The structure and propagation of laminar flames under autoignitive conditions

Combustion and Flame 188 (2018) 399–411

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The structure and propagation of laminar flames under autoignitive conditions Alex Krisman a,∗, Evatt R. Hawkes b,c, Jacqueline H. Chen a a

Combustion Research Facility, Sandia National Laboratories, Livermore, CA 96551-0969, USA School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, NSW 2052, Australia c School of Photovoltaic and Renewable Energy Engineering, The University of New South Wales, Sydney, NSW 2052, Australia b

a r t i c l e

i n f o

Article history: Received 2 May 2017 Revised 12 June 2017 Accepted 10 September 2017

Keywords: Laminar flame speed Internal combustion engines Autoignition Flame propagation Multi-stage ignition

a b s t r a c t The laminar flame speed sl is an important reference quantity for characterising and modelling combustion. Experimental measurements of laminar flame speed require the residence time of the fuel/air mixture (τ f ) to be shorter than the autoignition delay time (τ ). This presents a considerable challenge for conditions where autoignition occurs rapidly, such as in compression ignition engines. As a result, experimental measurements in typical compression ignition engine conditions do not exist. Simulations of freely propagating premixed flames, where the burning velocity is found as an eigenvalue of the solution, are also not well posed in such conditions, since the mixture ahead of the flame can autoignite, leading to the so called “cold boundary problem”. Here, a numerical method for estimating a reference flame speed, sR , is proposed that is valid for laminar flame propagation at autoignitive conditions. Two isomer fuels are considered to test this method: ethanol, which in the considered conditions is a singlestage ignition fuel; and dimethyl ether, which has a temperature-dependent single- or two-stage ignition and a negative temperature coefficient regime for τ . Calculations are performed for the flame position in a one-dimensional computational domain with inflow-outflow boundary conditions, as a function of the inlet velocity UI and for stoichiometric fuel–air premixtures. The response of the flame position, LF , to UI shows distinct stabilisation regimes. For single-stage ignition fuels, at low UI the flame speed exceeds UI and the flame becomes attached to the inlet. Above a critical UI value, the flame detaches from the inlet and Lf becomes extremely sensitive to UI until, for sufficiently high UI , the sensitivity decreases and Lf corresponds to the location expected from a purely autoignition stabilised flame. The transition from the attached to the autoignition regimes has a corresponding peak dLf /dUI value which is proposed to be a unique reference flame speed sR for single-stage ignition fuels. For two-stage ignition fuels, there is an additional stable regime where a high-temperature flame propagates into a pool of combustion intermediates generated by the first stage of autoignition. This results in two peaks in dLf /dUI and therefore two reference flame speed values. The lower value corresponds to the definition of sR for single-stage ignition fuels, while the higher value exists only for two-stage ignition fuels and corresponds to a high  temperature flame propagating into the first stage of autoignition and is denoted sR . A transport budget analysis for low- and high-temperature radical species is also performed, which confirms that the flame  structures at UI = sR and UI = sR do indeed correspond to premixed flames (deflagrations), as opposed to spontaneous ignition fronts which do not have a unique propagation speed. © 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction The laminar flame speed sl is an important quantity in combustion which provides a unique reference value for the rate of unstretched laminar flame propagation at a given thermochemical state. The validation of chemical kinetic models, the analysis of

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (A. Krisman).

scientific measurements, and the engineering design process all benefit from accurate measurements of sl at relevant conditions. Experimental measurements of sl require that the ignition delay time of the fuel/air mixture τ is much longer than the corresponding mixture residence time τ f , which is not satisfied for autoignitive conditions where τ becomes very small. For conditions which have a sufficiently large τ , numerical calculations of sl are usually performed for a freely propagating, adiabatic, un-stretched flame moving into an un-reacted upstream mixture [1]. With this

https://doi.org/10.1016/j.combustflame.2017.09.012 0010-2180/© 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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approach, sl is an eigenvalue solution, corresponding to the inlet velocity. However, at autoignitive conditions, which exist for many combustion devices such as internal combustion (IC) engines, this method does not produce a unique solution. The “cold boundary problem” is a long recognised concept [2] that identifies the contradiction between mathematical models for steady flame propagation and the underlying physical properties of combustable mixtures. Flammable mixtures have a finite ignition delay time, τ , and therefore the reactant mixture is changing ahead of the flame and a steady-state solution does not, in principle, exist [3]. In practice, for a sufficiently cold reactant mixture, τ is essentially infinite compared to the characteristic flame time τ f and a steady-state may be measured from experiment or calculated from numerical simulation. However, this approach fails where the reactant mixture is not sufficiently cold and τ ≈ τ f , leading to domain size (residence time) dependent solutions. Nonetheless, calculations of sL have been performed for steady [4–8] and transient [9–11] premixed flames at autoignitive conditions. The transient calculations approximate sL by initiating a hotspot that transitions to a flame and the speed of the flame front with respect to the unburnt mixture is taken to be sL . While this provides an estimate of sL , this method cannot define a unique value as the result depends upon many factors including the details of the initial hot-spot and the duration of front propagation used to estimate sL . Steady state calculations of sL at autoignitive conditions using solvers such as, e.g. PREMIX [12] or Cantera [13], usually do not address the cold boundary problem, and so the interpretation of those solutions is unclear. However, recently Sankaran performed a study of freely propagating laminar hydrogen flames at autoignitive conditions using Cantera [6] that did address this issue. In that study, the dependence of the solution on the domain size was used to investigate transitions in the flame structure. For sufficiently long domains, the steady-state solution was an autoignition front due to the long residence times. For sufficiently small domains, a premixed flame was observed with little chemical reactions ahead of the front. The results highlighted the effect of domain size on the eigenvalue problem at autoignitive conditions, and explored the possible flame structures that may arise for a single-stage ignition fuel. The distinctions between deflagrative flames (in which conduction and diffusion play a major role in flame propagation) and spontaneous ignition fronts (in which propagation is driven by gradients of ignition delay time) [14] have been numerically investigated for constant volume reactors [4,7–10,15–25]. These studies considered the transient ignition process subject to temperature [4,8,9,15–17,19,21–23,25] and/or velocity [8,16,21,22,25] and/or composition [17,19,21,24] fluctuations in physical space. For instance, temperature fluctuations dictate spatial gradients in τ , which control the speed of spontaneous ignition fronts [9,14,15], such that higher (lower) temperature fluctuations lead to slower (faster) spontaneous ignition front speeds and a lesser (greater) contribution to the overall ignition compared to the contribution from deflagrations. This picture is modified by turbulence, which produces a competition between mixing and chemistry and a regime diagram has been proposed to explain the various combustion modes that may occur [8,22]. Many other insights have been gained from these studies, however, the present study focusses on the behaviour of steady-state flames within a spatially developing domain and not on the temporally evolving ignition process. IC engines operate at elevated pressures and temperatures. At these conditions, the distinction between autoignition and flame propagation [14] is physically meaningful from a modelling and design perspective, but very challenging to quantify due to the extreme conditions. For instance, a critical parameter in diesel combustion is the lift-off length (LOL) which affects the mixing and

pollutant formation processes [26]. The prediction of the LOL depends upon the underlying method of flame stabilisation and so distinguishing between autoignition and flame propagation is important. However, in the absence of resolved flame measurements, it is difficult to distinguish between these modes of combustion. Reference flame speed estimates at these conditions would place some bounds on the possible turbulent flame speeds, and may therefore assist in distinguishing between ignition and flame propagation stabilisation modes. At IC engine conditions, the fuel may exhibit single or two-stage autoignition depending on the exact thermochemical state. A single stage ignition fuel, such as ethanol (C2 H5 OH) (at the conditions considered in the present study), has a monotonic decreasing relationship of τ with the initial mixture temperature. For two-stage ignition fuels, such as dimethyl ether (CH3 OCH3 ) (DME) considered in the present study, and n-dodecane which is widely used in the engine combustion community for diesel-relevant conditions [27,28], low-temperature chemistry (LTC) leads to a firststage of autoignition that is distinct from the second, main stage of autoignition due to high-temperature chemistry (HTC). The temperature-dependent interaction between the first- and secondstages of autoignition produces a regime of negative temperature coefficient (NTC) behaviour for τ . Many recent studies have been performed for DME at NTC conditions [7,17,18,29–34]. In particular, the behaviour of the LTC has received much attention. Under certain conditions, the LTC can establish a cool flame which propagates like a wave. Stable premixed [32] and nonpremixed [29] cool flames have been experimentally observed and numerical simulations have investigated the complex transitions and flame structures that can exist due to the interactions between the cool flame and the high-temperature flame [7,30,31,34–36]. These interactions may be important for IC engines, as the cool flame can influence the ignition delay time and most reactive mixture location for diesel combustion [36–39] and engine knock in spark ignition engines, e.g., Refs. [7,40,41]. It has been shown that the propagation of partially premixed DME flames at IC engine conditions can lead to a range of possible flame structures depending upon the oxidiser temperature, TOX [30], and inlet velocity, UI [31]. At lower UI and lower TOX , conventional tribrachial (triple) flames are observed that are stabilised primarily by premixed flame propagation (deflagration) and flow divergence due to gas expansion. As UI and TOX increase, additional branches appear upstream of the main tribrachial flame due to the first-stage of autoignition (due to LTC) and/or the second stage of autoigntiion (due to HTC); these polybrachial flames are stabilised by a combination of autoignition and flame propagation [30,31]. For sufficiently high TOX or UI , the flame is stabilised purely by autoignition. In other words, the flame propagation speed is directly related to the mode of combustion and it is ambiguous as to how to define a reference value. Recent DNS studies of turbulent, partially premixed combustion at NTC conditions have measured the displacement speed of polybrachial edge flames [11,35,42]. Minamoto and Chen observed that the partially reacted mixture ahead of the high-temperature flame (due to LTC) affects the estimated flame speed, similar to the analysis for single stage ignition fuels with temperature non-uniformities [9,10] and experimental observations of a turbulent slot flame featuring two-stage ignition chemistry [5]. Experimental studies have been performed on laminar partially premixed flame stablisation for iso-octane [43] and n-heptane [44] fuels at elevated temperatures and atmospheric pressure. In those studies, measurements of the flame height from the inlet, Lf , versus UI were presented which, alongside visualisations of the flame structure, identified transitions from edge-flame propagation to autoignition with increasing UI . For iso-octane, plots of Lf versus UI identified transitions from burner-stabilised, to edge-flame

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stabilised, to autoignition stabilised modes [43]. In those studies, resolved measurements of the flame structure were not available and so the change in reaction front structure during the transitions was not investigated further. A recent numerical investigation of flame speeds at autoignitive conditions performed domain-dependent flame speed calculations for the NTC fuel n-heptane [7]. In that study, the inlet velocity was adjusted to maintain the flame position in a 1D domain with respect to variations in the the inlet temperature, similar to the procedure used for 2D partially premixed DME flames at NTC conditions [30]. As has been previously noted [7,30], this procedure does not produce a unique value, but instead reflects the competition between the residence time, the ignition delay time, and the premixed flame propagation speeds for a given domain size. Furthermore, calculations for the flame speed obtained values in excess of 50 ms−1 , which are almost certainly associated with autoignition fronts, and therefore do not represent a laminar flame speed. In the present study, a numerical method is described for defining and estimating a unique reference flame speed, denoted here as sR , which is valid for autoignitive conditions. Here, sR is the speed of an unstrained, adiabatic, laminar premixed flame at autoignitive conditions in the limit of an unreacted upstream composition. sR differs from the typical “laminar flame speed” definition for non-autoignitive conditions, sL , by accounting for the role of residence time on the flame speed. sR represents an ideal case for autoignitive conditions, where the reactant mixture just ahead of the flame has vanishing residence time, i.e., the flame propagation is not affected by autoignition. To estimate sR , a 1D inflowoutflow domain is simulated using numerical simulations over a range of IC engine relevant thermochemical conditions, including the single stage ignition fuel ethanol and the NTC fuel DME. Similar to Refs. [43,44], the response curve of Lf to UI is reported for a fixed thermochemical condition. For a single-stage fuel, one inflection point exists which is identified by the location of the peak value of dLf /dUI with respect to UI , which corresponds to the transition from a burner-stabilised flame to a lifted flame, such that the reaction zone is positioned just downstream of the inlet (with negligible heat transfer to the inlet boundary and negligible residence time ahead of the flame). Additionally, the special case of two-stage ignition fuels is addressed here, which differs from the one-stage ignition due to the influence of LTC, which leads to two local peaks in dLf /dUI versus UI . The first peak is located ahead of the LTC and is equivalent to the single stage ignition case, whereas the second peak, at a higher value of UI , corresponds to the case of a premixed flame propagating into the LTC reaction zone denoted  by the symbol sR . This second speed is of practical relevance for diesel engine combustion, where high temperature flames propagate into regions containing LTC products [27,45]. Calculations of freely propagating flames are also performed using Cantera at identical inlet conditions and with variations in the domain size, similar to the method used in Ref. [6]. It is found that the response of the eigenvalue solution to changing domain size is nearly equivalent to the response of the flame position to changing inlet velocity in the 1D inflow-outflow simulations. The results and relative merits of the two approaches are discussed. In order to confirm that the measurements of sR presented here do indeed correspond to premixed flames and not to autoignition fronts, a transport budget analysis is presented for each case at UI = sR in terms of the HTC species hydroxyl radical (OH), which is collocated with the high-temperature reaction zone. Furthermore, a transport budget analysis is performed for the LTC species methoxy-methyl peroxide radical (CH3 OCH2 O2 ), which is collocated with the low-temperature reaction zone for a case featuring two-stage ignition. A complex transition occurs which complements previous findings for 2D DME lifted flames [30,31]. A regime diagram is presented and discussed with respect to inlet

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temperature and velocity for the special case of a stoichiometric premixed 1D flames. 2. Configuration and methodology The simulations are performed using the direct numerical simulation code S3D [46], which solves the compressible Navier– Stokes, conservation of energy, and conservation of species massfraction equations using higher-order accurate numerical methods [46]. S3D solves the time-dependent conservation equations by integrating in time using a fourth-order, six-stage explicit Runge– Kutta method. In the present simulations, the steady state solution is reached by integrating in time until the transient solution converges. Transport terms are evaluated using mixture-averaged properties and chemical reaction rates are evaluated using validated chemical kinetic models for ethanol [47] and DME [48]. A schematic of the 1D computational domain is shown in Fig. 1. At the inlet, the reactant composition, equivalence ratio φ , inlet velocity UI , and inlet temperature Ti are prescribed with a hard inflow boundary condition. Stoichiometric mixtures (φ = 1.0) are prescribed for all cases. The effect of variations in φ are complex and lie beyond the scope of this paper and will be addressed in future work. The extent of the 1D domain in the x direction is Lx = 2.004 mm and the outflow condition is non-reflecting, implemented with a Navier–Stokes non-reflecting characteristic boundary conditions (NSCBC) method [49]. A computational grid of 4008 grid points with a uniform spacing of 0.5 μm was used to resolve the thinnest flame scales, commensurate with prior simulations at these thermochemical conditions using S3D [30,35,36]. A time step size of 0.5 ns was required for adequate temporal resolution, due to the acoustic CFL constraint imposed by the use of a compressible formulation of the governing equations. For low Mach number conditions, future studies should consider an incompressible formulation which would allow larger time steps to be taken. The entire domain is initially identical to the inlet boundary condition, i.e., there is no forced ignition or imposed flame and the mixture is allowed to autoignite and transition to a steady state position. For solutions that have a flame solution within the domain (not blown-off), the flame location Lf is defined as the point of maximum heat release rate. The flame time scale is defined as L τ f = 0 f dx/u(x ), where u(x) is the axial velocity profile in the domain. Four cases are considered at a common pressure of 40 atmospheres and a common τ of 0.427 ms. Figure 2 shows τ versus T for both ethanol and DME at 40 atm, and the cases A, B, C (DME) and D (ethanol) are marked. DME is a single- or two-stage ignition fuel at φ = 1.0 (depending on TI ) and has a negative temperature coefficient (NTC) regime with respect to τ . This is evident in Fig, 2 as there are three crossings of τ = 0.427 ms. These intercept values are selected to compare the autoignitive sl values for identical fuel–air mixtures with identical τ values, but with variations in temperature. This choice was made to rule out the influence of variations in τ on the estimates of sL , which should not depend on the autoignition time scale, but which should be very sensitive to the initial temperature. Case A (T = 1052.9 K) is above the NTC region and has one stage of autoignition due to hightemperature chemistry, case B (T = 909.1 K) is in the NTC regime and has distinct first- (low temperature) and second- (high temperature) stages of autoignition, case C (T = 821.8 K) is below the NTC region but also has two-stage autoignition. For case D, ethanol is the fuel, which exhibits single-stage autoignition without NTC. Cases A and D have identical TI and τ , and ethanol and DME are isomers, furthermore, ethanol and DME have similar sl values at non-autoignitive conditions [50] and similar adiabatic flame temperatures (which strongly influence sL [50]) at the present

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Fig. 1. Schematic of the 1D domain with physical extent, Lx . The left boundary is a hard inflow with a fixed velocity, UI , temperature, TI , and equivalence ratio φ I , for a given fuel. The right boundary is a non-reflective characteristic outflow. The flame location, Lf is defined by the reaction zone where the heat release rate is maximum. The flame time, τ f , is the integral of velocity from the inlet to Lf .

response may be measured. The response of Lf to UI (the eigenvalue solution in this case) is then used to calculate the peak in the dLf dUI profile with respect to UI , which corresponds to the sR definition proposed here. The solutions obtained with Cantera are then directly compared to those obtained with S3D. A transport budget analysis [30,51] is performed in the present study, for the S3D results, to identify the reaction zone structure and to infer the combustion mode. Within the reaction zone, the transport budget analysis considers the transport equation for a thermochemical variable of interest, e.g. for a species  mass  fraction YI the corresponding budget is: ∂∂t [ρYi ] = − ∂∂x ρYi u j − j  ρYiV j,i + ωi , where ρ is density, Vi, j is the diffusion velocity, and ωi is the species mass reaction rate. For a steady state solu-

∂ ∂x j



tion the left hand side is zero and the terms of the right hand side correspond to (in order): convection (C), diffusion (D), and reaction (R). The transport budget analysis for a premixed flame corresponds to a D-R balance, whereas an autoignition front corresponds to a C-R balance. 3. Results 3.1. Response to inlet velocity Fig. 2. Upper: homogeneous ignition delay times, τ , calculated in a constant pressure reactor with CHEMKIN for stoichiometric mixtures of ethanol (black) and dimethyl-ether (dark gray). The circle markers show cases A, B, and C for dimethyl ether and the star marker shows case D for ethanol. The black dashed horizontal line marks the constant ignition delay τ = 0.427 ms value for all cases. Lower: temperature versus time corresponding to cases marked in the upper sub-figure.

conditions. It is therefore expected that cases A and D should have similar sR values. For each case, a range of independent simulations were performed by varying only UI and measuring the response of Lf and τ f . This is a similar procedure to experimental studies of pre-heated, lifted edge flames at atmospheric pressure [43]. The sensitivity of the flame location to the inlet velocity is measured by dLf /dUI . For single-stage ignition fuels, the response of dLf /dUI versus UI has a single peak value and the corresponding UI is chosen here to be the reference flame speed, sR . For two-stage ignition fuels there are two peaks in the response of dLf /dUI versus UI . The first peak corresponds to the value obtained for single-stage ignition fuels and the second peak only exists for two-stage ignition fuels and is de noted as sR . sR calculations are also performed in Cantera for a freely propagating flame with identical inlet conditions, identical thermodynamic and transport data, and with a mixture-averaged transport model. By varying the size of the domain [6], the flame position with respect to the domain inlet (Lf ) may be varied and the sR

The response of each case to increasing UI , as calculated with S3D, is reported in Fig. 3. For cases A and D, a single transition is observed. At low UI , the flame is attached to the inlet and τ f /τ = 0. As UI increases beyond a critical value, Lf increases rapidly and there is a corresponding peak in dLf /dUI . As UI increases further, Lf becomes less sensitive to UI and for sufficiently high UI , τ f /τ = 1 and dL f /dUI = τ , indicating that the flame response may be predicted by the properties of autoignition. The response curve of τ f /τ to UI has a sigmoid shape with a steep transition. The peak in dLf /dUI corresponds to this transition location and is taken here to define a unique reference flame speed sR for single-stage autoignitive conditions. For UI = sR , the flame is located just downstream of the inlet such that the temperature and species mass fraction gradients at the inlet are nearly zero, as shown in Fig. 4, and the flame is essentially unaffected by the inlet boundary. This definition of sR corresponds to the condition where there is vanishing residence time ahead of the flame and there is no transport from the flame through the inlet boundary. Due to the finite thickness of the flame, there is in fact some small but finite residence time ahead of the flame at the condition where the gradients at the inlet vanish. Therefore, the method proposed here provides a close estimate of the ideal (vanishing residence time) reference flame speed for autoignitive conditions, as opposed to an exact calculation. The behaviour of cases B and C is more complex. In addition to the overall transition from attached to lifted flame, there is a second transition observed between these limits. Figure 3 shows that

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Fig. 3. Flame response to UI for (Upper) flame position Lf , (middle) normalised flame time τ f /τ , and (lower) rate of change of τ f /τ with respect to UI . The vertical dashed lines mark sR for each case. The symbols in the lower sub-figure indicate key inlet velocity values that are referred to in later figures.

the first peak in dLf /dUI versus UI is followed by an intermediate region, where the sensitivity of Lf to UI decreases and the flame is somewhat stable. As UI increases further, a second, lower peak in dLf /dUI occurs, before decreasing again for sufficiently high UI . The response of τ f /τ to UI is plotted alongside the first-stage ignition delay times for cases B (τ 1, B and C (τ 1, C ). The correspondence between τ 1, B and τ 1, C and the intermediate stable regions (the intermediate flattened portion of the τ f /τ versus UI curves) suggests that this behaviour is caused by LTC. The secondary peak in the dLf /dUI versus UI plots for cases B and C indicates that a second reference flame speed exists for two-stage ignition fuels, which is  denoted here as sR . 

Table 1 reports all values for sR and sR calculated with S3D, alongside other important physical parameters for each case. As expected, sR increases with TI and the values for cases A and D are nearly identical. Included in Table 1 is a measure  of the flame thicknesses at reference speeds sR and sR , de

noted δ f and δ f , respectively, calculated with the definition δ f = (TMAX − TMIN )/max(dT /dx ). These quantities may be of use, e.g., in the calculation of the flame time scale required for estimating the

Damköhler number or Karlovitz number in turbulent flames under autoignitive conditions. Table 1 also includes the sR values obtained from Cantera by solving the freely propagating flame model with decreasing domain size until the inlet velocity converges. The solution converges due to the decreasing distance ahead of the reaction zone, which reduces the available induction time for autoignition to develop. This approach is similar to what was performed in Ref. [6]. The agreement between the S3D and Cantera calculations is within 2% for all cases. This comparison is quite good considering the many differences in terms of the numerical methods and the solution algorithms. In Fig. 5, the sR values for cases A, B, and C obtained with S3D are plotted alongside the Cantera solution from 400 to 1300 K and



α  β

T 1 also a power law expression sL = Sl,REF 298 , where Sl, REF is P the laminar flame speed at 1 atm and 298 K, obtained from Ref. [52] (targeting temperatures below 450 K). All S3D cases fall close to the Cantera solution. The Cantera solution rapidly diverges from the power-law curve beyond 450 K. This highlights the need to obtain reliable estimates of sR at autoignition conditions, rather

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A. Krisman et al. / Combustion and Flame 188 (2018) 399–411 Table 1 Key physical parameters for each case. Case

A B C D

Fuel

DME DME DME Ethanol

T

τ1

(K)

(ms)

sR (S3D) (ms−1 )

1052.9 909.1 821.8 1052.9

– 0.12 0.26 –

2.17 1.43 1.07 2.21

Fig. 4. Response of the gradient of temperature, the fuel mass fraction, YH2 O2 , YCH3 OCH2 O2 (cases B and C only), at the inlet boundary with respect to varying UI for all cases. The gradient values are normalised by the maximum value in the domain. The vertical lines indicate the reference flame speed for each case.

Fig. 5. Comparison of sR values calculated with S3D for cases A, B, and C (cross symbols) and Cantera (solid black line) for DME. A power law expression taken from Ref. [52] for DME at non-autoignitive conditions, is also plotted in the black dashed line.

than extrapolating sL values from non-autoignitive conditions using commonly employed power law expressions. The Cantera values were much more efficient to compute compared to S3D. Unfortunately, the response of Lf to UI obtained with Cantera is not very smooth and the profiles of dLf /dUI are not well

resolved, particularly for cases B and C which feature sR values that are not clearly identified. For this reason, the remainder of the analysis in this study uses the S3D results, which allow for a detailed characterisation of the flame structures and transitions. However, when simply estimating a sR value, it should be sufficient to use the methodology applied here with Cantera, or with a similar program.





sR (Cantera) (ms−1 )

sR (S3D) (ms−1 )

δf

δf

(S3D) (μm)

(S3D) (μm)

2.19 1.42 1.07 2.25

– 1.70 1.50 –

8.8 10.1 11.2 9.0

– 10.5 12.2 –

The transitions in flame structure for cases A and D with S3D are examined with respect to a reference homogeneous autoignition case in Fig. 6 for the HRR and the intermediate species mass fraction YCH2 O and the HTC radical species mass fraction YOH . In each sub-figure, the solid black line is the solution from a zero-dimensional (0D), constant pressure reactor, calculated using SENKIN [53]. The 0D solution shows that after almost no delay, pre-ignition chemistry commences and steadily increases until a rapid increase occurs just prior to autoignition. At this point YCH2 O is consumed and YOH is produced. The corresponding 1D flames for selected UI values are also plotted, including the solution for UI = sR (square symbols). The 1D solutions are mapped to the homogeneous solution in the time domain by the operaX tion: t (X ) = 0 dx/u(x ). The results show that for UI = sR , the HTC reactions occur much earlier than expected from the 0D simulations. With increasing UI , the 1D solutions rapidly converge to the 0D solution, which reflects the sharp transition apparent from Fig. 3. For UI ≥ 2.75 ms−1 , where the curve τ f /τ versus UI begins to asymptote to unity, the 1D solutions in Fig. 6 become insensitive to further increases in UI and approach the 0D solution. This suggests that the transition occurs due to a flame propagating ahead of where the autoignition front would have developed. As UI increases, the flame position moves downstream and, first suddenly and then gradually, the reaction zone approaches the homogeneous autoignition solution. The same analysis was performed for cases B and C, featuring two-stage autoignition, for HRR, YOH , YCH2 O , and the LTC radical species mass fraction YCH3 OCH2 O2 , shown in Fig. 7. The 0D HRR results show two peaks which correspond to the first and second stages of autoignition with increasing time. The first peak is coincident with the peak in YCH3 OCH2 O2 followed by increased YCH2 O which is later consumed at time τ . The 1D HRR for UI = sR (square markers) have a single peak which is located ahead of most of the 0D chemistry. This result is similar to that for cases A and D with single stage ignition. The profiles of YCH3 OCH2 O2 (square markers) peak just ahead of the HRR peak and have a peak value an order of magnitude lower than the corresponding peak from the 0D solution. This suggests that the high-temperature reaction zone has overtaken both the second and first stages of autoignition. The results for UI corresponding to τ f /τ = τ1 are also plotted (cross markers). At this UI , there is a single peak in HRR which is coincident with the first peak in the homogeneous profile and also to τ 1 (marked in the vertical dashed line) and the homogeneous peak in YCH3 OCH2 O2 . This UI value also corresponds to the local minimum in dLf /dUI versus UI where the flame position is relatively stable. This suggests that at this UI , that a high-temperature flame exists, which has overtaken the second stage of autoignition and is located at or just before the completion of the first stage of autoignition. The stability of the flame at this point may be due to the increased concentration of radical species, such as CH3 OCH2 O2 , HO2 , and OH, which are destroyed for higher UI values following  the completion of the first stage of autoignition. At UI = sR (triangle markers), there are two peaks in HRR, the first corresponds to the 0D solution and the second occurs just after this point where YCH3 OCH2 O2 has mostly been consumed. This corresponds to the

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405

Fig. 6. 1D solutions for a range of UI values (thin coloured lines with markers) mapped to the time domain and compared with the 0D autoignition calculated in a homogeneous constant pressure reactor at the same pressure and initial temperature and composition (bold black line). Left column is case A and right column is case D. Top row is HRR, middle row is CH2 O, and bottom row is OH. The value of τ for each case is marked in the vertical dotted line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

Fig. 7. 1D solutions for a range of UI values (thin coloured lines with markers) mapped to the time domain and compared with the 0D autoignition calculated in a homogeneous constant pressure reactor at the same pressure and initial temperature and composition (bold black line). Left column is case B and right column is case C. Rows correspond to (top to bottom): HRR, CH2 OCH2 O2 , CH2 O, OH. The values of τ 1 and τ for each case are marked in the vertical dashed and dotted lines, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

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Fig. 8. Transport budget analysis for all cases at UI = sR for YOH . The solid black lines are the YOH profile measured on the right y-axes and the coloured lines are the transport terms on the left y-axes. Each profile is plotted with respect to the distance from the inlet. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

second peak in dLf /dUI . Sensitivity with respect to UI and may be interpreted as the point at which the high-temperature flame is lifted beyond the LTC radical pool, which is destroyed by the completion of the first stage of autoignition at time τ 1 . As with cases A and D, the 1D solutions asymptote to the 0D solution for sufficiently high UI , indicating here a gradual transition from a hightemperature flame propagating into LTC products towards a twostage autoignition front with no flame propagation. 3.2. Transport budget analysis The reaction zone of a premixed flame is characterised by a balance between diffusion (D) and reaction (R), wherein gradients are sufficient such that upstream diffusive/conductive transport of radicals/heat assists ignition of unburned mixtures, leading to a propagating flame. In the reaction zone of a premixed flame, convection (C) plays a secondary role. This is in contrast to a autoignition front, which propagates purely due to gradients in τ , and where gradients of species and temperature are insufficient for diffusion/conduction to contribute to flame propagation. As such, for a configuration with steady inflow-outflow boundary conditions such as the present study, a balance is expected between C and R. This second case corresponds to the build up of radicals and enthalpy that occurs prior to autoignition which are transported downstream by C and consumed by R. Due to the difficulty in distinguishing between flame propagation and autoignition fronts at autoignitive conditions, a transport budget analysis is performed for the high-temperature reaction zones in each case for UI = sR in Fig. 8. Figure 8 shows that for each case that the high-temperature reaction zone consists predominantly of a R-D balance, with a lower contribution from C that convects OH downstream, but mostly within the reaction layer itself. The reaction zone is very thin at approximately 10 μm, which is also consistent with expectations of premixed flames, as compared to a spontaneous ignition front. There is consistency between the transport budget analysis results for the different cases, which occur over a wide range of temperatures and for different fuels, which is encouraging for the general use of this method for estimating a reference flame speed at autoignitive conditions. A transport budget analysis was also performed for case C (featuring two-stage autoignition) at various UI values in Fig. 9, to

further investigate the transitions observed in Figs. 3 and 7. The analysis was conducted for the LTC radical species CH3 OCH2 O2 to consider the low-temperature reaction zone, in addition to OH for the high-temperature reaction zone. At UI = sR (top row), the CH3 OCH2 O2 budget is predominantly R-D, with a lower, but still substantial, contribution from C. The low-temperature reaction zone is very thin, on the order of 10 μm, and it is directly adjacent to the high-temperature reaction zone as judged by the OH budget. The magnitude of CH3 OCH2 O2 is very low at this inlet velocity, because the high-temperature flame has overtaken almost all of the LTC, which therefore has little influence on the overall flame. As UI increases, the flame moves rapidly downstream, before stabilising at τ f ≈ τ 1,C (corresponding to UI = 1.25 ms−1 in Fig. 9). The CH3 OCH2 O2 budget is very different at this inlet velocity: firstly, it is broadly distributed in physical space, on the order of 100 μ m; secondly, the budget is mostly a C-R balance with the gradual build up of CH3 OCH2 O2 ahead of its destruction in a thin region with a C-D-R balance that is adjacent to the hightemperature reaction zone. This is consistent with the prior observation that for τ f ≈ τ 1, C , there exists a high-temperature flame which is propagating into a pool of LTC radicals that are produced by the first stage of autoignition but consumed (or partially consumed) by the high-temperature flame. The presence of the LTC radical pool adjacent to the high-temperature flame for UI corresponding to τ f ≈ τ 1, C , assists the propagation of the hightemperature flame and helps to stabilise it, as judged by the local reduction in dLf /dUI versus UI . This result is consistent with Ref. [7]. In that study, domain size dependent flame speed calculations were performed for n-heptane fuel at 40 atm , where an increase in flame speed is observed for temperatures in the NTC region and this effect is attributed to the build up of LTC reactive intermediate species.  For UI = sR = 1.50 ms−1 (corresponding to the second peak in dLf /dUI versus UI ), the transport budgets for OH and CH3 OCH2 O2 are spatially separated , indicating that the high-temperature flame is “lifted” from LTC radical pool, and is propagating into the products of the first stage of autoignition (such as CH2 O). The budget for CH3 OCH2 O2 is almost entirely C-R indicating that both the production and destruction of the low-temperature reaction zone is due to autoignition. The budget for OH is predominantly D-R, which shows that the high-temperature reaction zone remains a propagating flame. As UI increases to 3.50 ms−1 , the transport budgets for both OH and CH3 OCH2 O2 change slowly, with less contribution from D and a broadening of the reaction zones. It is interesting to note that even at UI = 3.5 ms−1 , where τ f /τ = 0.99, that the hightemperature reaction zone has a thin C-D-R structure. This suggests that the transition from flame propagation to autoignition front occurs very gradually over a wide range of inlet velocities, which was also noted in Ref. [6] in the context of freely propagating hydrogen flames at atmospheric pressure. This is also consistent with previous studies on the stabilisation of partially premixed DME edge flames [30,31]. In those studies, an order of magnitude increase in inlet velocity was required to transition from an edge flame stabilised by flame propagation to one stabilised by autoignition. Figure 10 also presents the normalised spatial profiles of temperature and mass fractions for species CH3 OCH3 , CH3 OCH2 O2 , H2 O2 , CH2 O, CO, and OH, for the same inlet velocities presented in Fig. 9 for case C. The results show the response of the flame structure to increasing UI . At UI = 1.07 (sL ) and 1.25 ms−1 , the low and high temperature reaction zones are adjacent to one and other. The separation of reaction zones is seen at UI = 1.50 ms−1 , which resembles the “double flame” structure previously observed with 1D numerical simulation for very lean n-heptane [54] and dimethyl ether [55] fuels at lower temperatures and pressures. The “dou-

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Fig. 9. Transport budget analysis for case C in terms of YCH3 OCH2 O2 (left column) and YOH (right column) for a range of inlet velocities. Each row corresponds to a fixed inlet velocity, increasing from the top to the bottom sub-figure. The solid black lines are the species mass fraction profiles measured on the right y-axes and the symbols indicate the UI values as given in Figs. 3 and 8. The coloured lines are the transport terms measured on the left y-axes. Each profile is plotted with respect to the distance from the inlet. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

ble flames” consisted of spatially separated cool- and hot-flames. While sharing some similar features, the separated flame structure observed here is essentially different to the “double flame” since the low-temperature reaction zone in the present case is an autoignition feature, rather than a diffusively-supported flame which was demonstrated for the DME case [55]. This difference may be due to the very different thermochemical conditions in that study, but the underlying causes are unclear. It may be that the double flame structure requires that the cool-flame propagates faster than the hot flame, and that the hot flame can only be sustained downstream of the cool-flame where the increased temperature enhances the local flame speed.

3.3. Regime diagram The results presented here identity several transitions that occur for stoichiometric, 1D premixed flames at high temperature and pressure. The sensitivity of the flame position Lf to the inlet velocity UI corresponds to changes in the underlying flame stabilisation mechanisms that occur due to the importance of residence time for autoignitive conditions, which are further complicated by the NTC regime and two-stage ignition. In order to summarise and conceptualise the present findings, a regime diagram is presented and discussed here. In Ref. [31], a regime diagram was presented for DME partially premixed flames with respect to inlet velocity and inlet

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Fig. 10. Normalised flame structure for case C at inlet velocities corresponding to those shown in Fig. 9. The black line with symbols shows the temperature and the blue lines show key species mass fraction profiles. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

temperature. In that study, regimes were delineated between nonautoignitive and autoignitive and attached and lifted. The lifted flames were further classified based on the method of flame stabilisation; propagation, autoignition, or hybrid autoignition and propagation. However, there were several unquantified effects, such as flow divergence, cross-stream scalar gradients, and inhomogeneous autoignition which do not apply here. Figure 11 shows the regime diagram based on the present results for a finite domain size and a hypothetical fuel featuring twostage ignition. The ignition delay times are shown in sub-figure a) for the first and second stages of ignition. Key temperature are also marked for later discussion. T1 corresponds to a sufficiently low temperature such that the non-reacting residence time, τNR = Lx /UI equals the corresponding value of τ . T1 therefore represents the boundary between autoignitive and non-autoignitive temperatures for the given domain size. Temperatures T2 and T3 bound the temperature range where two-stage autoignition can occur. In subfigure b), the temperature–velocity parameter space is divided into five regimes, labelled A through E. Regimes A through D correspond to cases where a flame exists within the domain: • Regime-A – Burner-stabilised flame, the high temperature flame propagates faster than UI and overtakes the entire autoignition process. This regime exists for both autoignitive conditions (one- or two-stage) and non-autoignitive conditions, where sR corresponds to the laminar flame speed sL . The flame is stabilised by heat and species mass transfer. The boundary of this regime with respect to increasing UI corresponds to the peak value of dLf /dUI , the location where sR is defined in the present study. • Regime-B – Low-temperature chemistry assisted flame propagation, a high temperature flame exists which is stabilised at or near a pool of LTC radical and intermediate species. This regime only exists for fuels with low- and high-temperature chemistry. In this study, we only observe Regime-B due to LTC autoignition, as opposed to diffusively-supported cool flames [32,36]. It is conceivable that for some conditions a cool flame may exist ahead of a hot flame similar to the “double flame” structure observed by Ju et al. [55] and provide another mechanism to stabilise the hot flame. The boundary of this regime with respect to increasing UI corresponds to the second peak values of  dLf /dUI which is where sR is defined in this study.

Fig. 11. A regime diagram for a 1D premixed flame in temperature-velocity space for a hypothetical NTC fuel. a) The first- and second- ignition delay times for a twostage ignition fuel. Key temperatures: T1 is a sufficiently low temperature such that τ is long compared to the residence time, T2 and T3 correspond to the lowest and highest temperatures, respectively, for which two distinct stages of autoignition exist. b) The regime diagram, labelled as A, B, C, D, or E (see text for details). Key temperatures T1 , T2 , and T3 from a) are indicated in the horizontal dashed lines. Also indicated are temperatures TCOLD , TNTC , and THOT in the solid horizontal lines. c) Evaluations of dLf /dUI with respect to UI for temperatures THOT (upper), TNTC (middle), and TCOLD (lower). The transitions in regime with respect to UI and the laminar flame speeds are also shown. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article).

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• Regime-C – Autoignition assisted flame propagation, a hightemperature flame exists that is predominantly propagating due to a diffusion-reaction balance. The flame location is downstream of significant pre-ignition chemistry (HTC for singlestage fuels, LTC and HTC for two-stage fuels) which modifies the reactant stream ahead of the reaction zone and assists flame propagation. The flame location becomes less sensitive to increasing UI in this regime as the reactant mixture ahead of the flame asymptotes to the products of homogeneous autoignition. The transition from regime-C to regime-D is gradual, as judged by the slow transition of the transport budget from a diffusion-reaction to a convection-reaction balance. • Regime-D – Autoignition stabilisation, no premixed flame structure exists. The reaction zone is spatially distributed and there is little contribution from diffusion and conduction. For sufficiently high UI , the spatial evolution of the reaction zone maps exactly to the corresponding homogeneous, constant pressure autoignition solution. Finite values of Lx constrain the available residence time and blow-off will occur for UI > UI, MAX , where UI, MAX is the maximum inlet velocity such that the flame is not blown-off for a given domain and thermochemical conditions. The maximum exact  L =Lx flame time τ f = 0 f dx/u(x ) may be approximated as τ f ≈ τNR = Lx /UI if gas expansion ahead of the flame is neglected. Assuming also that blow-off occurs where τ f /τ ≈ 1 (valid for autoignitive conditions), the blow-off velocity may therefore be estimated as UI,MAX = Lx /τ . That is, for finite domain sizes at autoignitive conditions, blow-off is inversely proportional to the ignition delay time, as expected. For non-autoignitive conditions, it is clear that UI, MAX becomes smaller than sR and this physical argument breaks down the flame speed should be calculated as sL for non-autoignitive conditions. The influence of finite domain size is represented on the regime diagram by: • Regime-E – Blow-off, for autoignitive conditions (TI > T1 ), this regime occurs where UI, MAX > Lx /τ . The transition curve to regime-E (dark blue line) will shift to lower values of UI with decreasing Lx and retain its shape. This would cause regime-E to cut-off regimes D, C, B (if present), and A, in that order, as Lx approaches zero. For T2 < TI < T3 , it is possible for only the HTC to be blown-off and for LTC to exist in the domain. The effect of regime cut-off highlights the requirement for a sufficiently large domain size in order to measure the response of Lf to UI and this requirement will depend upon the ignition time scales of the given thermochemical condition. For non-autoignitive conditions (TI < T1 ), there is no solution for the present method of measurement where the mixture is first allowed to autoignite and then to stabilise within the domain. Therefore, with the present method at TI < T1 all values of UI would result in a blown-off flame. However, if a flame was initialised within the domain then a transition from regime-A to regime-E with increasing UI would be expected, and this transition would correspond to sL , similar to the experimental method of measuring sL using burner-stabilised fames with plug-flow [50]. While Fig. 11 illustrates a regime diagram for a NTC fuel, the corresponding regime diagram for single-stage ignition fuels at autoignitive conditions would be essentially identical to this after the  removal of regime-B and the solution curve for sR , and changing the shape of the autoignition boundary curves (gold dashed line and dark blue solid line) to reflect the non-monotonic τ curve for single-stage ignition fuels. Also shown in Fig. 11 are the evaluations of dLf /dUI with respect to UI , annotated by regime, for temperatures TCOLD , TNTC , and THOT that correspond to non-autoignitive, autoignitive with twostage autoignition, and autoignitive with one-stage autoignition

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conditions, respectively. For TCOLD , the inlet temperature is sufficiently low that autoignition does not occur within the domain. Therefore, a transition from regime-A to regime-E occurs and dLf /dUI approaches infinity in the limit as UI approaches sL . This is conceptually similar to the method for measuring sR experimentally with plug-flow burner-stabilised flames, except that there is no flow-divergence in the present 1D simulations and hence the flame can not be stabilised downstream of the inlet. For TNTC , the solution first transitions from regime-A to regime-B, where dLf /dUI is a maximum and sR is defined. This is followed by a transition  from regime-B to regime-C where sR is defined. Another transition occurs gradually from regime-C to regime-D as the diffusionreaction balance shifts to a convection-diffusion balance. Finally, a transition from regime-D to regime-E occurs where UI > UI, MAX , leading to blow-off. The TNTC example is similar to that observed for cases B and C with DME in the present study. For THOT , there is a transition from regime-A to regime-C, where sL is defined, which gradually transitions to regime-D and then regime-E. The THOT example is similar to cases A (DME) and D (ethanol) in the present study. An order of magnitude estimate of the typical velocity scales are also indicated in Fig. 11, where values of sR are in general expected to be on the order of 1 ms−1 and the transition to autoignition stabilised occurs for velocities on the order of 10 to 100 ms−1 , depending upon the inlet temperature. The transition to regime-E (blow-off) depends upon the domain size. 4. Conclusions Motivated by the desire to calculate a reference flame speed at autoignitive conditions, a series of one-dimensional simulations were conducted at a pressure of 40 atmospheres and at elevated temperatures for the fuels ethanol and dimethyl ether, representative of fuels exhibiting single- and two-stage ignition, respectively. In order to rule out the influence of the ignition delay times on the solution, three cases for dimethyl ether and one case for ethanol were selected which had an identical ignition delay time. The dimethyl ether case with the highest initial temperature, case A, and the ethanol case D featured a single-stage ignition. The dimethyl ether cases B and C featured two-stage ignitions. For each case, a series of independent simulations were performed, varying only the inlet velocity, in an inflow-outflow configuration. The response of the flame location and flame time were measured with respect to the inlet velocity. Transitions in the sensitivity of the flame location to the inlet velocity were identified. For single-stage ignition, one peak in the flame location sensitivity exists, and two peaks exist for the two-stage ignition. A comparison of the one-dimensional flame structure to a homogeneous, constant pressure autoignition solution, in addition to a transport budget analysis of the low- and high-temperature reaction zones, confirmed that these local peaks correspond to flames stabilised by propagation just downstream of the inlet (sR ) and, for two-stage ignition only, propagation into the products of the first stage of au toignition (sR ). These flame speed definitions provide a method for measuring a characteristic laminar velocity scale for flame propagation at autoignitive conditions. The reference flame speeds identified here may prove useful for characterising combustion at autoignitive conditions where hybrid deflagration/autoignition flame structures exist [5,11,30,56]. There may also be a connection with flame propagation at nonautoignitive conditions subject to intense turbulence, local extinction, and recirculation and mixing of partially burnt product gases. In such a scenario, the dilution of the fresh gases with partially burnt products may enhance flame propagation, and/or lead to hybrid deflagration/autoignition flames. A comparison was also made with calculations of freely propagating, adiabatic, un-stretched laminar flames at identical inlet

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conditions, using Cantera. It was found that the domain size dependent solution [6] could be used as an equivalent means to estimate sR . The results obtained with Cantera were in good agreement with the results obtained in the inflow-outflow simulations. This suggests that although the freely propagating flame model is, in general, ill-posed at autoignitive conditions due to the domain size dependence, that it may nonetheless produce a good approximation of sR when the domain size dependence of the solution is accounted for. This approach is much more computationally efficient compared to solving the time-dependent inflow-outflow system, and it is the recommended method if only an estimate of sR is required. A regime diagram was also presented, which identified possible flame stabilisation regimes and transitions with respect to variations in the inlet temperature or velocity. The regime diagram is consistent with the present results, and prior simulations of partially premixed flames at autoignitive conditions [30,31]. Acknowledgments Conversations with Ramanan Sankaran and Bruno Savard improved the quality of this study. 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 U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0 0 03525. References [1] D.M. Smooke, J.A. Miller, R.J. Kee, Determination of adiabatic flame speeds by boundary value methods, Combust. Sci. Technol. 34 (1–6) (1983) 79–90. [2] J. Hirschfelder, C. Curtiss, D.E. Campbell, The theory of flames and detonations, Symp. (Int.) Combust. 4 (1) (1953) 190–211. [3] F. Williams, Combustion theory: the fundamental theory of chemically reacting flow systems, Combustion science and engineering series, Perseus Books Group, 1985. [4] J.H. Chen, E.R. Hawkes, R. Sankaran, S.D. Mason, H.G. Im, Direct numerical simulation of ignition front propagation in a constant volume with temperature inhomogeneities: I. Fundamental analysis and diagnostics, Combust. Flame 145 (2006) 128–144. [5] S.H. Won, B. Windom, B. Jiang, Y. Ju, The role of low temperature fuel chemistry on turbulent flame propagation, Combust. Flame 161 (2) (2014) 475–483. [6] R. Sankaran, Propagation velocity of a deflagration front in a preheated autoigniting mixture, 9th U.S. National Combustion Meeting, 2015. Paper 114LF-0349 [7] J. Pan, H. Wei, G. Shu, Z. Chen, P. Zhao, The role of low temperature chemistry in combustion mode development under elevated pressures, Combust. Flame 174 (2016) 179–193. [8] P. Pal, M. Valorani, P.G. Arias, H.G. Im, M.S. Wooldridge, P.P. Ciottoli, R.M. Galassi, Computational characterization of ignition regimes in a syngas/air mixture with temperature fluctuations, Proc. Combust. Inst. 36 (3) (2017) 3705–3716. [9] C.S. Yoo, T. Lu, J.H. Chen, C.K. Law, Direct numerical simulations of ignition of a lean n-heptane/air mixture with temperature inhomogeneities at constant volume: parametric study, Combust. Flame 158 (9) (2011) 1727–1741. [10] J.B. Martz, G.A. Lavoie, H.G. Im, R.J. Middleton, A. Babajimopoulos, D.N. Assanis, The propagation of a laminar reaction front during end-gas auto-ignition, Combust. Flame 159 (6) (2012) 2077–2086. [11] Y. Minamoto, J.H. Chen, DNS of a turbulent lifted DME jet flame, Combust. Flame 169 (2016) 38–50. [12] R. Kee, J. Grcar, M. Smooke, J. Miller, A fortran program for modeling steady laminar one-dimensional flames, 1985. Technical Report SAND85-8240 [13] D.G. Goodwin, H.K. Moffat, R.L. Speth, Cantera: an object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes, 2017, (http://www.cantera.org). Version 2.3.0. 10.5281/zenodo.170284. [14] Y.B. Zel’dovich, Regime classification of an exothermic reaction with nonuniform initial conditions, Combust. Flame 39 (2) (1980) 211–214. [15] R. Sankaran, H.G. Im, E.R. Hawkes, J.H. Chen, The effects of non-uniform temperature distribution on the ignition of a lean homogeneous hydrogen–air mixture, Proc. Combust. Inst. 30 (1) (2005) 875–882.

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