. Ion–ion: O

−2 + H 3 O

+ − > O 2 + H + H 2 O

31. Ion-neutral: H 2 O + N

+ 2 − > N 2 H

+ + OH

32. Ion-neutral: H 2 + N

+ 2 − > N 2 H

+ + H

33. Ion–ion: H

− + N 2 H

+ − > 2 H + N 2

34. Ion–ion: O

−2 + N 2 H

+ − > O 2 + H + N 2

35. Ion–ion: O

− + H 3 O

+ − > O + H + H 2 O

36. Ion–ion: O

− + N 2 H

+ − > O + H + N 2

37. Ion–ion: N 2 H

+ + H 2 O − > N 2 + H 3 O

+

(5)

.3. Electron Energy distribution function

The electron energy distribution function is governed by a

oltzmann equation, which is crafted in a common form as [30] ,

∂

∂ε

(˜ W F 0 − ˜ D

∂ F 0 ∂ε

)=

˜ S . (6)

214 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

W

C

Table 1

Rate parameters for (10) .

Reaction No. α cm

3 /s β γ

3 7 . 51 × 10 −8 –1/2 0

6 4 . 32 × 10 −9 –0.39 39.40

11a 2 . 7 × 10 −10 1/2 5590

11b 1 . 9 × 10 −12 1/2 4990

11c 2 × 10 −28 M –3 0

12a 2 × 10 −25 M –3/2 0

12b 4 . 2 × 10 −7 –1/2 0

13 2 × 10 −7 –1/2 0

14 7 × 10 −10 0 0

15 7 . 3 × 10 −10 0 890

18 1 . 704 × 10 −11 0.934 –973

20 4 . 6 × 10 −10 0 0

23a 3 . 9 × 10 −9 –1/2 0

23b 3 . 4 × 10 −9 –1/2 0

24 2 . 3 × 10 −9 –1/2 0

27 2 . 1 × 10 −9 –1/2 0

28 6 . 4 × 10 −10 0 0

29 7 . 51 × 10 −8 –1/2 0

30 7 . 51 × 10 −8 –1/2 0

31 5 × 10 −10 –1/2 0

32 2 × 10 −9 0 0

33 7 . 51 × 10 −8 −1/2 0

34 7 . 51 × 10 −8 −1/2 0

35 7 . 51 × 10 −8 −1/2 0

36 7 . 51 × 10 −8 −1/2 0

37 2 . 60 × 10 −9 −1/2 0

a

c

t

t

n

t

c

Z

a

w

c

t

R

w

s

t

p

T

e

c

[

R

f

t

r

r

t

w

w

r

This is deemed active where E / N > 1 Td; thermal equilibrium is

assumed elsewhere. In (6) , ε is the electron energy, ˜ W is the flux

coefficient in the energy coordinate, the diffusion coefficient ˜ D in-

cludes the effect of the heating field | E ( x )/ N | 2 , and

˜ S the reac-

tion source modeled as cross sections of the electron collisions

with neutrals (H 2 , O 2 , N 2 , H 2 O). The low-concentration radicals are

neglected. Mixture rules are based on particle number averaging,˜ =

∑ 4 k =1 x k W k , where x are the mole fractions of the background

species. The 78 cross sections used were taken from the sources

listed in Section 4 .

The electron energy distribution function (eedf) is represented

with a two-term spherical harmonic expansion dependent on the

local electric field. Such a representation is acceptable in the limit

of uniform electric field and collision probabilities over a mean

free-path [63] . The mean free path is l ≡ k B T √

2 πd 2 p � 1 × 10 −7 m,

which is smaller than any essential scales of advection, diffusion,

and reaction of the neutrals. Per the discussion of Section 3.2 , we

neglect transport in (6) , which leads to exponential growth of the

electron number density at a rate reflecting the difference between

ionization and attachment, each weighted by the particle fraction

of each target species. The popular BOLSIG+ package makes this

same approximation [30] , which has been examined critically in

previous plasma-coupled combustion studies [4,5] .

Our implementation was confirmed to reproduce measured

Townsend coefficients for electron impact ionization of pure N 2 ,

H 2 and O 2 [63] . We note one difference from the BOLSIG+ [30] for-

mulation: For the case with equal sharing of energy between the

created pair, our implementation sets

˜ 0 ,k d ε = −γ x k [ εσk ( ε) F 0 ( ε) d ε − 2 ( ε1 ) σk ( ε1 ) F 0 ( ε1 ) d ε1 ] ,

with ε1 = 2 ε + u k for constant energy loss u k . Expanding the dif-

ferential d ε1 = 2 d ε, we find that a factor 4, rather than 2 [30] ,

multiplies the last term in the normalized source term. With this

correction, the relative difference

�ν ≡ 2

νi, BLSG − νi, PLSCM

νi, BLSG + νi, PLSCM

, (7)

in the ionization coefficient obtained using the BOLSIG+ formula

ν i , BLSG is of the same magnitude as the error between measure-

ments and the present theory: relative errors between 10 and 20%

in the interval 100 Td ≤ E / N ≤ 200 Td and insensitivity to the ion-

ization degree α.

4.4. Numerical solution

4.4.1. Ionization rates and chemical time scales

The Boltzmann analysis shows that relevant ion production

1 /νi � 10 −8 s (for E / N � 100 Td) and destruction 1 /νr � 10 −6 (for

E / N ≈ 0) are much faster than the induction time of hydrogen-air

explosions, which have τi ≈ 1 × 10 −5 s [47] . Similarly, flame resi-

dence time over the dielectric surface is approximately equal to

the flow residence time τ r ≈ 1 ms, which is slower still and also

much longer than the electric period T act = 0 . 03 ms , which in turn

is much smaller than the gas heating time [ 11 , p. 219]. Thus, the

neutral concentration and temperature are taken to be fixed during

any particular microdischarge. More specifically, the simulations of

Section 8 suggest ˙ ω E ≈ 5 × 10 8 W / m

3 , thus the gas heating time

is approximately 50 times the electric half-period. Thus, the ion

chemistry can be represented as a system of ordinary differential

equations with coefficients dependent on the slow-varying neutral

chemistry.

4.4.2. Coupled kinetic system

The previous subsection shows that mixing and combustion

times are much longer than the 30 kHz DBD actuation period, so

these concentrations are effectively static with regard to ionization

nd recombination. The ion chemistry reactions do couple with the

ombustion reactions as sources of radicals and energy in the neu-

ral species and energy conservation equations.

The ion reactions of Sections 4.2.1 –4.2.4 lead, as usual, to a sys-

em of ordinary differential rate equations with temperature and

eutral target species compositions approximately constant over

he electric half-period. The represented particle-per-volume ion

oncentrations are

≡[N e − , N H − , N O − , N O −

2 , N O +

2 , N N +

2 , N H 2 O + , N N 2 H + , N H 3 O + , N H +

2

], (8)

nd governed by

d Z k dt

=

∑

j

(ν ′′

j,k − ν ′ j,k

)R j

∏

i

Z ν ′

j,i

i ,

here ν is the stoichiometric matrix and R i are the rate coeffi-

ients. The R i for reactions with electrons depend on the local elec-

ron energy distribution function,

i = x p i N γ

∫ εσi F 0 d ε, (9)

here the N accounts for the neutral concentrations, which is as-

umed constant within the ion system per our time scale separa-

ion. Additionally, in (9) , σ i are the cross sections, γ ≡√

2 e/m , and

i references the corresponding target neutral species for process i .

he eedf F 0 depends upon the time-dependent electric field and

lectron density and upon the effectively time-invariant chemical

omposition and temperature.

Rates not involving electrons are written with the UMIST

40] schema

i = αi

(T

300 K

)βi

exp

[ −γi

T

] . (10)

or parameters summarized in Table 1 , where M in 11c and 12a is

he concentration of the target species expressed in cm

−3 . All pa-

ameters are from UMIST, with two exceptions. The reaction 18 pa-

ameters were determined by directly minimizing the deviation of

he fit (10) from the data of Dotan et al. [64] . The reaction 13 rates

ere developed by Sakiyama et al. [65] , who assumed the rates

ithout citing detailed measurements. We note, however, that our

esults are insensitive to these particular reaction rates.

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 215

4

t

m

l

c

[

s

t

p

c

o

ν[

d

c

t

E

d

p

f

fi

a

t

o

n

p

v

T

Z

w

n

N

s

c

O

d

�

n

s

s

4

fl

r

b

s

s

S

i

m

c

t

s

n

t

t

s

〈

T

l

p

w

[

l

t

〈w

t

ω

w

b

c

t

ω

4

e

[

s

δ

m

c⟨w

N

d

w

d

a

1

5

t

t

d

a

o

d ∑

1

r

.4.3. Initial condition and the breakdown criterion

The final model component is the breakdown condition

hat governs the transition from the seemingly intermittent

icrodischarge initiation on the DBD surface, typical of the

ow-temperature experiments, to the apparently repetitive mi-

rostreamers supporting significant anodic current with the flame

8] . The definition of breakdown electric field at atmospheric pres-

ure is imprecise [ 11 , p. 135], because it can occur either by ei-

her a low-pressure route (i.e., the Townsend criterion) or by by-

assing this to a filamentary plasma. The high-pressure breakdown

ondition for an electronegative gas is typically assigned in terms

f a local balance between attachment and ionization frequencies

i = νa with an overvoltage factor on the resulting electric field E k 11,27] . Here, we introduce a self-sustained high-pressure break-

own condition that obviates initiation via wall emissions and ac-

ounts for the ratio between electric half-period and characteris-

ic particle removal rate by attachment. This leads to a breakdown

k dependent on temperature and concentrations appropriate con-

ition for a self-sustained pulsed discharge in the current high-

ressure, short-pulse regime. Particle losses and emissions to and

rom the surface are neglected. Functionally, it yields breakdown

elds that share characteristics with both the Paschen condition

nd a local balance between ionization and attachment.

The description is motivated, in part, by the common observa-

ion that microstreamers recur at the same location at each change

f polarity [19] . Even in the absence of secondary emission and

eglecting photoionization, a recurrence can be supported if the

lasma density reduction in the afterglow is slow compared to the

oltage period, which seeds the opposite polarity streamers.

Streamer recurrence justifies solution of (4.4.2) with periodicity

act /2

(0) = Z(T act / 2) ≡ B, (11)

here the half-period is sufficient since this particular system has

o symmetry breaking in the exposed versus covered electrode.

umerical solutions confirm that periodic solutions are indeed

trongly attracted to the periodic cycles supported by (11) , with

onvergence to three significant digits in three half periods.

These assumption and the condition (11) allow us to solve the

DE system independently with an adaptive fifth-order backward

ifferentiation formula method [66] . The requisite time step is

t � 10 −12 s. Still, this solution does depend on the time-evolving

eutral field, thus (11) is solved for a range of target state compo-

itions and temperatures and approximated (accurately) with the

ource term maps described in Section 5 .

.5. Source terms

Fast, non-LTE plasma effects are represented in the slow, LTE

ow and transport equations as source terms in the energy and

adical mass conservation equations. Corresponding terms in the

ackground species are determined by imposing atomic mass con-

ervation, yet they are found to be unimportant because of the

mall degree of ionization (cf. [ 67 , p. 281]). As discussed in

ection 4.1 , only non-equilibrium H radicals are considered in the

gnition of hydrogen in air at atmospheric condition. This does not

ean that the processes leading to molecular excitation and disso-

iation of other species are ignored. They are represented, though

heir products (vibrationally, electronically and rotationally excited

pecies, atomic oxygen, etc.) and assumed to equilibrate with the

eutral thermochemistry quickly on the ignition time scale. Thus,

hese processes provide an energy source in the flow equations.

Consistent with the T act τ r assumption of Section 4.4.1 (ac-

uation periods are slower than flow/flame residence times), any

ource f is applied as period averages

f (t) 〉 T ≡ 1

T act

∫ T act

0

f (t ) d t . (12)

his neglects any direct effect of the microdischarges on turbu-

ent fluctuations, which is justified based on the anticipated weak

ower coupled by any particular micro-pulse. Schlieren images

ith and without DBD actuation at 15 m/s supports this assertion

9] .

The energy dissipated and the species produced by electron col-

isions are convolutions of the corresponding cross sections σ i with

he eedf F 0 ,

σi ( ε) 〉 n ≡ γ

∫ εn σi ( ε) F 0 ( ε) d ε, (13)

here n = 1 for species and n = 2 for energy. The radical produc-

ion source term is thus

˙ H =

∑

j| ν ′ j,e

=1

x p j x H 2

ν ′′ j,H

− ν ′ j,H

2

⟨⟨σ j

⟩1 N e −

⟩T

+

∑

j| ν ′ j,e

=0

1

Nx H 2

ν ′′ j,H 2

− ν ′ j,H 2

2

⟨

R j

∏

i

Z ν ′

j,i

i

⟩

T

, (14)

here the number densities Z i are functions of time as determined

y solving (4.4.2). The corresponding mass source in the species

onservation equations is ˙ ω H ρH 2 . Similarly, the energy transfer be-

ween plasma and gas leads to the power density source term

˙ E = Ne

⟨

N e −

( ∑

k = inelastic

x k 〈 σk ( ε) 〉 2 +

∑

k = elastic

2

m e

M k

x k 〈 σk ( ε) 〉 2 ) ⟩

T

.

(15)

.6. Pulse waveform parameters

The waveform (3.2.2) contains three principal empirical param-

ters. The first, δD , adjusts the phase of the actuator. Experiments

8] show that microdischarges are formed between the times of

witching polarity and the peak voltage, which is reflected by

D = 0 . 15 × T act . The other parameters ( δc and t f ) are calibrated by

atching the measured absorbed power per cycle for cold, zero-

ross-flow conditions, ∫ ∞

−∞

∫ ∞

0

˙ ω E d y 2 π r d r

⟩T

= W exp = 8 W , (16)

here y and r are the axial and radial coordinate, respectively.

ote, the cold, zero-cross-flow conditions are axisymmetric. In ad-

ition, we impose a consistency condition based on observations:

hen increasing the air temperature without cross-flow, break-

own occurs for surface temperature T ≈ 750 K [8] . This constraint

nd (16) lead to δc = 1 × 10 −4 T act = 10 / 3 ns and t f = 1 × 10 −5 T act = / 3 ns.

. Energy and radical production maps

Practical implementation of the model in a large-scale simula-

ion depends upon multidimensional maps for the ions–neutrals

ime-scale separation model motivated in Section 4.4.1 and intro-

uced in Section 4.5 . Their goal is to provide ˙ ω Radical from (14) (for

ll radicals in chemical non-equilibrium) and ˙ ω E from (15) based

n local conditions. This is a six-parameter map with indepen-

ent variables x O 2 , x O 2 /x N 2 , x H 2 O , x H , T , and E / N , with constraints

x i = 1 and p = 1 atm . The present simulations used 16, 16, 16, 4,

6, and 30 entries to represent x O 2 , x O 2 /x N 2 , x H 2 O , x H , T , and E / N ,

espectively.

216 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 5. Energy source (15) map: log 10 ˙ ω E

W / m 3 for the local conditions as indicated in the titles and x H = 0 , P = 1 atm plus x H 2 + x O 2 + x N 2 + x H 2 O = 1 . Note: the vertical axis

scale changes between plots, while the color scale is fixed and ranges between −5 (blue) and 12 (yellow). (For interpretation of the references to color in this figure legend,

the reader is referred to the web version of this article.)

Fig. 6. Dimensionless price per radical: log 10 ( P ) (see (17) ) for the local conditions as indicated in the titles and x H = 0 , P = 1 atm plus x H 2 + x O 2 + x N 2 + x H 2 O = 1 . Note: the

vertical axis scale changes between plots, while the color scale is fixed and ranges between 1 (blue) and 4 (yellow). (For interpretation of the references to color in this

figure legend, the reader is referred to the web version of this article.)

i

i

t

b

t

t

Example maps are shown in Figs. 5 and 6 to illustrate the gen-

eral behavior of (15) and (14) and their sensitivity the local con-

ditions. The conditions in Fig. 5 were selected to illustrate three

key sensitivities: (1) a notable increase in ˙ ω E with decreasing x O 2 for x H 2 O = 0 , showing oxidizer sensitivity; (2) a decrease in ˙ ω E with x H 2 O reflecting a decreasing region of filament coverage with

ncreasing water content; and (3) an only weak effect of x O 2 /x N 2 n the E / N –T plane. The radical coupling term ˙ ω H 2

follows closely

he energy coupling: each plot shows a threshold-like increase of

oth ˙ ω E and radical coupling ˙ ω H with increasing E / N and T due

o the reduction of the electron-molecule collision frequency with

he increase of target gas temperature. In Fig. 6 we show the

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 217

d

a

P

w

o

e

i

(

c

a

6

t

d

T

v

Q

w

i

m

f

a

o

N

O

w

c

i

s

r

d

s

p

F

φ

w

p

7

e

j

a

s

i

b

u

a

t

s

t

p

p

Fig. 7. Schematic of the simulation domain matching a corresponding wind tunnel

configuration [9] , which includes a section of the flat windtunnel wall, a sandpaper

turbulence trip, a turbulent cross-flow, 4.826 mm diameter hydrogen jet, and a DBD

plasma actuator. The simulation domain width is 4.76 cm, with spanwise periodicity

used to model as section of wind tunnel. The domain is 50.8 cm long.

F

T

8

u

a

e

t

l

s

t

a

p

l

m

t

8

m

v

s

i

o

a

d

s

(

j

t

d

1

i

e

e

s

F

i

t

a

c

e

imensionless price P per H radical generated by the electric actu-

tion, which is defined

≡ ˙ ω E ρH 2 ˙ ω H 2 �H

◦f , H

, (17)

here �H

◦f , H

is the specific (i.e., mass based) heat of formation

f the radical. The price is lowest for pure H 2 at high reduced

lectric field and low temperature (top-right panel), and largest

n H 2 O −rich mixtures at low reduced field and large temperature

bottom-left panel). The price is consistently lowered by an in-

rease of the reduced electric field, while an increase in temper-

ture at constant E / N has an opposite (though weaker) effect.

. Governing equations and neutral chemistry

The source terms developed in the previous sections carry

he effect of the plasma and plasma chemistry into the three-

imensional chemically reacting compressible flow equations.

hese are expressed in general curvilinear coordinates for the state

ariables

= [ ρ ρu 1 ρu 2 ρu 3 ρE ρY 1 ρY 2 . . . ρY N ] � , (18)

here ρ is the gas density, ρu i is the momentum density in the

th direction, ρE is the total energy density, and ρY k is the species

ass density for k = 1 , 2 , . . . , N. Additional details regarding this

ormulation and its discretization with high-order finite differences

re available elsewhere [68] .

Neutral chemistry is modeled with the 12-steps H 2 –O 2 model

f Boivin et al. [47] , which includes 9 species H 2 , O 2 , H, H 2 O, HO 2 ,

2 , OH, O, and H 2 O 2 and invokes a quasi-steady state reduction for

H, O, H 2 O 2 .

The computational mesh is formed by the overlap of three grids

ith interpolation conditions at the mutual boundaries. The grid

ontains approximately 30 millions nodes. Resolution insensitivity

s established for both the turbulence and the ignition [17] .

The electric field is evaluated as a solution of a screened Pois-

on equation. A linearized approximation for the charge on the

ight-hand side of the Poisson equation leads to a sinusoidal time-

ependence of the electric field, so that the Poisson equation is

olved once per simulation. The right-hand side of (3.2.1) is ex-

ressed following Corke et al. [23] ,

( φ) =

φ − φ0

λ2 N

, (19)

0 = V exposed +

V covered − V exposed

K

for K ∈ [1 , ∞ ) (20)

ith K = 4 and model parameter λN evaluated as previously re-

orted [17] .

. Flow configuration

The specific ignition experiment for which we demonstrate and

valuate our model is report on in full elsewhere [9] . It is a fuel

et in a for 15 m/s crossflow with a developed turbulent bound-

ry layer. The turbulence was seeded with the sandpaper trip

hown on the left-hand side of Fig. 7 , which was represented

n model geometric detail by 1600 randomly Gaussian-shaped

umps on the same scale as the sandpaper grains [17] making

p a 1.64 mm thick strip on the windtunnel floor. The inter-

ction of the trip with the step affects the bypass process and

he turbulence characteristics at the hydrogen jet. A compari-

on between computed and measured Reynolds stress shows that

his model well reproduces turbulence at the hydrogen injection

ort [9] .

The flow is ignited by a thermal hot spot, deposited in the ex-

eriment via the optical breakdown of a focused laser, as shown in

ig. 10 . The focal point is h i above the bottom of the wind tunnel.

he igniter is described in more detail elsewhere [17] .

. Results

We first consider some essential components of the full config-

rations: the DBD plasma and the igniter gas hot spot, the inter-

ction of the flame front with the plasma, and the non-symmetric

missions from the plume. In Section 8.2, we demonstrate the con-

ributions of plasma-combustion coupling to the ignition of the

aminar jet in the turbulent cross-flow, then in Section 8.3 we

how that including the plasma-combustion contribution improves

he prediction of the ignition probability data. In Section 8.4 , we

nalyze the effect of the coupling on the spatially integrated power

redictions and plasma extension, in Section 8.5 we analyze the

ink between plasma-combustion coupling and the filtered experi-

ental light emissions, and, finally, in Section 8.6 we discuss how

he coupling can support two-stage ignition.

.1. DBD body force

Although they lack the direct comparison of direct electric-field

easurements, PIV data allow for the validation of the body-force

ector, which is proportional to the divergence of the electrostatic

tress tensor [69] . The flow is compared with the correspond-

ng stand burner experiment [8] in Fig. 8 . The major uncertainty

f the experiments is due to thermophoretic effects in the flame

nd the correction of the particle velocities due to thermal gra-

ients. No correction has been applied to the experimental data

hown in Fig. 8 , therefore particle velocities in the flame sheet

i.e., where thermal gradients are large) are not reported. The ma-

or uncertainty of the computations relates to modeling the vor-

icity added by the discharge. A streamline representation of the

ata in Fig. 8 shows significant vorticity at the flame edge ( r ≈0 mm). A model like the one proposed in (20) introduces vortic-

ty only due to misalignment of electric field and density gradi-

nt, ˙ �DBD ∝ −F

′ /ρ2 ∇ φ × ∇ ρ, which is concentrated at the flame

dge. Computational analyses with vorticity corrections [70] have

hown that the disagreement between data and simulations in

ig. 8 at y = 10 mm and y = 5 mm is likely due to lack of vorticity

n the model due to its irrotational specific force.

The same standburner configuration also allows us to validate

he computational model by comparing the predicted thermal field

gainst CARS wave spectroscopy measurements [71] . The flat-flame

ase obtained with a DBD forcing of 8 kV was selected for this

xperiment because the strong body forces appear to remove the

218 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 8. PIV measurement and current prediction for the DBD stand burner operated at 5 kV. The curves marked as “no-model” refer to computed results without a body-force

model.

Fig. 9. Unperturbed reduced electric field ˆ E /N 0 1 Td

: the solution of the screened Pois-

son equation (3.2.1). The red rectangle is the exposed electrode, while the black one

is the covered analog; the quartz dielectric is white. (For interpretation of the ref-

erences to color in this figure legend, the reader is referred to the web version of

this article.)

Fig. 10. The laser breakdown height h i .

8

o

S

c

c

c

t

g

b

e

a

w

v

i

t

o

t

o

i

n

n

t

t

t

n

i

flickering instability flames [72] . The errors in the temperature pre-

diction

∫ 2 D H −2 D H

| T (r, y ) − T CARS (r, y ) | d r

max r ( T CARS (r, y ) ) 4 D H

,

is 4%, 3%, 2%, and 6% at the measurement locations above the

dielectric surface of 1.5 mm, 2 mm, 3 mm and 4 mm, respectively.

This disagreement is comparable to the measurement uncertainty

quantified estimated based upon deviation of the data from axial

symmetry.

The unperturbed reduced electric field magnitude ˆ E /N 0 corre-

sponding to this comparison is shown in Fig. 9 for a value of

N 0 = P atm

/ (k B T ∞

) and T ∞

= 300 K. Because the model parameters

were chosen independently and a priori of the data measurement,

they suggest that the predicted distribution of the unperturbed

electric field

ˆ E is correct.

.2. Coupling at ignition

This section describes the effect of plasma combustion coupling

n the ignition of the hydrogen jet in a cross-flow introduced in

ection 7 . It is shown that the DBD actuation of the ignition pro-

ess is more efficient in the temperature regime where the dis-

harge is thermally coupled to the combustion. When plasma and

ombustion are decoupled, which is the condition typical of gas

emperatures significantly lower than those supported by hydro-

en flames, the augmentation of ignition by plasma is inefficient

ecause energy and radical addition to the flow occur (in gen-

ral) away from the ignition kernel. However, when the discharge

nd ignition are coupled, power is selectively added to the region

here both fuel concentration and temperature are high, in the

icinity of the induction region. This selective process is especially

mportant in cross-flow configurations because the advection of

he kernel moves the induction zone away from where breakdown

ccurs in the cold flow.

The computational setup is that described in Fig. 10 . The con-

rol variable is the laser-induced breakdown height h i and we focus

n successful ignition, for which a flame anchors itself at the lead-

ng edge of the fuel port. The size and shape of the ignition ker-

el have been deduced from luminosity measurements [17] . Those

umerical experiments have shown that if total energy transfer to

he fluid and the first moment of its distribution with respect to

he stoichiometric surface are maintained constant, this parame-

er does not play a significant role in the determination of the ig-

ition boundary. For h i = 2 . 68 mm, we predict no such sustained

gnition. Moreover, this is close to the predicted ignition bound-

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 219

Fig. 11. Ignition of the jet in cross-flow without DBD: red surface showing T = 1500 K and blue surface showing X H 2 = 0 . 1 . The labeled times are relative to the laser-induced

breakdown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

a

t

o

s

t

k

t

t

b

m

d

w

t

t

s

c

n

d

g

o

t

s

i

t

o

n

c

ry discussed in Section 8.3 , thus this is a useful case for testing

he marginal influence of the radical model. The thermal history

f this unsuccessful ignition is visualized in Fig. 11 . High-speed

chlieren movies of failed ignition experiments similarly close to

he sustained-ignition boundary show a similar dynamics for the

ernel: the heat addition by the DBD distorts the interface be-

ween low- and high-density gases, but does not penetrate inside

he slow boundary layer flow. It thus cannot propagate upstream

ecause the laminar flame speed of hydrogen S L ≈ 3 m/s is so

uch slower than the free-stream velocity 15 m/s [9] .

In order to analyze the proposed coupling mechanism in more

etail, we compare to an artificially decoupled ignition model

ithout the fundamental coupling mechanism of Section 4.5 . To do

his we simply evaluate the source terms at T = 298 K even though

he gas temperature of the laser igniter exceeds 50 0 0 K [73] . This

fi

uppresses energy and radical transfers because of both the in-

reased three-body attachment cross section and the increased

umber density leading to a smaller reduced electric field and a

iminished electron energy. Thus, the majority of the radicals are

enerated by the plasma close to the electrode, where breakdown

ccurs (because of the lower breakdown potential of hydrogen and

he higher electric field; see Fig. 9 ). The H radical concentration

upported by the DBD in the cold flow is visualized on the X H 2

so-surface in Fig. 12 . This figure shows a large radical concentra-

ion close to the electrode, and the computations agree with the

bserved emissions presented in Fig. 3 a showing no plasma lumi-

escence above the dielectric surface in cold conditions.

The radical and energy fields supported by the artificially de-

oupled model do not affect the ignition probability. The thermal

eld supported is shown in Fig. 13 , and is similar to the no-DBD

220 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 12. H radical stream generated by the DBD in a cold flow: the X H 2 = 0 . 1 isosur-

face is colored by the H radical molar fraction. The scale limit at 5 × 10 −6 highlights

the important region outside the fuel tube. The DBD turns on at t = 0. (For inter-

pretation of the references to color in this figure legend, the reader is referred to

the web version of this article.)

[

e

q

[

t

s

d

a

a

i

t

b

a

t

a

m

w

c

d

b

T

a

I

r

8

s

c

t

d

S

e

t

n

e

s

A

f

6

fi

d

b

S

n

p

a

a

h

t

t

s

b

y

q

t

t

p

c

w

c

t

case Fig. 11 . Such a result might be expected based upon the power

densities involved. The laser igniter transfers approximately 18 mJ

to a fluid volume that at ignition covers approximately 1 mm

3 ac-

cording to measurements [17] , whereas the DBD transfers approx-

imately 8 W to a volumetric flow rate of 1070 cm

3 /min, leading

to an energy density of only about 1/40th that of the laser. Thus,

without thermal coupling, we would indeed expect no direct effect

of the DBD-generated radicals.

Including the full coupling leads to the significant changes and

successful ignition visualized in Fig. 14 . Although the initial phase

is similar to the DBD-off analog (i.e., panels with t < 1 ms in

Figs. 11 and 14 ), the ignition kernel becomes wider as it advects

downstream and interacts with the external electric field seen in

Fig. 9 . The electric breakdown and Joule heating forms a region of

sustained heat release. The subsequent penetration of the hot gas

into the boundary layer ( t ≈ 1.47 ms) allows the ignition front to

propagate upstream ( t > 1.5 ms). The continued increased temper-

ature in the high electric field layer induces stronger energy and

radical concentration production by the electron impact collisions,

leading to a two-way coupled propagation of the ignition front.

Such thermally-coupled discharges are particularly efficient in sup-

porting thermo-chemical induction because the discharge current

density is selectively concentrated where the temperature and fuel

concentration are large, as they are in the ignition kernel. Such a

selective amplification is also responsible for the plasma filament

formation in the experiments of Savelkin et al. [7] : the tempera-

ture increase caused by the direct current discharge augments the

conductivity in selective locations of the flow field, which reduces

the gap voltage below the cold breakdown value, which in turn

concentrates current density in the filaments.

Therefore, the coupled ignition process is aided by the transfer

of energy to the gas by the discharge. The predicted spatial distri-

bution of the energy transferred ˙ ω E is shown in Fig. 15 , where the

surface ˙ ω E = 1 W / m

3 is superimposed to the hydrogen concentra-

tion surface X H 2 = 0 . 1 . Figure 15 shows that the ignition hot spot

generates a plasma kernel that is advected by the flow in synchro-

nized motion with the hot gas. The synchronization between ther-

mal and plasma fields is due to their two-way coupling so that

plasma can exist only where the temperature is large and the hot

spot is sustained by the presence of the plasma. This mechanism

is, in our opinion, also the underlying physics that controls the ad-

vection of plasma filaments in the experiments of Savelkin et al.

7] . Our model neglects the plasma time-scales associated with

lectron drift and diffusion. Although this assumption yields results

ualitatively similar to those from the analysis of Savelkin et al.

7] , the physics underlying our simulation is very different from

hat hypothesized by the authors of that study. Here the plasma

cales are much faster than the fluid scales, so that the electron/ion

ensity immediately adjusts to the background gas temperature

nd concentration, there the assumption of “local ionization bal-

nce” [7] in a spatially varying electric field effectively translates

n plasma being advected as a passive scalar, i.e., decoupled from

he thermal state of the fluid.

When the flame anchors ( t � 2 ms in Figs. 14 and 15 ), the com-

ined energy budget of combustion and discharge is locally bal-

nced by thermal losses, with statistically stationary plasma condi-

ions. Once this occurs, these visualizations of the coupled energy

re qualitatively similar to the emissions measured in the experi-

ents (see Fig. 3 b). The widening of region with plasma emission

hen the flame is on (compare Fig. 3 a and 3 b) is due to the de-

rease in breakdown electric field due to temperature increase in-

uced by the combustion, consistent with the obvious breakdown

oundary in Fig. 5 . A similar effect was noted by Savelkin et al. [7] .

his is also tied with the power coupled in the fluid and the time-

veraged plasma extension which will be analyzed in Section 8.4 .

n the next section we consider the ability of the coupled model to

eproduce the measured sustained-ignition boundary.

.3. Ignition probability predictions

Here we apply plasma-combustion coupling mechanism de-

cribed in Section 8.2 on the ignition probability of the jet in a

ross-flow Fontaine et al. [9] . While the measured ignition dis-

ance is a stochastic variable such that the associated probability

ensity function is obtained by ensemble-averaging 50 trials (cf.

ection 2.2 ), the simulations indicate that the stochasticity inher-

nt with the computed turbulent field has a marginal effect on

he ignition of the jet [17] . We, therefore, infer that the random-

ess of the measurements is mainly due to the variation in laser

nergy absorbed by the gas, which is estimated to be nearly Gaus-

ian with 95% of the data within ± 6.12 mJ of the average 17.64 mJ.

s a consequence, the simulations obtain a (for our purposes) ef-

ectively deterministic ignition boundary, as confirmed with up to

repeated attempts. The representative snapshot of the turbulent

eld used as the initial state for all the simulations discussed in

etail in this section is visualized in Fig. 16 .

A comparison between the computed and measured ignition

oundaries is shown in Fig. 17 . The uncoupled plasma model of

ection 8.2 is again used here for comparison, and shifts the ig-

ition boundary towards lower values of h i . The body forces sup-

orted by the discharge act to “flatten” the stoichiometric surface

t the edge of the H 2 −Air mixing layer. Such a flattening causes

reduction of the chemical reactions close to the port and, thus,

inders the ability of the ignition kernel to propagate upstream.

The coupled model shifts the ignition boundary of an amount

hat is consistent with the measurements [9] . The major uncer-

ainties of the experiments are related to the previously discussed

hot-to-shot variability of the power deposition and the distance

etween optical focal point and laser breakdown location. An anal-

sis of the schlieren images immediately after breakdown leads to

uantify such distance as 0.2 mm [9] . The experiments show that

he DBD-on case ignites with probability greater than 0 at a dis-

ance 0.6 mm greater than the DBD-off case. The computations

redict 0.4 mm. The experiments predict that the maximum lo-

ation of the slope of the curve P ( h i ) is displaced by 0.32 mm

hen the DBD is on. The computations still predict 0.4 mm. Be-

ause the radicals are generated in close proximity to the dielec-

ric surface, one of the possible causes of the over-prediction of P

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 221

Fig. 13. Ignition of the jet in cross-flow with thermally uncoupled DBD sources: red surface showing T = 1500 K and blue surface showing X H 2 = 0 . 1 . (For interpretation of

the references to color in this figure legend, the reader is referred to the web version of this article.)

i

t

M

f

8

W

i

m

p

c

I

p

T

t

fl

i

w

“

a

c

l

F

r

p

c

t

s

d

a

f

e

g

n the DBD-on case is the absence of the radical recombination at

he quartz surface. These effects have been recently quantified by

ackay et al. [74] . Future analysis by the authors will include sur-

ace effects.

.4. Power predictions

The total power coupled into the gas phase per DBD cycle,

˙ ≡

∫ ˙ ω E d V. (21)

s shown in Fig. 18 , as it depends on time and compared with the

easured (using the monitor-capacitor method) averaged over ap-

roximately 100 periods. The measurements are shown for both

old (before ignition) and hot (after flame anchoring) conditions.

nterestingly, Retter et al. [8] indicate that the measured power-

er-cycle varies significantly during the anchored-flame timeframe.

he simulations indicate a qualitatively similar result, which is due

o the action of the turbulence in changing the position of the

ame in the electric field layer.

Figure 18 shows the “Total” power, which is sum of the value

ntegrated above the dielectric surface and in the hydrogen tube,

hich would better correspond to the experiment, as well as the

Above the Surface” power that includes only the amount absorbed

bove the dielectric. The simulation results indicate a strong in-

rease in the latter value between t = 1 . 5 and 2.5 ms from the

aser breakdown. This “peak-energy” time-frame corresponds (see

ig. 14 ) to the advancement of the flame-front in the hydrogen-

ich gas. After the ignition front reaches the port at t ≈ 2 ms, the

roduction of water in the flame significantly reduces the power

oupled as discussed with regard to the maps in Section 5 .

The predicted, time-averaged power slightly exceeds 30 W in

he hot-period ( t > 2.5 ms). Contrasted with cold value of 8 W, this

hows the substantial contribution of the flame in changing the

ischarge structure. We note that also Savelkin et al. [7] measured

substantial increase in power absorbed by the gas when injecting

uel in the combustion chamber (burning cases), also noting the

xtension of the plasma filaments and consequently increases the

ap voltage. Our model supports the link between power absorbed

222 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 14. Ignition of the jet in cross-flow with thermally coupled DBD sources: red surface showing T = 1500 K and blue surface showing X H 2 = 0 . 1 . (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of this article.)

t

t

t

t

p

a

t

i

r

b

r

d

t

r

m

t

and plasma extension. Yet, our model does not account for the

solid-state electrical circuit, thus the gap voltage is constant. Our

model is, instead, based on a similarity law for the microstream-

ers, whereby the main limiting mechanism to the power coupled

is the quenching effect of the charge deposited by the microdis-

charges on the dielectric surface.

The proposed model is complex and was parameterized inde-

pendently of the current measurements, which makes this level of

agreement acceptable. Limiting the spatial scale of the pulses at

high temperature as described in (3.2.2) is an important ingredi-

ent of this prediction, without which the simulated power signif-

icantly overestimates the measurements. However the current pa-

rameterization does somewhat overpredict the measured power in

Fig. 18 . We can anticipate potential causes, but leave refinement of

it to future studies, likely with additional supporting experiments.

Because the model was calibrated in low temperature hydrogen, a

possible missing mechanism is that of water vapor on the power

ransferred to the gas in the hot burnt plume. Water increases

he rate of electron attachment at high E / N ; the hydration reac-

ion helps this process by reducing the detachment from O

−2 [75] ,

hus increasing the production of anions in the high temperature

lume. The kinetics of the superoxide anion in high temperature

tmospheric flames is not well understood.

We can estimate the potential effect of water vapor increasing

he exponent of the hydration reaction 11.c in (3) from its nom-

nal value of ζ = −3 . 5 to the upper-end value of −2.5. It indeed

educes power, as shown in Fig. 19 . However, the ignition proba-

ility prediction (i.e., Fig. 17 ) is only marginally dependent on the

ate constant of this reaction, because water concentration is small

uring both the advection of the ignition kernel and the propaga-

ion of the front ( t < 2 ms in Figs. 14, 15 and 18 ) . We find that the

elationship between coupled power and hydration exponent is al-

ost linear in the interval of interest. We have selected the hydra-

ion reaction for this test because its temperature dependence is

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 223

Fig. 15. Ignition of the jet in cross-flow with thermally coupled DBD sources: red surface showing ˙ ω E = 1 W / m

3 and blue surface showing X H 2 = 0 . 1 . (For interpretation of

the references to color in this figure legend, the reader is referred to the web version of this article.)

p

i

o

f

u

c

d

o

i

v

d

b

a

r

fl

m

t

s

D

a

t

o

c

l

m

e

a

8

p

c

t

articularly uncertain and the sensitive derivative with respect to

ts temperature exponent is the largest component of the gradient

f the coupling source-terms (cf. Fig. 5 ) in parameter space, there-

ore the variation in Fig. 19 provides an estimate for the parametric

ncertainty of the power predictions.

The substantial effect of the hydration exponent on the power

oupled is thus due to the quenching effect of water on the plasma

ensity. The results of Fig. 19 help us explain the marked variation

f the measured power against the cross-flow velocity discussed

n [ 9 , Fig. 10]: an increase in cross-flow velocity bends the water

apor plume and increases its interaction with the plasma at the

ielectric wall downstream of the port. Conversely, in the standing-

urner experiments [8] the water vapor is removed by buoyancy

nd discharged vertically. In fact, the contribution of buoyancy in

emoving hot products from the flame region is highlighted by the

ickering motion observed in the actuated stand-burner experi-

ents. Therefore, the interaction of the water vapor plume and

he dielectric walls explains why the standing burner supports a

ignificantly more energetic coupling between the plume and the

BD plasma.

Figure 20 shows the spatial distribution of power coupled aver-

ged in time over an interval of 2 ms within the quasi-stationary

urbulent flame ( t > 2.5 ms). Of particular note is that the plasma

nly marginally shrinks with the hydration reaction exponent in-

rease. Thus, as the intensity of the energy coupling increases,

eading to a larger power transfer, the overall plasma extension re-

ains approximately constant. The scope of this conclusion will be

xpanded in the next section by comparing the simulation results

gainst the brightness and size of the experimental emissions.

.5. Temperature profiles

The marked influence of the hydration exponent ζ on the

ower transferred to the fluid is due to the strong two-way

oupling between plasma and thermal processes. Chemical reac-

ions increase the gas temperature, which increases the reduced

224 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 16. The Q-criterion Q = 300 s −2 isosurface colored by the vorticity magnitude using a colormap linear color map between 0 (blue) and 100 s −1 (red). (For interpretation

of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 17. Comparison between prediction and experiment for the ignition distance

probability h i .

Fig. 18. Power coupled in the gas by the DBD plasma.

Fig. 19. Power coupled in the gas by the DBD plasma plotted versus the hydration

reaction exponent ζ for (3) , reaction 11.c.

e

s

p

t

w

o

t

s

i

n

g

d

fl

fi

t

g

l

a

n

lectric field leading to a larger electron temperature and thus

tronger collisional energy transfers. The reduced momentum loss

er electron-pair produced (see Section 3.1 ) only partially offset

his bootstrapping mechanism and the exothermal reactions in the

eakly ionized plasma can “launch” the system. The nonlinearity

f the temperature change versus the gap voltage is rooted in the

hermal budget of the gas phase. Temperature is, therefore, a very

ensitive indicator of coupling and will be analyzed in more details

n this section.

Measured thermal emissions from water molecules at 720

m [ 9 , Fig. 10] show that the actuated flames support two re-

ions of significant emissions. A first one is elongated in the

ownstream direction and is associated with a H 2 /O 2 diffusion

ame, the second, located immediately downstream of the ori-

ce, is wider, shorter and brighter. Anticipating that the lat-

er region is due to plasma-combustion coupling, we investi-

ate the ability of the model to simulate its spatial extent and

ocation.

The computed thermal field supported by the actuated flames

re shown in Fig. 21 for values of the hydration reaction expo-

ent equal to −3.5, −3.0 and −2.5; the top left panel also shows

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 225

Fig. 20. Energy coupling versus the hydration reaction temperature exponent ζ : red surface showing the energy coupled in the gas by the plasma ˙ ω E = 1 × 10 −2 W / mm

3

and blue surface showing the hydrogen molar fraction for X H 2 = 0 . 1 . (For interpretation of the references to color in this figure legend, the reader is referred to the web

version of this article.)

Fig. 21. Mean temperature in the electric field layer versus the hydration reaction rate temperature exponent.

Fig. 22. Integrated temperature versus the radial distance from the orifice center

for three values of the hydration reaction exponent. The ordinate for the “emission

peak location” has been selected based on ease of visualization.

t

t

3

e

T

w

s

a

a

c

o

t

s

t

t

p

t

i

b

c

w

a

[

t

c

he unactuated analog for reference. In these plots the tempera-

ure has been averaged in the wall normal direction over a layer

mm thick, corresponding, approximately, to the thickness of the

lectric-field layer over the dielectric in Fig. 9 ,

ˆ ( x, y ) ≡

∫ 3 mm

0 T ( x, y, z ) d z

3 mm

,

here z = 0 corresponds to the windtunnel wall.

The simulations predict the formation of a well-defined hot

pot starting approximately 10 mm downstream of the injection

xis. On the one hand, the size and shape of this high temper-

ture region is weakly dependent on the power coupled, as we

an infer by its variation against the hydration exponent. On the

ther hand, the maximum temperature is roughly proportional to

he power coupled. This result agrees qualitatively with the mea-

urements described by Fontaine et al. [9] , who correlated the in-

ensity of the water vapor emissions from the region directly above

he dielectric surface and downstream of the jet with the electrical

ower absorbed by the DBD. That experimental study reports that

he “characteristic” spanwise half-width of the thermal emissions

s equal to 5.84 mm for a DBD power of 26.80 W. The horizontal

lack, thick line superimposed to the colormaps in Fig. 21 indi-

ates the extension of the experimental measurements; the thin

avy line is a reconstruction of the experimental emission im-

ge at ˙ W = 26 . 80 using a morphological reconstruction algorithm

76] with pixel threshold set equal 0.5. This result, together with

he observed weak correlation of plasma extension against power

oupled in Fig. 20 , supports the idea that hot-spot brightness not

226 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 23. Upstream propagation of the flame in actuated conditions: red surface showing the energy coupled in the gas by the plasma ˙ ω E = 1 W / mm

3 and blue surface

showing the hydrogen molar fraction for X H 2 = 0 . 1 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this

article.)

T

v

p

c

a

i

t

c

t

t

8

i

c

size correlates with the absorbed power, which agrees with the

trend shown in the experiments [9] .

The streamwise location of the strongest plasma–gas energy ex-

change depends on various factors, including the unperturbed elec-

tric field, the water content and the temperature in the plume.

The experiments [9] have shown a distinct peak in thermal emis-

sions located approximately 15 mm from the port axis. We test the

model by analyzing the variation of the temperature field against

the radius. In order to that, the temperature is averaged in the cir-

cumferential direction by integrating ˆ T defined in (8.5) over the

angle θ ∈ [ − arctan ( 5 . 84 / 10 ) , arctan ( 5 . 84 / 10 ) ] ≈ [ −π/ 6 , π/ 6 ] , i.e.,

evaluating

˜ T ( r ) ≡∫ π/ 6

−π/ 6 ˆ T ( r cos θ, r sin θ ) d θ

1 / 3 π. (22)

he values of ˜ T ( r ) are plotted against the radius in Fig. 22 for three

alues of the hydration exponent. Although the variation in tem-

erature is markedly dependent on the power absorbed, the lo-

ation of the peak is marginally affected by it, and it is located

round 15 mm from the port, in good agreement with the exper-

ments. The peak temperature in the simulations is located near

he maximum of the external electric field in Fig. 9 . Therefore, this

omparison confirms that the use of the screened Poisson equation

o determine the reference electric in the pulse approximation of

he microstreamer, i.e., Eq. (3.2.2), is correct.

.6. Two-stage ignition

One of the most interesting aspects of the plasma-combustion

nteraction, is that it leads to a closed-loop feedback pro-

ess, whereby a larger temperature leads to a stronger energy

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 227

Fig. 24. Effect of direct and indirect paths on H radical production: log 10 ˙ ω H

1kg / m 3 −s for the local conditions as indicated in the titles and P = 1 atm plus x H 2 + x O 2 + x N 2 + x H 2 O =

1 .

a

t

i

o

o

i

F

t

w

p

t

r

i

9

a

c

j

p

d

T

a

c

t

a

e

c

p

c

t

a

n

t

d

t

o

s

p

t

o

T

s

l

a

n

l

e

b

t

p

s

t

t

p

t

m

l

m

f

i

i

t

t

R

c

i

p

c

g

t

s

i

i

bsorption from the discharge, thus resulting in an increase in the

emperature. This means by which the discharge can sustain itself

n an electric field that is below the cold-flow breakdown thresh-

ld is the probable mechanism the underlying physical mechanism

f the observed two-stage ignition process [9] .

In order to illustrate this process we consider the DBD-actuated

gnition close to the sustained flame boundary, h i = 3 . 07 mm in

igs. 17 and 10 . Six solution snapshots are shown in Fig. 23 where

he energy absorbed by the fluid is superimposed to a surface

ith constant hydrogen molar fraction equal to 0.1. High tem-

erature plasma self-sustains with small localized power absorp-

ion, ˙ W ≈ 16 W, for several milliseconds, before H 2 –Air chemistry

eaches the chemical runaway condition, which, then, leads to an

gnition kernel propagating upstream.

. Conclusions

Based upon independent measurements, we have designed

model to analyze the mechanisms of the observed plasma-

ombustion coupling in a DBD-mediated ignition and combustion

et in cross-flow system. We show that plasma-combustion cou-

ling is a two-way interaction rooted in the temperature depen-

ent terms appearing in the electron and neutral energy equations.

here are two aspects of the coupling. An increase in gas temper-

ture due to exothermal neutral reactions reduces the plasma spe-

ific collisional energy loss and leads to a larger electron tempera-

ure and number density. Likewise, an increase in electron temper-

ture increases the Joule heating contribution to the neutral en-

rgy by thermalization of the energy released in electron impact

ollisions. Because the neutral energy is proportional to the tem-

erature, this coupling creates a closed-loop feedback process that

an yield a nonlinear, super-adiabatic temperature increase against

he gap voltage. Because the coupling results in a selective mech-

nism that concentrates the absorbed power to the region of high

eutral temperature, it is an important, beneficial contribution of

he interaction of ignition hot spots with electric field layers over

ielectric surfaces.

The two-way mechanism supports non-LTE energy and radical

ransfers to the neutrals, which we model by solving a quasi-steady

ne-dimensional Boltzmann equation and corresponding charged

pecies mass conservation with an imposed electric field. The

lasma generated by an atmospheric DBD is sustained by stochas-

ic, recurrent, self-similar microstreamers that occur on a nanosec-

nd timescale in the gas pre-ionized by previous microdischarges.

he electric field supported by this microstructure is the superpo-

ition of the external, space charge and surface charge components,

eading to a pulses in the field strength. We assume that the time

nd length scales of the pulses affect the coupling terms but ig-

ore the details of its shape to obtain an assumed-shape scaling

aw. The periodicity of the streamers and the smallness of the en-

rgy relaxation time compared to the electric period lead to the

reakdown condition being imposed in terms of the balance be-

ween time-integrated attachment and ionization with no trans-

ort losses. Thus, the plasma-fluid coupling depends on the gas

tate, for which we presented a practical tabulation map.

The neutral composition and temperature significantly modify

he coupling. Three-body attachment dominates ionization in low

emperature air (see Figs. 5 and 6 ), which effectively limits the

lasma extension to the envelope of the diffusion flame and leads

o the non-symmetric plasma emissions observed in the experi-

ents. Because the three-body attachment cross section increases

inearly with the pressure, we expect that the plasma enhance-

ent will be more important at a reduced pressure, while the ef-

ect of plasma-combustion coupling will be less pronounced lead-

ng to a quasi-one-dimensional ignition [14] . The energy coupled

n the plasma is also sensitive to the degree of mixing between

he weakly electro-positive hydrogen and the strongly electronega-

ive oxygen. This effect was also observed in the experiments [71] .

esults presented in Figs. 21 and 22 show that changes in plasma-

ombustion coupling due to temperature are dominant over mix-

ng because the fluid internal energy supports a two-way cou-

ling with the electron energy. Yet, our computations show that

omposition has a non-marginal effect: H 2 supports a more ener-

etic interaction than mixtures of air and H 2 O in agreement with

he experiments of Yin et al. [42] . This is demonstrated by the

trong peak in power during the advancement of the ignition front

n Fig. 18 . Therefore, we expect that more energetic turbulence

ntensities will introduce a similar level of power augmentation

228 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Fig. 25. Fractional contributions of elementary processes to the time-integrated radical maps at X O 2 = 0 . 15 X H 2 O

= 0 . Frame (a) includes only predissociation to H 2 ( b 3 �u +) ,

the contribution of Lyman, Werner, and metastable transitions is smaller.

Fig. 26. Effect branching ratio γ c in on H radical production: log 10 ˙ ω H

1kg / m 3 −s for P = 1 atm and x H 2 + x O 2 + x N 2 + x H 2 O = 1 .

r

fl

t

s

i

t

during the flame period and a decreased coupling during ignition

due to better mixing. This might best be considered in a configu-

ration such at that of Bedat and Cheng [77] .

Water production in the atmospheric hydrogen flames supports

a strong removal rate of the superoxide anion through the hydra-

tion reactions, leading to an increase in the energy absorbed per

adical produced and a sharp decrease in energy coupled after the

ame anchors. This phenomenon controls the interaction between

he flame-plume and the electric field layer, and explains the mea-

ured decrease of power absorbed when the cross-flow velocity

s increased. The present model predicts the electrical power in

he anchored-flame period satisfactorily, with most of the power

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 229

c

t

h

p

t

h

r

n

u

w

D

f

s

v

a

w

t

n

c

A

G

p

d

A

o

a

n

e

m

n

H

t

t

(

e

t

I

fi

c

a

e

m

≡N

O

t

s

t

s

t

c

a

t

s

i

n

fi

b

4

b

t

a

v

p

a

a

p

(

r

r

t

b

h

H

(

t

i

j

l

t

t

e

S

e

o

i

f

d

w

t

[

e

H

a

[

r

t

H

g

b

[

w

h

c{

w

t

t

a

X

t

i

l

oupled in burning conditions away from the exposed electrode

ube through which the hydrogen is delivered.

Plasma-combustion coupling plays a role in the ignition of the

ydrogen jet in the cross-flow. The temperature and species de-

endence of the amplification mechanism selectively concentrates

he energetic and chemical plasma contributions to the region with

igh temperature and low water content, essentially the induction

egion. This mechanism is responsible for the penetration of the ig-

ition kernel in the slow boundary layer fluid and the consequent

pstream flame propagation; its inclusion improves the agreement

ith the measured augmentation of the ignition probability by the

BD. Moreover, our simulations have shown that the closed-loop

eedback process at the root of plasma-combustion interaction can

elf-sustain high-temperature kernels in fuel-lean air flows with

elocities larger than the local adiabatic flame speed. This mech-

nism enables the two-stage ignition in DBD assisted conditions,

here high temperature kernels were observed to anchor above

he covered electrode for time intervals long compared to the ig-

ition time, and then slowly move upstream due to velocity and

oncentration fluctuations in the turbulence eddies.

cknowledgments

We thank Ryan Fontaine, Jon Retter, Greg Elliott and Nick

lumac for providing and discussing their experimental results.

This material is based in part upon work supported by the De-

artment of Energy, National Nuclear Security Administration , un-

er Award Number DE-NA0 0 02374 .

ppendix A. Effect of excited nitrogen on radical production

The chemistry scheme of Section 4 includes the contribution

f electronic excitation of nitrogen and oxygen molecules, but

ssumes that these energy exchanges are thermalized instanta-

eously, thus they cannot affect radical production through non-

quilibrium kinetics. Nagaraja et al. [43] shows that a similar ther-

alization of all the electron impact collisions has a small but not

egligible effect on ignition of H 2 at very low pressures (25 Torr).

ere, we analyze the non-equilibrium paths of N

∗2 , and confirm

hat their omission is justified for our condition. This is impor-

ant to consider because nitrogen has a large dissociation enthalpy

about twice that of oxygen), and a discharge can load significant

nergy in its bound electronic states. These states in turn poten-

ially support production of H via dissociative quenching [43,49] .

n the following, we generalize the model to include this, and con-

rm that in our system the large peak reduced field in the mi-

rodischarges in flames and the high pressure makes this mech-

nism small compared to the direct dissociation of hydrogen by

lectron impact.

For this assessment, we include ten levels of excited

olecular nitrogen N 2 , two levels each for O 2 , O and N : Z e [ N 2 (A

3 �u

+) , N 2 (B

3 �g ) , N 2 (W

3 �u

) , N 2 ( B

′ 3 �−u

) , N 2 ( a ′ 1 �−

u

) ,

2 ( a 1 �g ) , N 2 ( w

1 �u

) , N 2 ( C

3 �u

) , N 2 (E 3 �+ g ) , N 2 ( a

′′ 1 �+ g ) ,

2 ( a 1 �g ) , O 2 ( b

1 �+ g ) , N( 2 D) , N( 2 P) , O( 1 S) , O( 1 D) ] . The reac-

ion set used in this test includes contributions from several

ources [49,78,79] and is shown in Table A.2 .

The implementation matches that described in Section 4.4.3 for

he charged species, periodicity is imposed on Z e . The charged

pecies Z in (11) is computed independently because the concen-

ration of the targets does not change during a periodic microdis-

harge. In fact, the peak mole fraction of excited species produced

re of the order of 1 × 10 −8 . Therefore, the electron concentra-

ion from the Boltzmann analysis ( Section 4 ) provides a set of

ource terms independent of Z e . The solution of Z e (0) = Z e (T act / 2)

s straightforward because the quenching rates are only weakly

on-linear with the concentration of the excited states. We use a

xed point iteration to solve for the periodic Z e .

Radical production is shown in Fig. 24 for different neutral

ackground conditions with the nitrogen to oxygen ratio fixed at

. We designate direct path production of H as the one supported

y (14) , rather the indirect path is supported primarily by reac-

ions A80, A84, A85 and A86 in Table A.2 . In pure H 2 ( Figs. 24 a

nd 24 d), radical production through the indirect path is zero (ob-

iously) and the direct path production is the strongest among the

lots on the first row. The addition of O 2 ( Fig. 24 b and 24 c) causes

drop in the energy coupled into the gas by the discharge and

concomitant reduction of radical production through the direct

ath. Conversely, Fig. 24 e and 24 f show that an increase in X O 2

i.e., a decrease in equivalence ratio) results first in an increase in

adical production through the indirect path and then a drop; the

adical contribution will obviously go to zero at X O 2 = 1 / 5 . Overall,

he indirect path contribution is small compared to the direct path,

ecoming stronger at larger reduced electric fields and weaker at

igher temperature. While excitation to the lowest repulsive state

2 ( b 3 �u

+) peaks at an energy comparable with the N

∗2 excitation

at about 20 eV [38] ) the Lyman- α emission peak is at 40 eV [35] ,

hus large fields, E / N > 300 Td, lead to a large ratio of direct to

ndirect radical production rates, ˙ ω H ,D / ̇ ω H ,I

.

In the simulations described in Section 8.2, we expect the ma-

ority of the radicals to be produced at high gas temperature and

arge E / N because the high temperature at the ignition kernel leads

o proportionally high reduced electric fields and selective concen-

ration of the power coupled at the hot spot. Similarly, large fuel

quivalence ratios support a stronger coupling (see discussion in

ection 8.4 and the experiments of Yin et al. [42] ) and stronger en-

rgy transfers [8,80] , so the majority of radicals will be produced at

r above the stoichiometric fuel-to-air ratio, where the direct path

s dominant over the indirect one. These considerations justify our

ocus on the direct path in the results presented in Section 8 .

In the present study (i.e., Table A.2 ), we have not included hy-

rogen dissociation by collisional quenching of N 2 (C

3 �u ), which

arrants additional discussion since in some regimes it can cer-

ainly be important. For example, Starikovskiy and Aleksandrov

81] mention that this channel is an important path of non-

quilibrium radical production in plasma-assisted combustion of

2 : O 2 : N 2 mixtures at high reduced fields. Still, the kinetic rates

nd branching ratios are not well known [81] , it is often neglected

14,48,80,82] , and we might anticipate that it has a subservient

ole at the current pressures. Starikovskaia et al. [48] assume that

he only products of the collisional quenching of N 2 (C

3 �u ) with

2 in the pressure range 1–8 Torr are molecular hydrogen and

round-state nitrogen. Higher pressures are expected to favor the

ound state stabilization over the dissociation to radical channel

83] , thus at the present atmospheric pressures the branching to-

ards the radical production channel will be small. In fact, a recent

igh-pressure study [82] also does not include this channel.

The sub-mechanism in question here can be written parametri-

ally as

N 2 ( C

3 �u

) + H 2 ⇒ N 2 + 2 γc H + ( 1 − γc ) H 2

A = 3 . 2 × 10

−10 cm

3

molecules −s , E = 0 , n = 0 ,

(A.97)

here γ c allows for parametric variation of the branching ratio of

he collisional quenching of N 2 (C

3 �u ) with hydrogen relative to

he overall conversion rate of Pancheshnyi et al. [84] . To make this

ssessment, we adjust γ c and examine H radical production for

O 2 = 0 . 15 .

We evaluate the fractional contributions of the most impor-

ant direct processes [ (14) with (2) –(5) ] and collisional quench-

ng ones (reactions A80, A84, A85, A86 and A97). Results for se-

ected processes are shown in Fig. 25 . For γc = 1 , we see that the

230 L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232

Table A.2

Reaction mechanism for triplet nitrogen quenching.

Reaction A (cm-molecules-sec-K) n E (K)

A1. N 2 ( A 3 �u +) + O 2 ⇒ N 2 + O + O 2.54e–12 0 0

A2. N 2 ( A 3 �u +) + O 2 ⇒ N 2 O + O 8.68e–16 0.55 0

A3. N 2 ( A 3 �u +) +O ⇒ NO + N ( 2 D ) 7e–12 0 0

A4. N 2 ( A 3 �u +) + N 2 O ⇒ N 2 + N + NO 1e–11 0 0

A5. N 2 ( A 3 �u +) + N 2 ( A

3 �u +) ⇒ N 2 ( B 3 �g ) + N 2 7.7e–11 0 0

A6. N 2 ( A 3 �u +) + N 2 ( A

3 �u +) ⇒ N 2 ( C 3 �u ) + N 2 5.54e–4 –2.64 0

A7. N 2 ( A 3 �u +) + N 2 ⇒ N 2 + N 2 3e–18 0 0

A8. N 2 ( A 3 �u +) + O 2 ⇒ N 2 + O 2 ( a

1 �g ) 8.68e–15 0.55 0

A9. N 2 ( A 3 �u +) + O 2 ⇒ N 2 + O 2 ( b

1 �+

g ) 8.68e–15 0.55 0

A10. N 2 ( A 3 �u +) +N ⇒ N 2 + N ( 2 P ) 1.79e–19 –0.67 0

A11. N 2 ( A 3 �u +) +O ⇒ N 2 + O ( 1 S ) 2.1e–11 0 0

A12. N 2 ( A 3 �u +) +NO ⇒ N 2 + NO 7e–11 0 0

A13. N 2 (B 3 �g )+ N 2 ⇒ N 2 ( A

3 �u +) + N 2 5e–11 0 0

A14. N 2 (B 3 �g ) ⇒ N 2 ( A

3 �u +) 1.5e5 0 0

A15. N 2 (B 3 �g )+NO ⇒ N 2 ( A

3 �u +) +NO 2.4e–10 0 0

A16. N 2 (B 3 �g )+ O 2 ⇒ N 2 + O + O 3e–10 0 0

A17. N 2 (W

3 �u )+ O 2 ⇒ N 2 + O + O 3e–10 0 0

A18. N 2 ( B ′ 3 �−

u ) + O 2 ⇒ N 2 + O + O 3e–10 0 0

A19. N 2 ( a ′ 1 �−

u ) + N 2 ⇒ N 2 + N 2 2.2e–11 0 0

A20. N 2 ( a ′ 1 �−

u ) + O 2 ⇒ N 2 + O + O 3e–10 0 0

A21. N 2 (a 1 �g )+ O 2 ⇒ N 2 + O + O 3e–10 0 0

A22. N 2 (w

1 �u )+ O 2 ⇒ N 2 + O + O 3e–10 0 0

A23. N 2 ( a ′′ 1 �+

g ) + O 2 ⇒ N 2 + O + O 3e–10 0 0

A24. N 2 (B 3 �g )+O ⇒ NO + N ( 2 D ) 3e–10 0 0

A25. N 2 (W

3 �u )+O ⇒ NO + N ( 2 D ) 3e–10 0 0

A26. N 2 ( B ′ 3 �−

u ) +O ⇒ NO + N ( 2 D ) 3e–10 0 0

A27. N 2 (C 3 �u )+O ⇒ NO + N ( 2 D ) 3e–10 0 0

A28. N 2 ( E 3 �+

g ) +O ⇒ NO + N ( 2 D ) 3e–10 0 0

A29. N 2 ( a ′ 1 �−

u ) +O ⇒ NO + N ( 2 D ) 3e–10 0 0

A30. N 2 (a 1 �g )+O ⇒ NO + N ( 2 D ) 3e–10 0 0

A31. N 2 (w

1 �u )+O ⇒ NO + N ( 2 D ) 3e–10 0 0

A32. N 2 ( a ′′ 1 �+

g ) +O ⇒ NO + N ( 2 D ) 3e–10 0 0

A33. N 2 (a 1 �g )+ N 2 ⇒ N 2 ( B

3 �g ) + N 2 2e–13 0 0

A34. N 2 (a 1 �g )+NO ⇒ N 2 + N + O 3.6e–10 0 0

A35. N 2 (C 3 �u ) ⇒ N 2 (B

3 �g ) 3e7 0 0

A36. N 2 (C 3 �u )+ N 2 ⇒ N 2 ( a

1 �g ) + N 2 1e–11 0 0

A37. N 2 (C 3 �u )+ O 2 ⇒ N 2 + O + O ( 1 S ) 3e–10 0 0

A38. N 2 ( E 3 �+

g ) + O 2 ⇒ N 2 + O + O ( 1 S ) 3e–10 0 0

A39. O 2 (a 1 �g )+ O 3 ⇒ O 2 + O 2 + O 9.7e–13 0 1564

A40. O 2 (a 1 �g )+N ⇒ NO + O 2e–14 0 600

A41. O 2 (a 1 �g )+ N 2 ⇒ O 2 + N 2 3e–21 0 0

A42. O 2 (a 1 �g )+ O 2 ⇒ O 2 + O 2 2.3e–20 0.8 0

A43. O 2 (a 1 �g )+O ⇒ O 2 + O 7e–16 0 0

A44. O 2 (a 1 �g )+NO ⇒ O 2 + NO 2.5e–11 0 0

A45. O 2 ( b 1 �+

g ) + O 3 ⇒ O 2 + O 2 +O 1.8e–11 0 0

A46. O 2 ( b 1 �+

g ) + N 2 ⇒ O 2 ( a 1 �g ) + N 2 4.9e–15 0 253

A47. O 2 ( b 1 �+

g ) + O 2 ⇒ O 2 ( a 1 �g ) + O 2 4.3e–22 2.4 241

A48. O 2 ( b 1 �+

g ) +O ⇒ O 2 ( a 1 �g ) + O 8e–14 0 0

A49. O 2 ( b 1 �+

g ) + O ⇒ O 2 + O ( 1 D ) 6e–11 –0.1 4201

A50. O 2 ( b 1 �+

g ) +NO ⇒ O 2 ( a 1 �g ) + NO 4e–14 0 0

A51. N( 2 D)+ O 2 ⇒ NO + O 8.66e–14 0.5 0

A52. N( 2 D)+ O 2 ⇒ NO + O ( 1 D ) 3.46e–13 0.5 0

A53. N( 2 D)+NO ⇒ N 2 O 6e–11 0 0

A54. N( 2 D)+ N 2 O ⇒ NO + N 2 3e–12 0 0

A55. N( 2 D)+ N 2 ⇒ N + N 2 2e–14 0 0

A56. N( 2 P)+ O 2 ⇒ NO + O 2.6e–12 0 0

A57. N( 2 P)+NO ⇒ N 2 ( A 3 �u +) +O 3.4e–11 0 0

A58. N( 2 P)+ N 2 ⇒ N ( 2 D ) + N 2 2e–18 0 0

A59. N( 2 P)+N ⇒ N ( 2 D ) + N 1.8e–12 0 0

A60. O( 1 D)+ N 2 ⇒ O + N 2 1.8e–11 0 –107

A61. O( 1 D)+ O 2 ⇒ O + O 2 ( b 1 �+

g ) 2.56e–11 0 –67

A62. O( 1 D)+ O 2 ⇒ O + O 2 6.4e–12 0 –67

A63. O( 1 D)+ O 3 ⇒ O + O + O 2 2.3e–10 0 0

A64. O( 1 D)+O 3 ⇒ O 2 +O 2 1.2e–10 0 0

A65. O( 1 D)+NO ⇒ N + O 2 1.7e–10 0 0

A66. O( 1 D)+ N 2 O ⇒ NO + NO 7.2e–11 0 0

A67. O( 1 D)+ N 2 O ⇒ N 2 + O 2 4.4e–11 0 0

A68. O( 1 S)+ N 2 ⇒ N + NO 5e–17 0 0

A69. O( 1 S)+ O 2 ⇒ O ( 1 D ) + O 2 1.333e–12 0 850

A70. O( 1 S)+ O 3 ⇒ O ( 1 D ) + O + O 2 2.9e–10 0 0

A71. O( 1 S)+ O 3 ⇒ O 2 + O 2 2.9e–10 0 0

A72. O( 1 S)+ NO ⇒ O + NO 2.9e–10 0 0

( continued on next page )

L. Massa, J.B. Freund / Combustion and Flame 184 (2017) 208–232 231

Table A.2 ( continued )

Reaction A (cm-molecules-sec-K) n E (K)

A73. O( 1 S)+NO ⇒ O ( 1 D ) + NO 5.1e–10 0 0

A74. O( 1 S)+ N 2 O ⇒ O + N 2 O 6.3e–12 0 0

A75. O( 1 S)+ N 2 O ⇒ O ( 1 D ) + N 2 O 3.1e–12 0 0

A76. O( 1 S)+ O 2 ( a 1 �g ) ⇒ O ( 1 D ) + O 2 ( b

1 �+

g ) 3.6e–11 0 0

A77. O( 1 S)+ O 2 ( a 1 �g ) ⇒ O + O + O 3.4e–11 0 0

A78. O( 1 S)+ O 2 ( a 1 �g ) ⇒ O + O 2 ( a

1 �g ) 1.3e–10 0 0

A79. O( 1 S)+O ⇒ O ( 1 D ) + O 5e–11 0 301

A80. N 2 ( A 3 �u +) + H 2 ⇒ N 2 + H + H 4.4e–10 0 3500

A81. N 2 (B 3 �g )+ H 2 ⇒ N 2 ( A

3 �u +) + H 2 5.e–11 0 0

A82. N 2 (W

3 �u )+ H 2 ⇒ N 2 ( A 3 �u +) + H 2 2.5e–11 0 0

A83. N 2 ( B ′ 3 �−

u ) + H 2 ⇒ N 2 ( A 3 �u +) + H 2 2.5e–11 0 0

A84. N 2 ( a ′ 1 �−

u ) + H 2 ⇒ N 2 + H + H 2.6e–11 0 0

A85. N 2 (a 1 �g )+ H 2 ⇒ N 2 + H + H 2.6e–11 0 0

A86. N 2 (w

1 �u )+ H 2 ⇒ N 2 + H + H 2.6e–11 0 0

A87. O 2 (a 1 �g )+ H 2 ⇒ OH + OH 2.8e–09 0 17,906

A88. O 2 (a 1 �g )+ H 2 ⇒ O 2 + H 2 2.6e–19 0.5 0

A89. O 2 (a 1 �g )+ H ⇒ O + OH 1.3e–11 0 2530

A90. O 2 (a 1 �g )+ H ⇒ O 2 + H 5.2e–11 0 2530

A91. O 2 (a 1 �g )+ HO 2 ⇒ O 2 + HO 2 2.0e–11 0 0

A92. O 2 ( b 1 �+

g ) + H 2 ⇒ O 2 + H 2 1.0e–12 0 0

A93. N( 2 D)+ H 2 ⇒ NH + H 4.6e–11 0 880

A94. N( 2 P)+ H 2 ⇒ N + H 2 2.0e–15 0 0

A95. O( 1 D)+ H 2 ⇒ H + OH 1.1e–10 0 0

A96. O( 1 S)+ H 2 ⇒ O + H 2 2.6e–16 0 0

c

c

3

C

i

F

d

t

d

b

f

a

o

f

r

s

b

q

t

A

i

t

p

[

o

o

R

[

[

[

[

[

ontribution of collisional quenching of N 2 (C

3 �u ) does indeed be-

ome comparable with the production via the direct path for E / N ≈00 Td at low gas temperatures with equal composition mixtures.

onversely, when γ c ≈ 0, the contribution of collisional quench-

ng is mainly via de-excitation of N 2 ( A

3 �u

+) and negligible (see

ig. 25 c).

However, for the purposes of the present study, the total pro-

uction maps in Fig. 26 a–e show insensitivity. When the reac-

ion in Eq. is included alongside those in Table A.2 , radical pro-

uction through the collisional quenching channel is influenced

y γ c . Even though the addition of A97 significantly changes the

ractional contributions to H, the collisional quenching production

t X O 2 = 0 . 15 is two orders of magnitude smaller than the direct

ne evaluated in fuel-rich mixtures (see Fig. 24 a and b). There-

ore, H 2 :air jet flames will support a selective energy coupling in

egions of high fuel concentration typical of non-premixed atmo-

pheric flames [7] , where the majority of radical production will

e through dissociation by electron impact. Thus, the collisional

uenching contribution can be neglected for the present condi-

ions.

Finally, we compare Table A.2 against other reported models.

prediction with the model of Shcherbanev et al. [82] is shown

n Fig. 26 f. A comparison between Figs. 26 f and 24 f show that the

wo models predict a similar radical production over the range of

arameters analyzed. Moreover, the model of Shcherbanev et al.

82] predicts a lower radical production via collisional quenching

f N

∗2 than that proposed in the present work over the entire range

f conditions of interest.

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