Influence of chemistry on the steady solutions of hydrogen gaseous detonations with friction losses

Abstract

The problem of the steady propagation of detonation waves with friction losses is revisited including detailed kinetics. The derived formulation is used to study the influence of chemical modeling on the steady solutions and reaction zone structures obtained for stoichiometric hydrogen-oxygen. Detonation velocity-friction coefficient (D − c f) curves, pressure, temperature , Mach number, thermicity and species profiles are used for that purpose. Results show that both simplified kinetic schemes considered (i.e., one-step and three-step chain-branching), fitted using standard methodologies, failed to quantitatively capture the critical c f values obtained with detailed kinetics; moreover one-step Arrhenius chemistry also exhibits qualitative differences for D/D CJ ≤ 0.55 due to an overestimation of the chemical time in this regime. An alternative fitting methodology for simplified kinetics is proposed using detailed chemistry D − c f curves as a target rather than constant volume delay times and ideal Zel'dovich-von Neumann-Döring profiles; this method is in principle more representative to study non-ideal detonation propagation. The sensitivity of the predicted critical c f value, c f,crit , to the detailed mechanisms routinely used to model hydrogen oxidation was also assessed; significant differences were found, mainly driven by the consumption/creation rate of the HO 2 radical pool at low postshock temperature.

Full text

Influence of chemistry on the steady solutions of hydrogen gaseous
detonations with friction losses
Fernando Veiga-L´opeza,∗
, Luiz M. Fariab, Josu´e Melguizo-Gavilanesa
aInstitute Pprime, UPR 3346 CNRS, ISAE-ENSMA, 86961, Futuroscope-Chasseneuil, France
bPOEMS, CNRS-INRIA-ENSTA Paris, Institut Polytechnique de Paris, Palaiseau, France
Abstract
The problem of the steady propagation of detonation waves with friction losses is revisited
including detailed kinetics. The derived formulation is used to study the influence of chemi-
cal modeling on the steady solutions and reaction zone structures obtained for stoichiometric
hydrogen-oxygen. Detonation velocity - friction coefficient (D−cf) curves, pressure, tem-
perature, Mach number, thermicity and species profiles are used for that purpose. Results
show that both simplified kinetic schemes considered (i.e., one-step and three-step chain-
branching), fitted using standard methodologies, failed to quantitatively capture the critical
cfvalues obtained with detailed kinetics; moreover one-step Arrhenius chemistry also ex-
hibits qualitative differences for D/DCJ ≤0.55 due to an overestimation of the chemical
time in this regime. An alternative fitting methodology for simplified kinetics is proposed
using detailed chemistry D−cfcurves as a target rather than constant volume delay times
and ideal Zel’dovich-von Neumann-D¨oring profiles; this method is in principle more repre-
sentative to study non-ideal detonation propagation. The sensitivity of the predicted critical
cfvalue, cf,crit, to the detailed mechanisms routinely used to model hydrogen oxidation was
also assessed; significant differences were found, mainly driven by the consumption/creation
rate of the HO2radical pool at low postshock temperature.
Keywords: hydrogen, detonation, friction, chemical mechanisms.
∗Corresponding author
Email address: fernando.veiga-lopez@ensma.fr (Fernando Veiga-L´opez )
Preprint submitted to Combustion and Flame February 4, 2022

1. Introduction
In the upcoming global energy transition, hydrogen (H2) has positioned itself as a promis-
ing fuel for several stationary and mobile energy conversion systems, such as fuel cells or
direct combustion applications. Chemical reactions between pure H2and oxygen (O2) re-
lease large amounts of energy and do not produce any CO2, making H2very attractive to
decarbonize industrial sectors such as long-haul transport, chemicals, and iron and steel
production, where it is proving difficult to reduce emissions considerably [1]. However, there
are still issues that need to be overcome before widespread use of H2becomes a reality [2].
The most pressing include: (i) the small size of molecular H2which makes it very prone to
leaks in comparison to more conventional fuels (e.g., propane or natural gas); (ii) its low ig-
nition energy and wide flammability limits which increase the risk of accidental combustion
events; (iii) its low energy density which favors storage at very high pressures (70 MPa) to
make it competitive/attractive for transport applications but increases the risk of unintended
releases and explosion.
For instance, in the case of a fuel leak, very small concentrations of H2in air/O2are
required to produce flammable layers which results in an increased likelihood of acciden-
tal ignition thereby posing a serious safety hazard. Such leaks usually occur in confined
obstacle-laden geometries where upon ignition, and even without obstacles, accidentally-
ignited flames may accelerate and potentially transition to detonation [3, 4]; a much more
destructive combustion mode that may cause significant structural damage.
Gaseous detonations propagating in tubes are always subject to dissipation mechanisms,
such as momentum, heat and curvature losses. Their steady structure and dynamics differ
considerably from the more commonly studied ideal case. In this manuscript, we restrict
our attention to the influence of momentum losses (i.e., friction) on one-dimensional det-
onations. Firstly addressed by Zeldovich [5] and revisited by many other researchers in
subsequent works [6–11], it continues to be a relevant and an interesting problem in deto-
2

nation theory. From a practical point of view, accounting for losses is important to predict
detonation propagation limits in tubes. Despite the rather strong simplifications typically
required to develop a theory and/or low-order models, this approach should in principle
provide reasonable and inexpensive predictions on whether or not a detonation can propa-
gate given appropriate initial and boundary conditions. Thereby, anticipating the operating
envelopes of systems powered by any reactive mixture and, in particular, by H2. While we
recognize that detonations are inherently unsteady and multidimensional, the propagation
limits predicted by the proposed model may yield faster and conservative [12] estimates
that are relevant for engineering risk assessment, which allow to cover a larger region of the
parameter space at a fraction of the cost of running detailed transient two-dimensional simu-
lations. The latter statement is supported by previous work using transient one-dimensional
simulations with simplified kinetics and friction losses [12] which result in earlier failure than
a quasi-steady model predicts (i.e., larger cross-sections/tube diameters for failure); current
work in our group suggests a similar trend for detailed thermochemistry.
At any rate, it is known that friction plays two interesting roles on the physics of detona-
tions: it simultaneously acts as a sink of momentum and a heat source. The former makes
steady detonations subject to friction propagate at velocities lower than the ideal Chapman-
Jouguet (CJ) value, DCJ , whereas the latter is the sustaining mechanism at large velocity
deficits –mixture dependent– (D/DCJ ≤0.56 for H2-O2), that is to say, when weak shocks
are present. In such scenarios, the thermodynamic jump induced is not strong enough to
trigger chemical reactions immediately behind the leading shock, and heating due to fric-
tion appears as a necessary mechanism for these self-sustained detonations to survive. At
low post-shock temperatures, the aforementioned heating may become even more important
than the exothermic chemistry itself.
Both physical effects are important on the two regimes that a detonation experiences
when friction losses are included (see Fig. 1). On the one hand, for small velocity deficits
3

Figure 1: Schematic of a detonation propagating steadily at velocity Dfrom right to left in a rough tube.
The two possible solutions admitted by the governing equations are depicted (i) with a sonic point; (ii)
without a sonic point. Note the stark differences in the length of the reaction zones.
(D/DCJ ≥0.56) the wave structure is similar to a Zel’dovich-von Neumann-D¨oring (ZND)
profile except that the presence of friction losses results in a slight temperature/pressure
increase and associated density/velocity decrease before chemical reactions are activated.
Under these conditions, a sonic point exists somewhere downstream the shock. On the other
hand, for large velocity deficits (D/DCJ ≤0.56) the sonic point ceases to exist. In such
cases, the wave structure resembles that of over-driven detonations supported by a piston
[13]. In the absence of a sonic point the wave/reaction zone is affected by the downstream
boundary condition which renders their structure significantly different from classical ZND
detonations. However, the combination of chemical heat release and heating due to friction
seems to be enough to sustain their propagation.
Most previous work on detonation propagation with friction losses, steady or unsteady,
have considered the simplest description of the chemistry available [10–12, 14–20], i.e., one-
step Arrhenius kinetics, which is thought to provide good qualitative agreement with real
detonations [11], in addition to being particularly well-suited for theoretical studies. Simpli-
fied kinetic schemes, however, have recently been shown to perform poorly when quantitative
4

agreement is sought [21]. To our knowledge, only the work of Agafonov and Frolov [22], and
that of Kitano et al. [23] and Suboi et al. [24] made use of detailed kinetics. The authors
computed the minimum tube diameter for detonation propagation using a semi-empirical
friction coefficient derived from Blasius boundary layer theory, and compared their results
with experimental data; reasonable predictions were obtained. We note that in these arti-
cles, the authors restricted their attention to the critical diameters and not to an in-depth
analysis of all the possible propagation regimes given by the locus of steady-states in D−cf
space, nor to the reaction zone structures that appear at sub-CJ conditions. The objective
of this paper is thus two-fold: (i) to assess the effect of chemistry modeling on D−cfcurves
and on the reaction zone structure of steady detonations with friction losses; (ii) to deter-
mine/quantify the detailed kinetics induced uncertainties on the predicted critical friction
factors, cf,crit. To the best of our knowledge, no previous studies have addresses these issues.
The paper is structured as follows. Section 2 introduces a general mathematical formu-
lation used to compute detonation waves with friction losses and detailed chemistry. The
different chemical modeling and numerical method used are described in Section 3. Section 4
discusses the main results of our study via D−cfcurves and spatial profiles of pressure,
temperature, flow Mach number, thermicity and species mass fractions; an alternative fitting
methodology for simplified kinetics using the D−cfcurve obtained with detailed chemistry
as a target is also described. Closing remarks are given in Section 5.
2. Mathematical formulation
2.1. Conservation form
One-dimensional detonations with friction losses [13] in tubes (sketched in Fig. 1) can
be generally represented by the reactive Euler equations with a sink term in the momentum
equation (2); a generic loss function of the form f=P/(2Aϕ)˜cfρ|u|u=cfρ|u|uwhere P
represents the perimeter of the tube, Aits cross-sectional area, ϕits porosity and ˜cfdenotes
5

a dimensionless skin-friction coefficient of the rough walls of the channel. For a tube of
circular cross-section and diameter, d, or a channel of square cross-section and side, L, this
quotient P/A yields 4/d and 4/L, respectively. In a generic way, we may obtain a priori
the friction coefficient cf=P/(2Aϕ)˜cfwith units m−1, and later look for the dimensionless
friction coefficient of a tube given its characteristics. Note that we perform a very simple
approach for the porosity, considering a direct proportionality with the wetted area P/A
with ϕ[11].
The system of equations reads:
∂ρ
∂t +∂ρu
∂x = 0,(1)
∂ρu
∂t +∂
∂x ρu2+p=−f, (2)
∂
∂t ρe+|u2|
2+∂
∂x ρu h+|u2|
2= 0,(3)
∂ρYk
∂t +∂ρuYk
∂x =Wk˙ωk, k = 1, ..., N, (4)
where, ρ,u,p,e,h,xand tare the mixture density, axial velocity in the laboratory frame,
pressure, specific internal energy, enthalpy (including the chemical contribution), and the
spatial coordinate and time, respectively. The mass fraction, molecular weight and net
production/consumption rate per unit mass of species kare given by Yk,Wkand ˙ωk. Next,
equations (1) - (4) are expressed in terms of the thermicity parameter ˙σwhich is a more
convenient form for the sought after numerical solutions [25].
6

2.2. Thermicity form
After some rather tedious algebra, shown in Appendix A for completeness, the system
becomes:
Dρ
Dt +ρ∂u
∂x = 0,(5)
ρDu
Dt +∂p
∂x =−f, (6)
Dp
Dt −a2
f
Dρ
Dt = (γ−1)uf +ρa2
f˙σ. (7)
DYk
Dt =Wk˙ωk
ρ, k = 1, ..., N, (8)
where the material derivative, D/Dt, the frozen speed of sound, af, and the thermicity
parameter, ˙σ, are used for a more compact notation:
D
Dt =∂
∂t +u∂
∂x ;a2
f=∂p
∂ρ s,Y
;
˙σ=
N
X
k=1 W
Wk−hk
cpTDYk
Dt .
(9)
The system above is closed with the ideal equation of state:
p=ρRu
WT;W= 1/
N
X
k=1
(Yk/Wk) (10)
with Ruthe universal gas constant and Tand Wthe temperature and average molecular
weight of the mixture.
2.3. Wave-fixed frame of reference / steady structure
For a frame of reference moving at a constant speed, −D, (i.e., the shock travels from
right to left), unknown a priori and to be determined as a solution of a nonlinear eigenvalue
problem, the following two variables are introduced, ξ=xs−xand w = D−u, which
7

represent a new spatial coordinate and velocity measured relative to the shock location, xs,
and wave speed, D, respectively. Substituting in the material derivative and seeking steady
solutions only (i.e., ∂/∂t = 0) the mapping is D/Dt →wd/dξ. Furthermore, noting that
w = dξ/dt, the system of equations reads,
dρ
dt=−ρ˙σ+Fq+ (η−1)F
η,(11)
dw
dt= w ˙σ+Fq+ (η−1)F
η,(12)
dp
dt=−ρw2˙σ+Fq−F
η,(13)
dξ
dt= w,(14)
dYk
dt=Wk˙ωk
ρ, k = 1, ..., N, (15)
In Eqns. (11) to (15), 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. The
main advantage of expressing the system using time, t, as the independent variable is that
it simplifies the implementation in the Shock and Detonation Toolbox (SDT) [25]. The
functions Fqand Fare given by:
Fq=(γ−1)
a2
f
cf(D−w)2|D−w|;
F=cfD
w−1|D−w|,
(16)
where γ=cp/cvis the ratio of the specific heats. It can be readily shown that for cf=
0 m−1,Fq=F= 0 which reverts the formulation to the ideal case included in Browne
et al. [25]. The only changes required in the are thus adding the functions Fqand F,
making it rather straight forward, and most importantly, allowing us to investigate arbitrary
chemical mechanisms written in Cantera format [26] (i.e., .cti files); the complete derivation
8

is included in Appendix A.
The system of ordinary differential equations (ODEs) obtained is a two-point boundary
value problem (BVP); the boundary condition at the shock (t= 0 s or ξ= 0 m) is computed
using the shock jump conditions at a fixed initial pressure, temperature and composition,
known as the von Neumann (vN) state. The boundary condition downstream depends on
the nature of the problem. There are two possibilities: solutions with/without sonic points
downstream. This is discussed at length in the following sections.
2.4. Chemistry modeling
One of the most important steps of this study is the choice made to model the interaction
between the chemical heat release and the gas dynamics. Next, we describe the different
approaches used to subsequently analyze its influence on the steady solutions admitted by
the system of ODEs just derived.
2.4.1. 1-step chemistry
The simplest approach is to consider that reactants are directly converted into prod-
ucts via one irreversible reaction, and model its consumption/production rate following an
Arrhenius law
˙ωF=−k(ρ/W )YFexp −Ea
RuT,(17)
whose kinetic parameters Ea/Ruand krepresent the activation temperature and the pre-
exponential factor of the reaction; the subscript Fdenotes the fuel. These parameters are
typically determined from fitting procedures using detailed chemistry as a target to match
constant volume delay times over a temperature range of interest and/or ideal ZND profiles;
see Taileb et al. [21] and Yuan et al. [27] for details. To specify the mixture using a simplified
scheme three additional quantities need being defined: the heat release Q, molecular weight
9

W, and γ. For a stoichiometric H2-O2mixture fitted to the detailed mechanism of M´evel et
al. [28], Ea/Ru= 14160 K, k= 6 ×109s−1,W= 12 g/mol, γ= 1.33 and Q= 4.8 MJ/kg.
2.4.2. 3-step chain-branching chemistry
Chemical reactions usually take place as a sequence of intermediate stages that include
chain-initiation, chain-branching and termination steps. This sequence, known to mimic H2
chemistry well, can be introduced by a slightly more complex model with three reactions;
subscripts I,Band Cdenote, respectively, the initiation, branching and termination steps.
The scheme reads:
Initiation: F(Fuel) →R(Chain-carriers),
rI=kI(ρ/W )YFexp −EI
RuT,
(18)
Branching: F+R→2R,
rB=kB(ρ/W )2YFYRexp −EB
RuT,
(19)
Termination: R→P(Products),
rC=kC(ρ/W )YR,
(20)
where rI,rBand rCrepresent the reaction rates, 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 about which the fitting with
detailed chemistry is done. The net production/consumption rates of fuel, F, and chain
carriers, R, are ˙ωF=−rI−rBand ˙ωR=rI+rB−rC, respectively. This kind of simplified
schemes can be applied to fuels that exhibit a change of slope in induction delay time, τind, vs.
inverse temperature, 1/T , semilogarithmic plots such as H2(see Fig. 3 (a)) or ethylene [29].
The possibility of including additional physics with a marginal increase in computational
cost makes them particularly attractive. The kinetic and mixture parameters for this model
10

are: ρvN = 2.684 kg/m3,EI/Ru= 25000 K, EB/Ru= 8500 K, TI= 2431 K, TB= 1350 K,
kc= 2 ×107s−1,W= 12 g/mol, γ= 1.33 and Q= 4.99 MJ/kg. These were determined
following the same methodology described in the previous subsection.
2.4.3. Detailed chemistry
Different detailed mechanisms have been developed over the years to reproduce H2oxida-
tion. They are complex, include numerous intermediate steps, and provide the best available
modeling for this fuel. The added complexity brings about an increase in computational cost,
as more species need being included and, therefore, the number of equations to be solved
becomes larger.
The chemical mechanism of M´evel et al. [28] is used as our reference to compare against
the simplified kinetic schemes described above. It includes 9 species and 21 reactions and
has been widely validated against experimental measurements of ignition delay times, flame
speed, detonation speeds and cell sizes, and has shown good predictive capabilities in deto-
nation quenching [21] and transmission [30] studies. Three additional detailed mechanisms,
´
O Conaire et al. [31], San Diego [32] and GRI 3.0 [33], are used to assess differences in their
cf,crit predictions. Every single one of the detailed mechanism includes pressure-dependent
reaction rates and are widely used in the combustion community. Finally, all the chem-
ical mechanisms, including those for simplified kinetics, were implemented using .cti files
(included as supplementary material).
3. Numerical integration and root-finding algorithm
The numerical integration of the system of ODEs was performed using the SDT algorithm
as a base. The following changes were made in the ZND system class: (i) the momentum
equation was included; not required in the ideal case as the induction zone velocity is constant
and determined via the shock jump conditions alone. (ii) the right hand side (RHS) of the
equations were modified to include the functions Fqand F. Note that the default integration
11

method used in the solve ivp function of the Scipy package [34], was also changed from
Radau (implicit Runge-Kutta) to BDF (Backward Differentiation Formula). A very robust
stiff solver is needed because the RHS of the system can become very large as one approaches
the sonic point (which is itself a critical point of the system of ODEs).
In contrast to the ideal steady solutions found with the ZND model, detonations with
friction losses admit two (or more) steady solutions for a particular value of cf. The solution
methodolody, irrespective of whether we are dealing with ideal or non-ideal cases, consists of
marching downstream from the postshock state until the flow Mach number relative to the
leading shock is unity, M= w/af= 1. For this to occur the numerator and denominator
of the RHS should vanish simultaneously; these are referred to as removable discontinuities
or singularities of the 0/0 type. At large velocity deficits, D/DCJ ≤0.56, a sonic point is
not attained and a steady solution is reached when the flow comes to rest in the laboratory
frame. The numerical procedure used to obtain the aforementioned solutions follows. For
a given value of the shock velocity, D, the system of ODEs is integrated varying cf, until
the optimum value is found, cf,opt. Once the integration is finished, we can discern if the
solution is valid by checking the spatial distribution of the Mach number downstream of the
leading shock as follows:
•Solution with a sonic point (Fig. 2(a)-left). If cf< cf,opt (blue line), the Mach number
reaches unity with an increasing slope that approaches infinity, related to the presence
of a strong singularity in the flow. If cf> cf,opt (red line), the loss term is too strong
and the flow does not reach sonicity. The sought for solution, cf=cf,opt, is obtained
when M= 1 is approached with a slowly varying slope (green line).
•Solution without a sonic point (Fig. 2(b)-left). The first possible outcome is the same
as that described above for cf< cf,opt (blue line). If cf> cf,opt, the Mach number
will progressively decrease and drop abruptly after chemical reactions take place (red
12

0.0 0.1 0.2
0.4
0.6
0.8
1.0
M
(a)
cf= 334.1 m−1
cf= 335.1 m−1
cf= 336.1 m−1
334 335 336
−1
0
1
fobj
0 200 400 600
ξ[mm]
0.4
0.6
0.8
1.0
M
(b)
cf= 107.6 m−1
cf= 108.6 m−1
cf= 109.6 m−1
108 109
cf[m−1]
−1
0
1
fobj
Figure 2: Left. Flow Mach number spatial distribution obtained for a steady detonation moving at (a)
D≈0.9DCJ and (b) D≈0.475DC J for three different friction coefficients, cf.Right. Sign change in
objective function, fobj , for varying cf; the sought for solutions –diamonds– occur when fobj (cf) = 0.
Conditions: stoichiometric H2-O2at p0= 100 kPa and T0= 300 K. The chemistry was modeled using the
mechanism of M´evel et al. [28].
line). A trial solution is considered valid if the Mach number remains constant for at
least, ξ≈0.5 m (green line).
For both cases, the solution is found using the same objective function
fobj =Mmax −(1 −δM )
|Mmax −(1 −δM )|,(21)
with δM a small value on the order of 10−4; reducing δM further did not result in appreciable
changes in the cf,opt values obtained. The objective function, fobj , was derived based on
the work of Klein et al. for detonations with curvature losses [35]. We found useful and
straight-forward to use the Mach number profile to define our convergence metric. The
validity of a pair (D, cf) is verified by looking at the M(ξ) profile around the sonic point
or where the flow velocity in the laboratory frame, u, approaches zero. For a given D, if
cf< cf,opt,Mmax is always unity and the numerator of fobj is always positive. By scaling
13

it by its absolute value, fobj = 1. If cf> cf ,opt,Mmax <(1 −δM ) yielding a negative the
numerator and fobj =−1. fobj vs. cfis shown in Fig. 2-right for two representative cases: (a)
D≈0.9DCJ and (b) D≈0.475DC J , both computed with the detailed mechanism of M´evel
et al. [28]. The desired value of cfoccurs when fobj (cf) = 0 obtained by recursively dividing
an initial closed interval [cf,min,cf,max] bracketing a sign change of the objective function
with a tolerance around 10−5; that is, the root of fobj is found using the bisection method.
In Semenko et al. [11] an alternative way of finding the steady solutions was presented.
The authors suggested a change of variables that effectively removes the singularity in the
governing equations. While it does not seem straight-forward to extend their formulation to
more general cases with temperature dependent thermodynamics and/or complex chemistry
we used their work and results to validate our implementation; see Appendix B for details.
4. Results and discussion
An extensive analysis of the quasi-steady one-dimensional solutions for stoichiometric
H2-O2detonations with friction losses was performed. For all the results shown below the
thermodynamic conditions ahead of the shock (fresh gases) were kept constant at p0=
100 kPa and T0= 300 K, respectively.
4.1. D−cfcurves
4.1.1. Simplified kinetics vs. detailed chemistry
Figure 3 (a) shows the D−cfcurves obtained with one-step, three-step chain-branching
kinetics and detailed chemistry. Relevant detonation and thermodynamic properties pre-
dicted by the simplified and detailed mechanisms used in the present study are given in ta-
ble 1. The results speak for themselves. There are significant differences between simplified
schemes that were designed to reproduce the constant volume delay times of detailed chem-
istry (Fig. 3 (b)), yielding very different detonation velocities for a given friction coefficient,
14

0 200 400
cf[m−1]
0.2
0.4
0.6
0.8
1.0
D/DC J,det
D=ab
D=a0
(a)
0.5 0.8 1.1
1000/T [K−1]
10−8
10−6
10−4
10−2
τind [s]
(b)
0.00 0.01 0.02
cf×lind
0.2
0.4
0.6
0.8
1.0
D/DC J,det
0.83
0.77 0.765
D=ab
D=a0
(c)
0.2 0.6 1.0
D/DC J,det
10−8
10−6
10−4
10−2
τchem [s]
(d)
cf
cf,crit
0
1
M´evel et al. [28]
1-step [21]
3-step [21]
Figure 3: (a) D−cfcurves, (b) τind, induction times at constant volume (ρvN ) for stoichiometric H2-O2
at p0= 100 kPa and T0= 300 K. The range of temperature and pressure considered are 900 K < T <
2000 K and 1.67 MPa <p<3.72 MPa, (c) D−cfcurves scaled with the ideal induction length and (d)
integrated chemical times, τchem, obtained with the single-step and three-step chain-branching kinetics [21]
and the detailed mechanism of M´evel et al. [28]. The horizontal dotted lines denote the limits –upper/lower
bounds obtained with simplified/detailed kinetics– for the quasi-detonation (ab≤D≤DC J ) and choking
(a0≤D≤ab) regimes. The D/DC J, det value at which the first turning point occurs for each mechanism
is included in (c). The intensity change of the lines in (d) represent the scaled friction coefficient, cf/cf,crit ,
for a given value of D/DC J,det .
15

Table 1: Detonation properties predicted for a stoichiometric H2-O2mixture with different simplified kinetic
schemes and the detailed mechanism of M´evel et al. [28]. Initial conditions of the reactive mixture: p0=
100 kPa and T0= 300 K.
DCJ [m/s] TvN [K] pvN [MPa] γ(0 - vN - CJ) lind [µm] E/RuT0
M´evel et al. [28] 2839.9 1768.7 3.29 1.4 - 1.315 - 1.218 41.0 27.5
1-step [21] 2801.5 1674.8 3.25 1.33 87.9 34.5
New 1-step 2836.9 1769.5 3.31 1.35 57.7 34.0
3-step [21] 2850.4 1723.7 3.37 1.33 46.8 30.5
New 3-step 2836.2 1768.7 3.30 1.35 25.2 30.1
cf. The differences are both qualitative (1-step vs. 3-step/det. chem.) and quantitative
(1-step/3-step vs. det. chem.), and do not seem to be simply scaled by the ZND induction
length (Fig. 3 (c)), lind, defined as the distance from the leading shock to the peak in ther-
micity, ˙σ. This makes it difficult to perform a meaningful comparison of the peculiarities of
each model. Similar observations regarding the inadequacy of simplified kinetic schemes to
qualitatively and quantitatively reproduce the outcomes obtained with detailed chemistry
were also discussed by Liberman et al. [36] but in the context of detonation initiation by
temperature gradients. At this point, we just provide an overview of the possible reasons
behind the found discrepancies.
Before discussing the D−cfcurves further, it is instructive to go over the most salient
findings of previous studies. Zeldovich et al. [5] analyzed the eigenvalue problem posed above
in the limit of strong heat release. The authors restricted their investigation to relatively high
Dvalues close to the curve’s first turning point (D > ab;ab= 0.58DC J, det with the simplified
mechanisms and ab= 0.56DCJ, det with detailed mechanisms). Brailovsky and Sivashinsky [6]
extended the solutions in [5] after realizing the possibility of having an entirely subsonic
flow field behind the detonation wave; their D−cfcurves showed the emergence of a second
turning point within the velocity interval 0.185DCJ, det =a0< D < abwhere subscripts 0
and brefer to fresh and burnt gases, respectively. This outcome suggested the existence of
an additional stable detonation regime including planar as well as galloping and spinning
16

waves [6]. The cfvalues associated to the special points described are denoted cf,0and cf,b.
For cf> cf,0the wave does not quench but propagates shockless (sustained by drag-induced
diffusion of pressure and adiabatic compression); the gas flow is subsonic throughout the
entire structure. The latter solution referred to as subsonic detonations [6, 37, 38] were not
pursued in the present study. Figure 3 (a) thus includes the quasi-detonation and choking
regimes bounded by ab≤D≤DCJ and a0≤D≤ab, respectively. The D−cfcurve
for detailed chemistry shows only the existence of one turning point, cf,crit = 429 m−1at
D/DC J,det ≈0.77 beyond which no possible steady solutions are found. For cf< cf,crit two
(or more) steady solutions are possible at low and high D/DC J,det values. At D/DC J,det <0.4
the curve asymptotically reaches cf→0 m−1as D→0 m/s.
One-step Arrhenius kinetics, being it the most common choice of chemical model in
similar studies, fails to reproduce the cf,crit value predicted by detailed chemistry, underes-
timating it by 39%. Moreover, it exhibits a second turning point at low velocities that leads
to the Z-shaped curve, usually reported in the literature; see Brailovsky and Sivashinsky [6]
for a detailed discussion. The qualitative difference with the curve obtained with detailed
chemistry suggests a potential inadequacy of one-step kinetics to reproduce the expected
behavior of stoichiometric H2-O2non-ideal detonations, particularly in the choking regime
where the chemical times are very likely being under-predicted. Additionally, the absence of
an induction length results in an evenly distributed heat release (see ˙σprofiles in Figs. 6 and
7) in contrast to the abrupt and thin profile predicted with detailed chemistry. In [6] it was
shown that the second turning point is less pronounced as the effective reduced activation
energy, E/RuT0, increases or may be fully suppressed in the strong heat release limit thereby
ruling out the existence of subsonic detonations.
While 3-step chain-branching chemistry shows a similar qualitative behavior to that
of detailed chemistry, as some of the shortcomings described above are partially addressed,
differences are still present. Similarly to one-step kinetics, the normalized detonation velocity
17

(D/DC J,det ) initially decreases at a faster rate, and cf,crit is underpredicted compared to
detailed chemistry. D(cf,crit) is also 8% higher, and the asymptotic approach to cf= 0 m−1
occurs at a higher Dvalue than the detailed chemistry predictions. The latter behavior
may be due to the fact that the 3-step scheme does not include a pathway to replenish the
radical pool for T < TB, shown to be important in H2detonation chemistry [39], and/or
assuming a termination step with a null activation energy. We emphasize that fundamental
improvements to the simplified schemes are outside of the scope of this work and we restrict
our analysis to reporting the differences present among the chemical modeling typically used
in detonation research when applied to the study of non-ideal detonations.
Both simplified models assume a constant molecular weight, W, and ratio of specific
heats, γ, which for a stoichiometric H2-O2mixture undergo strong changes across the det-
onation structure. Wvaries from 12 g/mol to 18 g/mol as the products are mostly water
vapor (H2O); γgoes from γ0= 1.4 in fresh gases to γvN = 1.31 at the vN -state, further
decreasing to γCJ = 1.21 at the CJ -state. All these thermodynamic changes are neglected
by the two simplified models considered and may be responsible for some of the discrepancies
observed. Again, the fitting of these models in Taileb et al. [21] were performed by matching
the detailed chemistry induction times at constant volume (ρvN ) and spanning a tempera-
ture range of interest, despite providing acceptable results at ideal conditions, as shown in
this section, this fitting procedure does not seem to be appropriate to capture the locus of
steady solutions when losses are included (i.e., friction and/or curvature). Fig. 3-(d) shows
the chemical reaction times, τchem, obtained from the integration system (11)–(15) plotted as
a function of D/DC J,det (following the D−cfcurve). τchem defined as the time required for a
fluid parcel to travel from the leading shock to the location where ˙σmax occurs, accounts for
the actual thermodynamic changes in the reaction zone therefore is a more representative
reaction metric of the process. The significant differences observed in τchem for the kinetics
considered, provide clues about the qualitative and quantitative differences described above
18

for the D−cfcurves.
Seeking to improve the predictive capabilities of simplified kinetic schemes for non-ideal
detonations, and to enable a meaningful comparison among the chemical modeling tech-
niques tested we introduce an alternative approach in the next section.
4.1.2. Modified simplified kinetics vs. detailed chemistry
Taking the D−cfcurve obtained with detailed chemistry as a target we slightly modified
both simplified kinetic schemes. We kept a constant Wand γ, the same activation and
cross-over temperatures (Ea/Ru,EI/Ru,EB/Ru,TIand TB) as those defined previously,
but modified γ(always within the bounds of the real mixture), the total heat release, Q,
and the pre-exponential factors, kand kC, as follows: (i) vary γand Qsimultaneously to
find a combination that reproduces TvN and DCJ as best as possible; (ii) check whether the
given combination provides reasonable values for D(cf,crit); (iii) Modify k(one-step) and kC
(three-step) until their respective cf,crit approaches the value predicted by detailed chemistry
within an arbitrarily prescribed tolerance; increasing/decreasing kor kCshifts the curves
right/left.
The methodology described above yields the following parameters: Ea/Ru= 14160 K,
k= 6.735 ×109s−1,γ= 1.35 and Q= 4.606 MJ/kg for one-step; ρvN = 2.684 kg/m3,
EI/Ru= 25000 K, EB/Ru= 8500 K, TI= 2431 K, TB= 1350 K, kC= 3.35 ×107s−1,
γ= 1.35 and Q= 4.613 MJ/kg for three-step chain-branching kinetics. We include in table 1
the ideal detonation properties obtained with the modified chemical schemes. Figure 4 shows
the same plots as in Fig. 3 but using the updated simplified schemes. The D−cfcurves are, as
expected, in much better agreement with the detailed mechanism but a few differences persist
(see Fig. 4-(a)). For 1-step kinetics, the second turning point is still present but captures
rather well all the quasi-detonation regime. For 3-step chain-branching chemistry despite
the additional physics included that results in the correct qualitative behavior (captures the
change in activation energy at low temperatures) its quantitative performance is subpar.
19

The initial decay from D=DCJ to D=D(cf,crit ) occurs at a lower rate and D≤D(cf,crit)
is appreciably overestimated. As mentioned above, more fundamental refinements are thus
required to improve the predictive capabilities of this scheme. In addition, relaxing the
constant Wand γassumption, modifying the activation energies of both the initiation and
branching steps to better fit the slopes in the τchem plot, introducing a crossover temperature
for the termination step, separating the reaction rates for each step or releasing partial heat
during branching –when dealing with hydrocarbons– may be worth examining. The updated
constant volume induction times, shown in Fig. 4-(b), do not match those of the detailed
mechanism, providing further evidence that this fitting procedure is not representative of
the reaction times and thermodynamic changes that take place when dealing scenarios with
losses. While the updated mechanisms provide improved τchem (Fig. 4-(d)) the scaling with
lind continues to provide unsatisfactory results (Fig. 4-(d)).
Note that there exists a close relationship between τchem and the D−cfcurves predicted
with the different kinetic schemes. Closer inspection of Figs. 4-(a) and (d) shows that for
a given D/DCJ, det , whenever τchem computed with the simplified models is longer than the
detailed, the corresponding cfis lower.
4.1.3. Detailed chemistry induced uncertainties
Table 2 includes ideal detonation properties obtained with the detailed mechanisms listed
in subsection 2.4.3. The thermodynamics among the mechanisms seem to be in good agree-
ment, showing maximum deviations of around 0.2% for DCJ , the vN-state or γ, whereas
kinetic related properties such as lind and E/RuT0, have maximum deviations of 25% and
19%, respectively. The D−cfcurves are shown in Fig. 5-(a). While their qualitative behav-
ior is expectedly similar, the quantitative differences found among the mechanism are rather
surprising. Particularly, in their cf,crit predictions yielding values that range from 247 m−1
(GRI 3.0 mechanism [33]) to 429 m−1(M´evel et al. [28]), i.e., a 42% increase. There are
also minor differences in D(cf,crit), only on the order of a few percent (2.6%), between GRI
20

0 200 400
cf[m−1]
0.2
0.4
0.6
0.8
1.0
D/DC J,det
A
B
C
D=ab
D=a0
(a)
0.5 0.8 1.1
1000/T [K−1]
10−8
10−6
10−4
10−2
τind [s]
(b)
0.00 0.01 0.02
cf×lind
0.2
0.4
0.6
0.8
1.0
D/DC J,det
0.815 0.77 0.77
D=ab
D=a0
(c)
0.2 0.6 1.0
D/DC J,det
10−8
10−6
10−4
10−2
τchem [s]
(d)
cf
cf,crit
0
1
M´evel et al. [28]
New 1-step
New 3-step
Figure 4: (a) D−cfcurves, (b) induction times at constant volume, τind, (c) D−cfcurves scaled with
the ideal induction length and (d) integrated chemical times, τchem, obtained with the updated single-step
and three-step chain-branching kinetics [21] and the detailed mechanism of M´evel et al. [28]. The horizontal
dotted lines denote the limits –upper/lower bounds obtained with simplified/detailed kinetics– for the quasi-
detonation (0.59DCJ, det =ab≤D≤DC J ) and choking (a0≤D≤ab) regimes. The round markers in (a)
indicate the points at which the p,T,M, ˙σand Ykprofiles are analyzed. The D/DCJ, det value at which
the first turning point occurs for each mechanism is included in (c). The intensity change of the lines in (d)
represent the scaled friction coefficient, cf/cf,crit, for a given value of D/DCJ, det .
21

Table 2: Detonation properties predicted for a stoichiometric H2-O2mixture with different detailed chemistry
mechanisms. Initial conditions: p0= 100 kPa and T0= 300 K.
DCJ [m/s] TvN [K] pvN [bar] γ(0 - vN - CJ) lind [µm] E/RuT0
M´evel et al. [28] 2839.9 1768.7 32.9 1.4 - 1.315 - 1.218 41.0 27.5
´
O Conaire [31] 2840.0 1768.8 32.9 1.4 - 1.315 - 1.219 41.3 27.4
San Diego [32] 2835.0 1764.1 32.7 1.4 - 1.316 - 1.214 49.7 27.9
GRI 3.0 [33] 2835.7 1764.2 32.8 1.4 - 1.316 - 1.213 51.1 32.6
(0.79) and M´evel (0.77). The differences in induction times, τind, (Fig. 5-(b)) and chemical
times, τchem, (Fig. 5-(d)) among the mechanisms may partly explain the dissimilar cf,crit
predictions. Fig. 5-(c) shows an inverse relation between lind given by the detailed mech-
anism and the critical friction coefficient cf,crit (i.e., smaller lind results in larger cf ,crit); a
dependence that does not hold for the simplified schemes. To fully understand the reported
discrepancies between the detailed mechanisms, which are of the same order of magnitude
than those predicted by the original simplified mechanisms used [21], requires a thorough
chemical analysis which is outside of the scope of this study.
4.2. Reaction zone structures
p,T,M, ˙σand Ykprofiles are computed for values of D/DC J, det and cfwhere (some of )
the curves coexist (Points A, B and C in Fig. 4-(a)). Note that the axes may vary between
different figures for the sake of clarity.
4.2.1. Modified simplified kinetics vs. detailed chemistry
Figure 6 shows the profiles calculated for a detonation in the quasi-detonation regime
(D≈0.8DCJ, det ) with cf≈422 m−1(point A in Fig. 4-(a)). The profiles somewhat resemble
the ideal ZND structures except that pressure/temperature increase in the induction zone,
and the flow Mach number, M, decreases as result of heating due to friction. The postshock
state is ps/pvN = 0.63 and Ts/TvN = 0.71. The 3-step chain-branching scheme and the
detail mechanism do not react until ξ∼4lind det, single-step and its inability to reproduce
an induction length results in slow consumption of the mixture immediately after the shock.
22

0 200 400
cf[m−1]
0.2
0.4
0.6
0.8
1.0
D/DCJ
D=ab
D=a0
(a)
0.5 0.8 1.1
1000/T [K−1]
10−8
10−6
10−4
10−2
τind [s]
(b)
0.00 0.01 0.02
cf×lind
0.2
0.4
0.6
0.8
1.0
D/DCJ
0.79 0.77
D=ab
D=a0
(c)
0.2 0.6 1.0
D/DCJ
10−8
10−6
10−4
10−2
τchem [s]
(d)
cf
cf,crit
0
1
M´evel et al. [28]
O Conaire et al. [31]
San Diego [32]
GRI 3.0 [33]
Figure 5: (a) D−cfcurves scaled with their respective DC J , (b) induction times at constant volume, τind , (c)
D−cfcurves scaled with the ideal induction length and (d) integrated chemical times, τchem, obtained with
all detailed mechanisms considered. The horizontal dotted lines denote the limits for the quasi-detonation
(ab≤D≤DCJ ) and choking (a0≤D≤ab) regimes. The D/DCJ value at which the first turning point
occurs for each mechanism is included in (c). The intensity change of the lines in (d) represent the scaled
friction coefficient, cf/cf,crit , for a given value of D/DC J .
23

10
20
30
p/p0
p=pvN
3
7.5
12
T/T0
T=TvN
0.4
0.6
0.8
1.0
M
0
10
20
30
˙σ[µs−1]
0 0.15 0.3
0
0.5
1
Yk/Yk,0
H2
F
F
0 0.15 0.3
0
0.5
1
Yk/Yk,max
H
R
ξ[mm] ξ[mm]
M´evel et al. [28]
New 1-step
New 3-step
Figure 6: p,T,M, ˙σand scaled Ykprofiles obtained for a shock velocity D= 0.8DCJ, det and a friction
factor of cf≈422 m−1with the 1-step, 3-step and the detailed chemical mechanism of M´evel et al [28].
Once reactions are fully activated, heat release takes place at a rate given by the shape of
the thermicity profiles. The detailed mechanism exhibits the fastest and narrowest profile,
whereas the one-step shows an evenly distributed and wider profile. The higher temperature
predicted by the detailed mechanism, is due to the variation of the molecular weight Wand
γas the reaction takes place which in not included in the simplified schemes. Contrary to
detailed kinetics, the fuel mass fraction profiles show that the simplified schemes deplete all
the available fuel at the end of the reaction zone. Notably, after all the chemical heat is
deposited into the flow, all the three mechanisms predict the appearance of a sonic region
at distances around ξ∼0.33 mm behind the shock.
The flow conditions change for larger deficits (point B in Fig. 4-(a)) as we move into
the choking regime (D= 0.55DCJ, det and cf= 240 m−1), where only the curves for 1- step
24

10
20
30
p/p0
p=pvN
3
7.5
12
T/T0
T=TvN
0.4
0.6
0.8
1
M
0
10
20
30
˙σ[µs−1]
0123
0
0.5
1
Yk/Yk,0
H2
F
0123
0
0.5
1
Yk/Yk,max
H
ξ[mm] ξ[mm]
M´evel et al. [28] New 1-step
Figure 7: p,T,M, ˙σand scaled Ykprofiles obtained for a shock velocity D= 0.55DCJ, det ) and a friction
factor of cf= 240 m−1with the 1-step and the detailed chemical mechanism of M´evel et al [28].
25

and detailed chemistry intersect (Fig. 7). The shock is weak with postshock pressure and
temperature of ps/pvN = 0.29 and Ts/TvN = 0.42. It is at these conditions when the influence
of heating due to friction should be comparable to that of the chemical energy release. The
shocked but unreacted mixture heats up at a slow but constant rate, which makes the
reaction zone move far downstream, at distances of the order of millimeters (ξ∼40 lind, det).
The shorter induction times predicted by the 1-step mechanism result in faster ignition of
the mixture (see ˙σprofiles). In spite of this, the overall reaction zone structure is similar for
both chemical models except for a significantly thicker main heat release zone. The mass
fraction profiles show these qualitative differences clearly. Note that the flow does not reach
sonicity and the boundary condition is satisfied at infinity (not shown here for clarity).
Figure 8 compares the profiles obtained with detailed chemistry and 3-step chain-branching
kinetics for D= 0.325DCJ, det , close to the speed of sound in the fresh mixture. The general
outline is very similar to the one described above, but with significantly lower postshock
states (ps/pvN = 0.11 and Ts/TvN = 0.25), slower pressure build-up and frictional heating
as a result of a cf→0 m−1. This leads to a main heat release zone located at distances
of around half a meter downstream the shock. The applicability of an inviscid model to
adequately capture the reaction zone structure of waves propagating at such large deficits is
questionable. It is plausible that if thermal/mass diffusion were to be included in the model
the computed structures would differ, however, the fact that shockless solutions sustained
by drag-induced diffusion of pressure and adiabatic compression were shown to exist in [6]
for larger deficits may suggest otherwise.
A quick order of magnitude analysis of the different time/length scales present in our
physical system may help substantiate this argument. First, given the slowest computed
shock speed (i.e., D= 0.21DC J, det ∼600 m/s) and an axial characteristic length of Lx∼6 m
(i.e., reaction zone length at these conditions), the convective transient time yields τconv ∼
Lx/D ∼10 ms. Second, the characteristic chemical time from the computation of the steady
26

3
11.5
20
p/p0
p/pvN = 32.9
3
7.5
12
T/T0
T=TvN
0.3
0.5
0.7
M
0
5
10
˙σ[µs−1]
434 436
0
5
10
459 461
0 200 400 600
0
0.5
1
Yk/Yk,0
H2
F
0 200 400 600
0
0.5
1
Yk/Yk,max
HR
434 436
0
0.5
1
459 461
ξ[mm] ξ[mm]
M´evel et al. [28] New 3-step
Figure 8: p,T,M, ˙σand scaled Ykprofiles obtained for a shock velocity D= 0.325DCJ, det) and a friction
factor of cf= 3.75 m−1with the 3-step and detailed chemical mechanism of M´evel et al [28].
27

10
20
30
p/p0
p=pvN
3
7.5
12
T/T0
T=TvN
0.4
0.6
0.8
1
M
0
10
20
30
˙σ[µs−1]
0 0.25 0.5 0.75
0
0.5
1
Yk
O2
H2×7
H2O
0 0.25 0.5 0.75
10−9
10−7
10−5
10−3
10−1
Yk
H
OH
HO2
ξ[mm] ξ[mm]
M´evel et al. [28]
O Conaire et al. [31]
San Diego [32]
GRI 3.0 [33]
Figure 9: p,T,M, ˙σand Ykprofiles obtained for a shock velocity D= 0.77DCJ ) and its correspondent
friction factor of cf=cf,crit m−1with the detailed chemical mechanisms. The species are only depicted for
M´evel et al. [28] and GRI 3.0 [33] for the sake of clarity.
solution is τchem ∼20 ms. The diffusive length scale computed using the thermal diffusivity
at postshock conditions (Ts∼320 K and Ps∼1.3×105Pa), αs∼7.7×10−5m2/s, is
Lα∼√τconvαs∼1 mm. These results suggest that thermal diffusive proccesses in the
wave propagation direction play a minor role, and convective and chemical terms drive the
reaction zones at near-sonic conditions. However, radial heat losses may still be important
which our one-dimensional model does not account for.
4.2.2. Detailed chemistry induced uncertainties
Figure 9 shows the profiles obtained with all the detailed mechanisms used for D=
0.77DCJ and their corresponding friction factors which are cf,crit for M´evel and O’ Conaire,
and slightly below for GRI and San Diego. Since all the detailed mechanisms share the same
thermodynamic database, the postshock conditions are identical. However, as mentioned
28

before, there are significant disprepancies in the predicted criticality with cf,crit ranging from
247 to 429 m−1, which expectedly results in differences in their reaction zone structures. It
seems to be mostly a spatial shift and higher rates of pressure/temperature increase for higher
cfvalues. This results in the thermicity profiles peaking at different locations and attaining
different maxima. However, the total amount of energy deposited by the exothermic reaction
is approximately the same; how fast/much and the location where chemical heat release
occurs in the flow plays a key role on the reaction zone structures.
The way in which the intermediate elementary chemical reactions proceed is important.
The mass fraction profiles show similar behaviors for the consumption of fuel/oxidizer
and the production of H2O. However, differences in the production/consumption of radicals
in the reaction zone are evident. First, the net production/consumption rate of both OH and
H is fastest when using the mechanism of M´evel et al. [28] (only the GRI 3.0 [28] and M´evel
are plotted for clarity). These differences may be reinforced by the fastest heating rate due to
friction for M´evel when the pathways that create these radicals are activated (at ξ≈50 µm).
Second, the behavior of HO2is peculiar. Just behind the shock, GRI 3.0 favors a higher
HO2production rate. Soon after, its production rate decreases and M´evel et al. overcomes
it, resulting in an overall faster chemical times. Note that the behavior described may be
an artifact of having the main reaction zones occurring at different locations downstream.
However, a preliminary sensitivity analysis on the pre-exponential factors of the elementary
reactions (O + H2↔H + OH and H + H2O2↔HO2+ H2) that were found to have
the largest differences between GRI and M´evel (10 and 6 orders of magnitude, respectively)
brings the cf, crit of GRI to closer to that of M´evel. The rates of production/consumption
of HO2and OH seem to be a key aspect for adequate prediction of the steady reaction
zones of detonation with friction losses; the competition for radicals has also been shown to
be important in multidimensional detonations [39]. Given the large differences among the
detailed mechanisms tested, including the mechanism induced uncertainties to the critical
29

diameter estimates such as those reported in [22–24] will certainly give a more fair assessment
of the predictive capabilities of the model.
5. Conclusions
We revisited the problem of one-dimensional steady detonations with friction losses and
analyzed the influence of the chemistry modeling making use of both simplified (one-step
and three-step chain-branching) and detailed kinetics. First, we compared the results ob-
tained with simplified schemes, fitted using conventional methods such as matching the
constant volume ignition delay times, to those of detailed mechanisms commonly used in
the literature for hydrogen oxidation. Both simplified schemes failed to capture the critical
cfvalues predicted with detailed kinetics. While the three-step chain-branching successfully
reproduced the qualitative trend of the reference D−cfcurve, the one-step scheme showed
qualitative differences in the choking regime likely due to the under-prediction of the H2-O2
ignition delay times at postshock temperatures below the chain-branching cross-over tem-
perature. Second, an alternative approach to fit simplified schemes was introduced aiming
to reproduce the D−cfcurves obtained with a reference detailed mechanism. The result-
ing modified schemes were expectedly capable of better reproducing the reaction zones of
detonations with friction losses. In particular, the 1-step mechanism was found to be in
agreement for the entire quasi-detonation regime whereas the choking regime continued to
be over predicted (i.e., larger cfvalues for a fixed velocity deficit D/DCJ ). The 3-step mech-
anism retains its good qualitative agreement but predicts lower deficits at fixed cfin the
latter regime. Suggestions to improve the predictive capabilities of the simplified schemes
were proposed. Additionally, the detailed mechanism induced uncertainties in the prediction
of cf,crit were quantified. We found differences of up to 42% between the lowest (GRI) and
highest (M´evel et al.) cf,crit predicted; this difference is somewhat reconciled by scaling the
curves using their respective ideal induction lengths, lind. We identified the importance of
30

the consumption/creation rates of the HO2radical pool in the postshock region as a po-
tential culprit for the discrepancies observed; a more in-depth analysis will nonetheless be
required to verify this.
Future efforts will be directed to further understand: (i) the transition between low
and high velocity solutions at a fixed cfwith a transient solver and (ii) the influence of
heat losses on the reaction zone structures of detonations with friction losses, both, using
detailed kinetics; comparisons with available experimental data will be possible with this
model. Determining transient D−cfcurves and characterizing their behaviors close to
failure may also be an avenue worth exploring.
6. Acknowledgements
The authors want to acknowledge the financial support from the Agence Nationale de la
Recherche Program JCJC (FASTD ANR-20-CE05-0011-01).
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34

Appendix A. Detailed formulation algebra
Appendix A.1. Thermicity form
Starting from Eqs. (1)-(4) and for convenience in subsequent derivation, the energy
equation (3) can be explicitly expressed in terms of the specific enthalpy, h, and entropy,
s, of the mixture. Recalling the themodynamic identity dh=Tds+ 1/ρ dp+PN
k=1 gkdYk,
where gkis the specific Gibbs free energy of species kand Tis the gas temperature, the
changes denoted by dh, ds, etc. are those taking place within a given fluid parcel, and can be
written using the material derivative, D/Dt. In these variables the energy equation reads:
ρDh
Dt −Dp
Dt =uf, (A.1)
Ds
Dt =1
T"uf
ρ−
N
X
k=1
gk
DYk
Dt #,(A.2)
For gaseous reacting mixtures pressure is a function of the density, the specific entropy and
the species mass fractions p(ρ, s, Y), with Yrepresenting a vector that includes all species
k. Expanding yields:
Dp
Dt =∂p
∂ρ s,Y
Dρ
Dt +∂p
∂s ρ,Y
Ds
Dt +
N
X
k=1
∂p
∂Ykρ,s,Yi̸=k
DYk
Dt .(A.3)
Replacing the expression for Ds/Dt (Eq. (A.2)),
Dp
Dt =∂p
∂ρ s,Y
Dρ
Dt +∂p
∂s ρ,Y"1
T uf
ρ−
N
X
k=1
gk
DYk
Dt !#+
+
N
X
k=1
∂p
∂Ykρ,s,Yi̸=k
DYk
Dt .
(A.4)
35

Rearranging,
Dp
Dt =∂p
∂ρ s,Y
Dρ
Dt +∂p
∂s ρ,Y1
Tuf
ρ+
+
N
X
k=1 "−gk
T
∂p
∂s ρ,Y
+∂p
∂Ykρ,s,Yi̸=k#DYk
Dt .
(A.5)
where ∂p
∂ρ s,Y=a2
fis the frozen sound speed of the mixture. Assuming the mixture to behave
as an ideal gas, the second term in the right hand side of Eq. (A.5) simplifies to:
∂p
∂s ρ,Y
1
ρT =1
∂s
∂p ρ,Y
1
ρT =p
T cvρ=Rg
cv
=γ−1
with p
ρ=RgT,
(A.6)
all symbols in Eq. (A.6) are defined in the main body of the text. Introducing the thermicity,
˙σ, in general form,
˙σ=
N
X
k=1
σk
DYk
Dt ,(A.7)
where σkis given by,
σk=−gk
T
∂p
∂s ρ,Y
+∂p
∂Ykρ,s,Yi̸=k
.(A.8)
and restricting it to ideal gases, yields:
σk=W
Wk−hk
cpT.(A.9)
36

Replacing Eq. (A.6) in (A.5) and using definitions (A.7) and (A.9), Eq. (A.5) becomes:
Dp
Dt =a2
f
Dρ
Dt + (γ−1)uf +ρa2
f˙σ. (A.10)
which replaces the energy equation (3) in the system of equations (5)-(8).
Appendix A.2. Wave-fixed frame of reference
For conciseness, the derivation is continued from Eq. (10) of the main body of the
manuscript. Using Eq. (9) in system (5)-(8) yields:
wdρ
dξ+ρdw
dξ= 0,(A.11)
wdw
dξ+1
ρ
dp
dξ=f
ρ,(A.12)
wdp
dξ−wa2
f
dρ
dξ= (γ−1)uf +ρa2
f˙σ, (A.13)
wdYk
dξ=Wk˙ωk
ρ, k = 1, ..., N. (A.14)
Restricting the system to seek for steady solutions (i.e., ∂/∂t = 0) the time derivatives
vanish, resulting in the mapping below:
D
Dt →wd
dξ.(A.15)
37

Combining like terms, rearranging and introducing the variable η= 1−M2where M= w/af,
is the frozen flow Mach number gives:
dρ
dξ=−ρ
w˙σ+Fq−F
η+F,(A.16)
dw
dξ=˙σ+Fq−F
η+F, (A.17)
dp
dξ=−ρw˙σ+Fq−F
η,(A.18)
wdYk
dξ=Wk˙ωk
ρ, k = 1, ..., N. (A.19)
Finally noting that for a detonation wave propagating at a constant speed d/dt= w(d/dξ),
the system of equations given by Eqs. (11)-(16) is recovered.
0 0.003 0.006
cf×l1/2
0.2
0.6
1.0
D/DCJ
(a)
Semenko et al. [11]
Z code
0 0.01 0.02
cf×l1/2
(b)
Mod. SDT code
Z code
Figure A.10: Comparison of the D/DC J −cfcurves obtained by (a) Semenko et al. [11] (ϕ= 0.4, E= 30,
Q= 20 and γ= 1.2) with an in-house implementation of their framework (Z code); (b) the Z code and the
updated SDT algorithm using the 1-step kinetics of Taileb et al. [21] for H2-O2ideal detonations (ϕ= 1,
γ= 1.33). The x-axes are multiplied by their respective half-reaction lengths, l1/2.
Appendix B. Numerical code validation
The validation of the numerical implementation was carried out as follows: First, the
algorithm and mathematical techniques of Semenko et al. [11] –named Z code– were im-
38

plemented and compared against their results (Fig. A.10-(a)). Second, taking the standard
SDT code as a base, the friction losses terms were added. D−cfcurves were then computed
using the updated SDT algorithm and the Z code for the 1-step mechanism presented in
Taileb et al. [21]. The results are shown in Fig. A.10-(b); the agreement is evident.
39