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Phil. Trans. R. Soc. A (2012) 370, 567–596
doi:10.1098/rsta.2011.0342
Gas-phase detonation propagation in mixture
composition gradients
BY D. A. KESSLER*, V. N. GAMEZO
AND
E. S. ORAN
Laboratory for Computational Physics and Fluid Dynamics,
Naval Research Laboratory, Washington, DC, USA
The propagation of detonations through several fuel–air mixtures with spatially varying
fuel concentrations is examined numerically. The detonations propagate through twodimensional channels, inside of which the gradient of mixture composition is oriented
normal to the direction of propagation. The simulations are performed using a twocomponent, single-step reaction model calibrated so that one-dimensional detonation
properties of model low- and high-activation-energy mixtures are similar to those
observed in a typical hydrocarbon–air mixture. In the low-activation-energy mixture,
the reaction zone structure is complex, consisting of curved fuel-lean and fuel-rich
detonations near the line of stoichiometry that transition to decoupled shocks and
turbulent deflagrations near the channel walls where the mixture is extremely fuellean or fuel-rich. Reactants that are not consumed by the leading detonation combine
downstream and burn in a diffusion flame. Detonation cells produced by the unstable
reaction front vary in size across the channel, growing larger away from the line of
stoichiometry. As the size of the channel decreases relative to the size of a detonation
cell, the effect of the mixture composition gradient is lessened and cells of similar sizes
form. In the high-activation-energy mixture, detonations propagate more slowly as the
magnitude of the mixture composition gradient is increased and can be quenched in a
large enough gradient.
Keywords: detonation; cellular structure; concentration gradient; triple points;
partially premixed; methane
1. Introduction
Gas-phase detonations are inherently unstable to transverse perturbations, which
means that a nominally planar detonation wave is actually a dynamic multidimensional structure including multiple transverse waves interacting with a
leading shock. The points of intersection of these waves are called triple shock
configurations or, more commonly, triple points. Many detailed descriptions of
the structure of triple points exist in the literature (see the paper by Thomas
[1] in this Issue and references therein for more details); so here we introduce
only the most basic concepts relevant to our topic of interest. Examples of the
pressure distribution in the vicinity of the reaction front of a simulated planar
*Author for correspondence ([email protected]).
One contribution of 12 to a Theme Issue ‘The physics, chemistry and dynamics of explosions’.
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D. A. Kessler et al.
Figure 1. Magnified view of the pressure field along a detonation front computed using the reaction
model discussed in §3 for a uniform stoichiometric low-activation-energy mixture at (a) 3.5143 ms,
(b) 3.5464 ms, (c) 3.5773 ms, (d) 3.6063 ms, (e) 3.6303 ms and (f ) 3.6600 ms after ignition. The
detonation moves from left to right in the figures as time progresses at the Chapman–Jouget
detonation velocity. Several triple points are labelled A–J, and their trajectories are mapped by the
dashed lines spanning each figure. The numerical labels in (a) denote (1) an overdriven detonation,
(2) an inert shock and (3) a transverse wave.
two-dimensional detonation as it propagates through a uniform fuel–air mixture
are shown in figure 1a–f . Details of the simulation that produced this detonation
will be discussed later in §3. Ten unique triple points (labelled A through J)
have been marked on the figure. Each triple point marks the intersection of
(1) an overdriven detonation, (2) an inert normal shock, and (3) a transverse
wave as labelled in figure 1a. Material directly behind the overdriven detonations
has been fully consumed, while that behind the inert normal shocks has been
compressed but not yet reacted. When two of the triple points collide (points B
and E in figure 1c), a region of extremely high pressure forms that ignites a new
overdriven detonation, which locally propagates faster than the Chapman–Jouget
(CJ) detonation velocity DCJ (e.g. the wave between B and E in figure 1c,d).
Overdriven detonations eventually slow to speeds below DCJ , and the leading
shocks decouple from the reaction zones (e.g. the wave between points C and E
between figure 1b,c) until the next triple-point collision.
The trajectories of the triple points, represented by the dashed lines in each
image shown in figure 1, form a regular cellular pattern composed of a number
of ‘detonation cells’ between triple point collisions. These cellular structures
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Detonations in concentration gradients
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Figure 2. Experimental soot traces generated by detonations of mixtures of natural gas and air in
a 1.05 m diameter and 73 m long, partially closed tube. The direction of detonation propagation
was from right to left in the figure. Photo courtesy of R. Karl Zipf (NIOSH).
were first observed by Denisov & Troshin [2] and Voitsekhovsky [3] in patterns
in the soot left behind on the wall of a tube through which a detonation
had propagated. In more recent experiments, special soot-covered plates are
purposely added to the system prior to detonation initiation in order to record
detonation cells. An example of the cellular patterns generated by a stoichiometric
methane–air detonation and recorded using this ‘smoke foil’ technique is shown
in figure 2 [4]. Detailed experiments, starting at least in the 1970s, and numerical
simulations, starting over 10 years later, revealed a great deal of information
on how detonation cells form and how their size, shape and regularity depend
on the thermodynamic properties of the gas mixture. Although warranted, an
extensive review of this topic is beyond the scope of the current discussion. The
interested reader is again referred to the paper in this Issue by Thomas [1] and
references therein. For detonations propagating through large enough systems,
the properties of the cellular structures are now believed to be independent of
boundary conditions and can be considered a property of the kinetics of energy
release for a particular gas mixture. The characteristic size of a detonation cell, l,
is used in a number of empirical criteria for the critical conditions for detonation
propagation. Ultimately, the quenching behaviour of a detonation depends on
the behaviour of the triple points that create detonation cells. This behaviour
has been documented quite extensively for many uniform fuel–air mixtures (for
example, the recent literature surveys by Shepherd [5] and Lee [6]).
In industrial-scale explosions, however, the gas mixtures that ignite are often
not uniform. One common scenario occurs in unventilated, sealed tunnels in
underground coal mines. The leaching of methane gas into the tunnel through
porous rock walls and the transfer of air into or out of the tunnel through leaks in
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D. A. Kessler et al.
the tunnel seals can cause the mixture composition to vary from locally fuel-rich
to fuel-lean before diffusion can effectively mix the reactants. At various locations
in space and time, the mixture can become explosive, and accidental ignition
may occur.
Compared with uniform mixtures, relatively little is known about how a
detonation may propagate through systems with spatial gradients in mixture
composition. In real systems, gradients will likely be oriented in arbitrary
directions relative to the direction of propagation and may change in time
and from location to location. Since the properties of a detonation, such as
its propagation speed and length scales associated with transverse instabilities,
depend strongly on equivalence ratio, f, its behaviour in a mixture with nonuniform f can be quite complex. We begin by considering two simpler systems
in which the gradients are either aligned with the direction of propagation of the
detonation or perpendicular to it.
Consider first the propagation of a detonation through a simple non-uniform
mixture in which the gradient of fuel concentration is parallel to the direction
of propagation. This case has been studied extensively for the problem of
transmission of a detonation from a more to a less reactive mixture or through a
gap filled with an inert gas [7–16]. In these studies, the concentration gradients
were formed by diffusion after two initially segregated gas mixtures were abruptly
brought into contact with one another after a diaphragm burst or removal of a
sliding plate. Thomas et al. [11], for instance, created well-defined gradients in
a vertical detonation tube by varying the length of time between the removal
of a horizontal sliding plate that separated two different gas mixtures and the
initiation of the detonation. In all of these situations, at any given instant the
detonation ‘sees’ a uniform (or nearly uniform) mixture as it propagates through
the gradient. The rate at which the properties of the mixture change depends
on the speed of the detonation and the steepness of the concentration gradient.
The steeper the gradient, the less time the detonation has to adapt to its new
environment and the less likely it is to survive.
The second scenario, a system in which a detonation propagates in a direction
perpendicular to the gradient in mixture composition, has been studied less
extensively. Some of the initial work in this area was done on the problem
of detonation transmission through layers of different gas mixtures [17–19].
A detonation was first ignited in one uniform mixture and then allowed to
abruptly come into contact with a secondary uniform mixture above or below it
(in the direction perpendicular to its direction of propagation). This is a limiting
case of a nearly infinite mixture composition gradient inside a thin mixing layer
that is small compared with the thickness of the detonation. Oran et al. [20]
computed detonations propagating through two initially segregated mixtures of
diluted hydrogen and oxygen. They found that along the line of intersection of
these mixtures, several different shock structures can exist, the details of which
depend on the relative reactivity of the upper and lower mixture and whether or
not the detonation in the primary mixture was overdriven.
Layered gas mixtures are separated by sharp interfaces that are thin compared
with reaction length scales, and thus their internal structure is inconsequential to
the behaviour of the detonation. The situation is more complex when the gradient
in mixture composition spreads over a larger area. Ishii and co-workers [21–23]
were the first to consider this type of system by studying the propagation of
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Detonations in concentration gradients
571
a detonation through several non-uniform hydrogen–oxygen mixtures. In their
experiments, the concentration gradients were formed by allowing a diluent,
nitrogen or oxygen gas, to diffuse into a test section containing stoichiometric
mixture of hydrogen and oxygen from below. How long the diluent was allowed
to diffuse into the test section determined the size of the resulting concentration
gradient. After the specified diffusion time, a detonation was ignited in the test
section. They found that the propagation speed was less than the ideal CJ velocity
(DCJ ) in a stoichiometric mixture for all concentration gradients that were tested.
Schlieren images showed that the detonation front was curved in the region where
the fuel was most diluted. Curvature was found to be smaller for longer diffusion
times (smaller effective concentration gradients). They also recorded detonation
cells using the smoke foil technique. Near the bottom of the channel (inside the
concentration gradient), the detonation cells resembled parallelograms and were
larger than the regular, diamond-shaped cells that formed near the opposite wall
of the test section where the mixture was nearly stoichiometric.
Lieberman & Shepherd [24] studied the propagation of detonations through
non-uniform mixtures of ethylene and oxygen. In their experiments, a combustible
fuel-rich ethylene–oxygen mixture was separated from a diluent (nitrogen or
oxygen gas) by a vertical sliding plate. When the sliding plate was removed, a
gravity current caused the gas with larger molecular weight to slide beneath the
lighter gas. Then, mixing and diffusion caused a gradient in fuel concentration
to form along the interface of the two gases, the thickness of which could be
controlled by the time interval between initiating the gravity current and igniting
the detonation. When the mixing time was large, the gradient of the mixture
composition was nearly perpendicular to the detonation propagation direction.
In these cases, high-speed schlieren images showed a curved detonation front,
which was more pronounced than any of the images shown by Ishii & Kojima [23].
Sufficiently far from the line of stoichiometry, the leading shock and reaction front
decoupled. This was attributed to the sharp rise in induction times in regions
where the diluent was the dominant species. A simple model of the impulse
produced by burning in the decoupled shock-turbulent-mixing zone predicts that
this secondary burning contributed approximately 1–5% of the total impulse,
which was in accordance with the pressure data obtained in the experiments.
Calhoon & Sinha [25] were the first to calculate the structure of a stable,
two-dimensional detonation in the mixing layer of initially non-premixed coflowing streams of fuel and air. A diagram of the complex wave structure
they computed is shown in figure 3, which the authors refer to as a tripleflame detonation wave (TFDW). The TFDW is the high-speed analogue to the
triple flame found in low-speed mixing layers [26–28]. Fuel-lean and fuel-rich
detonations form along the leading edge of the reaction zone near the centre of
the channel where the reactants are more thoroughly mixed. The excess reactants
not consumed by these detonations combine downstream and react in a laminar
diffusion flame. Variation in local mixture composition leads to a variation in
the propagation velocity along the leading shock. This causes the front to curve
in a manner similar to that observed in experiments [23,24]. Eventually, where
the mixture becomes too rich or too lean to support a detonation, the shock
and reaction zone decouple, leaving an inert oblique shock and a premixed flame
similar to the decoupled shock-turbulent-mixing zone described by Lieberman &
Shepherd [24].
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D. A. Kessler et al.
oblique
shock
pure fuel
fuel-rich
premixed flame
diffusion flame
fuel-rich
detonation
fuel-lean
detonation
fuel-lean
premixed flame
oblique
shock
mixing layer
pure oxidizer
Figure 3. Schematic of the triple-flame detonation wave calculated by Calhoon & Sinha [25].
The propagation velocity of this TFDW structure was found to vary only
slightly with the thickness of the mixing layer. For very thick mixing layers,
it approached a constant value about 1.5 per cent larger than DCJ for the
stoichiometric mixture. This small increase in velocity is similar in magnitude
to the increase in impulse generated by the decoupled shock and reaction
zone reported by Lieberman & Shepherd [24]. As the thickness of the mixing layer
was decreased, the propagation speed also decreased. For very thin mixing layers
about five times the thickness of a one-dimensional stoichiometric detonation
computed using their reaction model, the velocity fell to 94 per cent of CJ. It was
impossible to sustain a stable detonation in mixing layers smaller than this.
While the simulations performed by Calhoon & Sinha [25] were able
to reproduce some of the features of detonations propagating normal to a
concentration gradient, they considered only the stable detonation structure
and did not compute or resolve transverse instabilities. Such instabilities are
characteristic of gas-phase detonations, and the question remains as to how
the motions of these triple points affect the large-scale propagation and quenching
behaviour of a detonation in a gradient. Preliminary work on this topic has been
presented [29,30].
In this work, we attempt to address the question of how unstable gasphase detonations propagate through non-uniform fuel–air mixtures in which
the gradient of mixture composition is aligned perpendicular to the direction
of propagation. To do this, we require a model for the chemical kinetics that can
reproduce the behaviour of detonations in uniform mixtures over a wide range
of equivalence ratios. A suitable model is developed by extending concepts used
previously for reactions in single-component mixtures [31,32] to mixtures with
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Detonations in concentration gradients
573
two independent reactants. This model is used to compute detonation propagation
through two different non-uniform fuel–air mixtures. The first is a low-activationenergy mixture that produces very regular cellular structures similar to those
found in highly diluted mixtures. The second is a high-activation-energy mixture
that produces very irregular detonation cells and detonation properties that are
similar to those of methane–air mixtures.
2. Computational model
(a) Chemical model and governing equations
We consider a single-step reaction model, nF F + nO O → nP P, in which nF moles
of fuel (F ) combine with nO moles of oxidizer (O) to form nP moles of product (P).
The local mass fractions of the fuel and oxidizer can be scaled by their maximum
values in the unburned mixture, Ỹ ∞ and X̃ ∞ , so that the scaled mass fractions,
Y and X , respectively, range between 0 and 1. For a reactive flow, these mass
fractions are governed by
v(rY )
+ V · (rY v) − rU̇ = 0
vt
(2.1)
and
v(rX )
+ V · (rX v) − rS U̇ = 0,
(2.2)
vt
where v is the velocity and r is the density of the gas mixture. In the above
equations,
dY
(2.3)
= U̇ = −YX rA∗ e−Ea /RT
dt
is the reaction rate (in units of s−1 ), Ea is the activation energy, R is the ideal
gas constant and A∗ = nF X̃ ∞ A/WO is the pre-exponential factor (A) in units of
cm3 g−1 s−1 scaled by X̃ ∞ . The factor S = s Ỹ ∞ /X̃ ∞ is a global equivalence ratio
based on a mixture containing the maximum amounts of fuel and oxidizer in the
unburned mixture, and s = nO WO /nF WF is the stoichiometric coefficient of the
reaction. Note that for a uniform mixture, S is exactly equal to the equivalence
ratio of the unburned mixture.
Equations (2.1)–(2.2) are coupled with the reactive Euler equations,
vr
+ V · (rv) = 0,
vt
(2.4)
v(rv)
+ V · (rvv) + VP = 0
vt
(2.5)
vE
+ V · ((E + P)v) + q ∗ rU̇ = 0,
vt
using the equations of state for an ideal, calorically perfect gas
and
E=
Phil. Trans. R. Soc. A (2012)
rv · v
P
+
g−1
2
(2.6)
(2.7)
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574
D. A. Kessler et al.
and
rRT
,
(2.8)
M
where E and M are the total energy density and the molecular weight of the gas
mixture, respectively, and P is the thermodynamic pressure. Energy is released by
combustion according to dE/dt = q ∗ rU̇, where q ∗ = q Ỹ ∞ is the chemical energy
release per unit mass of fuel scaled by Ỹ ∞ .
The system is partially premixed, and the local f in the unburned mixture
varies both spatially and temporally. For a two-component mixture, we define a
local mixture fraction Z ,
P=
Z=
SY − X − SYlean + 1
,
1 + S − Xrich − SYlean
(2.9)
where Ylean is the scaled mass fraction of fuel in the leanest region of the flow
(where X̃ = X̃ ∞ ), and Xrich is the scaled mass fraction of oxidizer in the richest
region of the flow (where Ỹ = Ỹ ∞ ). The local mixture fraction Z is a conserved
quantity across the reaction zone and can be related to the fuel and oxidizer mass
fractions in the unburned mixture just upstream of the reaction zone, Yunb and
Xunb , respectively [33], according to
Yunb = Z (1 − Ylean ) + Ylean
and
Xunb = (1 − Z )(1 − Xrich ) + Xrich .
(2.10)
The local f is proportional to the ratio of these unburned mass fractions and is
computed using
SYunb
f=
.
(2.11)
Xunb
The details of the particular reaction model must still be specified. Here, we
consider two different fuel–air systems: one for which the activation energy is
low and one for which Ea is high and has properties similar to those found
for methane–air detonations. These two systems will be discussed in §§3 and 4,
respectively.
(b) System geometry and solution details
The canonical geometry used in the two-dimensional simulations described
here is a plane channel of height h and length L, closed at the left end (x = 0)
and open at the right end (x = L), as shown in figure 4. The channel walls are
assumed to be no-slip, chemically inert and adiabatic with boundary conditions,
v = 0, vT /vn = vY /vn = vX /vn = 0, where n is the direction normal to the wall
surface and v is the velocity vector. Zero-gradient conditions are applied at the
open end of the channel.
The reactive Euler equations (2.4)–(2.6) and equations (2.1) and (2.2) are
solved concurrently using an explicit operator-splitting technique on a structured
adaptive grid. Mesh refinement, based on the fully threaded tree method [34],
is used to achieve high spatial resolution in regions of the flow where there
are strong gradients of density, temperature or reactant mass fraction. The
convective fluxes are computed using a second-order, Godunov-type numerical
method incorporating a Riemann solver, and the conserved variables are partially
advanced forward in time using an explicit Euler integration method. The reaction
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Detonations in concentration gradients
L
initial
shock
Ms
h
ignited
detonation
fr
outflow
Ls
y
f1
x
adiabatic
walls
Figure 4. Schematic of computational domain used for §§3 and 4.
terms are then integrated using a quasi-steady-state method [35]. Parameters for
the reaction model are computed everywhere in the domain at the beginning of
each time step based on local values of f. The time-step subcycling procedure
divides each global time step (based on a Courant number of 0.7 and the size of
a grid cell at the finest level of resolution) into 10 smaller time steps to improve
the numerical stability of the integration.
3. Detonations in a non-uniform low-activation-energy mixture
(a) Reaction model
First consider a fuel–air mixture with a low activation energy, which means that
the transverse waves and triple points along the detonation front form regular
cellular patterns [36,37]. The particular choice of Ea used in this model will be
described in more detail later. In this model reaction, we assume that fuel and
oxidizer are consumed in the same proportion as in the oxidation of methane, so
that nF = 1, WF = 16 g mol−1 , nO = 2, WO = 32 g mol−1 and s = 4. The process for
choosing the remaining reaction parameters g, M , q ∗ , Ea and A∗ is described later.
First, we assume the ratio of specific heats and the molecular weight of the
mixture are constant over the entire domain and equal to those of air at a
temperature of approximately 1300 K. That is, g = 1.33 and M = 27 g mol−1 .
Using these values, we next calibrate the heat release parameter q so that
computed DCJ are similar to those of methane–air mixtures over a wide range
of f. Values of DCJ for five different values of f (0.6, 0.72, 1.0, 1.2 and 1.43)
are listed in table 1. For each of the five mixtures, we find values of q so that
the computed DCJ matches these values. Based on these calibration points, we
construct a piecewise continuous function for q,
qM
−0.5766f2 + 0.3991f + 1.1775 f ≤ 1,
= 395.5248
(3.1)
f > 1,
−0.4244f2 + 1.03f + 0.3943
RT0
that gives approximate values of the heat release for the range of equivalence
ratios used in this study. Here, T0 = 298 K is the temperature of the unburned
gas mixture.
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D. A. Kessler et al.
Table 1. Target values of ideal one-dimensional detonation properties calculated using a complex
methane–air reaction mechanism [38] and measured experimentally [4,39] and the corresponding
single-step reaction model parameters that reproduce them.
f
0.6
0.72
1
1.2
1.43
target quantities
DCJ (m s−1 )
l (cm)
xd (cm)
1587
200
4
1691
85
1.7
1820
23
0.46
1827
28
0.56
1800
56
1.12
model parameters
qM /RT0
Ea /RT0
g
nF A/WO (cm3 g−1 s−1 )
Ea /RTZND
478.336
26.72
1.333
1.29 × 109
6.23
461.155
26.72
1.333
1.94 × 109
5.63
395.5248
26.72
1.333
4.90 × 109
5
403.155
26.72
1.333
3.08 × 109
4.97
395.33
26.72
1.333
1.37 × 109
5.09
We choose Ea so that in a stoichiometric mixture, Ea /RTZND = 5, where TZND is
the post-shock temperature computed using the one-dimensional Zeldovich, Von
Neumann, Döring (ZND) model. This value of Ea /RTZND is sufficiently small
to ensure the formation of regular detonation cells [36,37]. In this model, Ea is
assumed to be independent of local f and is thus taken to be a constant in the
simulations that follow. For the values of f shown in table 1, Ea /RTZND varies
from 4.97 to 6.23.
For each of the mixtures described in table 1, the final parameter in the reaction
model, A, is then calibrated to give physically reasonable approximations of the
length of the induction zone in a stable one-dimensional ZND detonation for q
given by equation (3.1). We choose target values for the half-reaction length of the
ZND detonation, xd , inferred from experimental measurements of the detonation
cell size l of methane–air mixtures [4,39] based on a correlation l/xd = 50 [40]. For
each value of f, A was systematically varied until the values of xd computed from
the one-dimensional ZND equations matched the target values given in table 1.
A least squares method was then used to fit a smooth function to the values of
A computed for the five calibration mixtures:
nF A
= 4.9 × 109 exp[−8.6011f2 + 17.594f − 9.0] cm3 g−1 s−1 .
WO
(3.2)
We will show later that the initial guess of the correlation factor l/xd = 50 was
too large, and the actual values of l computed in the two-dimensional simulations
using this reaction model are smaller than those listed in table 1. Nonetheless,
the same general trend of increasing detonation cell size as mixtures deviate from
stoichiometric still holds. Values of DCJ and xd computed using this model are
shown in figure 5a.
(b) Geometry and initialization
We consider both uniform and non-uniform fuel–air mixtures described by
the reaction model given in the previous section. For cases where the mixture
is non-uniform, the equivalence ratio varies from f = 1.95 (Ỹ ∞ = 0.10204) at
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Detonations in concentration gradients
(a)
1750
12
1700
10
1650
8
DCJ (m s–1)
1800
6
1600
xd (cm)
y
(b) h/2
14
0
4
1550
2
1500
0.6
0.8
1.0
f
1.2
1.4
1.6
–h/2
0.5
1.0
f
1.5
Figure 5. (a) Variation of DCJ (solid line) and xd (dashed line) computed using the reaction model
described in §3a as a function of f and (b) distribution of the local equivalence ratio in the unburned
mixtures across the channel for the non-uniform test cases discussed in §3c.
the top wall to f = 0.05 (X̃ ∞ = 0.2342) at the bottom wall according to the
distribution shown in figure 5b. The mixture is ignited directly by placing a
slug of hot material compressed by a strong shock (with Mach number of
seven) to the right of the closed end of the channel. In all cases considered,
a total area of 256 cm2 of compressed material was sufficient to initiate a
detonation that then propagates from left to right through the domain. The
minimum computational cell size created by the adaptive mesh refinement
is 1/32 cm along shocks and reaction zones, which corresponds to seven
computational cells per xd . Past simulations in low-activation-energy mixtures
have shown this level of grid resolution to be sufficient to capture transverse
instabilities [36,37]. Initial temperature and pressure are taken to be 298 K and
0.101325 MPa, respectively.
(c) Results and discussion
We first discuss the behaviour of a detonation propagating through a uniform,
stoichiometric (f = 1) mixture. The width of the channel is taken to be h =
128 cm, which is large enough to accommodate a multitude of detonation cells.
The length of the channel is set to 1048 cm, which is long enough for the velocity
of the initially overdriven detonation to decay to DCJ and for detonation cells
to grow to a constant size. A magnified view of the reaction front showing the
behaviour of several triple points over a short interval of time was given in figure 1
and discussed in §1. The trajectories of these triple points are shown in figure 6,
where the maximum pressure encountered at each location in the domain has
been recorded and plotted. The detonation is, on average, planar and propagates
at DCJ = 1820 m s−1 . The cellular pattern it leaves behind is fairly regular, with
the sizes of each individual cell differing by no more than a factor of two, so that
approximately 15 detonation cells span the width of the channel. This suggests
that the average cell size is nearly 9 cm. As stated earlier, the initial guess of
the ratio of average detonation cell size to the half reaction thickness of a ZND
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D. A. Kessler et al.
Figure 6. Numerical smoke foil computed for a detonation propagating through a uniform
stoichiometric mixture for the parameters given in table 1.
detonation, l/xd = 50, used in the calibration of A was too large. On the basis
of computed cell size, l/xd ≈ 20, which is still consistent with the calculations of
Gavrikov et al.
In a non-uniform mixture in which there is a gradient in f, the behaviour of
the detonation can be significantly more complex. If we restrict our attention
to the situation where f varies only in the direction normal to the direction
of propagation of the detonation, the detonation velocity and cell size can vary
across the channel. Variable propagation velocities can lead to the formation
of a curved detonation front, as computed by Calhoon & Sinha [25] and
recorded experimentally by Ishii & Kojima [23] and Lieberman & Shepherd [24].
Variation in detonation cell sizes indicates that induction times and reaction
zone thicknesses vary along the detonation front. This can lead to local shock–
flame decoupling, such as that observed for the TFDW (cf. figure 3) [25]. We can
expect a global detonation extinction if the shock–flame decoupling near channel
walls damps transverse waves. Here, we investigate these issues by considering the
propagation of unstable detonations through a number of non-uniform mixtures in
channels of varying heights ranging from h = 4 cm to 512 cm. In each case, f varies
according to figure 5b. Near the top and bottom of the channel, the composition
of the gas mixture lies outside the flammability and detonability limits and is
functionally inert. The mixture in the centre of the channel is reactive, and the
width of this region varies with the size of the channel. Thus, for fixed l, the
number of detonation cells that fit within the detonable region decreases with
decreasing channel size.
Numerical smoke foils computed for each of the mixtures are shown in figure 7.
These are drawn to scale in order to show how the sizes of the computed
detonation cells change with the size of the channel. Magnified versions of
figure 7a,c are shown later in figures 11a and 10a, respectively. In the larger
channels, the trend is for the size of the detonation cells to increase as the width
of the detonable region decreases until only one and a half cells span the entire
channel (64 cm). A subsequent decrease in the channel width to 32 cm gives rise
again to 1.5 cells spanning the channel that are half the size of those found in the
64 cm channel. Only two triple points and a single detonation cell fit in the 16 and
8 cm channels. Further reduction to 4 cm causes one of the triple points to die out,
leaving behind a single triple point and only a half cell within the channel. The
instantaneous velocities of the reaction fronts for several of these cases are shown
in figure 8a,b. Figure 8c shows the average detonation velocity as a function of
the channel width. Also shown in figure 8c are DCJ for a uniform stoichiometric
Phil. Trans. R. Soc. A (2012)
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Detonations in concentration gradients
579
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 7. Numerical smoke foils computed for detonations propagating through non-uniform
mixtures in channels of width (a) 4 cm, (b) 8 cm, (c) 16 cm, (d) 32 cm, (e) 64 cm, (f ) 128 cm,
(g) 256 cm and (h) 512 cm.
∗
∗
mixture and a mixture averaged detonation velocity, DCJ
. We compute DCJ
using
the average value of the heat release across the channel q̄ for the equivalence ratio
distribution shown in figure 5b, i.e.
1
q̄ =
h
Phil. Trans. R. Soc. A (2012)
h/2
−h/2
q ∗ (y)dy.
(3.3)
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580
D. A. Kessler et al.
(a) 4500
(b)
front velocity (m s–1)
4000
3500
3000
2500
2000
1500
1000
0.020
0.025
0.030
time (s)
0.035
0.040
0.001 0.002 0.003 0.004 0.005 0.006
time (s)
(c) 1850
DCJ
front velocity (m s–1)
1800
1750
1700
D*CJ
1650
1600
1550
1500
101
102
channel width (cm)
Figure 8. Instantaneous velocities for detonations propagating through non-uniform mixtures in
(a) 4 (red dashed line), 8 (green dashed-dotted line) and 16 (blue solid line) cm channels and
(b) 128 (black dashed line), 256 (purple dashed-dotted line) and 512 (blue solid line) cm channels
along with DCJ (solid black line) for a uniform stoichiometric mixture, and (c) average propagation
velocities in the non-uniform mixtures as a function of channel width. (Online version in colour.)
The detonation structure formed in the 512 cm channel is similar to Calhoon &
Sinha’s [25] stable detonation wave structure. Contours of reaction rate,
temperature and fuel mass fraction for this case are shown in figure 9a. The
detonation front is curved, and in the lean region (near the bottom of the channel)
the reaction zone and shock are decoupled. In the rich mixture (near the top of
the channel), the detonation survives almost all the way up to the wall. Near the
centre of the channel, excess fuel (from the rich portion of the channel) and excess
oxidizer (from the lean portion of the channel) combine and react in a turbulent
diffusion flame. A time history of the instantaneous reaction front velocity is
shown in figure 8a. There is little fluctuation in this computed velocity, indicating
that the detonation propagation is relatively unaffected by quenching near the
walls. The average propagation velocity is, however, less than DCJ for a uniform
stoichiometric mixture.
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Detonations in concentration gradients
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Figure 9. Contours of reaction rate (left column), temperature (centre column) and fuel mass
fraction (right column) computed for a detonation propagating through (a) a non-uniform mixture
in a 512 cm channel, (b) a non-uniform mixture in a 256 cm channel and (c) a non-uniform mixture
in a 128 cm channel. (Online version in colour.)
The most notable difference between the detonation structure shown here and
the TFDW (figure 3) is the presence of Mach stems near the top and bottom of
the channel that form along the bounding walls. Oblique parts of the detonation
front near walls deflect the reactive flow towards the walls. As a result, the reactive
material mixes with practically inert fuel or oxidizer layers and forms weak
reaction zones along the channel walls. Temperature and fuel mass fraction plots
in figure 9a show that very little material is consumed and only a small amount
of heat is released in these regions. The trajectories of the triple points attached
to these Mach stems can also be seen on the numerical smoke foil (figure 7h). The
Mach stem at the bottom of the channel reaches a nearly constant size and leaves
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D. A. Kessler et al.
behind a band of high pressure. Triple points that propagate into this region
are partially reflected and partially transmitted by the Mach stem. The reflected
triple points are relatively weak and are only barely visible on the numerical
smoke foil. As they propagate into more reactive regions in the mixture, the
triple points slowly strengthen as more energy is released in the reaction zone.
The Mach stem near the bottom of the channel never grows large enough to
propagate into the reactive portion of the mixture, but the Mach stem near the
top of the channel does and propagates towards the centre of the channel. This
is because DCJ in rich regions of the mixture is larger than that in lean regions
of the mixture of the same relative difference in f from stoichiometric. Figure 5a
indicates that DCJ peaks around f = 1.1 and slowly decreases with increasing f,
causing the two-dimensional reaction zone to be asymmetric.
A multitude of other triple points form along the detonation front, giving rise to
a large number of detonation cells across the width of the channel. The cells that
are formed near the centre of the channel are similar in size to those computed
for the uniform stoichiometric mixture (figure 6). Cells that are closer to the
channel walls are larger and more distorted than those near the centre of the
channel, which is consistent with experimental observations [23]. The distortion
of the cells at the edges of the detonable mixture can be explained by considering
the propagation of a triple point along a curved reaction front. Those moving
away from the centre are travelling backward (in the frame of the leading shock)
and those moving towards the centre are moving forward in the shock frame of
reference. This causes their trajectories in the laboratory frame to bend closer to
the vertical and horizontal, respectively.
The curvature of the detonation front that forms in the 256 cm channel
(figure 9b) is somewhat less pronounced than that of the detonation in the 512 cm
channel (figure 9a), and there are fewer triple points that form along the leading
reaction zone. Again, the shock and reaction zone decouple near the bottom of
the channel. The average detonation velocity is 10 per cent lower than DCJ in the
stoichiometric mixture, and the cellular structures that form near the centre of
the channel (figure 7g) are larger than those shown in figures 6 and 7h. Since there
are fewer triple points along the reaction front, each propagates through layers of
material in which f can vary substantially between collisions. This can exacerbate
the sensitivity of reaction rate to temperature and result in more pronounced
decoupling of the shock and reaction zone, especially near the walls. The layers
of fuel that pass through the detonation unreacted in the near-wall region are
larger in the 256 cm channel than those in the 512 cm channel (figure 9b).
The behaviour of a detonation in the 128 cm channel is different from that in
larger channels. Shortly after initiation of the detonation all but two triple points
disappear, and the detonation nearly dies. The collision of these two remaining
triple points ignites a new detonation that then propagates through the channel.
This collision leads to a temporary sharp spike in the reaction front velocity that
quickly decays to a value below DCJ . Subsequent collisions between triple points
cause smaller spikes in the velocity, and on average the detonation propagates
more slowly than in the larger channels. This cycle of near detonation failure
and reinitiation by triple point collisions occurs frequently in all of the channels
smaller than 128 cm. After the detonation settles into a quasi-steady propagation
pattern, the sizes of the cellular structures formed in the 128 cm channel are larger
and more irregular than those in the 256 and 512 cm channels, and eventually only
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Detonations in concentration gradients
583
several cells exist in the channel (figure 7f ). For this case, the Mach stem that
forms along the bottom of the channel does extend into the reactive portion of
the mixture and becomes a propagating triple point as does the Mach stem at
the top of the channel.
Contours of reaction rate, temperature and fuel mass fraction in the vicinity of
the detonation front for the 128 cm channel are shown in figure 9c. The leading
edge of the detonation is still curved, albeit to a lesser degree than that in the
256 cm channel. The turbulent diffusion flame in the centre of the channel and the
reaction zones near the channel walls occupy nearly the entire channel. Again,
only a small amount of energy is released in these regions, and large bands of
inert material pass through the detonation near the channel walls. The extent to
which this affects the average propagation speed is not known, but the detonation
propagates more slowly for this case than in the 512 and 256 cm channels.
In even smaller channels, the detonation propagation is controlled entirely by
the behaviour of a few triple points. There is no sustained curved detonation
front, and triple points leave behind fairly regular cellular patterns that resemble
those found in uniform mixtures. Both the 64 and 32 cm channels house 1.5 cells,
the sizes of which are dependent on the height of the channel. The cells appear to
be asymmetric because the strength and the velocity of the triple points varies
across the channel, with the strongest collisions occurring away from the walls.
Only single detonation cells can form in the 8 and 16 cm channels. For these
cases, two triple points coexist, and periodic collisions sustain the propagation of
the detonation. This behaviour can be seen in the numerical smoke foils. A larger
version of figure 7c is shown in figure 10a. Unlike detonations propagating
through a uniform mixture, collisions between a triple point and a channel wall
occur in a practically inert material and are weak compared with collisions of
triple points with each other. The width of the reaction zone broadens and the
detonation slows in between triple point collisions that cause the formation of a
new overdriven detonation.
Fields of temperature and fuel mass fraction near the reaction zone of the
detonation in a 16 cm channel at several instants in time are shown in figure 10.
The sequence starts (figure 10b) just after the collision of triple points with
channel walls. The reflected triple points are practically inert, and the flame is
decoupled from the leading shock across the entire height of the channel. Triple
points remain inert even when they move into a more reactive mixture near the
centre of the channel (figure 10c). The reaction at the shock front is reignited
only when triple points collide at the centre of the channel (figure 10d). The
collision creates a powerful reactive Mach stem (figure 10e) that weakens as it
grows to the size of the channel (figure 10f ). Triple points remain reactive almost
until they collide with channel walls, and the collision cuts off large pockets of
unburned material near the walls that persist behind the leading shock. Since
the material at the walls is practically inert, the reflected triple points do not
induce reaction, weaken, and remain inert until the next collision at the centre of
the channel ignites a new overdriven detonation. Thus, the entire reaction zone
passes through a cycle of decay and reignition. This is reflected in the periodic
fluctuations in instantaneous velocity observed for this case (figure 8b).
The smallest channel considered in this study (4 cm) is too small for more
than one triple point to exist. After an initial transient period, a single triple
point propagates back and forth between the channel walls and leaves behind the
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D. A. Kessler et al.
Figure 10. (a) Close-up view of numerical smoke foil and (b–g) temperature (left column) and fuel
mass fraction (right column) maps of a detonation propagating through a non-uniform mixture
in a 16 cm channel over the course of one cycle of a triple point crossing the channel after it has
reached a quasi-steady propagation routine. (Online version in colour.)
half-cell pattern in the numerical smoke foil shown in figure 11a. The mode of
propagation of the resulting detonation is different in character from those in the
larger channels. Again, as observed in the 8 cm and 16 cm channels, the reaction
front progresses through a periodic cycle, which is shown in figure 11. Here, the
only collisions that occur are between a triple point and a wall. These take
place in the inert layers of the mixture, which precludes the formation of new
Phil. Trans. R. Soc. A (2012)
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Detonations in concentration gradients
585
Figure 11. (a) Close-up view of numerical smoke foil and (b–f ) temperature (left column) and fuel
mass fraction (right column) maps of a detonation propagating through a non-uniform mixture
in a 4 cm channel over the course of one cycle of a triple point crossing the channel after it has
reached a quasi-steady propagation routine. (Online version in colour.)
overdriven detonation kernels of the type that drive the detonation in the 16 cm
channel. Instead reaction takes place behind the leading shock after an induction
delay. The single triple point shown in figure 11b is moving towards the top of
the channel. As it propagates, it forces reactive mixture from the centre of the
channel towards the wall, where it mixes with the material in the inert layer.
After the transverse shock attached to the triple point compresses this material
that has already been compressed once by the leading shock, the temperature
is sufficiently high to ignite a reaction. A kernel of burning forms in figure 11c.
After the triple point reflects from the top of the channel, the transverse shock
passes back through this kernel, strengthening the reaction that consumes a
large portion of the unreacted material near the top of the channel (figure 11d).
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D. A. Kessler et al.
As the triple point propagates towards the bottom of the channel, the same
process occurs, consuming the unburned material near that wall (figure 11e,f ).
In between these periods of burning, pockets of unreacted material again pass
through the detonation near the top and bottom of the channel. Since the
main energy release in this regime occurs well behind the leading shock and
the transverse wave induces mixing in the stratified material before it burns, the
burning can be more complete than in larger channels, where the main reaction
occurs at the shock front. This effect could explain the slight uptick in average
propagation velocity observed in the 4 cm channel (figure 8c). However, sorting
out the various competing effects that contribute to the velocity of the detonation
is difficult for this complicated reaction zone.
We should note that the detonation regime observed in a 4 cm channel, and
the absence of detonation extinction in our simulations, may be specific to the
low activation energy used in this model. The relatively weak dependence of the
reaction rate on the temperature allows the detonation to propagate at velocities
well below DCJ with the main energy release occurring far behind the leading
shock. This may not be possible for systems with higher activation energies.
4. Detonations in a non-uniform high-activation-energy mixture
While the fuel–air mixture discussed in §3 was a convenient one for studying
some fundamental properties of detonation propagation in a gradient of mixture
composition, it cannot be directly compared with most real fuel–air mixtures.
In this section, we calibrate a reaction model that more closely reproduces
properties of methane–air mixtures and make a first attempt at modelling the
propagation of a detonation through a non-uniform mixture of methane and air.
The primary difference between this reaction model and that described in §3 is
that the activation energy is variable and much larger, which means the reaction
rate is more sensitive to small variations in temperature. The implications of this
on the behaviour of a detonation will be discussed in this section.
(a) Methane–air reaction model
We consider a partially premixed methane–air system, for which the values of
nF , nO , WF and WO are the same as those used in §3 and listed in table 2. The
mass fraction of fuel is small compared with that of air and, for simplicity, we
assume that the molecular weight of the mixture is equal to that of air. In an effort
to improve the capabilities of the reaction model, and with an eye towards future
simulations of explosions and deflagration-to-detonation transitions (DDTs) in
similar non-uniform mixtures, the reaction parameters are calibrated so that
the model yields reasonable estimates of both detonation and laminar flame
properties. To do so, we must specify four additional parameters (m, K , DF
and DO ) that affect the speed and structure of laminar flames. These transport
coefficients are assumed to vary with temperature according to
n
T
,
(4.1)
m = m0
T0
n
T
,
(4.2)
K = K0
T0
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Detonations in concentration gradients
587
Table 2. Thermodynamic properties of methane–air mixtures used in the partially premixed
reaction model.
nF
nO
WF
WO
s
M
m0
P0
T0
r0
1.0
2.0
16.0
32.0
4.0
27 g mol−1
1.94 × 10−5 Pa s
0.101325 MPa
298 K
1.1042 × 10−3 g cm−3
rDF = D̄F ,0
and
rDO = D̄O,0
T
T0
T
T0
n
(4.3)
n
,
(4.4)
where the parameters m0 , K0 , D̄F ,0 and D̄O,0 are the values of the viscosity, thermal
conductivity, fuel mass diffusion coefficient and oxidizer mass diffusion coefficient,
respectively, at the reference temperature T0 and pressure P0 . Here we take T0 =
298 K, P0 = 0.101325 MPa and the temperature exponent n = 0.7. The viscosity
does not affect the properties of a laminar flame or a stable, one-dimensional
detonation; so we choose m0 to be that of air at T0 and P0 . This value is listed
in table 2.
For this chemistry model, it is thus necessary to specify values for A, Ea ,
g, q, K0 , D̄F ,0 and D̄O,0 . Specifically, we attempt to match the adiabatic
flame temperature Tb , laminar flame speed Sl , laminar flame thickness xl , DCJ
and xd computed with this reaction model to values measured experimentally or
computed using a full methane–air chemical reaction mechanism.
We first calibrate the reaction parameters for a stochiometric (f = 1) mixture.
The procedure is similar to that described for a single-component reaction model
by Kessler et al. [32]. We solve the steady-state laminar flame equations and the
one-dimensional ZND equations for a propagating detonation wave [32] using the
two-component reaction model, equation (2.3), to obtain a set of parameters for
which Tb , Sl , xl , DCJ and xd match the target values given in table 3. The first
step is to determine q and g. Unique values of q that yield a specific value of Tb
and a specific value of DCJ exist for each g. We search for g such that these two
values of q that give the desired Tb and DCJ are the same. These values, listed
in table 3, are then used in the model.
The next step is to calibrate A and Ea based on the Sl and xd given in table 3.
Although methane–air detonation cells are highly irregular, the average cell size
has been measured in experiments to be about 23 cm [4,39]. The value of xd
correlates with l within a range of 50 < l/xd < 150 [40] for the range of activation
energies in which we are interested. We assume l/xd = 100 and compute values
of A and Ea that give xd = 0.23 cm. For fixed q and g, Sl depends on three
parameters, however, A, Ea and K0 . There is not a unique locus of points in
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D. A. Kessler et al.
Table 3. Target laminar flame properties [41] and one-dimensional detonation wave properties of
methane–air mixtures at various values of f and the input model parameters used to attain these
values. Values of DCJ are computed using a chemical equilibrium code [38], and xd are calculated
based on measured detonation cell sizes [4,39].
f
0.6
0.72
1
1.2
1.43
target quantities
DCJ (m s−1 )
l (cm)
xd (cm)
Tb (K)
Sl (cm s−1 )
xl (mm)
1587
200
2
1770
15
3
1691
85
1
1971
23.7
1.7
1820
23
0.23
2210
40
1.0
1827
28
0.28
2110
38
1.3
1800
56
0.34
1990
16
2.2
model parameters
qM /RT0
g
nF A/WO (cm3 g−1 s−1 )
Ea /RT0
K0 (W cm−1 K−1 )
D̄F ,0 = D̄O,0 (g cm−1 s−1 )
996.45
1.172
4.75 × 1011
42.0
1.73 × 10−3
8.25 × 10−4
913.96
1.18
3.13 × 1012
52.55
2.29 × 10−3
1.13 × 10−3
708.19
1.197
8.08 × 1013
67.55
3.28 × 10−3
1.75 × 10−3
569.16
1.244
3.48 × 1015
95
5.01 × 10−3
3.19 × 10−3
497.79
1.27
4.34 × 1013
77.55
4.10 × 10−3
2.83 × 10−3
the A–Ea parameter space that gives the target Sl . In fact, a different locus of
points exists for each value of K0 . We can make a selection among these curves
by comparing the computed values of xl to the target value listed in table 3 for
each set of parameters. Ideally, we could choose K0 so that Sl = 40 cm s−1 for the
same set of values of A and Ea that give xd = 0.23 cm and the target xl = 1 mm.
Doing so requires using an unrealistically large value of Ea . Instead, we choose
A so that Sl = 40 cm s−1 and xd = 0.23 cm for Ea /RT0 = 67.55, the same value
used in a previous study of detonation propagation in uniform stoichiometric
methane–air mixtures [42]. The value of nF A/WO at this intersection point is
8.081 × 1013 cm3 g−1 s−1 , and K0 = 3.28 × 10−3 W cm−1 K−1 . The resulting value
of xl for this parameter set is 2.13 mm, which is slightly larger than the average
experimental measurement shown in table 3. The set of parameters given in
table 3 thus represents a compromise between physically realistic transport
properties and the laminar flame thickness; however, it does give correct values of
Tb , DCJ , Sl and xd to within the uncertainty with which they are known. Finally,
we make the assumption of unity Lewis number and set D̄F ,0 = D̄O,0 = K0 /cp ,
where cp = gR/(g − 1)M is the specific heat of the mixture.
We next use the same methodology to calibrate the reaction parameters for
four additional lean and rich methane–air mixtures: f = 0.6, 0.72, 1.2 and 1.43.
The target Sl , Tb , xl , DCJ and xd for each mixture are listed in table 3. As was
the case for the stoichiometric mixture, we found it impossible to obtain a set of
parameters with realistic values of activation energy that produces xd , Sl and xl
that are all close to the accepted values shown in table 3. Instead, we adjusted
A, Ea and K0 to match xd and Sl and maintain the correct qualitative behaviour
of xl in relation to the values obtained for the other control points. In this study,
we are not concerned with this mismatch of flame thicknesses since we model
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Detonations in concentration gradients
(a)
1850
2.0
(b) 16
12
8
1750
1.5
1700
1.0
1650
1600
xd (cm)
y (cm)
DCJ (m s–1)
1800
0.5
1550
0.6
0.8
1.0
f
1.2
1.4
4
0
–4
–8
–12
–16
0.8
1.0
f
1.2
Figure 12. (a) DCJ (solid lines) and xd (dashed lines) computed using one-dimensional ZND
equations and model parameters given by equations (4.5)–(4.9) (square symbols) compared with
target values (no symbols). (b) Variation of f as a function of position in the channel for each of
the three non-uniform methane–air cases considered (solid line, case 1; dashed-dotted line, case 2;
dashed line, case 3).
the propagation of detonations using the Euler equations, which do not admit a
laminar flame as a solution. The implications of using artificially large values of
xl for flame acceleration and DDT will be discussed in a future publication. The
transport and reaction coefficients obtained for all five calibration mixtures are
shown in table 3.
Finally, we construct best-fit fourth-order polynomials for each of the reaction
parameters:
nF A ∗
= 1.78 × 1013 exp[−225.6201f4 + 836.2564f3
WO
− 1131.2621f2 + 674.6915f − 154.0657],
Ea
= 67.55(−24.565f4 + 92.888f3 − 127.59f2 + 76.484f − 16.216),
RT0
q ∗M
= 40.36(2.3585f4 − 7.8387f3 + 8.4459f2 − 2.9394f + 0.9737),
RT0
and
g = 1.197(−1.5509f4 + 5.9854f3 − 8.2683f2 + 4.9247f − 0.091)
g
K0 = 5.398 × 10−4
(−42.707f4 + 161.04f3
g−1
− 219.07f2 + 129.13f − 27.391).
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
These particular choices of fitting functions have no physical basis. They appear
as a result of trying to force a single reaction model to describe two disparate
burning regimes: low-speed flames and high-speed detonations. Taken collectively
and used in conjunction with equation (2.3), these parameters produce DCJ and
xd shown in figure 12a that exhibit the desired features of accepted values of
DCJ and xd for methane–air mixtures. Similarly, good agreement is obtained for
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D. A. Kessler et al.
Tb and Sl (not shown), and the variation in xl is qualitatively correct although
each individual value is larger than the value shown in table 3. We caution that
equations (4.5)–(4.9) should not be used outside of the range f = 0.6–1.43 without
further calibration.
(b) Geometry and initialization
As in §3, we consider the propagation of a detonation through a concentration
gradient perpendicular to its direction of propagation in a planar channel closed
at the left end (x = 0) and open at the outflow (x = L) (cf. figure 4). The width of
the channel is chosen to be h = 32 cm, which is nominally larger than the size of
a single detonation cell in a stoichiometric methane–air mixture, and the length
is taken to be 1024 cm. In the calculations that follow, the channel is filled either
with a uniform methane–air mixture (f = const.) or with a mixture that varies
linearly from rich (f = fr ) at the top of the channel to lean (f = fl ) at the bottom
of the channel as shown in figure 12b.
Fully resolving the reaction zone of high-activation-energy detonations requires
a very large number of computational cells at the detonation front. In the
calculations presented in §4c, the maximum computational cell size (away from
shocks and reaction zones) is set to be 1/4 cm, and the smallest cells are 1/256 cm,
which corresponds to approximately 57 computational cells per half-reaction
length of a CJ detonation computed using the parameters given in table 3 for
a stoichiometric methane–air mixture. This level of resolution was found to be
necessary to compute the multi-dimensional instability of the detonation front
that gives rise to the formation of cellular structures. In order to reduce the
computational effort of the calculations and to be consistent with the reaction
model used in §3, we solve the reactive Euler equations and hence do not use the
transport coefficients m0 , K0 , DF ,0 and DO,0 .
(c) Results and discussion
We first consider the propagation of detonations through a variety of uniform
mixtures with f ranging from 0.7 to 1.1. The resulting time histories of the
reaction front propagation velocities are shown in figure 13a. For the two
lowest values of f, −0.7 (red dashed line) and 0.8 (green dashed-dotted line),
a detonation cannot be sustained, and the propagation velocity falls below
1500 m s−1 . In these cases, longitudinal instabilities, which are known to exist
for high-activation-energy mixtures [6], cause significant temperature fluctuations
across the induction zone of the detonation. The induction zone broadens until
the reaction zone decouples from the leading shock. Collisions between triple
points and transverse waves, which may serve to reignite the detonation, are too
few and cannot support the detonation propagation. For f = 0.9, 1.0 and 1.1, the
velocity of the reaction front approaches DCJ for each particular mixture. DCJ for
richer mixtures, 1.1 < f < 1.3, are similar to that for f = 1.1 (cf. figure 13a), and
a detonation can also propagate through these mixtures in the 32 cm channel.
Next, we consider the propagation of detonations through non-uniform
mixtures of methane and air in the same two-dimensional channel where f
varies linearly from the bottom to the top of the channel as shown in figure 12b.
Three different cases are investigated: (1) fl = 0.7 and fr = 1.3, (2) fl = 0.8 and
fr = 1.2, and (3) fl = 0.9 and fr = 1.1. The propagation velocities of detonations
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591
Detonations in concentration gradients
(b)
reaction front velocity (ms –1)
(a)
x (m)
x (m)
Figure 13. Velocity of the leading edge of the reaction front as a function of its distance from the
closed end of the channel in (a) uniform methane–air mixtures with various f ranging from 0.7 to
1.1 and (b) methane–air mixtures with concentration gradients. In (a), f = 0.7 is denoted by red
dashed line, f = 0.8 green dash-dotted line, f = 0.9 blue dotted line, f = 1 black dash double-dotted
line and f = 1.1 purple solid line. The horizontal lines of the same type and colour represent DCJ
for that particular mixture. In (b), case 1 (fr = 1.3 and fl = 0.7) is indicated by red dashed line,
case 2 (fr = 1.2 and fl = 0.8) green dash-dotted line, and case 3 (fr = 1.1 and fl = 0.9) blue solid
line. (Online version in colour.)
ignited in each of the three test mixtures are shown in figure 13b. For case 1, the
propagation velocity abruptly changes from ≈ 1700 to ≈ 1000 m s−1 , indicating
complete detonation failure. For cases 2 and 3, the detonations are able to survive
and propagate at speeds near DCJ . The average speeds for the two cases are quite
similar, with case 2 propagating only slightly slower.
The same basic reaction zone structure forms for cases 2 and 3, and is shown in
figure 14. The leading edge of the reaction zone is an unstable detonation front.
Away from the centre of the channel, which is also the line of stoichiometry f = 1,
excess fuel and oxidizer pass through the propagating detonation wave. These
reactants mix downstream and burn in a turbulent diffusion flame that trails
behind the leading detonation. As we noted in §3, any burning in this diffusion
zone is only qualitative, since we are solving the Euler equations and not the
Navier–Stokes equations. A large degree of turbulent mixing occurs in this region,
which helps in keeping the trailing diffusion flame active in the place of physical
diffusion. Some differences between this detonation structure and those computed
in §3 are evident. First, the non-uniform methane–air mixtures considered in this
section are reactive across the entire channel. No inert layers exist near the walls
of the channel, and the corresponding curvature of the reaction front is not as
pronounced. In fact, no global curvature is found for any of these configurations.
The y-position of the leading edge of the reaction front fluctuates significantly as it
propagates, and any curvature that is caused by the variation in f throughout the
mixture is negated by these fluctuations caused by the motion of the transverse
instabilities. Another notable difference from the structure shown in figure 9 is
how thin the reaction zone is in figure 14. This is another consequence of the
increased temperature sensitivity of the reaction rate caused by the use of large
activation energies.
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D. A. Kessler et al.
Figure 14. Maps of the logarithm of the reaction rate for a detonation propagating through a
32 cm wide channel with the case 3 concentration gradient (fr = 1.1 to fl = 0.9). (Online version
in colour.)
(a)
(b)
(c)
Figure 15. Numerical smoke foils produced by detonations propagating through concentration
gradients of (a) case 1 (fr = 1.3 and fl = 0.7), (b) case 2 (fr = 1.2 and fl = 0.8) and (c) case 3
(fr = 1.1 and fl = 0.9).
For each of the three cases, we also recorded numerical smoke foils in the
same manner as we did for the detonations in §3. A portion of the smoke foil for
each channel is shown in figure 15. The smoke foil generated by the detonation
propagating through the case 1 mixture before it quenches is shown in figure 15a.
The detonation is initially overdriven, and the very small detonation cells are
not visible in the figure. As the detonation slows, cells generated by the motion
of triple points along the detonation front begin to grow. At one point in the
process, near the halfway mark in the portion of the channel shown in figure 15a,
a distinction among the sizes of the cells across the width of the channel can be
made. Cells near the bottom (where the mixture is lean) are larger than those
near the top (where the mixture is rich). This is physically consistent with the
behaviour shown in figure 12b, noted in §3 and measured experimentally by Ishii
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593
& Kojima [23]. Cells continue to grow and the number of triple points in the
channel decreases. Eventually, triple point collisions with the wall are no longer
able to sustain the detonation, and it fails.
The smoke foil for case 2 (fl = 0.8, fr = 1.2) shows a system in which
the survival of the detonation is tenuous and is, to some extent, controlled
by triple point reflections from the channel walls (figure 15b). For this case,
strong triple points that are reflected from the top wall propagate into the less
reactive mixture near the bottom of the channel. These reflections also ignite
new overdriven detonations. As they propagate, new triple points form along the
unstable detonation front and form smaller cellular patterns. This behaviour is
similar to that observed for a marginal detonation in a uniform mixture [43].
Reflection of the triple points at the bottom wall only sometimes ignites a
new detonation, however, since the mixture is less reactive there. When the
collision is inert, the triple point appears to ‘disappear’ into the wall in the figure.
This irregular propagation behaviour causes drastic distortions of the detonation
surface as large regions of reaction zone decouple from the leading shock at any
particular instant only to be consumed later by a newly ignited detonation.
The detonation propagating through the case 3 mixture (fl = 0.9, fr = 1.1)
is more robust than cases 1 and 2. On the basis of behaviour observed for
uniform mixtures (figure 13a), the mixture throughout the channel is everywhere
sufficiently reactive for detonation propagation to occur. The cellular patterns
shown in figure 15c are similar to those obtained for the case 2 mixture
(cf. figure 15b). The strongest triple points again propagate from top to bottom
in the channel. These are formed both by collisions with the top wall and by
collisions with other triple points. For this case, these collisions take place more
frequently than with the case 2 detonation. This is due in part to stronger
triple point reflections from the bottom wall, where f = 0.9 is an appreciably
reactive mixture. The collision of the strong reflected waves with triple points
travelling from the top to the bottom of the channel helps to sustain the primary
detonation, which produces a hierarchy of cellular structures, the largest of
which is incomplete. This behaviour is similar to that observed for a uniform
stoichiometric mixture [42].
5. Conclusions and future directions
Fuel–air mixtures that accumulate in large spaces, such as those that may
be encountered in industrial settings, are likely to be highly non-uniform.
A detonation that propagates in such a complex environment will encounter a
wide variety of conditions, some favourable to its survival and ability to propagate
and some adverse to it. The predominant question is how does the combined
effect of these non-uniformities affect the global behaviour of a detonation?
We have considered this question for a limited set of mixtures in which the
gradient in mixture composition is normal to the direction of propagation.
In §3, we investigated the effects of system size on detonations propagating
through a low-activation-energy mixture, and in §4, we considered how the
magnitude of concentration gradients in channels of fixed height affects the
propagation and quenching behaviour of detonations in high-activation-energy
mixtures reminiscent of methane–air.
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D. A. Kessler et al.
The computations show that the structure of the propagating detonations
depends on the width of the stratified reactive layer relative to the size of a single
detonation cell. For all of the test cases computed in §4 and those computed
in smaller channels in §3, this ratio was small since the width of the channel
was only slightly larger than the nominal cell size for a stoichiometric mixture.
Detonations propagating through non-uniform mixtures in these small channels
were similar to detonations in uniform mixtures. Fluctuations in the position of
the leading edge of the detonation obscured the mean curvature of the front.
For smaller concentration gradients (larger channels in §3), the structure of the
computed detonation was similar to the stable TFDW structure depicted in
figure 3, and its velocity approached DCJ for the stoichiometric mixture in the
centre of the channel. The TFDW of Calhoon & Sinha [25] is a limiting case
for the computations presented in §3 of infinite channel size, where there are no
triple point reflections from the bounding walls and the detonation is completely
self-sustaining.
For the systems of finite size discussed in §§3 and 4, triple point reflections
and collisions played a crucial role in determining not only the shape of the
detonation but also the propagation speed and, for the high-activation-energy
mixture presented in §4, the quenching behaviour. In the low-activation-energy
mixture described in §3, detonations were able to survive in all channels
considered, even when the channel was small enough so that only one triple
point could exist. In contrast, complete quenching was observed for the case
in §4 with the largest composition gradient (fl = 0.7, fr = 1.3). Fluctuations
in the local temperature caused by the motion of the transverse instabilities
led to severe drops in the local reaction rate and decoupling of the shock and
reaction zone.
For the detonations computed in §4 that did not quench (cases 2 and 3),
the global propagation velocities varied with the magnitude of the concentration
gradient, but were roughly 5–10% less than the stoichiometric DCJ . These relative
differences in velocity fall in line with measurements taken by Ishii & Kojima in
non-uniform hydrogen–oxygen mixtures. Also, the propagation speeds computed
by Calhoon & Sinha [25] dropped to a minimum of about 94 per cent of DCJ
before quenching was observed. The detonation velocities computed for the model
mixture in §3 reached even smaller values, approaching 84 per cent of DCJ in the
smallest channels. In more realistic mixtures, for which the effective activation
energies are larger, it seems likely that it will not be possible to sustain such
low-speed detonations.
There are still a number of open questions regarding detonation propagation in
mixtures with concentration gradients that need to be addressed. For example,
we need to better quantify how the shape and size of the detonation gradient
affects the behaviour of a detonation. In particular, the question of whether the
presence of mixture composition gradients always proves to be an impediment to
detonation propagation should be carefully considered. The results presented in
§4 show that a detonation can survive and propagate through a mixture of which
a significant portion is not sufficiently reactive to support detonation on its own
in this particular configuration. Another question is how the detonation would
propagate in an unconfined mixture with a concentration gradient. There is also
a related issue of DDT in a non-uniform explosive mixture, which has important
safety implications.
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