Towards predictive simplified chemical kinetics for hydrogen detonations

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Towards predictive simplified chemical kinetics for hydrogen
detonations
Fernando Veiga-L´opeza,∗, Said Tailebb, Ashwin Chinnayyac, Josu´e Melguizo-Gavilanesc,d
aAerolab, Research Institute of Physics and Aerospace Science, University of Vigo, Campus de Ourense,
Ourense, 32004, Galicia, Espa˜na
bSafran Tech, Magny-Les-Hameaux, France
cInstitute Pprime, UPR 3346 CNRS, ISAE-ENSMA, Universit´e de Poitiers, 86961,
Futuroscope-Chasseneuil, France
dShell Global Solutions B.V., Major Hazards Management, Energy Transition Campus, 1031 HW
Amsterdam, The Netherlands
Abstract
A methodology to develop predictive simplified kinetics schemes (one-step/three-step chain-
branching) is presented in which detonation velocity-curvature (D−κ) curves computed
with detailed thermochemistry are used as the fitting target aiming to capture the turning
point of the curve (κcrit). This was motivated by the similar trend observed between the κcrit
values obtained using the simplified schemes of Taileb et al. [1], fitted using conventional
methods, and the critical reactive layer heights for detonation propagation under yielding
confinement (hcrit) reported by the same authors. Both updated schemes satisfactorily
reproduce the target D−κcurves and are used to (re)compute multidimensional cellular
detonations propagating in channels and confined by inert layers. Simulations show a much
better agreement with the results obtained with detailed kinetics for the detonation flow
fields, cell sizes distributions, and hcrit. Moreover, it is observed that the average curvatures
of the computed fronts are in line with those predicted by the D−κformulation, providing
supporting evidence of the applicability of reduced order models for fast and inexpensive
estimates of detonation limiting behaviors in safety studies.
Keywords: Detonation, Curvature, Simplified Kinetics, Hydrogen
∗Corresponding author
Email address: fernando.veiga@uvigo.gal (Fernando Veiga-L´opez)
Preprint submitted to Combustion and Flame October 8, 2024

Information for Editors and Reviewers
1) Novelty and Significance Statement
The novelty of this research lies in proposing and testing an alternative methodology
to determine the kinetics parameters of simplified kinetics schemes (1−step and 3−step)
so that these retain the predictive capabilities of detailed kinetics. This is achieved by
using detonation velocity-curvature curves (D−κ) as a fitting target. Conventional fitting
procedures, such as targeting ignition delay times and/or matching ZND profiles have been
shown to perform poorly in previous research. The new methodology shows significant
improvements in the prediction of dynamic detonation parameters in ideal detonations such
as cell sizes distributions, as well as in the prediction of the critical reactive layer heights
for detonation propagation under yielding confinement (hcrit); a canonical configuration of
interest to propulsion and industrial safety. For the latter, the new methodology yields hcrit
for H2-O2detonations of hcrit = 12 mm instead of 24 mm (with conventional methods), and
hcrit = 8 mm instead of 20 mm, for the 1-step and 3-step models, respectively. The Dκ
results are much closer to those obtained with detailed kinetics (hcrit = 6 mm) and to the
experimental measurements (hcrit = 4.6 mm). The methodology is generic to be applied to
any mixture of interest.
2) Author Contributions
•F. Veiga-L´opez: performed the research, analyzed data, wrote the paper.
•S. Taileb: reviewed & edited.
•A. Chinnayya: reviewed & edited.
•J. Melguizo-Gavilanes: designed the research, analyzed data, reviewed & edited.
2

1. Introduction
One of the primary challenges in combustion modeling is the integration of chemistry.
For example, a detailed mechanism for a typical hydrocarbon fuel used in transportation may
include hundreds of species and thousands of reactions [2]. To address this issue, reduction
techniques have been developed, which aim to maintain their predictive capabilities while
minimizing the number of species and reactions. These techniques include quasi-steady
state, directed graph, manifold, and genetic algorithms methods [3–5], as well as simpler
systematic approaches based on error metrics [6].
The turbulent combustion community has led most of these efforts, likely driven by in-
dustrial needs [7, 8], both in stationary and mobile energy conversion systems, to conceive
more fuel-efficient and less polluting alternatives. On the contrary, the detonation commu-
nity has lacked such drivers and has been historically satisfied with a mostly qualitative
understanding of the phenomenon given by the simplest descriptions of the chemistry (i.e.,
one-step Arrhenius). This is due in part to the very stringent resolution requirements to
adequately resolve multidimensional detonation fronts. In actuality, when predictions/data
are needed for limiting behaviors (i.e., detonability limits, detonation initiation/diffraction,
and quenching), experimental databases continue to be the most reliable source [9], although
some success has been achieved recently when coupling inviscid hydrodynamic solvers with
H2-O2detailed kinetics (9 species/21 reactions) [1, 10].
The integration of more complex fuels is expected to remain prohibitively expensive
in the near future. Therefore, a good compromise would be to develop simplified kinetic
schemes that can reproduce quantitatively the limiting behaviors mentioned above. Some
past and recent efforts on simplified modeling for detonation applications include one- to
five-step schemes developed with varying degrees of sophistication [11–19]. One common
feature present in detonation initiation, propagation, and diffraction is that their fronts are
observed to be globally curved. Conventional fitting procedures, however, rely on using the
laminar planar ZND structure hoping to reproduce the complex thermodynamic changes
and associated chemical rates in the induction zone of multidimensional detonations. As
discussed below, kinetic schemes fitted in this way (or variations thereof such as targeting
constant pressure/volume induction times over a temperature and pressure/volume range
of interest) not only fail at predicting the minimum reactive layer height for detonation
propagation under yielding confinement (hcrit) [1] but also at capturing one of the simplest
extensions to the ZND model (i.e., quasi-steady weakly-curved detonation waves), the so-
called D−κcurves.
It is natural to expect that fitting the rates of simplified schemes to capture the critical
curvatures predicted by detailed mechanisms would result in simplified kinetics that repro-
duce more closely the expected chemical rates in the reaction zone at dynamic sub-DCJ
conditions. In this work, a methodology in which D−κcurves obtained with detailed ki-
netics are used as the fitting target is presented (Sect. 2). Results using the new approach
and conventional methods are compared and discussed for ideal and non-ideal detonation
propagation (Sect. 3). Closing remarks are included in Sect. 4.
3

2. Fitting methodology
2.1. Physical model
2.1.1. Governing equations
The generalized quasi-steady ZND model, which considers weakly curved leading shock
fronts, can be derived by adding source/sink terms to the reactive Euler equations as de-
scribed in [20] and other seminal works on the topic [21–24].
dρ
dx=−ρ
w
( ˙σ−w(1 −η)α)
η,(1)
dw
dx=( ˙σ−wα)
η,(2)
dp
dx=−ρw( ˙σ−wα)
η,(3)
dYk
dx=Wk˙ωk
ρw(k= 1, ..., N ),(4)
˙σ=
N
X
k=1 W
Wk−hk
cpTdYk
dt,
α=1
A
dA
dx=κD
w−1,
where, ρ, w, p, and tare the mixture density, axial velocity in the wave-attached frame,
pressure, and time, respectively. The mass fraction, molecular weight and net produc-
tion/consumption rates per unit mass of species kare given by Yk,Wkand ˙ωk. Here
η= 1 −M2is the sonic parameter and M= w/afis the Mach number relative to the
leading shock computed using the frozen speed of sound, af. ˙σis the thermicity, αthe axial
area change representing the flow divergence that is related to the shock curvature. Wis
the mean molar mass of the mixture, cpthe mixture specific heat at constant pressure, and
hkis the specific enthalpy of species k. Finally, κis the curvature of the wave front given
by 2/Rcfor spherical waves, and 1/Rcfor cylindrical waves with Rcbeing the local radius
of curvature; see [20] for additional details. It can be readily shown that setting α= 0
reverts the formulation to the ideal case included in Browne et al. [25]. The implementation
in the Shock and Detonation Toolbox (SDT) [25] only entails adding the terms containing
α, making it rather straight forward, and most importantly, allowing us to investigate ar-
bitrary chemical mechanisms written in Cantera format [26] (i.e., .cti files). The numerical
root-finding algorithm developed by Veiga-L´opez et al. [27] for detonations with friction
was adapted to handle weakly curved detonations. The model has been used successfully
in previous work that investigated the influence of low temperature chemistry on steady
detonations with curvature losses [28, 29].
4

2.1.2. Chemistry modeling
The chemistry is modeled using the same methodology presented in Taileb et al. [1],
namely, chemical schemes of increasing complexity: single-step, three-step chain-branching
and detailed kinetics [30]. For completeness, the most salient features and assumptions made
in the simplified schemes are briefly described next.
In the single-step model, the fuel, F, is directly converted into products following a single
irreversible Arrhenius reaction, F→P, occurring at a rate k=Asexp(−Ea/RuT). In the
three-step chain-branching model, initiation,branching and termination steps are accounted
for that mimic the initial thermally neutral decomposition of the fuel (F) to produce active
radicals (R) at rate rI, the subsequent abrupt increase of the radical pool at rate rB, and
their final conversion into products (P) at rate rTaccompanied by heat release, respectively.
Initiation: F(Fuel) →R(Chain-carriers),
rI=kI(ρ/W )YFexp −EI
RuT,(5)
Branching: F+R→2R,
rB=kB(ρ/W )2YFYRexp −EB
RuT,(6)
Termination: R→P(Products),
rT=kC(ρ/W )YR,(7)
where EI/Ruand EB/Ruare the activation temperatures, and kI=kCexp (EI/RuTI),
kB=kC(W/ρvN ) exp (EB/RuTB) and kCare the pre-exponential factors; ρvN is the von
Neumann density.
The additional degrees of freedom in the latter model (i.e., initiation/branching acti-
vation energies and cross-over temperatures) allow for increased flexibility to reproduce
more complex chemical behaviors such as the sensitivity of hydrogen auto-ignition to initial
temperature/pressure, and the distinction between induction and reaction zone and their
respective length-scales. For all the simplified chemical schemes, the reactive mixture is as-
sumed to have a constant mean molar mass, W, ratio of specific heats, γ=cp/cv, and total
heat release, Q; see [1] for further details and the implications of these assumptions. The
detailed mechanism of M´evel et al [30] (9 species and 21 reactions) whose predictions had
been shown to provide sensible results when compared against experiments [1, 10] is used to
compute the reference curves. It should be noted that measurable variations are anticipated
based on the specific detailed chemical kinetics chosen [27] the so-called mechanism-induced
uncertainties which are outside of the scope of the current work. Therefore, the predictive
capabilities of the simplified models discussed here are restricted to those of the reference
mechanism selected.
5

2.2. D−κcurves
Figure 1 (left) shows the D−κcurves obtained for a stoichiometric H2-O2mixture at
ambient pressure and temperature using the chemical models described above, and the same
kinetic parameters as in Taileb et al. [1]: As= 1.1×109s−1, and Ea/Ru= 11277 K, for
single-step kinetics; and kC= 2×107s−1,EI/Ru= 25000 K, EB/Ru= 9300 K, TI= 2431 K,
TB= 1430 K, for three-step chain-branching kinetics. These are referred to as/labeled CV
1-step and CV 3-step, respectively. The γand Qvalues characterizing the reactive mixture
are given in Table 1.
0.5 1.0
1000/T [K−1]
10−8
10−6
10−4
10−2
τind [s]
0 100 200
κ[m−1]
0.7
0.8
0.9
1.0
D/DCJ,det
Detailed
CV 1-step
CV 3-step
Figure 1: Constant volume induction times, τind at ρvN (left) and D−κcurves (right) obtained with
detailed chemistry [30] and the CV schemes. DCJ,det is the steady propagation velocity computed with
detailed chemistry. Conditions: stoichiometric H2-O2at p0= 100 KPa and T0= 300 K.
Figure 1 clearly shows that simplified mechanisms fitted to reproduce the constant volume
induction times of detailed kinetics, as was done in [1] (Fig. 1-left), yield significantly different
turning points, i.e., κcrit (Fig. 1-right). More interesting is the fact that the κcrit obtained
(κcrit, 1-step = 67.4 m−1;κcrit, 3-step = 74.8 m−1;κcrit, det = 168.5 m−1) follows the same trend as
that of the 2-D simulations of Taileb et al. [1] where the critical reactive layer heights, hcrit,
for detonation propagation under yielding confinement were determined; a higher critical
front curvature yields a lower hcrit. To compare with their results a crude estimate for
the curvature defined as κeq ∝1/hcrit is used: κeq, 1-step = 41.7 m−1;κeq, 3-step = 50.0 m−1
κeq, detailed = 166.7 m−1. Note that in both cases, κ1-Step < κ3-step ≪κDetailed . This last
observation may imply, and will be confirmed later, that using κcrit as a fitting target for
the development of simplified schemes could result in improved predictive capabilities in
multidimensional simulations.
2.3. κcrit as fitting target
Figure 2 shows the effect of varying the thermodynamic and kinetic parameters of the
simplified schemes on the resulting D−κcurves. The size of the arrows qualitatively
indicates how sensitive the turning point of the curve is to changes in each parameter. Note
that this is consistent with [31, 32] for the 1-step case. For the modified mechanisms, the
initial conditions were improved over those in [1] by matching as best as possible the steady
detonation velocity, DCJ , and von Neumman temperature, TvN , computed with detailed
kinetics. This entailed finding the best combination of Qand γwithin realistic values
6

0 50 100
κ[m−1]
0.8
0.9
1.0
D/DCJ
k
γ
Q
CV 1-step
0 50 100
κ[m−1]
0.8
0.9
1.0
D/DCJ
kC
kI
γ
Q
kB
CV 3-step
Figure 2: Influence of the thermodynamic and kinetic parameters on the calculated D−κcurves for single-
step (left) and three-step chain-branching kinetics (right). The size of the arrows qualitatively indicates how
sensitive the turning point of the curve is to changes in each parameter.
Table 1: Detonation and thermodynamic properties for a stoichiometric H2-O2mixture for all chemical
schemes used. The pre-exponential factors for simplified kinetics are also shown. Initial conditions: p0=
100 kPa and T0= 300 K.
Chemical model DCJ [m/s] TvN [K] γ(0−vN−CJ) Q[MJ/kg] As;kC[s−1]lind [µm] Ea,eff/RuTo
Detailed kinetics [30] 2839.9 1768.7 1.4 - 1.315 - 1.218 - - 41 28.78
CV 1-step [1] 2801.5 1674.8 1.33 4.800 6.0×10987.9 33.99
Dκ 1-step 2836.9 1769.5 1.35 4.606 1.08 ×1010 36.2 35.92
CV 3-step [1] 2850.4 1723.7 1.33 4.990 2.0×10746.8 30.83
Dκ 3-step 2836.2 1768.7 1.35 4.613 4.0×10721.4 30.08
for stoichiometric H2-O2detonations while ensuring that the proper velocity deficits for
increasing κwere captured, as well as the turning point of the curves, κcrit.
The parameters found after the manual trial-and-error fitting exercise are included in
Table 1 as well as detonation properties of interest to the discussion. Bear in mind that only
Asand kCwere modified whereas the initiation/branching activation energies and cross-over
temperatures were left unchanged as these are more appropriately defined in ignition delay
time vs. inverse temperature plots. Note that the slopes in the high- and low-temperature
regimes and the temperature threshold at which the change of activation energy occurs do
not change, there is only a shift along the y−axis consistent with the increase of the rate
multipliers (see Fig. 3-left). Changes in the available degrees of freedom should always be
guided by the thermochemical behavior of the mixture of interest. This method ensures that
the induction time and reaction zone length are matched reasonably well to those obtained
with the detailed chemistry.
The updated D−κcurves are shown in Fig. 3-right. In agreement with the large activa-
tion energy asymptotics in [20]: (i) given fixed thermodynamic conditions and Ea/Ru, higher
reaction rates kresult in larger κcrit values; (ii) an increase in TvN, given by the new thermo-
dynamics (i.e., Qand γ), yields a decrease in D(κcrit) and an increment in κcrit. Both simpli-
fied schemes reproduce quite well the expected behavior for D/DCJ,det > D/DCJ,det (κcrit).
7

0.5 1.0
1000/T [K−1]
10−8
10−6
10−4
10−2
τind [s]
0 100 200
κ[m−1]
0.6
0.8
1.0
D/DCJ,det
Detailed
Dκ 1-step
Dκ 3-step
Figure 3: Constant volume induction times, τind at ρvN (left) and D−κcurves (right) obtained with the Dκ
schemes and compared with detailed kinetics results. Conditions: stoichiometric H2-O2at p0= 100 KPa
and T0= 300 K.
However, significant deviations are present for larger deficits. The differences in the com-
puted velocity deficits D/DCJ,det at fixed κusing detailed/three-step chain-branching kinet-
ics when compared with those of one-step in the range D/DCJ,det <0.9 are a consequence
of the known inability of the one-step models to adequately reproduce the H2-O2ignition
delay times at post-shock temperatures below the chain-branching cross-over temperature
(see Fig. 3- right). Nonetheless, this does not seem to be a first order effect for improved
predictions of hcrit in multidimensional simulations; capturing κcrit will be shown to provide
much better estimates than the reference simplified kinetics used in [1]. The schemes fitted
with the methodology described above will be referred to as/labeled Dκ 1-step and Dκ
3-step hereinafter.
3. Results and discussion
3.1. Ideal 1-D steady structure
The ZND profiles obtained with the CV (left) and Dκ (right) schemes are compared
with those using detailed kinetics in Fig. 4. It can be seen that the modified simplified
schemes reproduce much better the von Neumann state, which was one of the fitting ob-
jectives. Moreover, the induction lengths, lind, obtained with the updated reduced kinetics
vary considerably because the modified pre-exponential factors always increased during the
fitting, and the resulting combination of Qand γrequired to yield higher TvN, ultimately
sped up the chemical reaction. For instance, the Dκ 3-step scheme, resulted in a lind that
is ∼48% smaller than that predicted by the detailed mechanism. Note that matching lind
was not a target nor a constraint of the proposed fitting procedure, which aimed to find a
parameter combination that captured the critical curvature obtained with the quasi-steady
1-D model. The CJ state position (i.e., ˙σ→0) better matched that predicted by the de-
tailed chemistry. However, the thermodynamic conditions at this state strongly differ from
the detailed mechanism due to the constant γassumption. Therefore, all simplified models
do not account for changes in the molecular weight of the gas between the fresh mixture
and the products. The χparameter [33] given by χ= (Ea,eff/RuTvN)·lind ·˙σmax/uC J where
˙σmax denotes the maximum in the thermicity profile, can be computed solely based on ZND
8

15
25
35
p/p0
Detailed
CV 1-step
CV 3-step
15
25
35 Detailed
Dκ 1-step
Dκ 3-step
5
10
T/T0
5
10
0 0.1 0.2
0
15
30
˙σ[1/µs]
0 0.1 0.2
0
15
30
x[mm] x[mm]
Figure 4: ZND profiles - Pressure, temperature and thermicity obtained with the CV (left) and Dκ (right)
schemes. The ZND profiles computed with detailed chemistry [30] are included as a reference.
profiles, and is reported in Table 2. This metric is often used to characterize cellular struc-
ture regularity. Section 3.2 will show that χdoes not seem to be as relevant in our case for
predicting changes in cellular structure probably due to the fact that their values remain
relatively similar. The effective activation energy seems to better predict the changes in
the nature of the cellular structure (i.e., higher Ea,eff/RuTovalues yielding more irregular
flow fields). Furthermore, Sect. 3.3 will show that hcrit bears no correlation with χwhich
suggests a key role of multidimensional effects in its determination.
Finally, while it is plausible that several combinations of the fitting parameters (i.e.,γ,
Q,ASand kC) may lead to simplified schemes that would match κcrit in D−κcurves, it is
outside of the scope of this work to provide the optimal combination. Rather, a novel fitting
approach for simplified kinetics is presented and systematically evaluated.
3.2. Ideal 2-D propagation in channels
The parallel in-house code RESIDENT (REcycling mesh SImulations of DEtoNaTions)
was used to integrate the Euler equations in 2-D with a uniform mesh (∆x= ∆y=lind/10
pts). A detonation was initiated and allowed to propagate in a channel completely filled with
stoichiometric H2-O2at p0= 100 kPa and T0= 300 K until a quasi-steady structure was
achieved. A sliding window technique was used to follow the detonation as it propagated
into mixture at rest [34]. This allowed us to have a fixed number of cells in the computational
9

Table 2: χparameter [33] computed with the different chemical mechanisms.
Chemical model χ
Detailed kinetics [30] 3.187
CV 1-step [1] 2.270
Dκ 1-step 1.875
CV 3-step [1] 1.389
Dκ 3-step 1.177
domain at all times, irrespective of the length of the channel computed. Detailed descriptions
of the numerical methods employed in RESIDENT for spatial and temporal discretizations
as well as the parallelization strategy can be found in [34, 35]. The instantaneous flow
fields, the numerical soot foils and average and local front curvatures are analyzed once the
detonation reaches a velocity close to the ideal Chapmann-Jouguet speed, DCJ (to within
±2%). Following this approach, it is possible to avoid any influence of the initiation transient.
Note that this is the same numerical methodology used in the work by Taileb et al. [1] so
that a meaningful comparison can be performed. RESIDENT has been previously used in
fundamental detonation studies using simplified [36] and detailed chemical kinetics [1, 37]
for ideal and non-ideal gases [38].
3.2.1. Instantaneous flow fields
Figure 5 shows the instantaneous temperature and normalized density gradient fields of
the detonations obtained with the reference (CV ) and new (Dκ) simplified schemes; results
with the detailed chemical mechanism are also included as a reference. The Dκ schemes yield
fields that are qualitatively much closer to the detailed mechanism, albeit exhibiting a lower
final average temperature because of the constant molecular weight assumption discussed
earlier. The combined effect of (i) enhanced resilience to curvature, which allows the front
to locally fold further without quenching, and (ii) the presence of stronger transverse waves
that leads to a sharper density gradient behind the shock compared to the detonations
computed in [1], yield more homogeneous flow fields and reduce the amount of unburned
pockets present when the Dκ schemes are used. Furthermore, the visibly more irregular
detonation fields obtained with the Dκ 1-step are directly related to its higher effective
activation energy, Ea,eff/RuTo, when compared to that of detailed chemistry (see Table 1).
3.2.2. Numerical soot foils and cell size histograms
Figure 6 includes a schematic of the length that is typically referred to as the cell size
λ, the numerical soot foils and the cell size histograms for all the chemical schemes tested.
Oblong cells are measured from bottom-to-top vertex. The sample size is fixed to 90 and
the size of the cells is measured manually from the soot foils and counting the number
of times, i.e. frequency, a range of cell size appears. The histograms show that the cell
size distributions are significantly better predicted by the Dκ schemes. The latter yield
10

Figure 5: Instantaneous temperature and normalized density gradient fields obtained with detailed chem-
istry [30], CV and Dκ schemes.
smaller cells (λmax, CV , 1-step = 1.8 mm vs. λmax, Dκ, 1-step = 0.4 mm ; λmax, CV , 3-step = 1.2 mm
vs. λmax, Dκ, 3-step = 0.57 mm) that were in much closer agreement to those estimated by
the detail chemical mechanism (λmax, detailed = 0.6 mm). The reduction in cell sizes can be
explained by the faster rates required in the D−κschemes and associated reduction in lind.
3.3. Non-ideal 2-D propagation
Further analysis is carried out by computing 2-D detonations confined by an inert layer.
This canonical configuration is of interest to novel propulsion applications such as rotating
detonation engines (RDEs) [39–41], as well as safety since upon a fuel leak into open space
and subsequent ignition of the resulting reactive cloud a detonation may be initiated [42].
Again, the same numerical setup as in [1] is used and, for completeness, briefly summarized
next. Once a detonation at DCJ is stabilized in an ideal channel, the upstream conditions are
modified to include an inert layer in the domain. The size of the inert layer, measured from
the top of the domain, is progressively increased at a constant step (∆h= 2 mm) resulting
in a thinner reactive layer of thickness h. At a certain height the detonation quenches
before reaching xQ=x−x0,inert layer where x0,inert layer is the axial position at which the
ideal detonation meets the inert layer, and xQthe distance at which the cellular pattern
fades completely. The xQvalues reported in [1] at hcrit are used as a reference to determine
quenching. Once again, the reader is referred to [1] for further details about the initial and
boundary conditions used. It should be noted that one simulation is run every time the
reactive height his changed, which makes finding hcrit a computationally expensive process.
11

Figure 6: Numerical soot foils and cell-size histograms obtained with detailed chemistry [30], CV and Dκ
schemes. Note the differences in scale among the soot foils. A schematic showing the length that is typically
referred to as the cell size λis also included.
3.3.1. Critical heights prediction
The Dκ schemes hcrit predictions are discussed next. Table 3 shows the results of the
simulations. The Dκ schemes yield lower hcrit values than the CV schemes, i.e., hcrit =
12 mm instead of 24 mm, and hcrit = 8 mm instead of 20 mm, for the 1-step and 3-step
models, respectively. Note that the Dκ results are much closer to those obtained with
detailed kinetics (hcrit = 6 mm) and to the experimental measurements (hcrit = 4.6 mm).
Therefore, it seems that κcrit is a valid fitting target to develop simplified schemes with
improved predictive capabilities. Put differently, adding the physics expected to play an
important role as part of the fitting procedure helps to properly capture important gas
dynamical and chemical features thereby improving the predictive capabilities in simulations
of multidimensional detonations.
The numerical soot foils for representative cases are included in Fig. 7. These show
the complex dynamics of the interaction between the detonation and the inert layer prior
and during quenching. The presence of strong expansions partially quench and curve the
detonation close to the interface; transverse waves and their reflections at the lower wall
re-initiate the detonation. A more thorough description of the propagation and quenching
dynamics for this configuration was already provided in [1].
3.3.2. Average and local front curvature at h > hcrit
Due to the interaction with the inert layer and the lateral expansion of burnt gases,
the leading shock curves. Figure 8 depicts steady detonation fronts propagating through
a reactive layer slightly higher than hcrit (i.e, h=hcrit + ∆h; note that ∆hused here
differs from that used in [1]). The Dκ schemes allow more curved detonations to propagate
without quenching through thinner layers, because of faster reaction rates required to match
12

Figure 7: Numerical soot foils for detonation waves propagating through reactive layers heights of h=
hcrit + ∆hand h=hcrit for detailed chemistry [30], CV and Dκ schemes.
Table 3: Critical heights predicted by the different chemical schemes.
Chemical scheme hcrit [mm] xQ[mm]
Detailed kinetics [30] 6 60
CV 1-step [1] 24 200
Dκ 1-step 12 176
CV 3-step [1] 20 140
Dκ 3-step 8 12
Experimental [43] 4.6 -
13

κcrit than those in the CV schemes. The weak global curvature assumption invoked in the
D−κmodel, used to fit the chemistry herein, is justified by the 2-D flow fields obtained.
Interestingly, these fields also provide the means to assess to which extent the actual 2-D
front curvatures can be compared with the κcrit value given by the D−κmodel.
The 2-D detonation front average and local curvatures are obtained by fitting a second
and fifth order polynomial, respectively, to the plane curve f(x, y) given by the maximum of
the normalized density gradient which delineates the front. The curvature is subsequently
computed using κ=|x′y′′ −y′x′′|·(x′2+y′2)−3/2where the prime notation, ′and ′′ , represent
first and second derivatives. Analytical differentiation was used in the function evaluation.
Figure 9 shows the average and local curvature of the propagating fronts. A few things
are worth mentioning regarding the values of the local curvature as these are sensitive to: (i)
the smoothing used on the initially noisy line that delineates the detonation front, and (ii)
the order of the polynomial chosen to fit the detonation front. The results show that the local
curvatures can attain values as high as κloc, 2-D ∼250 m−1; this is ∼4 times larger than the
curvature given by the D−κmodel for a detonation propagating at D∼0.96DC J , that is,
κ1-D ∼70 m−1. Conversely, the average curvature κavg, 2-D ∼50 m−1is comparable to κ1-D.
This observation provides supporting evidence that κavg, 2-D serves as a more representative
metric to compare against the results of the D−κmodel. It is worth mentioning that
Reynaud et al. [35] used a more sophisticated method to compute global/average and local
curvatures, argued to be more relevant to general detonation dynamics, which entailed time-
averaging the flow field at the lower wall.
3.3.3. Curvature-driven quenching dynamics at h=hcrit
The fronts curve more close to the inert layer where the expansion waves originate from
than at the vicinity of the lower wall. As a result, the detonation wave slows down and
eventually quenches as h→hcrit. In Fig. 10, the curvatures of the detonation fronts prior to
quenching are given. The local curvatures are, as expected, higher than those for h > hcrit
(κloc, 2-D ∼500 m−1); the average curvatures (κavg, 2-D ∼70 m−1) are again similar to the
estimates of the D−κmodel for a velocity deficit of D∼0.95DCJ (κ1-D ∼100 m−1).
The temporal quenching dynamics of the Dκ schemes are analyzed in detail in Fig. 11 for
h≥hcrit. This is carried out measuring the variation of the average instantaneous curvature
of the fronts as a function of time, the diamond-shaped marker indicates the time at which
quenching occurred. Note that a similar convention to that used to determine the quenching
location xQabove, is used here to define the quenching time t−t0, that is, time is measured
relative to the moment the detonation wave meets the inert interface, t0.
For the Dκ 1-step scheme (Fig. 11 - left), the detonation propagating in a reactive layer
height of h=hcrit + ∆h= 14 mm experiences a smooth increase in curvature immediately
after interaction with the inert layer. Upon reaching a value of κ2-D ∼35 m−1the curvature
decreases and an oscillatory phase starts, in which folding and stretching of the front takes
place. At t−t0∼32.5µs, t−t0∼50 µs and t−t0∼82.5µs, the detonation exhibits a strong
sudden increase in its curvature, yielding local quenching at x∼110 mm, x∼170 mm and
x∼240 mm, respectively, in agreement with the soot foils of Fig. 7. In this case, the
detonation front re-initiates due to the presence of strong transverse waves/triple points
14

Figure 8: Normalized density gradient fields obtained for detonations propagating through reactive layer
heights of h=hcrit + ∆hcomputed with detailed chemistry [30], CV and Dκ schemes.
that reflect off the bottom wall, originating from the interface with the inert layer. For
h=hcrit = 12 mm, the dynamics are essentially the same: the front curves when the
detonation interacts with the inert layer, and an oscillatory phase in the curvature of the
front ensues, causing local extinction/re-initiation. However, at a certain point, the front
curvature reaches a value of κcrit, 2-D ∼110 m−1, which quenches the detonation fully; no
re-initiations were observed thereafter.
For the Dκ 3-step scheme (Fig. 11 - right), the detonation front exhibits similar dynamics
as that described above for propagation in a reactive layer height of h=hcrit +∆h= 10 mm.
In this particular case, the oscillations are not as regular. This is due to the very strong
curvature increase at around t−t0∼12 µs and t−t0∼40 µs, that results in locally
strong quenching events followed by re-initiations. For h=hcrit = 8 mm, the detonation
quenches shortly after interacting with the inert layer. The sudden increase of curvature
around ∆κ∼100 m−1, and associated expansion, effectively decouples the front and the
chemical energy release; no re-initiation attempts were observed in this case.
While the propagation and quenching dynamics observed for the Dκ 1-step and Dκ
3-step schemes were quite different at h=hcrit, they do share a common feature. The
detonations locally/globally quench whenever the average curvature reaches a value around
κ2-D >100 m−1provided that the transverse waves present are not strong enough to re-
activate chemical reactions and re-initiate the front. Note that the critical curvatures and
velocity deficits the D−κmodel yield, κcrit, 1-D ∼167 m−1and D/DCJ (κcrit ) = 0.9, are
15

Figure 9: Normalized density gradient fields obtained for detonations propagating through layers of h=
hcrit + ∆hfor the Dκ 1-step (top) and Dκ 3-step (bottom). Absolute |κ|and signed κvalue of local (blue)
and average (red) curvatures measured from the instantaneous fields are shown. 1D κ(green) denotes the
value of κcrit obtained from the D−κmodel.
of the same order of magnitude as those computed from the 2-D fields. The quantitative
discrepancies just reported are not necessarily surprising given the D−κmodel assumptions,
and indicates that unsteadiness/deceleration [44] of the front as well as its multidimensional
structure play an important role in the quenching dynamics. In spite of this, the results seem
to suggest that low-order models may still serve as fast and inexpensive tools to estimate
reactive layer heights (i.e., limiting behaviors) in RDE design and safety relevant scenarios,
at least for the case of H2detonations.
Finally, note that the hcrit predicted by the Dκ 1-step and Dκ 3-step differ even though
they share the same 1-D fitting target. Two reasons come to mind. First, the conception of
the simplified 1-/3-step schemes irrespective of their fitting target (i.e., CV or Dκ). 3-step
schemes provide a single pathway for the build-up radicals, if this pathway is stopped, there
will be very little heat release to sustain the detonation wave. The latter occurs when T < TB
suddenly stopping the production of chain-branching species. As previously explained, the
initial transient upon interaction with the inert layer results in very high local curvatures
that lead to thermodynamic states within the above mentioned temperature range ultimately
resulting in prompt decoupling of the leading shock and reaction zone. On the other hand, 1-
step schemes are less sensitive because the chemistry is always active in the postshock region.
As a result, these are not capable of reproducing the thermally neutral lind nor of mimicking
16

Figure 10: Normalized density gradient fields obtained for detonations propagating through layers of h=hcrit
for the Dκ 1-step (top) and Dκ 3-step (bottom). Absolute |κ|and signed κvalue of local (blue) and average
(red) curvatures measured from the instantaneous fields are shown. 1D κ(green) denotes the value of κcrit
obtained from the D−κmodel.
0 25 50 75
t−t0[µs]
0
25
50
75
100
125
|κ|[m−1]
Dκ 1−step
h= 14 mm
hcrit = 12 mm
0 25 50 75
t−t0[µs]
Dκ 3−step
h= 10 mm
hcrit = 8 mm
Figure 11: Temporal evolution of the average curvature of the detonation front profiles measured from
interaction with the inert layer (t=t0). The red diamond-shaped markers represent the quenching point
for reactive layer heights of h=hcrit.
the change in activation energy typical of H2mixtures. 1-step schemes always result in
unrealistically short induction times for T < TB; see Fig. 1-left. Second, the differences
in lind and the much slower energy deposition in the flow for the Dκ 1-step mechanism,
17

rendering it less effective to reproduce realistic detonation behaviors; see ˙σprofiles in Fig. 4.
The above may also explain the differences in the quenching distances, xQ, reported in
Table 3. Finally, while it could be argued that the reduction in hcrit may be fundamentally
due to a reduction in lind, it has been shown in previous work [1] that capturing lind alone
does not lead to an adequate prediction of hcrit . Note that the lind that is used as a fitting
target in conventional methods usually comes from a single ZND wave propagating at DCJ .
A key aspect that the current study reveals is the crucial role that curvature plays during
detonation quenching and the importance of properly capturing the reactivity changes as the
wave gets weakened and curved by expansion waves thereby decoupling the leading shock
from the reaction zone while propagating at sub-DCJ speeds. Therefore, the evidence so far
seems to suggest that it is the more adequate chemical response of the wave to curvature
losses that leads to better predictive capabilities. This is precisely what using the D−κ
model with κcrit as a fitting target allows us to do in a relatively straightforward fashion.
4. Conclusion
Simplified kinetics schemes obtained by fitting the critical curvature predicted by the
D−κcurves obtained with detailed chemistry are proposed (Dκ schemes). These were used
to compute ideal and non-ideal detonation propagation in 2-D. Of special interest was the
determination of the critical reactive layer height that leads to quenching, hcrit. The results
were compared to those reported in Taileb et al. [1] in which simplified kinetics fitted by
matching the constant volume induction times (CV schemes) given by the same detailed
mechanism were used. Results show that the Dκ schemes yield front structures that are
qualitatively and quantitatively more similar to those given by detailed chemistry (i.e., cell
size histograms and lower hcrit values) thus in better agreement with experimental measure-
ments. Including the physics expected to play a role in the quenching process (curvature
effects) as part of the fitting procedure, resulted in simplified schemes with improved pre-
dictive capabilities over standard methods. The methodology is generic and can be applied
to any mixture of interest. Furthermore, detonations interacting with inert layers are found
to follow an oscillatory change in their curvature when h→hcrit: the fronts stretch and fold
up to a critical curvature that impedes their propagation. Future work will include test-
ing/extending the proposed fitting procedure to H2-O2mixtures away from stoichiometry,
H2-air and hydrocarbons; the development of a thermodynamically consistent methodology
for predictive simplified kinetics is the object of current research efforts [19].
Acknowledgements
Financial support from the Agence Nationale de la Recherche Program JCJC (FASTD
ANR-20-CE05-0011-01) is gratefully acknowledged. Computations were carried out at the
supercomputer facilities of the M´esocentre de Calcul de Poitou-Charentes and using HPC
resources from GENCI-CINES (Grant A0152B07735). F. Veiga-L´opez is grateful to Univer-
sidade de Vigo for their financial support for Open Access publication.
18

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