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c Cambridge University Press 2017
J. Fluid Mech. (2017), vol. 824, pp. 5–41. doi:10.1017/jfm.2017.282
5
Two-stage autoignition and edge flames in a high
pressure turbulent jet
Alex Krisman1,2, †, Evatt R. Hawkes1,3 and Jacqueline H. Chen2
1 School of Mechanical and Manufacturing Engineering, University of New South Wales, Kensington,
NSW 2052, Australia
2 Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94550, USA
3 School of Photovoltaics and Renewable Energy Engineering, University of New South Wales,
Kensington, NSW 2052, Australia
(Received 17 August 2016; revised 21 February 2017; accepted 25 April 2017)
A three-dimensional direct numerical simulation is conducted for a temporally
evolving planar jet of n-heptane at a pressure of 40 atmospheres and in a coflow of
air at 1100 K. At these conditions, n-heptane exhibits a two-stage ignition due to lowand high-temperature chemistry, which is reproduced by the global chemical model
used in this study. The results show that ignition occurs in several overlapping stages
and multiple modes of combustion are present. Low-temperature chemistry precedes
the formation of multiple spatially localised high-temperature chemistry autoignition
events, referred to as ‘kernels’. These kernels form within the shear layer and core of
the jet at compositions with short homogeneous ignition delay times and in locations
experiencing low scalar dissipation rates. An analysis of the kernel histories shows
that the ignition delay time is correlated with the mixing rate history and that the
ignition kernels tend to form in vortically dominated regions of the domain, as
corroborated by an analysis of the topology of the velocity gradient tensor. Once
ignited, the kernels grow rapidly and establish edge flames where they envelop the
stoichiometric isosurface. A combination of kernel formation (autoignition) and the
growth of existing burning surface (via edge-flame propagation) contributes to the
overall ignition process. An analysis of propagation speeds evaluated on the burning
surface suggests that although the edge-flame speed is promoted by the autoignitive
conditions due to an increase in the local laminar flame speed, edge-flame propagation
of existing burning surfaces (triggered initially by isolated autoignition kernels) is the
dominant ignition mode in the present configuration.
Key words: combustion, flames, turbulent reacting flows
1. Introduction
The autoignition of turbulent fuel jets at elevated pressures and temperatures occurs
in compression ignition engines such as diesel engines. Ignition is an important
process in diesel combustion that influences flame stabilisation, which in turn affects
pollutant formation and fuel conversion efficiency. For conventional diesel engine
† Email address for correspondence: [email protected]
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6
A. Krisman, E. R. Hawkes and J. H. Chen
conditions, ignition occurs following the liquid injection of a high-velocity fuel jet
into a high pressure and high-temperature environment. A two-staged autoignition
then occurs, involving low-temperature chemistry (LTC) after the first-stage ignition
delay time, τ1 , and high-temperature chemistry (HTC) after the second-stage ignition
delay time, τ (Pickett, Siebers & Idicheria 2005; Idicheria & Pickett 2006; Pickett,
Kook & Williams 2009). Following autoignition, a pseudo-stable, lifted flame is
established (Dec 1997). The distance between the injector and the flame base of the
lifted flame is defined to be the lift off length (LOL). The location of autoignition
and flame propagation in mixture-fraction (ξ ) space is fuel and case dependent.
Measurements of optically accessible diesel engines and chambers have been
performed which have informed the development of conceptual models for diesel
combustion (Dec 1997; Idicheria & Pickett 2006; Musculus, Miles & Pickett 2013;
Maes et al. 2016). At conventional diesel engine conditions, pre-ignition reactions
due to LTC are observed, followed by the main ignition due to HTC. The HTC
ignition appears to occur as a distributed event, originating from an ensemble of
ignition locations (kernels) within the fuel rich and high velocity region of the jet
(Dec 1997). Ignition then proceeds towards leaner mixtures, and a non-premixed
(diffusion) flame is established as the ignition front crosses the stoichiometric mixture
fraction isosurface. The flame tends towards a pseudo-steady state such that the
LOL is statistically steady. Recent large-eddy simulations (LES) of diesel spray
combustion have reproduced these qualitative features and identified the effect of the
LTC causing HTC ignition to occur in a rich mixture within the jet core (Gong, Jangi
& Bai 2014). However, the stabilisation mechanism for diesel flames has not been
conclusively demonstrated.
Siebers & Higgins (2001) and Siebers, Higgins & Pickett (2002) observed that the
variation in LOL could be explained by power laws derived from flame propagation
scaling arguments for lifted flames at atmospheric conditions proposed by Peters
(2001). This implied that the diesel flame was stabilised essentially by a propagation
mechanism (either a premixed flame (Vanquickenborne & van Tiggelen 1966) or a
partially premixed edge flame (Müller, Breitbach & Peters 1994; Buckmaster 2002)).
However, a subsequent study by Pickett et al. (2005) using Arrhenius law expressions
for τ , successfully explained the same trends as an autoignition controlled process. It
is also possible that both stabilisation mechanisms contribute simultaneously, and/or
the mechanism is dependent upon the operating conditions. The actual behaviour of
the ignition and stabilisation processes is important to understand, as it would inform
the appropriate selection of a suitable modelling framework when investigating diesel
combustion.
For atmospheric conditions, high-resolution experimental and numerical observations
exist for non-premixed single-stage autoignition, e.g. see the review article by
Mastorakos (2009). Mastorakos, Baritaud & Poinsot (1997) identified important
properties of autoignition for laminar and turbulent non-premixed and partially
premixed conditions. For laminar conditions, there exists a most reactive mixturefraction value (ξMR ) (Mastorakos et al. 1997) which has the shortest τ and is therefore
the preferred location of autoignition in composition space. The autoignition occurs
first at ξMR and then propagates into richer and leaner mixtures, including the ξST
value. At turbulent conditions, where fluctuations in the scalar dissipation rates (χ)
conditioned upon ξ exist, ignition occurs first at locations near the ξMR value where
χ values are near the conditional minimum (Mastorakos et al. 1997; Im, Chen &
Law 1998; Cao & Echekki 2007). Direct numerical simulation (DNS) studies have
demonstrated that this phenomenon is related to the turbulent flow field such that
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Autoignition and edge flames in a high pressure turbulent jet
7
ignition kernels tend to form in low dissipation regions such as the interior of vortices
(Sreedhara & Lakshmisha 2000, 2002; Viggiano 2004, 2010). Echekki & Chen (2002)
also showed the establishment of lean and rich expanding premixed flames following
autoignition which led to edge flames when the stoichiometric mixture fraction was
crossed.
A series of experiments at atmospheric conditions were conducted by Markides
& Mastorakos (2005), Markides, De Paola & Mastorakos (2007), Markides &
Mastorakos (2011) for turbulent non-premixed jets featuring autoignition using
hydrogen, n-heptane and acetylene, respectively. A statistically stationary state was
achieved in each study, such that a rapid succession of autoignition kernels was
observed that did not lead to a continuously burning flame. This was termed the
‘random spots’ regime (Markides & Mastorakos 2005). Tracking of individual kernels
was conducted. Rapid kernel growth was observed leading to individual flamelets
that later merged or were extinguished. The flamelets resembled tribrachial (triple)
flames, which have a main crescent-shaped front composed of premixed lean and rich
branches and a trailing non-premixed (diffusion) flame branch. The results for the
hydrogen case were later reproduced in a DNS study by Kerkemeier et al. (2013).
The DNS results showed the following: that the ignition kernels formed near the ξMR
value, with low χ; that increasing levels of turbulence suppressed the formation of
the ignition kernels; and that autoignition and flame propagation modes could coexist
in a statistically stationary ignition case.
Numerical (Lyra et al. 2015) and experimental (Micka & Driscoll 2012; Fleck et al.
2013a,b; Sullivan et al. 2014) investigations have been performed for autoignition in
shear flows for the jet in cross-flow (JICF) configuration. Several of these studies
(Micka & Driscoll 2012; Sullivan et al. 2014; Lyra et al. 2015) considered the
stabilisation mechanism and time-averaged structure of autoignitive JICF flames,
which is not the focus of the present temporally evolving study. High-speed imaging
performed by Fleck et al. (2013a,b) at elevated pressure with hydrogen/nitrogen jets
observed a transient ignition processes with many independent ignition kernels. Some
kernels were advected out of the combustor (blow off), while others successfully
produced a stabilised flame. An overlap in the spatial distribution of successful and
unsuccessful ignition kernels was observed. This was attributed to fluctuations in:
mean turbulent structure, local mixing rates and local thermochemical fluctuations,
however it was not possible to directly measure these effects with respect to each
ignition kernel as resolved measurements of the ξ and χ were not available. Fleck
et al. (2013a) further distinguished between primary and secondary ignition kernels.
Primary ignition kernels were those that formed in the absence of other ignition
sources in the domain while secondary kernels occurred in the presence of combustion
elsewhere in the domain. For most cases it was observed that primary kernels did not
in themselves establish a stabilised flame but that they influenced the fluid mechanical
and mixing fields in a manner that promoted the formation and success of secondary
ignition kernels.
Resolved measurements of temperature and mixture fraction for impulsively forced
methane jets in a hot coflow burner have been performed by Papageorge et al. (2014)
and Arndt et al. (2016). The location of the formation of ignition kernels in ξ and χ
space was measured for an ensemble of injection events. It was observed that most of
the ignition kernels formed in the periphery of the jet, in mixtures with ξ < ξST and
experiencing low χ , which was in good agreement with prior DNS of autoignition
with single-stage ignition chemistry (Mastorakos 2009).
It is difficult to directly compare ignition studies conducted at low to moderate
pressures and with single-stage ignition chemistry to diesel-engine-relevant conditions
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8
A. Krisman, E. R. Hawkes and J. H. Chen
due to the weakness of two-stage ignition and negative temperature coefficient (NTC).
(Two-stage ignition can be observed at atmospheric and moderately elevated pressures,
however, over the temperature range it occurs the corresponding ignition delays are
very long, more than 10 ms for large n-alkanes.) Recently, several two-dimensional
(2-D) DNS studies were conducted at diesel-relevant thermochemical conditions using
the oxygenated fuel dimethyl ether (Deng et al. 2015a,b; Krisman et al. 2015, 2016,
2017), for a lifted laminar flame (Deng et al. 2015a,b; Krisman et al. 2015) and
pseudo-turbulent mixing layer (Krisman et al. 2016, 2017) configuration. Krisman
et al. (2015) identified polybrachial edge flames that exhibit characteristics of both
edge-flame propagation (a main tribrachial flame) and autoignition (observed in
additional upstream branches due to LTC and/or HTC autoignition). The branch
structure of the polybrachial flames is related to the homogeneous ignition delay
times of both the first stage (τ1 ) and the second stage (τ ) of autoignition, and their
respective most reactive mixture-fraction values (ξMR,1 and ξMR ). A subsequent study
by Deng et al. (2015a) used a chemical explosive mode analysis (CEMA) (Lu et al.
2010) to provide additional support for the hybrid premixed/autoignitive character of
the polybrachial flames.
The lifted laminar 2-D DNS studies did not consider turbulence effects, which may
influence the ignition and stabilisation process. An important effect of turbulence for
non-premixed conditions is that it produces conditional fluctuations of χ in ξ -space.
In order to study this effect, a simulation was performed for a 2-D dimethyl ether-air
mixing layer in isotropic pseudo-turbulence (Krisman et al. 2017) for thermochemical
conditions identical to the TOX = 900 K case from Krisman et al. (2015). In that
study, a mixing field with a Damköhler number (Da) of 0.4 was imposed over
an initially laminar mixing layer profile. The Da value was selected in order to
approximate the value calculated from simulations of igniting n-dodecane sprays
near the flame stabilisation location (Pei et al. 2016). An investigation of the LTC
behaviour identified that the first stage of autoignition transitions to a diffusively
supported cool flame that moves up the local mixture-fraction gradient towards
richer mixtures much faster than expected from spatial gradients in the first-stage
ignition delay time. The cool flame advances the timing of the LTC heat release,
shortening the main ignition delay time, τ , which is in good agreement with prior
LES observations (Gong et al. 2014), inferences from transported probability density
function models (Pei et al. 2016) and an independent study by Dahms et al. (2017)
in the context of turbulent n-dodecane ignition modelled with a Lagrangian flamelet
method. A detailed study of the overall ignition dynamics for the same dataset in
Krisman et al. (2017) was also performed (Krisman et al. 2016). A very complex
ignition was observed such that multiple stages of autoignition and multiple modes
of combustion were identified. The results suggested that both autoignition and
edge-flame propagation can both be prominent at diesel-relevant conditions, which
has implications for practical modelling of diesel engines.
Several other high-resolution studies have been performed for diesel-relevant
ignition (Sreedhara & Lakshmisha 2002; Viggiano 2004, 2010; Mukhopadhyay
& Abraham 2012b; Borghesi, Mastorakos & Cant 2013; Minamoto & Chen
2016). Sreedhara & Lakshmisha (2002) conducted a 3-D DNS of autoignition
at engine-relevant thermochemical conditions for a domain of decaying isotropic
turbulence. The simulations used a global chemical mechanism for n-heptane that
models both the LTC and HTC reactions (Müller & Peters 1992). The results
identified that ignition occurred in regions near ξMR where χ is low, which
corresponded to flow topologies that were judged to be vortically dominated, similar
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Autoignition and edge flames in a high pressure turbulent jet
9
to previous observations for the autoignition of non-premixed vortex-mixing layer
interactions at atmospheric pressure by Thévenin & Candel (1995). However, in
the study by Sreedhara & Lakshmisha (2002), individual ignition kernels were
not identified; rather, ignition was identified by ensemble-averaged statistics, which
showed a broad region of ignition in ξ space. Furthermore, the measurement of
ignition with respect to flow topology was conducted for stoichiometric mixtures,
not for the most reactive mixtures where autoignition was known to occur. Borghesi
et al. (2013) performed a DNS of n-heptane droplet autoignition at a pressure
of 24 atmospheres using a chemical mechanism with 18 global steps. A spotty
ignition pattern was observed and doubly conditioned statistics demonstrated that
kernel formation favoured ξMR locations where χ values are low. The importance
of macro-mixing was also emphasised, since the formation of regions with ξ ≈ ξMR
assist early ignition, which may be affected by many factors including mixing
time, evaporation rates and turbulence intensity. Recently Minamoto & Chen (2016)
conducted a 3-D DNS study of a turbulent lifted flame at NTC conditions. In order
to reduce the computational expense of the simulation, a partially reacted mixture
was imposed at the inlet in order to represent the products of the LTC reactions.
This approach reduced the residence time (and hence domain size) requirements,
which made the use of DNS tractable. The turbulent flow disrupted the laminar flame
structures observed by Krisman et al. (2015). However, appropriate conditioning of
the results revealed the same polybrachial flame structure observed in the laminar case.
Measurements of displacement speeds were also performed that indicated that the
presence of LTC reactions substantially enhances the displacement speeds. The main
limitation of this study was the use of the partially reacted inflow which precluded
the study of the two-stage ignition process.
To the best of the authors’ knowledge, no fully resolved studies exist that included
both diesel-engine-relevant thermochemical conditions (leading to two-stage ignition)
and realistic, 3-D turbulence. On the experimental side, this is due to the extreme
challenge of obtaining well resolved measurements at diesel engine conditions. For
experimental studies, the high pressure and temperature environment combined with
the extremely small spatial scales and fast time scales (in addition to other challenges
such as multiphase flow effects), inhibit the collection of fully resolved measurements.
For numerical studies, DNS studies are limited by computational cost, which scales
with the third power of the range of simulated scales and linearly with the range
of temporal scales. Consequently, any 3-D DNS study is computationally expensive
and existing databases are limited to statistically small computational domains with
physically unrepresentative isotropic, decaying turbulence (Sreedhara & Lakshmisha
2002; Borghesi et al. 2013) or do not include the LTC dynamics (Minamoto & Chen
2016).
The approach in the present study is to make appropriate simplifying assumptions
while focusing on interactions between the two-stage ignition, edge-flame propagation
and the flow topology. The major simplifying assumption is the use of a global
n-heptane chemical mechanism in place of a detailed mechanism with elementary
reactions, while three-dimensionality and realistic turbulence are retained. The
mechanism used here, by Müller & Peters (1992), has been previously used in
DNS of diesel-engine-relevant conditions in order to reduce the computational cost
(Sreedhara & Lakshmisha 2002; Viggiano 2004). Without this simplifying assumption
the simulation would not have been feasible, even on leading high performance
computing facilities.
In this study, a temporally evolving turbulent slot jet of n-heptane fuel is surrounded
by initially stagnant oxidiser. The initial conditions produce a two-stage autoignition
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10
A. Krisman, E. R. Hawkes and J. H. Chen
in the presence of shear-driven turbulence. The turbulence produces intense mixing
between the fuel and oxidiser which leads to large conditional fluctuations of χ
in ξ -space. The main aim of the simulation is to study the ignition dynamics in
the presence of both LTC and HTC, subject to realistic, shear-driven turbulence. In
particular, the connections between combustion modes, mixing and the flow topology
are explored. The use of Lagrangian tracer particles allows for an examination of the
ignition kernels time histories prior to autoignition. This is used here to investigate
the effects of mixing and flow topology on the autoignition event. The present study
details both the development of individual combustion features and the evolution of
the statistically 1-D flame.
2. Methodology
2.1. Numerical method
The DNS was conducted with the code S3D (Chen et al. 2009). S3D solves the
conservation equations for the compressible Navier–Stokes, continuity, total energy
and species mass fraction equations with a high-order accurate, non-dissipative finite
difference scheme. Spatial derivatives are approximated with an eighth-order central
finite difference scheme and temporal integration is performed with a fourth-order,
six-stage explicit Runge–Kutta method. Spurious high-wavenumber oscillations were
removed with a tenth-order explicit filter (Kennedy & Carpenter 1994) that is applied
once every 10 time steps. S3D has been used in many DNS studies of turbulent
combustion (Im et al. 1998; Echekki & Chen 2002; Sankaran et al. 2007, 2015;
Wang & Rutland 2007; Chen et al. 2009; Yoo et al. 2011; Chatakonda et al. 2013;
Karami et al. 2015, 2016), including studies of temporally evolving non-premixed
slot-jet flames (Hawkes et al. 2007; Lignell, Chen & Smith 2008), which is the
configuration used in the present simulation. The mixture specific heat and viscosity
transport properties were calculated based on the local temperature and composition,
assuming unity Lewis numbers for all species.
2.2. Configuration
A diagram of the domain is presented in figure 1. The domain consists of a slot
jet of fuel between stationary layers of oxidiser at a pressure of 40 atmospheres.
The fuel is pure n-heptane at 400 K and the oxidiser composition is 79 % N2
and 21 % O2 by volume at 1100 K. The profile of the jet is defined in terms of
the mixture fraction, ξ : ξ (y) = 0.5(tanh((y + HJET /2)/σ ) − tanh((y − HJET /2)/σ )),
where the profile thickness is σ = HJET /8, and where HJET is the jet width. HJET
and the jet velocity, UJET , are constrained by the target jet Reynolds number,
ReJET = (HJET UJET )/νJET , and jet Damköhler number, DaJET = (HJET /UJET )/τMR ,
where νJET is the kinematic viscosity of pure fuel, equal to 1.94 × 10−7 m2 s−1 ,
and τMR is the homogeneous ignition delay time of the most reactive mixture fraction,
equal to 0.28 ms. ReJET is constrained by computational cost and is set to 9000,
which approaches the ReJET of previous, moderately turbulent combustion DNS cases
(Hawkes et al. 2007; Yoo, Sankaran & Chen 2009; Yoo et al. 2011). DaJET is
set to 0.11, which was selected to approximately match the Da in the vicinity of
autoignition in diesel conditions as calculated from Reynolds-averaged Navier–Stokes
(RANS) simulations of n-dodecane flames (Pei et al. 2016), and to appropriately time
the autoignition with respect to the turbulent jet development in the present simulation.
The timing of autoignition was very sensitive to DaJET in preliminary, under-resolved
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Autoignition and edge flames in a high pressure turbulent jet
11
Periodic in
z-direction
OXID
FUEL
OXID
F IGURE 1. (Colour online) Diagram of the domain configuration, including specification
of the boundary and initial conditions. Blue circles represent the region initially seeded
with passive fluid tracer particles.
simulations of the present configuration. DaJET was selected to time the ignition after
the development of turbulence due to shear, but well before the spreading of the jet
reached the outflow boundaries. This allowed the jet to reach a burning state prior
to the statistical saturation of the domain. These constraints result in an initial jet
width of HJET = 0.233 mm and an initial centreline velocity of UJET = 7.49 m s−1 ,
which define the jet time of tJET = 31.1 µs. Non-dimensional parameters are defined
for time, t∗ = t/tJET , and spatial directions, x∗ = x/HJET , y∗ = y/HJET , and z∗ = z/HJET .
The extent of the domain in the x, y and z directions is chosen to be Lx = 12HJET ,
Ly = 18HJET , Lz = 8HJET , respectively. The domain size is adequate for the jet to
develop and to obtain a sufficient statistical sample in the periodic directions, while
minimising the computational expense. Superimposed on the initial condition is a
spectrum of low-amplitude isotropic turbulence in order to excite the unstable jet.
The isotropic turbulence has a velocity fluctuation scale of u0 /UJET = 0.05 and an
integral length scale of Lt /HJET = 0.33. The fluctuations satisfied a Passot–Pouquet
energy spectrum (Hinze 1975).
The boundary conditions are periodic in the streamwise (x) and spanwise (z)
directions and non-reflecting outflows in the cross-stream (y) direction, evaluated
using the Navier–Stokes characteristic boundary condition method (NSCBC) (Poinsot
1992). The grid count in each direction is nx = 1440, ny = 1472 and nz = 960, which is
selected to properly resolve the smallest chemical and turbulent length scales. There
are 81 points across the HJET , which is consistent with values selected in previous
DNS of slot jets at similar ReJET values (Hawkes et al. 2007; Yoo et al. 2011).
There are 0.72 grid points across the smallest Kolmogorov length scale, ηk . This is
sufficient as it exceeds suggested guidelines for DNS of turbulent flows (Pope 2000),
and the value reported from a previous, well-resolved slot-jet DNS case (Hawkes
et al. 2007). In order to ensure that the chemical structure of the flame was resolved,
a grid convergence test was conducted for a one-dimensional premixed flame, see
figure 2. The results showed that the selected grid was sufficient to correctly resolve
the premixed flame; in three dimensions, depending on the flame orientation, the best
and worst case spacing is between 2 and 4 microns, respectively.
Fluid tracer particles were also embedded in the flow at the start of the simulation
to aggregate Lagrangian statistics. Approximately 53.3 million particles were randomly
placed at a uniform density within the central half of the cross-stream direction.
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12
A. Krisman, E. R. Hawkes and J. H. Chen
10 6
Inset: HTC reaction zone
10 5
10 4
10 3
0
0.1
0.2
0.3
0.4
0.5
0.6
Distance (mm)
F IGURE 2. (Colour online) Grid convergence test result for lifted 1-D premixed flame
at the most reactive mixture fraction for the conditions in this study. The lower left
peak corresponds to the LTC ignition and the higher right peak corresponds to the
high-temperature reaction zone. The inset shows the high-temperature reaction zone in
detail. The grid resolutions noted in the legend have units of µm.
2.3. Chemistry model
A four-step, six-species global chemical mechanism for n-heptane was used, based on
that first proposed by Müller & Peters (1992). The choice of global chemical reaction
scheme was motivated by the need for a computationally inexpensive chemical
mechanism. Since diesel engine conditions involve very high pressures, the reaction
zone thicknesses and hence resolution requirements are onerous (e.g. requiring
approximately 1 micron resolution or less for chemical mechanisms that include
radical species and short lived intermediate species). For the target configuration,
it was not feasible to use even the smallest available detailed chemical mechanism
which reproduced the competing high- and low-temperature chemical processes of
diesel fuel ignition. The four-step n-heptane mechanism was selected as a minimal
set of global reactions and species which reproduces the two-stage ignition process
and competing chemical pathways.
The mechanism does not reproduce the NTC region for a fixed composition with
varying mixture temperature, however it does produce a flat region on the τ curve
that approximates the experimental data points (see figure 3). Despite the lack of
NTC in this mechanism, it does capture the two-stage ignition process for the fixed
thermochemical conditions considered in this study.
The global mechanism is described by the following reactions:
F −→ X,
X + 11O2 −→ P,
F + 2O2 ←→ I, and
I + 9O2 −→ P,
(R1)
(R2)
(R3)
(R4)
where F represents the fuel species (n-heptane), P is the lumped product species
(7CO2 + 8H2 O), X is a lumped high-temperature intermediate species (e.g. 3C2 H4 +
CH3 + H) and I is a lumped low-temperature intermediate species (e.g. HO2 C7 H13 O +
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Autoignition and edge flames in a high pressure turbulent jet
13
(b) 1.0
(a)
0.8
100
0.6
0.4
10–1
0.8
Model
Fieweger et al. (1997)
0.9
1.0
1.1
1.2
1.3
0.2
1.4
0.05
0.10
0.15
0.20
0.25
0.30
0.35
F IGURE 3. (Colour online) (a) Model validation (dashed line) with respect to shock-tube
data (Fieweger et al. 1997) (symbols) for ξST at a pressure of 40 atmospheres. (b) τ with
respect to ξ at the conditions used in this DNS. Key values marked for the stoichiometric
mixture (blue) and the most reactive mixture (red).
H2 O). Species X represents the incomplete combustion products of high-temperature
chemistry. Species I represents the intermediate species formed due to LTC.
Reactions R1 and R2 represent the HTC pathway and reactions R3 and R4
represent the LTC pathway. These two pathways compete for the consumption of the
fuel and are temperature sensitive. Importantly, reaction R3 is reversible and the rate
constants for the forwards and reverse directions are selected to produce the correct
temperature dependence with respect to the transition from low- to high-temperature
chemical pathways.
The chemical rate constants as presented by Müller & Peters (1992) produce a stiff
set of equations that are incompatible with the explicit Runge–Kutta time integration
used in S3D. In order to reduce the stiffness, the rate constants of the fastest reactions
(R3f and R3b) were capped at the rates corresponding to a temperature of 1100 K; i.e.
for higher temperatures the rate constants k3f and k3b were held fixed. This choice
was justified by the large separation in time scales between the rate constants for R3
and those for the other reactions, such that R3f and R3b remained very fast with
respect to R1, R2, and R4 above 1100 K, and hence are not rate limiting.
The final Arrhenius rates are presented in table 1, which were adjusted compared to
those presented in Müller & Peters (1992). The adjustments were performed due to the
stiffness reduction and to obtain an improved agreement with respect to experimental
ignition delay time data (Fieweger, Blumenthal & Adomeit 1997), see figure 3, which
were generated using the adjusted rates. Figure 3 also shows the τ values for a range
of ξ at the conditions simulated in the present DNS, with the values of ξST , ξMR , τST ,
τMR marked.
Apart from ignition delay times, it would be desirable for the mechanism to
reproduce other quantities such as the laminar burning velocity sL and the extinction
dissipation rate χC . Unfortunately, for the conditions in this study, there are no
experimentally available data for these quantities. Furthermore, detailed chemical
mechanisms are not suitable for validating this four-step scheme since they themselves
are not validated for flame propagation or strain response behaviour in the considered
thermochemical conditions. Therefore, the four-step mechanism has the potential to
introduce errors that may not be directly quantified. In particular, this is of concern
for the τ response to χ , since the four-step mechanism does not feature chemical
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14
A. Krisman, E. R. Hawkes and J. H. Chen
Reaction
R1
R2
R3f
R3b
R4
A (mol cm s K)
10
1.2 × 10
2.0 × 1012
3.0 × 1018
4.0 × 1022
3.8 × 1010
E (K)
19 710
7 317
21 650
34 500
6 100
TABLE 1. Arrhenius rates for four-step n-heptane mechanism.
chain branching due to the presence of radical species; ignition is purely thermally
driven. With these limitations in mind, reference values for sL and χC are calculated
and used to non-dimensionalise all results for ignition and flame propagation. The
results presented in this study should therefore be interpreted as qualitative trends,
relative to the defined reference values, rather than quantitatively accurate data, for
these nominal conditions.
To numerically calculate sL , the usual approach is to solve a 1-D, adiabatic, freely
propagating flame using software such as Chemkin that may efficiently determine the
steady state solution, for which sL (the inflow velocity) is an eigenvalue. However,
due to the cold boundary problem, this method degenerates as the inlet temperature
increases and the autoignition time scale approaches the residence time from the inlet
to the flame. For the present thermochemical conditions, a unique sL is not expected
for a freely propagating flame since the upstream reactant mixture state is not fixed
but will autoignite.
In order to determine a unique value for sL at the present conditions, an alternative
approach is taken using a 1-D lifted flame configuration with S3D. In this method,
premixed reactants are introduced at the inlet with a fixed velocity and composition.
For sufficiently large inlet velocities the flame will blow off (leave the domain). As
the inlet velocity is reduced the flame will reside in the domain at a location where
the flame residence time as calculated by integrating the velocity (u) profile, τflame =
R LOL
dx/u, is approximately equal to τ for the inlet condition. As the inlet velocity is
0
further reduced the flame will move upstream until such a point where the upstream
conduction of heat overcomes downstream convection; where this condition is satisfied
the steady state solution is for the flame to become attached to the inlet. Figure 4(a)
shows the flame residence time versus inlet velocity resulting from this test for a
stoichiometric mixture of fuel and air at the conditions considered in 3-D DNS. A
sharp transition is observed between the low inlet velocity regime where the flame is
attached, to the high inlet velocity regime where the flame location is determined by
τ . The transition curve resembles a sigmoid function and has a unique inflection point
which is used here to define a reference laminar flame speed, sL (ξST ) = 1.18 (m s−1 ).
A calculation for χC is performed by simulating a steady state non-premixed laminar
counterflow using S3D. The non-reacting flow field is first solved for a range of global
strain rate values that correspond to peak scalar dissipation rate values, χP . The results
are then used to initialise the reacting simulations, where the solution along the axial
centreline measures the ignition processes. The results of this test are presented in
figure 4(b), for the τ response to increasing χP . The value of χC is then approximated
to be the midpoint between the last point of ignition and the first point of non-ignition,
with increasing χP . This gives a value of χC = 785 ± 50 (s−1 ).
Table 2 contains the relevant physical parameters for the four-step chemical
mechanism at the conditions considered in this DNS.
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Autoignition and edge flames in a high pressure turbulent jet
(a)
(b) 8
1.2
7
1.0
0.8
15
Attached
flame
6
Autoignition
front
0.6
5
0.4
4
0.2
3
0
0.5
1.0
1.5
2.0
2.5
3.0
2
200
300
400
500
600
700
800
F IGURE 4. (a) Normalised flame residence time versus inlet velocity; sL marked at the
inflection point in the curve. (b) Normalised ignition delay time for a non-premixed
laminar counterflow reactor versus the peak value of the steady χ profile; χC marked in
the vertical dashed line.
ξST
ξMR
τST
τMR
sL (ξ = ξST )
χC
0.062
0.19
3.8 × 10−4
2.8 × 10−4
1.18
785
—
—
s
s
m s−1
s−1
TABLE 2. Physical properties of four-step n-heptane mechanism.
3. Results
The results are organised into five parts. In § 3.1 the main qualitative features of
ignition are identified and illustrated. Section 3.2 presents conditionally averaged
statistics in ξ -space and uses the results to identify and track distinct combustion
modes. Ignition kernels are defined and investigated in § 3.3 in terms of the mixing
history and in § 3.4 in terms of the flow topology in conjunction with conditionally
averaged statistics. Lastly, in § 3.5, the transition of the ξST surface from an unburnt
to a burnt state is analysed with a view to distinguish the contributions of autoignition
and edge-flame propagation.
3.1. Qualitative description
Figure 5 shows 2-D slices in the z∗ = 0 plane at t∗ = 18 and 27 (2 and 3 times
τMR ) for YI , YP and T. By t∗ = 18, the initially laminar jet profile has developed
sheared turbulence due to the initial perturbation. The hot oxidiser mixes with the
fuel jet and by t∗ = 18 broad regions with high values of YI and moderate values
of YP are observed at rich mixtures, indicating the presence of LTC. By t∗ = 27,
regions of high T and YP are observed, centred on the ξST surface, indicating the
HTC ignition has taken place. Figure 6 shows a blow up of an example region
of the domain at t∗ = 27. Multiple edge flames can be observed, centred on the
ξST surface. The edge flames have strong rich premixed and trailing diffusion flame
branches, and a much weaker lean premixed branch which is folded into the diffusion
flame. Ahead of the edge flames, regions of lower intensity heat release rate (HRR)
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A. Krisman, E. R. Hawkes and J. H. Chen
16
(a)
(c)
(b)
2000
T (K)
1500
1000
500
(d )
5
0.25
4
0.20
3
0.15
2
0.10
1
0.05
(e)
T (K)
0
0
(f)
2000
1500
1000
500
5
0.25
4
0.20
3
0.15
2
0.10
1
0.05
0
0
F IGURE 5. (Colour online) Instantaneous images of T (a,d), YI (b,e) and YP (c,f ) at
t∗ = 18 (a–c) before HTC ignition and at t∗ = 26 (e–f ) after HTC ignition. Dashed white
contour marks τST .
Colliding edge-flames
LTC
branch
LTC
Edge-flame branch
Edge-flame
F IGURE 6. (Colour online) Instantaneous image of HRR at t∗ = 27, evaluated on the z = 0
plane over the region 0 < x < 1, 0.2 < y < 0.7. The black dashed line shows the ξST surface.
are observed which correspond to the LTC reactions. The edge flames in this case
resemble the tetrabrachial laminar flames observed with the detailed dimethyl ether
(DME) mechanism (Krisman et al. 2015), albeit with a much weaker lean premixed
branch.
Overall, figure 5 shows two snapshots in time during ignition, the first during
the LTC reactions and the second after HTC ignition has occurred and multiple
combustion modes are simultaneously present. The LTC is established early in the
simulation and moves into increasingly rich mixtures within the jet core. This region
coincides with the location of high local χ and turbulence, resulting in a highly
contorted field. Later in the simulation, regions of high T and YP are observed,
centred on the ξST surface. The ξST surface resides at the jet periphery, and is not
strongly contorted by turbulence, since the region of high-velocity fluctuations occurs
at richer mixtures.
High-temperature autoignition develops from t∗ ≈ 19.8 as a spatially distributed and
temporally staged event. The HTC emerges in multiple locations that are detected
as localised maxima in temperature, HRR and YP . These maxima are referred to as
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Autoignition and edge flames in a high pressure turbulent jet
17
F IGURE 7. (Colour online) The evolution of the ξST surface, viewed from the y+ domain
boundaries, coloured by threshold of YP = 0.19. The threshold is selected to delineate
regions of burning (red) and non-burning (white) ξST .
kernels that are defined here to exist when the local temperature exceeds a temperature
threshold of THTC = 1400 K. The threshold was selected to rule out regions of LTC and
the detection of kernels was insensitive to the threshold temperature. Ignition kernels
form due to autoignition and are spatially distinct from pre-existing regions of the
domain that exceed THTC . Forty ignition kernels were identified via visual inspection
of the evolution of the THTC surface.
The ignition kernels originate in rich mixtures and rapidly expand into less rich
mixtures, engulfing the stoichiometric mixture-fraction isosurface. Where the ignition
kernels cross the ξST surface they establish edge flames (e.g. see figure 6) which
propagate along the ξST surface, similar to the behaviour observed by Domingo &
Vervisch (1996) with simple chemistry and Echekki & Chen (2002) with detailed H2
chemistry. Behind the edge flames a non-premixed flame is established, centred upon
the ξST surface, which demarcates burning and non-burning regions of the ξST surface.
Figure 7 shows the progress of combustion in the domain towards a fully burning
state, visualised by the portions of the ξST surface that exceeds a scalar threshold.
The ξST surface is coloured by YP , such that the white regions are not burning and
the red regions are burning (defined here as YP > 0.19). This threshold corresponds
to the YP value on the ξST surface where the HRR is maximum. As can be seen, the
burning regions originate as isolated pockets which spread along the ξST surface. These
isolated pockets originate from kernels that form in rich mixtures at earlier times
(not shown in figure 7). At t∗ = 21, the first burning region is observed due to the
first kernel. The burning region established by the first kernel expands rapidly, while
additional burning regions form (which are established by the out-of-plane expansion
of additional ignition kernels).
Visualisation of the formation and expansion of the first kernel is presented in
figure 8. The kernel forms in a region bounded by the ξST surface to the leaner side,
and the region of LTC to the richer side, which is illustrated here by a surface of
YI = 1 × 10−4 , a value that is approximately 1 % of the maximum of YI observed
and demarcates regions of the domain experiencing LTC. This shows that the kernel
forms in a rich mixture that has already undergone the first stage of ignition. The
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A. Krisman, E. R. Hawkes and J. H. Chen
18
(a)
–0.40
–0.45 y
–0.50
(b)
–0.55
0.40 0.34
z
(c)
0.28
0.78
0.83 x
0.88
(d)
F IGURE 8. (Colour online) Evolution of the first ignition kernel. The kernel is defined
as the surface of T = 1400 K, coloured in red. The kernel forms between the ξST surface,
coloured dark blue, and the region of LTC and richer ξ , coloured light blue (only shown in
(a)). The kernel rapidly expands, engulfing the ξST surface and spreading along ξST . Each
panel shows the same volume of space, viewed from the same perspective. The bounding
box (a) shows the grid values in mm.
kernel rapidly grows and crosses the ξST surface, expanding along the surface in all
directions. This example is typical of many kernels, however, some kernels merge
with pre-existing burning regions before reaching the ξST surface (not shown here).
It is the combined effect of multiple ignition kernels, which establish multiple edge
flames, that leads to the overall ignition of the jet and the progress towards a fully
burning jet by the end of the simulation.
3.2. Conditionally averaged statistics
The conditionally averaged statistics for the turbulent jet are presented in conjunction
with a series of laminar non-premixed counterflow ignition simulations performed with
steady scalar dissipation rate profiles. The laminar results were generated using 2-D
DNS in the configuration described in § 2.3.
Figure 9 compares the steady laminar χ profiles to the decaying mean and root
mean square (r.m.s.) values from the turbulent simulation. Four laminar cases are
considered, where the most strained case has χP /χC = 1.1, indicating that the HTC
ignition will never occur. The peak turbulent mean and r.m.s. profiles are initially far
higher that χC , and then decay during the simulation. By t∗ = 18, the mean χ profile
is below χC at all mixture fractions. The value of τMR is marked on each plot in the
vertical dashed line and this shows the ξ value where the HTC ignition is expected to
occur from homogeneous calculations. It is noted that the mean and r.m.s. χ profiles
are increasing as ξ exceeds ξMR , meaning that dissipation is preferentially higher at
richer mixtures.
Figure 10 presents temporally evolving statistics for YI , YP and T in increments of
t∗ = 9 (or τMR ) from the turbulent DNS, alongside laminar solutions corresponding to
the steady χ rates plotted in figure 9.
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Autoignition and edge flames in a high pressure turbulent jet
(a)
2.5
Lam. counterflow (steady)
(b)
2.5
Turb. mean (decaying)
(c)
2.5
2.0
2.0
2.0
1.5
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0
0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0
19
Turb. r.m.s. (decaying)
0.2
0.4
0.6
0.8
F IGURE 9. (Colour online) Conditional χ profiles. (a) Steady state profile for
non-premixed counterflow with variations in the bulk strain rate. (b) The decaying mean
profile from the 3-D DNS simulation. (c) The decaying r.m.s. profile from the DNS. The
solid vertical line in each panel marks ξST and the dashed vertical line marks ξMR .
At t∗ = 9, the LTC is located in very rich mixtures for both the laminar and turbulent
results. The scatter plot shows large conditional fluctuations in YI , possibly due to
the very large r.m.s. values of χ as shown in figure 9. The turbulent profile is more
broad in ξ space, which is to be expected from the turbulent field. It is also noted that
the laminar profiles are not very sensitive to values of χP /χC , even for values greater
than unity that prevent the HTC ignition. This result is consistent with prior studies of
two-stage ignition fuels that have observed the LTC ignition to be much more resilient
to strain than the HTC ignition (Liu et al. 2004). The YP and T profiles at t∗ = 9 show
all laminar solutions have progressed further towards ignition than the turbulent mean
profile.
By t∗ = 18, the YI profiles have shifted further towards pure fuel. This is more clear
for the laminar solutions, for which YI values are lower compared to the turbulent case
for ξ < 0.4, where the reduction in YI values is more pronounced for lower values of
χP /χC . At the same time, sensitivity to χP /χC in the laminar cases is observed for
YP and T for ξ < 0.4, and so it is seen that ignition is proceeding more rapidly for
lower values of χP /χC . The increase in T and YP is most prominent at rich mixtures
between ξ = 0.1 and 0.3. At this time the turbulent mean YP and T profile remains
below all of the laminar profiles. The scatter samples also show that HTC ignition
has not occurred within the domain by this point.
The onset of HTC in the turbulent case occurs at t∗ = 19.8 and by t∗ = 27 several
new features in the conditional statistics are evident. Most prominently, two branches
of T and YP have formed: the lower branch corresponds to the LTC chemistry and the
higher branch corresponds to the HTC chemistry, with few data points falling between
these branches. The laminar profiles are also split into two groups. For χP /χC =
0.4 and 0.6, HTC ignition has occurred, while the solutions at χP /χC = 0.9 and 1.1
sit astride the LTC branch. The mean turbulent profiles of YP and T are weighted
towards the lower branches, indicating that at this time most of the domain has not
undergone HTC. The YI profiles show a large decrease for the ignited laminar cases
and a reduction in the mean turbulent profile for ξ < 0.4.
At t∗ = 36, the mean turbulent profiles show a nearly complete HTC ignition for
ξ < 0.4, and a region of LTC persisting mostly in very rich mixtures. All cases with
χP /χC less than unity have ignited, while the χP /χC = 1.1 case has reached a steady
state solution on the LTC branch.
Overall, the conditional statistics support the notion of a two-stage autoignition
process involving an initial LTC (first stage) autoignition that moves into increasingly
(b)
0.015
(c)
0.3
0.010
0.2
0.005
0.1
Turb. samples
Turb. mean
Lam.
Lam.
Lam.
Lam.
2500
T (K)
(a)
2000
1500
1000
500
0
0
0.2 0.4 0.6 0.8
(e)
0.015
0
0.2 0.4 0.6 0.8
0
0.2 0.4 0.6 0.8
0
0.2 0.4 0.6 0.8
0
0.2 0.4 0.6 0.8
(f)
0.3
2500
0.010
0.2
2000
0.005
0.1
T (K)
(d )
0.2 0.4 0.6 0.8
1500
1000
500
(g)
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
(h)
0.015
(i)
0.3
0.010
0.2
0.005
0.1
2500
T (K)
0
2000
1500
1000
500
0
( j)
0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
(k)
0.015
(l)
0.3
2500
0.010
0.2
2000
0.005
0.1
T (K)
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A. Krisman, E. R. Hawkes and J. H. Chen
20
1500
1000
500
0
0.2 0.4 0.6 0.8
0
0.2 0.4 0.6 0.8
F IGURE 10. (Colour online) Conditional statistics for YI (a,d,g,j), YP (b,e,h,k) and T
(c,f,i,l). Each row represents an instant in time from (a–c) at t∗ = 9 to ( j–l) at t∗ = 36. The
grey dots are scatter samples from the domain (randomly selected 0.01 % of locations), the
solid black line is the conditional mean and the thin blue lines correspond to the laminar
profiles for the χP /χC values shown in figure 9.
rich mixtures, followed by a high-temperature ignition that establishes the HTC
mode of combustion, and finally moving back from rich mixtures to be centred on
ξST . The results also demonstrate the importance of turbulence, which is responsible
for conditional fluctuations of mixing rates and chemical reaction that produce
multi-modal distributions in ξ -space. These results are broadly in agreement with
prior LES (Gong et al. 2014), 2-D DNS with reduced dimethyl ether chemistry
(Krisman et al. 2017), transported probability density function modelling (Pei et al.
2016), flamelet modelling with detailed n-dodecane chemistry (Dahms et al. 2017)
and conceptual models derived from optical measurements of diesel spray flames
(Musculus et al. 2013).
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Autoignition and edge flames in a high pressure turbulent jet
(a)
21
(b)
1.0
1.0
7
6
0.8
0.20
0.8
0.15
5
0.6
4
0.6
0.4
3
0.4
0.10
2
0.2
0
10
20
30
0
(c)
0.05
0.2
1
0
10
20
30
0
(d)
1.0
2.0
0.8
1.5
0.6
1.0
1.5
0.8
1.0
0.6
1.0
0.4
0.4
0.5
0.2
0
10
20
30
0
0.5
0.2
0
10
20
30
0
F IGURE 11. (Colour online) Maps of conditional means for YI (a), YP (b), ω̇P (c) and
χ (d). Each panel maps the conditional mean value in time and ξ space. The thick solid
horizontal line marks ξST and the dashed line marks ξMR . The thin solid isocontour bounds
the region of χ/χC > 1 mixing rates are highest.
The temporal evolution of conditional mean quantities is mapped in figure 11 for
YI , YP , ω̇P and χ. The YI result shows that the LTC proceeds with almost no delay (a
limitation of the global chemical mechanism at these conditions) from rich mixtures.
The peak in YI closely corresponds to the ω̇P distribution from t∗ = 0 to about t∗ = 20.
The LTC peak rapidly moves into richer mixtures until about t∗ = 7, when the YI
and ω̇P profiles remain nearly stationary in ξ space and moderate in intensity. The
attenuation of LTC corresponds with the development of the χ profile, which peaks
over similar temporal and ξ values. This peak in χ, due to the development of the
shear turbulence, produces strong mixing which exceeds the χC value (delineated
by the while contour line) and slows the progress of the LTC into richer mixture
fractions.
In a prior 2-D DNS ignition study (Krisman et al. 2017), it was observed that the
movement of the LTC into richer mixtures is resilient to (or even, in some cases,
promoted by) intermediate to high levels of χ, although it is inhibited by very high
levels of χ. In comparison with the previous result, the LTC here appears to be
more strongly inhibited by χ. This result is not necessarily contradictory with the
previous DNS, due to several differences between the two simulations, including:
higher peak levels of χ in the present result; different chemical mechanism; and
two-dimensional versus three-dimensional turbulence. Further investigation of these
differences is required in future work.
It is also noted, and has been previously discussed by Borghesi et al. (2013) in
the context of droplet autoignition, that the initial specification of peak ξ equal
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A. Krisman, E. R. Hawkes and J. H. Chen
22
(a)
(e)
(b)
50
1.0
25
0.5
0
0
10 20 30 40
(c) 2.6
(d)
0.6
10
20
30
40
1.0
0.4
T
2
(K)
1.8
Location in -space of peak values
0.8
0.2
1
0
0
10 20 30 40
0
10
20
30
40
0
10
20
30
40
F IGURE 12. (Colour online) (a–d) Shows four panels of the evolution of the maximum
conditionally averaged values. (a) HRR, (b) YI , YX and YP , (c) temperature and (d) ξ .
(e) Shows the corresponding location in ξ space of the maximum values throughout the
simulation.
to unity within the jet may have implications for comparing the present results to
diesel-engine-relevant conditions. At diesel-engine-relevant conditions, ignition occurs
far downstream of the injector at which point entrainment has reduced the centreline
mixture-fraction value well below unity (Musculus et al. 2013). Therefore, rich
mixtures may disappear as a result of mixing before they ignite. This may be an
important effect, since the rich side boundary condition experienced by the flame is
not cold fuel but a reacting mixture having undergone LTC. However, experimental
planar laser-induced luminescence (PLIF) images of CH2O (an LTC marker) and OH
(a HTC marker), e.g. see Maes et al. (2016), show that near the flame base, HTC and
LTC still certainly overlap in typical heavy duty diesel engine operating conditions.
The attenuation of LTC in the present results is transitory. As mixing rates relax,
the YI and ω̇P profiles recover and continue to move into richer ξ . By approximately
t∗ = 22, an increased ω̇P is observed between ξST and the location of peak YI . The
HRR rapidly moves towards the stoichiometric location and increases dramatically in
magnitude. The timing and location of this HRR feature corresponds to the observed
formation of the ignition kernels and edge flames. Until the end of the simulation,
strong ω̇P remains centred on the ξST isoline, as the secondary low-temperature ω̇P
peak weakens and tracks into richer ξ values with the YI profile. Large amounts of
YP form over a wide range of ξ values while the mixing rates continue to relax.
Figure 12 presents the maximum values of the conditional mean of ω̇P , YI , T and χ
over the duration of the simulation, alongside the location of these maximum values
in mixture-fraction space over time. For the first stage of ignition, the results show
that the peak in ω̇P closely follows the LTC marker, YI . The peak in the mixing rates
corresponds with an approximate 15 % reduction in peak YI values, which recovers as
the mixing rates decline. By approximately t∗ = 22, a rapid increase in the temperature,
YX , YP and ω̇P profiles occur simultaneously, showing the timing of the second stage
of ignition. The location of the maximum temperature, ω̇P and YP profiles converge
to the ξST value.
3.3. Ignition kernel formation
An inspection of the temperature field identified forty distinct ignition kernels, defined
as isolated local maxima of temperature exceeding the threshold of THTC = 1400 K.
23
(b) 0.3
(a) 0.30
0.25
0.2
0.20
0.15
0.1
0.10
0.05
0
10
20
30
0
15
40
20
25
30
35
F IGURE 13. (Colour online) The ξ value (a) and χ value (b) for each ignition kernel at
the time of formation.
1.5
1.0
t (s)
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Autoignition and edge flames in a high pressure turbulent jet
0.5
0
0.1
0.2
0.3
0.4
F IGURE 14. (Colour online) Ignition delay times for each kernel, τk , denoted by the blue
circles, superimposed on levels of τ in black lines. All ignition kernels form in a region
of low τ gradient.
This threshold was selected in order to distinguish regions of rapidly increasing
HTC leading to thermal runaway and ignition. The time at which this threshold
is exceeded is defined to be the kernel ignition delay time, τk . All of the kernels
identified proceeded to fully ignited states, i.e. the ignition progress never failed once
it began.
The location of the kernels in ξ and χ space is presented in figure 13. All kernels
form in rich mixtures between ξ = 0.1 and 0.3 and at low χ values compared to the
conditional mean (see figure 10). Ignition delay time results from a homogeneous
reactor, see figure 14, show that the τ profile has a ‘U-shape’ with a broad
region of short τ values in ξ -space corresponding to the location of the ignition
kernel formation. The shallow gradient of τ may explain the wide distribution of
ignition kernels. The large number of ignition kernels and their broad distribution
in mixture-fraction space is consistent with previous 2-D DNS results with detailed
dimethyl ether chemistry (Krisman et al. 2017). However, in the present DNS the
ignition kernels also form in mixtures leaner than ξMR , which may be due to relatively
shorter τ values for ξ < ξMR for the present n-heptane global chemical mechanism
compared with the detailed DME chemical mechanism used in Krisman et al. (2017).
This difference may also be due to the χ profile in ξ space, which is much higher
in mixtures richer than ξMR compared to mixtures less rich than ξMR , see figure 9.
Another possible explanation is the lack of radical species in the four-step chemical
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24
A. Krisman, E. R. Hawkes and J. H. Chen
mechanism, which alters the ignition response of a mixture to strain rate. However, the
present result is broadly consistent with RANS-based simulations of a diesel engine
with detailed n-heptane chemistry (Fu & Aggarwal 2015; Pei et al. 2016), and DNS
studies by Sreedhara & Lakshmisha (2000, 2002), which identified autoignition to
emerge over a wide range of rich ξ values, although individual autoignition kernels
were not identified in these studies.
Experimental studies of autoignition at atmospheric conditions have observed
similar behaviour (Markides & Mastorakos 2005; Markides et al. 2007; Markides
& Mastorakos 2011). A quasi-steady, spatially evolving autoignition was observed
for hydrogen (Markides & Mastorakos 2005), n-heptane (Markides et al. 2007) and
acetylene (Markides & Mastorakos 2011) fuel jets in co-axially flowing, heated air.
In those experiments, the autoignition was sustained by a rapid series of autoignition
events. In the present results, and as was also observed for the 2-D results with
detailed DME chemistry of Krisman et al. (2017), the ignition occurs temporally
within a periodic box which therefore allows for the transition from isolated ignition
spots to a fully burning flame.
The ignition delay times of the kernels are plotted against multiples of the
homogeneous ignition delay times in figure 14. It is observed that τk /τ is between
1.8 and 3.2, which is consistent with the range of values reported in a previous
DNS study with global n-heptane chemistry (Sreedhara & Lakshmisha 2002) and
also with a recent experiment of an autoigniting turbulent jet in highly heated coflow
at atmospheric conditions, as determined by simultaneous measurements of ξ and
temperature (Papageorge et al. 2014).
In order to understand the difference between the τ and τk values, the ignition
kernel histories are extracted from the passive Lagrangian tracer particles embedded
within the simulation.
The kernels develop from point locations as local maxima in temperature and HRR
and the nearest tracer particle at the time of kernel formation is selected to represent
the kernel history. For each kernel, the most representative particle is selected as the
nearest particle to the local maxima of temperature (as judged from the Eulerian field).
From the number density of tracer particles in the simulation, the nearest particle
will (on average) be located within two grid points from any arbitrary location. By
extracting the thermochemical and mixing histories for the selected tracer particles,
comparisons can be made between the ensemble of ignition kernels.
Figure 15 shows the ensemble of selected tracer trajectories from the initial
condition up until the point of ignition, mapped to T-ξ and YP -ξ space. Given the
initially bimodal distribution of ξ (due to the initially thin mixing layer), most tracers
originate from near either ξ = 1 (pure fuel) or ξ = 0 (pure air). The tracers initially
move along the mixing line (in an adiabatic mixing process), followed by a gradual
increase in temperature and YP due the first stage of autoignition. The first stage of
autoignition only proceeds after the tracers reach an intermediate ξ value. After a
delay, the gradual buildup of YP and increase in temperature leads to thermal runaway
and the formation of the autoignition kernels. The tracer trajectories tend to converge
towards common regions in these phase spaces just prior to ignition, irrespective of
their initial state.
The temporal evolution of four example kernels are presented in figure 16. Kernels
1, 11, 24 and 40, (named in order of their time of formation) are selected as
representative examples of the overall trend. The normalised scalar dissipation rates
for each kernel show that the mixing process is extremely intermittent. Each kernel
experiences initially very low mixing. After some amount of delay, there is a rapid
(a) 1400
(b) 0.10
1200
0.08
1000
0.06
800
0.04
600
0.02
T (K)
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Autoignition and edge flames in a high pressure turbulent jet
400
0
0.5
1.0
0
0.5
25
1.0
F IGURE 15. (Colour online) The ensemble trajectories of ignition kernels in T-ξ (a) and
YP -ξ (b) space. Each series of grey lines with black markers represents an ensemble
member.
(a) 10
(b) 1.0
8
0.8
6
0.6
4
0.4
2
0.2
0
10
20
30
0
(c) 1.0
(d ) 1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
10
20
30
0
10
20
30
10
20
30
F IGURE 16. (Colour online) Kernel histories for χ (a), ξ (b), YI (c) and YP (d). Line
plots show example kernel histories and the circle markers show the values at the point
of ignition.
and intense period of mixing followed by a relaxation of χ to very low levels. The
timing of the intense mixing is positively correlated with τk , such that earlier intense
mixing leads to early kernel formation. The evolution of ξk is also presented and
shows that the period of intense mixing leads to the kernels rapidly converging
towards ξ values at which ignition occurs. The first stage of ignition, as judged by
the normalised YI plots, slightly lags the peaks in the mixing. As the kernels reach
appropriate ξ values, and the mixing rates sufficiently relax, the LTC rapidly proceeds
and leads to the steady buildup of YP .
The results suggest that, in the Lagrangian sense, that the mixing process for
igniting regions is rapid and intense, followed by a period with very low mixing
rates which allows for the buildup of temperature and product species from the first
stage of ignition, leading to thermal runaway. Kernels form in regions which are well
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A. Krisman, E. R. Hawkes and J. H. Chen
26
(a)
(b)
1.0
6
1.0
6
0.8
4
0.6
0.4
2
0.8
4
0.6
0.4
2
0.2
0
10
20
30
0
0.2
0
10
20
30
0
F IGURE 17. (Colour online) Ignition kernels (circle markers) mapped to the spatially
]C (b). The Favre averages exploit the
Favre-averaged jet (filled contours) for e
ξ (a) and χ/χ
symmetry about the jet centreline and the markers are located by the absolute value of
their cross-stream coordinate. For both figures, the solid white contour and the dashed
f
white contour correspond to ξf
ST and ξMR , respectively. The circle markers and filled
contours share a colour scale.
mixed and have low dissipation rates compared to the conditional mean and have
an appropriate mixture fraction. Kernels that experience earlier mixing and an earlier
relaxation of mixing rates, settle down to this condition earlier, and hence lead to an
earlier ignition event.
3.4. Influence of flow topology
In § 3.3 the importance of the mixing field in conditional space on the ignition process
was discussed. In this section, the link between ignition and the flow topology is
explored.
Figure 17 shows the kernels mapped to the temporally evolving and spatially
Favre-averaged jet, where the Favre-averaged quantities are represented with tildes.
Figure 17(a) shows e
ξ for the jet with the kernels superimposed, while figure 17(b)
]C . In both cases, the kernels are coloured
shows the same information in terms of χ/χ
by their local instantaneous values of ξ and χ , with the colour scales the same as
used for the corresponding Favre-averaged variables across the jet. The kernels are
widely dispersed in the y∗ direction and in time. All of the kernels form either in the
]C is near its maximum, or in the interior of the jet where χ]
shear layer where χ/χ
/χC
e
is elevated and ξ is much richer than ξMR . The colouring of the kernels also shows
that the local conditions at ignition are (in general) less rich and less dissipative
than the local spatial Favre average, indicating that there are isolated, protected
regions with favourable conditions for ignition. In other words, the Favre-averaged
fields of composition and mixing are not good predictors of the actual locations of
ignition. This discrepancy can in part be explained by the results § 3.3, where the
kernel histories showed that the ignition kernels dwell in regions with low χ and
appropriate ξ for an extended period prior to ignition. In this section, the ignition
behaviour will also be explained in terms of the topology of the jet.
A turbulent field may be decomposed into topological regions by considering
the velocity gradient tensor, ∇U = A. In the method proposed by Chong, Perry &
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Autoignition and edge flames in a high pressure turbulent jet
27
Cantwell (1990), the characteristic equation for A may be cast in terms of three
invariants named P, Q and R. The details of the PQR analysis may be found
elsewhere (Chong et al. 1990) and so only a brief overview is provided here. The
invariants PQR define a 3-D solution space which is divided between a region of
real solutions and complex solutions by the surface, S, defined by the equation:
27R2 + (4P3 − 18PQ)R + (4Q3 − P2 Q2 ) = 0. It can be shown (Chong et al. 1990)
that the PQR space intersected by S produces 27 possible topological classifications.
By everywhere evaluating the PQR invariants, the domain may be decomposed into
distinct topological regions. For incompressible flows, the dilatation invariant P is
zero and the flow topologies conform to a single QR plane. However, in the current
simulation, dilatation is significant and therefore P 6 = 0 due to: (i) heat release from
combustion and (ii) the steep spatial gradients in density due to the turbulent mixing
of the high density jet and the low density oxidiser (ρJET /ρOX ≈ 10).
Similarly to recent studies by Grout et al. (2011), Cifuentes et al. (2014) for
reacting and compressible turbulent flows, only a subset of the possible topological
classifications were observed in this case. These topologies may be categorised as
strain-dominated nodal regions:
(i)
(ii)
(iii)
(iv)
Classification
Classification
Classification
Classification
1, Node/Node/Node, stable;
2, Node/Node/Node, unstable;
11, Node/Saddle/Saddle, stable;
12, Node/Saddle/Saddle, unstable.
Or as vortically dominated foci regions:
(i)
(ii)
(iii)
(iv)
Classification
Classification
Classification
Classification
18,
19,
20,
21,
Foci/Stretching, stable;
Foci/Stretching, unstable;
Foci/Compressing, stable;
Foci/Compressing, unstable.
Illustrations of these classifications are summarised in figure 18, which has been
adapted from Cifuentes et al. (2014).
Figure 19(a) shows the joint probability density function (PDF) of the domain in
QR space just prior to ignition at t∗ = 18 (note: filled contours plotted with log scale).
The joint PDF has a classic tear drop shape with a very strong peak near the origin.
The temporal evolution of the joint PDF (not shown here) shows that this shape is
preserved throughout the simulation, but decreases in extent in QR space from a
maximum at t∗ = 9 (near the peak turbulent intensity) until the end of the simulation.
The extent of the joint PDF decreases rapidly following the HTC ignition at t∗ = 19.8
due to the increase in viscosity relative to inertial forces. Conditioning the joint PDF
on ξ (also not presented here), shows that the extent of the joint PDF increases with
increasing ξ , because low ξ values reside mostly at the jet periphery while high ξ
values exist within the shear layer and turbulent jet core.
The white dashed isocontour in figure 19(a) bounds the joint PDF conditioned on
0.1 < ξ < 0.3, where all of the kernels form. The doubly conditioned means in QR
space for ω˙P and χ/χC , further conditioned on 0.1 < ξ < 0.3 are also presented in
figure 19 (b and c, respectively). ω˙P increases away from the origin and is generally
higher for Q > 0, but peaks near Q ≈ 0 for positive R. χ increased to the upper left
and lower right of the joint PDF and is lower near the peak in ω˙P , revealing that
the anti-correlation of χ and ω˙P is also apparent in QR space. For samples with
P ≈ 0 (dilation free), the region of peak ω˙P and low χ correspond to classification
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A. Krisman, E. R. Hawkes and J. H. Chen
28
Unstable
Unstable
Unstable
Unstable
Stable
Stable
Stable
Stable
F IGURE 18. Cartoon illustrations of the flow topologies identified in this study.
Reproduced from Cifuentes et al. (2014) (doi:http://dx.doi.org/10.1063/1.4884555), with
the permission of AIP Publishing.
(a)
Joint PDF
0.6
(b)
(c)
0.4
0.4
6
0
3
–0.6
–0.03
0
0
0.03
4
7
0
–0.4
–0.02
5
3
0
0.02
0
–0.4
–0.02
2
0
0
0.02
F IGURE 19. (Colour online) Statistics at t∗ = 18, just before HTC ignition in the QR
plane: (a) joint PDF for ξ > 0.005 (coloured on a log scale) where the white dashed line
bounds the joint PDF conditioned on 0.1 < ξ < 0.3; (b) mean ω˙P ; and (c) mean χ/χC
corresponding to 0.1 < ξ < 0.3 (the region delineated by the dashed line in (a)).
21 (Foci/Compressing, unstable). However, the presence of dilatation means that
topological classifications cannot be generally inferred from the QR plane alone.
Furthermore, since ignition kernels form from small, spatially isolated locations,
it is possible that the conditionally averaged result in figure 19 could mask the
true location of ignition onset in topological space, due to tendency of averages to
suppress outliers. In order to rule out this possibility, figure 20 shows the evolution
of the distribution of classification types, conditioned on the samples with the highest
values of ω˙P . (Here, the top 0.01 % of samples are plotted. Different thresholds
were selected ranging from 1.00 % to 0.0001 % and the results were qualitatively
unaffected.) Figure 20(b) also shows the PDFs for ξ of the most reactive samples at
the times plotted in (a). In (b) the thickness profiles of the black regions at a given
time (x-axis) corresponds to the probability density profile in ξ space (y-axis).
At t∗ = 4.5, the most reactive samples are concentrating in the strain dominated
N/S/S steady and unsteady topologies. By t∗ = 9, when the turbulence becomes
developed, the foci topologies are dominant, particularly F/S stable and F/C unstable.
As the jet proceeds towards ignition, the concentration of pre-ignition chemistry
in the foci topologies increases. The ξ PDFs of the most reactive samples prior
to ignition are located in rich ξ values, where the LTC is concentrated. The grey
shading between t∗ = 19.8 and t∗ = 31.5 in figure 20 shows the duration of kernel
formation. This corresponds to a marked change from foci to nodal topologies, and
the most reactive locations shift to 0.05 < ξ < 0.3, which includes regions of both
edge flames and expanding ignition kernels. The most reactive regions correspond to
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Autoignition and edge flames in a high pressure turbulent jet
(a)
NNN-stable
NNN-unstable
NSS-stable
NSS-unstable
LTC ignition
100
First HTC
kernel
(b)
FS-stable
FS-unstable
FC-stable
FC-unstable
Last HTC
kernel
29
PDF for
0.6
0.4
75
0.2
50
25
0
9
18
27
36
0
9
18
27
36
F IGURE 20. (Colour online) (a) Evolution of the classification of the most reactive
locations (top 0.01 % of ω˙P values) in the domain (in increments of t∗ = 4.5); blue
bars represent the nodal type (strain dominated) modes and red bars represent foci type
(vortically dominated) modes. (b) PDFs for ξ for the locations and times shown in (a).
The profile thickness of the black shaded regions at each time (x-axis) correspond to the
probability density profile in ξ space (y-axis). The dashed line marks ξMR and the solid
line marks ξST . The grey shaded region in each panel marks the duration between the first
and last kernel formation.
nodal topologies for the remainder of the simulation, with the largest contributions
from Node/Saddle/Saddle unstable classification.
The value of ω˙P also depends on ξ , and so figure 21 presents the same analysis
for the following ranges of ξ : 0.0 < ξ < 0.1, which experiences little chemistry until
for formation of edge flames; 0.1 < ξ < 0.3, which experiences LTC before t∗ = 19.8
followed by ignition kernels and then rich premixed branches of edge flames; and
0.3 < ξ < 0.7 where the LTC is most prominent and HTC develops gradually from
t∗ ≈ 27 onwards. Figure 20 suggests that the HTC ignition develops from vortical
regions (as judged by the result just before ignition at t∗ = 18) as has been previously
suggested by Sreedhara & Lakshmisha (2002) and observed for non-premixed, vortexmixing layer autoignition at atmospheric pressure by Thévenin & Candel (1995). It is
also seen that the most reactive regions are strain dominated following ignition (for
ξ < 0.3). This is consistent with prior DNS Grout et al. (2011), Cifuentes et al. (2014)
that also showed flames to exist predominately within Node/Saddle/Saddle unstable
type topologies.
Finally, the topological classifications were also interpolated to the Lagrangian
tracers in order to extract the histories for the ignition kernels. This was performed
to ensure that the statistics from the Eulerian field were not masking the time history
behaviour for the ignition kernels. Figure 22 shows the Lagrangian result for a subset
of the ignition kernels, k, over time. For each kernel, the classification is plotted
between the time when mixing starts (as judged by χ) until the time when HTC
ignition occurs (when TOX exceeds 1400 K). The results confirm that most kernels
spend most of their history from mixing to ignition in vortical (foci) topologies.
3.5. Edge-flame speed and ignition mode analysis
Edge flames are an important feature in non-premixed combustion and are associated
with both ignition and extinction events. Prior DNS studies at non-autoignitive
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A. Krisman, E. R. Hawkes and J. H. Chen
30
(a)
(b)
100
(c)
100
100
75
75
75
50
50
50
25
25
25
0
9
18
27
0
36
9
18
27
0
36
9
18
27
36
F IGURE 21. (Colour online) Evolution of the classification of the most reactive locations
in the domain, conditioned on ranges of ξ : 0.0 < ξ < 0.1 (a), 0.1 < ξ < 0.3 (b), 0.3 < ξ <
0.7 (c). Colour bars and grey shading correspond to those presented in figure 20.
Kernel classification history
30
25
20
15
10
5
0
10
20
30
40
k
F IGURE 22. (Colour online) Topological classification, interpolated to the Lagrangian
tracers corresponding to ignition kernels, k. The kernels are plotted between the time of
the onset of mixing and the time of HTC ignition. The colours correspond to the legend
presented in figure 20.
conditions have identified χ as a key parameter which affects both the formation
of locally extinguished regions and the reignition process. In particular, it has been
observed that edge-flame propagation speeds, Se , are negatively correlated with χ
during extinction processes (Pantano 2004; Hawkes, Sankaran & Chen 2008; Karami
et al. 2016) but may have a non-monotonic correlation during reignition processes
under some conditions (Hawkes et al. 2008). A recent DNS study of edge-flame
statistics at non-autoignitive conditions with simple chemistry (Karami et al. 2017)
studied local extinction and reignition for holes formed near the base of a lifted
turbulent jet. In that study, the scalar dissipation rate and the strain field were found
to play an important role in the formation and growth of the extinction holes, which
later relaxed and allowed for the holes to heal principally by edge propagation.
For autoignitive conditions, few studies have been conducted. For forced ignition,
Chakraborty & Mastorakos (2008) showed that χ influenced the reactivity of the
expanding ignition kernels, which was also dependent on the forcing location in
ξ -space. Krisman et al. (2016) measured the displacement speed of a scalar surface
at the triple point of tetrabrachial edge flames (quadruple flames) in the context of
dimethyl-ether ignition at NTC conditions. High values of χ reduced the displacement
speed while fluctuations in the upstream mixture composition (due to varying upstream
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Autoignition and edge flames in a high pressure turbulent jet
31
Oxidiser
Fuel
F IGURE 23. (Colour online) Cartoon of edge-flame cross-section. The solid black line
represents the YP = 0.19 isosurface and the dashed red line shows the ξST isosurface.
progress in autoignition) were not found to be strongly correlated. Minamoto & Chen
(2016) also calculated the displacement speed for a turbulent lifted dimethyl-ether
flame propagating into a partially reacted mixture at NTC conditions. The partially
ignited reactant stream was found to increase the displacement speed compared to an
unreacted mixture. Experimental studies by Choi & Chung (2013), Al-Noman, Choi
& Chung (2015) for iso-octane and n-heptane lifted flames at autoignitive conditions
measured the propagation speed of edge flames for a range of inlet velocities and
ambient temperatures. Both attached and lifted flames were observed that propagated
via edge flames or autoignition, depending upon the boundary conditions. However,
to the best of the author’s knowledge, resolved turbulent edge-flame statistics during
ignition have not been calculated for autoignitive conditions, and so they are presented
and discussed here. This section focusses on the relationship between χ and Se and
its components, and on the non-dimensional values of se /sL . An analysis of the
alignment between the scalar and mixture-fraction isosurfaces is also performed in
order to infer the dominant ignition modes.
The edge-flame location is defined as the intersection of the ξST surface with a
surface of product mass fraction YP = 0.19, which is the triple point of the edge
flame that corresponds to the location of maximum HRR. Figure 23 shows a cartoon
example cross-section of an edge fame. The intersection of the YP and ξST surfaces
(a singular point in figure 23) defines the location of the edge flame. The edge-flame
speed (Se ) is defined in terms of the displacement speeds of the YP isosurface (SYP )
and the ξST isosurface (Sξ ) and inner product of their respective surface-normal vectors
(k), where k = 1 corresponds to parallel surfaces and k = 0 corresponds
to orthogonal
p
surfaces. The edge-flame speed is given by Se = (SYP − kSξ / 1 − k2 ), which may be
derived using the same methodology as presented by Karami et al. (2015, 2016). In
this paper the edge-flame speed and its components are presented normalised by the
laminar flame speed sL = 1.18 m s−1 calculated in § 2.3. It is important to note that
while the intersection of YP and ξST isosurfaces nominally identifies the edge-flame
location, it does not necessarily imply that the local flame structure corresponds to a
conventional edge flame. The local ignition may be due to other mechanisms such as
autoignition or turbulent flame folding.
A three-dimensional DNS of a non-premixed flame in decaying isotropic turbulence
featuring local extinction and reignition with single-step chemistry by Sripakagorn
et al. (2004) identified three distinct reignition modes. These modes were: edge-flame
propagation, which involves positive Se values moving the edge flame into previously
extinguished regions; flame folding, where turbulence brings together burning and
extinguished regions; and flamelet reignition, where reignition occurs in the absence
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32
A. Krisman, E. R. Hawkes and J. H. Chen
of an external source of heat and is only possible in regions that are only mildly
extinguished. It has been observed that: the edge-flame propagation speed exhibits a
negative correlation between Se and χ (Pantano 2004; Hawkes et al. 2008; Karami
et al. 2016) and occurs at low values of k (Hawkes et al. 2008); flame folding occurs
at relatively higher χ values and high values of k (Hawkes et al. 2008); and that
independent flamelet reignition is a very small contribution to the reignition process
(Sripakagorn et al. 2004) and is generally neglected.
A temporally evolving DNS of a syn-gas slot jet featuring extinction and reignition
was the first study to present time-evolving statistics for Se with respect to χ and
k (Hawkes et al. 2008). A key finding was that during the reignition phase a nonmonotonic relationship between Se and χ exists, such that peak Se values occurred at
relatively high χ values. The joint PDF of Se and χ, combined with k values doubly
conditioned upon Se and χ, identified flame folding as the dominant reignition mode
and that edge-flame propagation was of secondary importance.
A major difference in the present DNS is the absence of local extinction and the
discussion will therefore refer to ignition modes rather than reignition modes hereafter.
The autoignitive conditions produce the scenario of expanding autoignition kernels,
which begin at rich mixtures and expand to leaner mixtures, eventually igniting
the ξST surface, as illustrated in figure 8, which represents an additional mode of
ignition that does not exist in the non-autoignitive conditions previously reported.
The autoignition mode is expected to be associated with a high k value, due the
nearly parallel alignment of surfaces as the kernel crosses the ξST surface, and with
high values of χ, due to the peak in χ observed at the leading edge of an expanding
ignition kernel (as also observed by Mukhopadhyay & Abraham (2012a) and Krisman
et al. (2016)).
Statistics gathered on the edge flames, over the entire simulation are presented in
order to first consider the effects of χ on Se and its components. Here, all χ values
are normalised by the critical scalar dissipation rate, χC = 785 s−1 , as calculated in
§ 2.3.
Figure 24 presents the joint PDFs of the edge-flame speed, its components and k,
with the logarithm of the normalised scalar dissipation rate, ln(χ /χC ). The edge-flame
speed distribution is monomodal, centred at se /sL ≈ 2.4 (2.8 m s−1 ), with most values
lying between 1.8 and 3.5. In terms of χ , almost all samples are below χC and the
joint PDF mode is located at ln(χ/χC ) ≈ −1.5. The low values of χ on the ξST surface
are due to the small value of ξST and the initial centreline ξ value of unity, causing
the conditional ξ profile to peak in very rich mixtures and become small near ξST (see
figure 9). It is possible, that for the same nominal turbulent and chemical conditions
but with a conditional χ profile that peaks near ξST , that the results presented here
would differ greatly in terms of featuring: failed ignition kernels, local extinction of
burning surface, reduced edge-flame speeds and enhanced flame folding.
Regarding the correlations, for ln(χ/χC ) between −5 and −3.5, there is a positive
correlation with Se , but for ln(χ/χC ) between −3.5 and 0.5 (the most probable range
of values), Se is negatively correlated. At very high χ values a slight increase in Se
is observed. The conditional mean of Se peaks at approximately ln(χ /χC ) = −3.5
and the marginal PDF in Se space at this point is broad and skewed towards
high
p
values of Se . The first component of the edge-flame speed, (SYP /sL )/ 1 − k2 , is
positive and higher in magnitude than Se . At high levels of ln(χ /χC ), there is a
strong, positive
p correlation with increasing ln(χ /χC ). The second component of Se ,
(−kSξ /sL )/ 1 − k2 , has a more narrow joint distribution. At low levels of dissipation,
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Autoignition and edge flames in a high pressure turbulent jet
6
6
0.2
0.1
4
4
2
2
0
–5
–3
–1
1
0
–5
2
1.0
0
0.7
–2
–4
–5
0.6
0.4
0.2
0.2
0.1
–3
–1
–1
1
0
–5
1
0.9
0.6
0.3
0.3
–3
33
–3
–1
1
F IGURE 24. (Colour online)
of dissipation rates with Se /sL ,
p Joint PDF of the logarithm
p
2
2
its components (SYP /sL )/ 1 − k and (−kSξ /sL )/ 1 − k and k. The colour map shows
the joint PDF information and the grey dashed line indicates the mean ordinate value,
conditioned upon the abscissa value, ln(χ).
p
the magnitude of (−kSξ /sL )/ 1 − k2 is small and the most likely values are negative.
At high levels ofpdissipation the speed decreases rapidly, balancing the rapid increase
in the (SYP /sL )/ 1 − k2 component. The results for k reveal a positive correlation
between χ and k. The correlation is most pronounced for high values of χ where the
joint PDF becomes narrow and rapidly approaches k = 1. The joint PDF for k explains
the sharp ‘up tick’ and ‘down tick’ at high χ values for the components of
pSe . At
lower χ values the Se /sL statistics closely resemble those of the (SYP /sL )/p1 − k2
component, with a near-uniform reduction due to the negative (−kSξ /sL )/ 1 − k2
term.
The results indicate that the scalar dissipation rate has a non-monotonic impact on
the edge-flame speed. For low values of χ, the edge-flame speed is slightly promoted
p
by increasing χ. This can be attributed to the increase in the (SYP /sL )/ 1 − k2
component. At intermediate values of χ, both components of Se are reduced with
increasing χ, causing a modest attenuation in Se . At very high χ rates, the YP and
ξST surfaces become aligned, causing the components of Se to increase in magnitude
rapidly and with opposite sign. Overall, the effect on Se is a slight increase at very
high χ values, with the caveat that the sample size at very high dissipate rates is
small.
Two observations are made with regard to the magnitude of se /sL . Firstly, the flow
velocities near the stabilisation location in diesel flames are ≈10 m s−1 (Pei et al.
2016). This is only 2–3 times the observed (dimensional) local se in this DNS, and the
turbulence experienced at diesel engine conditions could potentially increase the net
overall propagation speeds over and above the local propagation speed by wrinkling of
24
3.5
Non-dimensional
speed
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A. Krisman, E. R. Hawkes and J. H. Chen
34
PDF
3.0
20
2.5
16
2.0
12 PDF
1.5
8
1.0
4
0.5
0
0.05
0.10
0.15
0
0.20
Ignition progress
F IGURE 25. (Colour online) Left y-axis: non-dimensional speeds for sL /sL,0 (circles),
√
ρu /ρb sL /sL,0 (pluses) and ρu /ρf sL /sL,0 (triangles) with respect to the ignition progress of
the reactant mixture. Right y-axis: PDF for the ignition progress measured 2 × δL ahead
of the edge flame on the ξST isosurface.
the edge-flame structure, similar to how in premixed turbulent flames the net burning
velocity is much larger than sL .
The second observation is that the non-dimensionalised speed se /sL exceeds unity.
This is expected for both hydrodynamic reasons due to flow expansion (Ruetsch,
Vervisch & Liñán 1995; Im & Chen 1999) and thermochemical reasons due to
enhanced flame propagation into a partially reacted mixture ahead of the flames
(Minamoto & Chen 2016). Flow expansion occurs across a premixed flame due to
the temperature increase from combustion. For this reason, displacement speeds are
often weighted by the ratio of the local density at the flame, ρf , to the unburnt
density, ρu (Echekki & Chen 1998; Im & Chen 1999). Here, this equates to the
expression se /sL = ρu /ρf = 1.88. Alternatively, the deviation between sL and se can
be explained by the divergence of streamlines across the edge flame (Ruetsch et al.
1995). The streamline divergence causes flow deceleration just ahead of the triple
point, and so displacement speed measurements made at this location underestimate
the true √
edge-flame speed. Taking this effect into account yields a correction factor of
se /sL = ρu /ρb = 1.47, where ρb is the burnt density. Since the displacement speeds
here are evaluated where the HRR is maximum on the product side, and not on the
reactant side of the triple point where the streamline divergence is greatest, it is argued
that the relevant correction factor here is ρu /ρf = 1.88. This factor is insufficient to
account for the values of se /sL here, which mostly lie between 1.8 and 3.5. Another
correction due to the partially reacted mixture ahead of the flame is required. As
discussed by Minamoto & Chen (2016), the value of sL for autoignitive conditions is
dependent upon the state of the reactant mixture, since pre-ignition reactions due to
LTC may occur ahead of the premixed flame. In order to evaluate this effect, figure 25
presents the response of non-dimensionalised flame speeds to the upstream ignition
progress, measured by YP /YP,b , where YP,b is the burnt value of YP and the laminar
flame speed at zero progress, sL,0 , is used to non-dimensional all values. Also plotted
on figure 25 is the PDF of ignition progress upstream of the edge-flame location. The
PDF is composed of all samples on the unburnt ξST isosurface during ignition that
are 2 × δL (Sensitivity to this distance was assessed for values between 1.5 and 3 × δL
and the result was not strongly affected as the spatial gradient of ignition progress
sufficiently far from the edge flame was small.) ahead of an edge-flame location. As
expected, sL /sL,0 increases with ignition progress but by itself does not account
for the
√
discrepancy with the observed values of se /sL . The correction factor ρu /ρb sL /sL,0
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Autoignition and edge flames in a high pressure turbulent jet
8
35
0.9
0.6
6
k
0.3
4
2
0
–5
–3
–1
1
F IGURE 26. (Colour online) Ensemble mean of alignment factor k, conditioned upon χ
and Se . Superimposed in white is the joint PDF of χ and Se and superimposed in black
is the mean value of Se conditioned on χ, as presented in figure 24.
is also lower than most observed values of se /sL . However, the correction factor
due to dilatation at the flame location, ρu /ρf sL /sL,0 , is close to the range of likely
values of se /sL as shown in figure 24. Even with this correction, there is still a
slight discrepancy between se /sL (higher) and ρu /ρf sL /sL,0 (lower) for the likely
range of ignition progress values upstream of the flame. This may be explained by:
(i) non-edge-flame ignition modes introducing a positive bias to the se /sL statistics,
which do not distinguish between ignition modes; (ii) turbulent enhancement of
se /sL , via wrinkling of the edge-flame front; (iii) the definition of sL used here (see
§ 2.3) may provide an underestimate of the true laminar reference speed for a given
reactant composition. Despite the slight discrepancy, this result provides evidence
that the ignition of the ξST isosurface is primarily explained by autoignition-assisted
edge-flame propagation, as opposed to the expansion of independent autoignition
kernels from rich mixtures engulfing the ξST isosurface, or through turbulent flame
folding, as these ignition modes could in principle take a wider range of values and
are not constrained by the physical arguments considered here. This interpretation is
also consistent with a visual inspection of the ξST isosurface over time. For example,
figure 7 showed the evolution of the state of the ξMR isosurface in terms of burning
and non-burning regions. Those images (and time resolved images not shown here)
support the notion that the independent burning ‘spots’ due to the expanding ignition
kernels are responsible for the initiation of ignition of the ξST isosurface, but that it
is edge-flame propagation that consumes the majority of the ξST isosurface.
In order to provide further evidence for the relative contribution of ignition modes,
the ensemble-averaged alignment factor k, doubly conditioned upon Sξ and χ, is
presented alongside the joint PDF of Sξ and χ in figure 26. For ln(χ /χC ) < −0.5 and
Se /sL < 4.5, the alignment factor takes a low to intermediate value, consistent with
an edge-flame propagation ignition mode (Hawkes et al. 2008). This corresponds
to the most probable region of the joint PDF where the se /sL values correspond
to those predicted by the ρu /ρf sL /sL,0 scaling, which provides further evidence that
autoignition-assisted edge-flame propagation is the dominant mode of ignition in this
case. These edge flames are therefore akin to the hybrid premixed/autoignitive edge
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36
A. Krisman, E. R. Hawkes and J. H. Chen
flames previously observed with detailed dimethyl-ether chemistry in two dimensions
(Krisman et al. 2015) and three dimensions (Minamoto & Chen 2016) DNS.
Two regions of high k value and low probability are also observed. The first region
occurs for ln(χ/χC ) > −0.5. This region may be attributed to either a flame-folding
mode, or an autoignition mode as both are associated with high values of k and
χ. Flame folding requires a strong interaction between the turbulent field and the
stoichiometric surface. However, due to the low value of ξST in the present case, the
ξST isosurface resides at the jet periphery and is only moderately distorted by the
turbulent jet. This suggests that flame folding in unlikely to be a significant ignition
mode, which is supported by visual inspection of the burning surface evolution. For
this reason, autoignition rather than flame folding is proposed to be responsible for
the high χ, high k, but low probability region of the joint PDF.
The second region occurs for Se /sL > 4. This region coincides with an extremely low
probability region of the joint PDF at intermediate χ values and very high edge-flame
speeds. Visual inspection of the domain shows that multiple edge-flame collisions
occurs as a larger proportion of the stoichiometric surface reaches a burning state.
As shown in figure 6, where the edge flames meet, the rich premixed branches can
collide before the triple points on the ξST surface do. This causes the YP surface to
become aligned with the ξST surface and therefore results in a high value of k, but
not necessarily a larger value of χ . Visual inspection also suggests that the edgeflame collisions are associated with an acceleration of the ignition of the ξST surface
due to vanishing scalar gradients during the collision. Based upon this evidence, it is
proposed that edge-flame collisions are responsible for the high Se , high k, but low
probability region of the joint PDF. This ignition mode therefore introduces a positive
bias to the mean values of se /sL reported here and may explain why some samples
have se /sL larger than expected from the ρu /ρf sL /sL,0 scaling argument.
4. Conclusions
A direct numerical simulation was performed of an igniting, three-dimensional,
temporally evolving n-heptane/air slot jet at 40 atm. A global chemical mechanism
was used for computational affordability while retaining the two-stage autoignition
behaviour of diesel fuel. Overall, a two-stage autoignition event was observed that
transitioned to edge-flame propagation.
The main findings of the study are summarised as follows.
(i) The first stage of autoignition developed in rich mixtures and moves up the
mixture-fraction gradient. However, the peak conditional scalar dissipation rates
are sufficient to inhibit the intensity of the conditionally averaged LTC. The
peak conditional scalar dissipation rates occurred due to the development of
shear-driven turbulence during the initial transition of the mean laminar jet
profile to fully developed turbulence. As dissipation rates relaxed, the LTC
recovers and leads to the second stage of autoignition.
(ii) The second stage of autoignition occurs as a distributed (in physical and
composition space), sequential event comprising multiple ignition kernels. The
ignition kernels form over a range of mixture fractions, both lower and higher
than the homogeneous most reactive mixture fraction.
(iii) A topological analysis during ignition revealed that the LTC and onset of HTC
ignition occurs preferentially in vortically dominated regions of the jet. After
the onset of HTC ignition, the strongest burning occurs in strain-dominated
topologies associated with premixed flame fronts.
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Autoignition and edge flames in a high pressure turbulent jet
37
(iv) The formation of ignition kernels in rich mixtures and at low scalar dissipation
rates and within vortical topology types is consistent with a previous autoignition
DNS in isotropic turbulence using a similar chemical mechanism (Sreedhara &
Lakshmisha 2002). In the present results, the exact location of the ignition
kernel formation is available, in addition to the Lagrangian history of each
igniting fluid element. Analysis of the kernel histories reveal that all ignition
kernels pass through a period of intense mixing followed by a rapid decrease
in mixing rates and a convergence to mixture fractions corresponding to short
homogeneous ignition delay times. The timing of kernel mixing is a parameter
that is positively correlated with the formation of the ignition kernel.
(v) An analysis of edge-flame speeds with respect to scalar dissipation rates was
conducted. Overall, the edge-flame speed is negatively correlated with dissipation
rates, which is consistent with studies of edge-flame extinction at atmospheric
conditions.
(vi) The edge-flame analysis was also used to evaluate the relative contributions of
ignition modes on the ξST isosurface. The following observations were made:
(1) The non-dimensionalised edge-flame speeds are in good agreement with
values predicted by the expression ρu /ρf sL /sL,0 , which assumes that the
ignition occurs via edge-flame propagation, and which takes into account
the local dilatation due to the premixed flame (ρu /ρf ) and the laminar flame
speed enhancement due to the autoignitive conditions ahead of the flame
(sL /sL,0 ).
(2) The edge-flame speed, which has components due to both the displacement
speed of a mixture-fraction isosurface and a product mass fraction isosurface,
is observed to be mostly controlled by the movement of the product mass
fraction surface. If autoignition is dominant, greater flame motion in the
mixture-fraction normal direction would be expected at the stoichiometric
location since ignition occurs first in richer mixtures.
(3) The isosurfaces are poorly aligned at most locations, which is expected for
edge-flame propagation and not for autoignition or flame folding.
These observations strongly suggest that the ignition of the ξST isosurface,
although initiated by isolated autoignition kernels, is predominantly due to
edge-flame propagation that is enhanced by the partially reacted mixture ahead
of the flame. This is therefore consistent with the hybrid autoignitive/premixed
polybrachial edge flames that have been observed for laminar (Krisman et al.
2015) and turbulent (Minamoto & Chen 2016) dimethyl-ether lifted flames with
detailed chemistry.
The conclusions of this study may be affected by case-specific choices such
as: (1) the global chemical mechanism, that would differ in particular in terms
of the autoignition response to scalar dissipation rates compared to a detailed
chemical mechanism; (2) the peak mixture fraction of the jet and the stoichiometric
mixture-fraction values, which determines the location of the ignition kernels and
edge flames with respect to the turbulence intensity and scalar dissipation profiles,
and may therefore influence the ignition/extinction behaviour; and (3) the oxidiser
temperature and O2 concentration, which for a given pressure influences the location
in mixture-fraction space and the time scales of the first and second stages of
autoignition and the laminar flame speed.
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38
A. Krisman, E. R. Hawkes and J. H. Chen
Acknowledgements
This work was supported by the Australian Research Council. The work at Sandia
National Laboratories was supported by the US Department of Energy, Office of
Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences.
Sandia National Laboratories is a multimission laboratory managed and operated by
National Technology and Engineering Solutions of Sandia, LLC., a wholly owned
subsidiary of Honeywell International, Inc., for the US Department of Energys
National Nuclear Security Administration under contract DE-NA-0003525. This
research used resources of the National Energy Research Computing Center (NERSC)
which is supported by the Office of Science of the US DOE under contract no.
DE-AC03-76SF00098. The research was also supported by computational resources
on the Australian NCI National Facility through the National Computational Merit
Allocation Scheme and Intersect Australia partner share and by resources at the
Pawsey Supercomputing Centre.
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