3.1 Qualitative description

Figure 5 shows 2-D slices in the $z^{\ast }=0$ plane at $t^{\ast }=18$ and 27 (2 and 3 times $\unicode[STIX]{x1D70F}_{MR}$ ) for $Y_{I},Y_{P}$ and $T$ . By $t^{\ast }=18$ , the initially laminar jet profile has developed sheared turbulence due to the initial perturbation. The hot oxidiser mixes with the fuel jet and by $t^{\ast }=18$ broad regions with high values of $Y_{I}$ and moderate values of $Y_{P}$ are observed at rich mixtures, indicating the presence of LTC. By $t^{\ast }=27$ , regions of high $T$ and $Y_{P}$ are observed, centred on the $\unicode[STIX]{x1D709}_{ST}$ surface, indicating the HTC ignition has taken place. Figure 6 shows a blow up of an example region of the domain at $t^{\ast }=27$ . Multiple edge flames can be observed, centred on the $\unicode[STIX]{x1D709}_{ST}$ 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) 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. Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2015), 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 $\unicode[STIX]{x1D712}$ and turbulence, resulting in a highly contorted field. Later in the simulation, regions of high $T$ and $Y_{P}$ are observed, centred on the $\unicode[STIX]{x1D709}_{ST}$ surface. The $\unicode[STIX]{x1D709}_{ST}$ surface resides at the jet periphery, and is not strongly contorted by turbulence, since the region of high-velocity fluctuations occurs at richer mixtures.

Figure 5. Instantaneous images of $T$ (a,d), $Y_{I}$ (b,e) and $Y_{P}$ (c,f) at $t^{\ast }=18$ (a–c) before HTC ignition and at $t^{\ast }=26$ (e–f) after HTC ignition. Dashed white contour marks $\unicode[STIX]{x1D70F}_{ST}$ .

Figure 6. Instantaneous image of HRR at $t^{\ast }=27$ , evaluated on the $z=0$ plane over the region $0<x<1$ , $0.2<y<0.7$ . The black dashed line shows the $\unicode[STIX]{x1D709}_{ST}$ surface.

High-temperature autoignition develops from $t^{\ast }\approx 19.8$ as a spatially distributed and temporally staged event. The HTC emerges in multiple locations that are detected as localised maxima in temperature, HRR and $Y_{P}$ . These maxima are referred to as kernels that are defined here to exist when the local temperature exceeds a temperature threshold of $T_{HTC}=1400$  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 $T_{HTC}$ . Forty ignition kernels were identified via visual inspection of the evolution of the $T_{HTC}$ surface.

Figure 7. The evolution of the $\unicode[STIX]{x1D709}_{ST}$ surface, viewed from the $y+$  domain boundaries, coloured by threshold of $Y_{P}=0.19$ . The threshold is selected to delineate regions of burning (red) and non-burning (white) $\unicode[STIX]{x1D709}_{ST}$ .

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 $\unicode[STIX]{x1D709}_{ST}$ surface they establish edge flames (e.g. see figure 6) which propagate along the $\unicode[STIX]{x1D709}_{ST}$ surface, similar to the behaviour observed by Domingo & Vervisch (Reference Domingo and Vervisch1996) with simple chemistry and Echekki & Chen (Reference Echekki and Chen2002) with detailed $\text{H}_{2}$ chemistry. Behind the edge flames a non-premixed flame is established, centred upon the $\unicode[STIX]{x1D709}_{ST}$ surface, which demarcates burning and non-burning regions of the $\unicode[STIX]{x1D709}_{ST}$ surface. Figure 7 shows the progress of combustion in the domain towards a fully burning state, visualised by the portions of the $\unicode[STIX]{x1D709}_{ST}$ surface that exceeds a scalar threshold. The $\unicode[STIX]{x1D709}_{ST}$ surface is coloured by $Y_{P}$ , such that the white regions are not burning and the red regions are burning (defined here as $Y_{P}>0.19$ ). This threshold corresponds to the $Y_{P}$ value on the $\unicode[STIX]{x1D709}_{ST}$ surface where the HRR is maximum. As can be seen, the burning regions originate as isolated pockets which spread along the $\unicode[STIX]{x1D709}_{ST}$ surface. These isolated pockets originate from kernels that form in rich mixtures at earlier times (not shown in figure 7). At $t^{\ast }=21$ , 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 $\unicode[STIX]{x1D709}_{ST}$ surface to the leaner side, and the region of LTC to the richer side, which is illustrated here by a surface of $Y_{I}=1\times 10^{-4}$ , a value that is approximately 1 % of the maximum of $Y_{I}$ 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 kernel rapidly grows and crosses the $\unicode[STIX]{x1D709}_{ST}$ 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 $\unicode[STIX]{x1D709}_{ST}$ surface (not shown here).

Figure 8. 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 $\unicode[STIX]{x1D709}_{ST}$ surface, coloured dark blue, and the region of LTC and richer $\unicode[STIX]{x1D709}$ , coloured light blue (only shown in (a)). The kernel rapidly expands, engulfing the $\unicode[STIX]{x1D709}_{ST}$ surface and spreading along $\unicode[STIX]{x1D709}_{ST}$ . Each panel shows the same volume of space, viewed from the same perspective. The bounding box (a) shows the grid values in mm.

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 $\unicode[STIX]{x1D712}$ 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 $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}=1.1$ , indicating that the HTC ignition will never occur. The peak turbulent mean and r.m.s. profiles are initially far higher that $\unicode[STIX]{x1D712}_{C}$ , and then decay during the simulation. By $t^{\ast }=18$ , the mean $\unicode[STIX]{x1D712}$ profile is below $\unicode[STIX]{x1D712}_{C}$ at all mixture fractions. The value of $\unicode[STIX]{x1D70F}_{MR}$ is marked on each plot in the vertical dashed line and this shows the $\unicode[STIX]{x1D709}$ value where the HTC ignition is expected to occur from homogeneous calculations. It is noted that the mean and r.m.s. $\unicode[STIX]{x1D712}$ profiles are increasing as $\unicode[STIX]{x1D709}$ exceeds $\unicode[STIX]{x1D709}_{MR}$ , meaning that dissipation is preferentially higher at richer mixtures.

Figure 9. Conditional $\unicode[STIX]{x1D712}$ 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 $\unicode[STIX]{x1D709}_{ST}$ and the dashed vertical line marks $\unicode[STIX]{x1D709}_{MR}$ .

Figure 10 presents temporally evolving statistics for $Y_{I}$ , $Y_{P}$ and $T$ in increments of $t^{\ast }=9$ (or $\unicode[STIX]{x1D70F}_{MR}$ ) from the turbulent DNS, alongside laminar solutions corresponding to the steady $\unicode[STIX]{x1D712}$ rates plotted in figure 9.

At $t^{\ast }=9$ , the LTC is located in very rich mixtures for both the laminar and turbulent results. The scatter plot shows large conditional fluctuations in $Y_{I}$ , possibly due to the very large r.m.s. values of $\unicode[STIX]{x1D712}$ as shown in figure 9. The turbulent profile is more broad in $\unicode[STIX]{x1D709}$ 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 $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}$ , 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. Reference Liu, Hewson, Chen and Pitsch2004). The $Y_{P}$ and $T$ profiles at $t^{\ast }=9$ show all laminar solutions have progressed further towards ignition than the turbulent mean profile.

By $t^{\ast }=18$ , the $Y_{I}$ profiles have shifted further towards pure fuel. This is more clear for the laminar solutions, for which $Y_{I}$ values are lower compared to the turbulent case for $\unicode[STIX]{x1D709}<0.4$ , where the reduction in $Y_{I}$ values is more pronounced for lower values of $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}$ . At the same time, sensitivity to $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}$ in the laminar cases is observed for $Y_{P}$ and $T$ for $\unicode[STIX]{x1D709}<0.4$ , and so it is seen that ignition is proceeding more rapidly for lower values of $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}$ . The increase in $T$ and $Y_{P}$ is most prominent at rich mixtures between $\unicode[STIX]{x1D709}=0.1$ and 0.3. At this time the turbulent mean $Y_{P}$ 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 $t^{\ast }=19.8$ and by $t^{\ast }=27$ several new features in the conditional statistics are evident. Most prominently, two branches of $T$ and $Y_{P}$ 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 $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}=0.4$ and 0.6, HTC ignition has occurred, while the solutions at $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}=0.9$ and 1.1 sit astride the LTC branch. The mean turbulent profiles of $Y_{P}$ and $T$ are weighted towards the lower branches, indicating that at this time most of the domain has not undergone HTC. The $Y_{I}$ profiles show a large decrease for the ignited laminar cases and a reduction in the mean turbulent profile for $\unicode[STIX]{x1D709}<0.4$ .

At $t^{\ast }=36$ , the mean turbulent profiles show a nearly complete HTC ignition for $\unicode[STIX]{x1D709}<0.4$ , and a region of LTC persisting mostly in very rich mixtures. All cases with $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}$ less than unity have ignited, while the $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}=1.1$ case has reached a steady state solution on the LTC branch.

Figure 10. Conditional statistics for $Y_{I}$ (a,d,g,j), $Y_{P}$ (b,e,h,k) and $T$ (c,f,i,l). Each row represents an instant in time from (a–c) at $t^{\ast }=9$ to (j–l) at $t^{\ast }=36$ . 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 $\unicode[STIX]{x1D712}_{P}/\unicode[STIX]{x1D712}_{C}$ values shown in figure 9.

Overall, the conditional statistics support the notion of a two-stage autoignition process involving an initial LTC (first stage) autoignition that moves into increasingly 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 $\unicode[STIX]{x1D709}_{ST}$ . 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 $\unicode[STIX]{x1D709}$ -space. These results are broadly in agreement with prior LES (Gong et al. Reference Gong, Jangi and Bai2014), 2-D DNS with reduced dimethyl ether chemistry (Krisman et al. Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2017), transported probability density function modelling (Pei et al. Reference Pei, Hawkes, Bolla, Kook, Goldin, Yang, Pope and Som2016), flamelet modelling with detailed n-dodecane chemistry (Dahms et al. Reference Dahms, Paczko, Skeen and Pickett2017) and conceptual models derived from optical measurements of diesel spray flames (Musculus et al. Reference Musculus, Miles and Pickett2013).

Figure 11. Maps of conditional means for $Y_{I}$ (a), $Y_{P}$ (b), $\dot{\unicode[STIX]{x1D714}}_{P}$ (c) and $\unicode[STIX]{x1D712}$ (d). Each panel maps the conditional mean value in time and $\unicode[STIX]{x1D709}$ space. The thick solid horizontal line marks $\unicode[STIX]{x1D709}_{ST}$ and the dashed line marks $\unicode[STIX]{x1D709}_{MR}$ . The thin solid isocontour bounds the region of $\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C}>1$ mixing rates are highest.

The temporal evolution of conditional mean quantities is mapped in figure 11 for $Y_{I}$ , $Y_{P}$ , $\dot{\unicode[STIX]{x1D714}}_{P}$ and $\unicode[STIX]{x1D712}$ . The $Y_{I}$ 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 $Y_{I}$ closely corresponds to the $\dot{\unicode[STIX]{x1D714}}_{P}$ distribution from $t^{\ast }=0$ to about $t^{\ast }=20$ . The LTC peak rapidly moves into richer mixtures until about $t^{\ast }=7$ , when the $Y_{I}$ and $\dot{\unicode[STIX]{x1D714}}_{P}$ profiles remain nearly stationary in $\unicode[STIX]{x1D709}$ space and moderate in intensity. The attenuation of LTC corresponds with the development of the $\unicode[STIX]{x1D712}$ profile, which peaks over similar temporal and $\unicode[STIX]{x1D709}$ values. This peak in $\unicode[STIX]{x1D712}$ , due to the development of the shear turbulence, produces strong mixing which exceeds the $\unicode[STIX]{x1D712}_{C}$ 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. Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2017), 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 $\unicode[STIX]{x1D712}$ , although it is inhibited by very high levels of $\unicode[STIX]{x1D712}$ . In comparison with the previous result, the LTC here appears to be more strongly inhibited by $\unicode[STIX]{x1D712}$ . This result is not necessarily contradictory with the previous DNS, due to several differences between the two simulations, including: higher peak levels of $\unicode[STIX]{x1D712}$ 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. (Reference Borghesi, Mastorakos and Cant2013) in the context of droplet autoignition, that the initial specification of peak $\unicode[STIX]{x1D709}$ equal 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. Reference Musculus, Miles and Pickett2013). 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. (Reference Maes, Meijer, Dam, Somers, Toda, Bruneaux, Skeen, Pickett and Manin2016), 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 $Y_{I}$ and $\dot{\unicode[STIX]{x1D714}}_{P}$ profiles recover and continue to move into richer $\unicode[STIX]{x1D709}$ . By approximately $t^{\ast }=22$ , an increased $\dot{\unicode[STIX]{x1D714}}_{P}$ is observed between $\unicode[STIX]{x1D709}_{ST}$ and the location of peak $Y_{I}$ . 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 $\dot{\unicode[STIX]{x1D714}}_{P}$ remains centred on the $\unicode[STIX]{x1D709}_{ST}$ isoline, as the secondary low-temperature $\dot{\unicode[STIX]{x1D714}}_{P}$ peak weakens and tracks into richer $\unicode[STIX]{x1D709}$ values with the $Y_{I}$ profile. Large amounts of $Y_{P}$ form over a wide range of $\unicode[STIX]{x1D709}$ values while the mixing rates continue to relax.

Figure 12 presents the maximum values of the conditional mean of $\dot{\unicode[STIX]{x1D714}}_{P}$ , $Y_{I}$ , $T$ and $\unicode[STIX]{x1D712}$ 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 $\dot{\unicode[STIX]{x1D714}}_{P}$ closely follows the LTC marker, $Y_{I}$ . The peak in the mixing rates corresponds with an approximate 15 % reduction in peak $Y_{I}$ values, which recovers as the mixing rates decline. By approximately $t^{\ast }=22$ , a rapid increase in the temperature, $Y_{X}$ , $Y_{P}$ and $\dot{\unicode[STIX]{x1D714}}_{P}$ profiles occur simultaneously, showing the timing of the second stage of ignition. The location of the maximum temperature, $\dot{\unicode[STIX]{x1D714}}_{P}$ and $Y_{P}$ profiles converge to the $\unicode[STIX]{x1D709}_{ST}$ value.

Figure 12. (a–d) Shows four panels of the evolution of the maximum conditionally averaged values. (a) HRR, (b) $Y_{I}$ , $Y_{X}$ and $Y_{P}$ , (c) temperature and (d) $\unicode[STIX]{x1D709}$ . (e) Shows the corresponding location in $\unicode[STIX]{x1D709}$ space of the maximum values throughout the simulation.

Figure 13. The $\unicode[STIX]{x1D709}$ value (a) and $\unicode[STIX]{x1D712}$ value (b) for each ignition kernel at the time of formation.

Figure 14. Ignition delay times for each kernel, $\unicode[STIX]{x1D70F}_{k}$ , denoted by the blue circles, superimposed on levels of $\unicode[STIX]{x1D70F}$ in black lines. All ignition kernels form in a region of low $\unicode[STIX]{x1D70F}$ gradient.

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 $T_{HTC}=1400$  K. 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, $\unicode[STIX]{x1D70F}_{k}$ . 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 $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D712}$ space is presented in figure 13. All kernels form in rich mixtures between $\unicode[STIX]{x1D709}=0.1$ and 0.3 and at low $\unicode[STIX]{x1D712}$ values compared to the conditional mean (see figure 10). Ignition delay time results from a homogeneous reactor, see figure 14, show that the $\unicode[STIX]{x1D70F}$ profile has a ‘U-shape’ with a broad region of short $\unicode[STIX]{x1D70F}$ values in $\unicode[STIX]{x1D709}$ -space corresponding to the location of the ignition kernel formation. The shallow gradient of $\unicode[STIX]{x1D70F}$ 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. Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2017). However, in the present DNS the ignition kernels also form in mixtures leaner than $\unicode[STIX]{x1D709}_{MR}$ , which may be due to relatively shorter $\unicode[STIX]{x1D70F}$ values for $\unicode[STIX]{x1D709}<\unicode[STIX]{x1D709}_{MR}$ for the present n-heptane global chemical mechanism compared with the detailed DME chemical mechanism used in Krisman et al. (Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2017). This difference may also be due to the $\unicode[STIX]{x1D712}$ profile in $\unicode[STIX]{x1D709}$ space, which is much higher in mixtures richer than $\unicode[STIX]{x1D709}_{MR}$ compared to mixtures less rich than $\unicode[STIX]{x1D709}_{MR}$ , see figure 9. Another possible explanation is the lack of radical species in the four-step chemical 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 Reference Fu and Aggarwal2015; Pei et al. Reference Pei, Hawkes, Bolla, Kook, Goldin, Yang, Pope and Som2016), and DNS studies by Sreedhara & Lakshmisha (Reference Sreedhara and Lakshmisha2000, Reference Sreedhara and Lakshmisha2002), which identified autoignition to emerge over a wide range of rich $\unicode[STIX]{x1D709}$ values, although individual autoignition kernels were not identified in these studies.

Experimental studies of autoignition at atmospheric conditions have observed similar behaviour (Markides & Mastorakos Reference Markides and Mastorakos2005; Markides et al. Reference Markides, De Paola and Mastorakos2007; Markides & Mastorakos Reference Markides and Mastorakos2011). A quasi-steady, spatially evolving autoignition was observed for hydrogen (Markides & Mastorakos Reference Markides and Mastorakos2005), n-heptane (Markides et al. Reference Markides, De Paola and Mastorakos2007) and acetylene (Markides & Mastorakos Reference Markides and Mastorakos2011) 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. (Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2017), 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 $\unicode[STIX]{x1D70F}_{k}/\unicode[STIX]{x1D70F}$ 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 Reference Sreedhara and Lakshmisha2002) and also with a recent experiment of an autoigniting turbulent jet in highly heated coflow at atmospheric conditions, as determined by simultaneous measurements of $\unicode[STIX]{x1D709}$ and temperature (Papageorge et al. Reference Papageorge, Arndt, Fuest, Meier and Sutton2014).

In order to understand the difference between the $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D70F}_{k}$ 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 $T$ - $\unicode[STIX]{x1D709}$ and $Y_{P}$ - $\unicode[STIX]{x1D709}$ space. Given the initially bimodal distribution of $\unicode[STIX]{x1D709}$ (due to the initially thin mixing layer), most tracers originate from near either $\unicode[STIX]{x1D709}=1$ (pure fuel) or $\unicode[STIX]{x1D709}=0$ (pure air). The tracers initially move along the mixing line (in an adiabatic mixing process), followed by a gradual increase in temperature and $Y_{P}$ due the first stage of autoignition. The first stage of autoignition only proceeds after the tracers reach an intermediate $\unicode[STIX]{x1D709}$ value. After a delay, the gradual buildup of $Y_{P}$ 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.

Figure 15. The ensemble trajectories of ignition kernels in $T$ - $\unicode[STIX]{x1D709}$ (a) and $Y_{P}$ - $\unicode[STIX]{x1D709}$ (b) space. Each series of grey lines with black markers represents an ensemble member.

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 and intense period of mixing followed by a relaxation of $\unicode[STIX]{x1D712}$ to very low levels. The timing of the intense mixing is positively correlated with $\unicode[STIX]{x1D70F}_{k}$ , such that earlier intense mixing leads to early kernel formation. The evolution of $\unicode[STIX]{x1D709}_{k}$ is also presented and shows that the period of intense mixing leads to the kernels rapidly converging towards $\unicode[STIX]{x1D709}$ values at which ignition occurs. The first stage of ignition, as judged by the normalised $Y_{I}$ plots, slightly lags the peaks in the mixing. As the kernels reach appropriate $\unicode[STIX]{x1D709}$ values, and the mixing rates sufficiently relax, the LTC rapidly proceeds and leads to the steady buildup of $Y_{P}$ .

Figure 16. Kernel histories for $\unicode[STIX]{x1D712}$ (a), $\unicode[STIX]{x1D709}$ (b), $Y_{I}$ (c) and $Y_{P}$ (d). Line plots show example kernel histories and the circle markers show the values at the point of ignition.

Figure 17. Ignition kernels (circle markers) mapped to the spatially Favre-averaged jet (filled contours) for $\widetilde{\unicode[STIX]{x1D709}}$ (a) and $\widetilde{\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C}}$ (b). The Favre averages exploit the 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 white contour correspond to $\widetilde{\unicode[STIX]{x1D709}_{ST}}$ and $\widetilde{\unicode[STIX]{x1D709}_{MR}}$ , respectively. The circle markers and filled contours share a colour scale.

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 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 $\widetilde{\unicode[STIX]{x1D709}}$ for the jet with the kernels superimposed, while figure 17(b) shows the same information in terms of $\widetilde{\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C}}$ . In both cases, the kernels are coloured by their local instantaneous values of $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D712}$ , with the colour scales the same as used for the corresponding Favre-averaged variables across the jet. The kernels are widely dispersed in the $y^{\ast }$ direction and in time. All of the kernels form either in the shear layer where $\widetilde{\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C}}$ is near its maximum, or in the interior of the jet where $\widetilde{\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C}}$ is elevated and $\widetilde{\unicode[STIX]{x1D709}}$ is much richer than $\unicode[STIX]{x1D709}_{MR}$ . 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 $\unicode[STIX]{x1D712}$ and appropriate $\unicode[STIX]{x1D709}$ 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, $\unicode[STIX]{x1D735}\boldsymbol{U}=\unicode[STIX]{x1D63C}$ . In the method proposed by Chong, Perry & Cantwell (Reference Chong, Perry and Cantwell1990), the characteristic equation for $\boldsymbol{A}$ 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. Reference Chong, Perry and Cantwell1990) 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: $27R^{2}+(4P^{3}-18PQ)R+(4Q^{3}-P^{2}Q^{2})=0$ . It can be shown (Chong et al. Reference Chong, Perry and Cantwell1990) 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 $P\neq 0$ 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 ( $\unicode[STIX]{x1D70C}_{JET}/\unicode[STIX]{x1D70C}_{OX}\approx 10$ ).

Similarly to recent studies by Grout et al. (Reference Grout, Gruber, Yoo and Chen2011), Cifuentes et al. (Reference Cifuentes, Dopazo, M. and Jimenez2014) 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:

1. (i) Classification 1, Node/Node/Node, stable;

2. (ii) Classification 2, Node/Node/Node, unstable;

3. (iii) Classification 11, Node/Saddle/Saddle, stable;

4. (iv) Classification 12, Node/Saddle/Saddle, unstable.

Or as vortically dominated foci regions:

1. (i) Classification 18, Foci/Stretching, stable;

2. (ii) Classification 19, Foci/Stretching, unstable;

3. (iii) Classification 20, Foci/Compressing, stable;

4. (iv) Classification 21, Foci/Compressing, unstable.

Illustrations of these classifications are summarised in figure 18, which has been adapted from Cifuentes et al. (Reference Cifuentes, Dopazo, M. and Jimenez2014).

Figure 19(a) shows the joint probability density function (PDF) of the domain in $QR$ space just prior to ignition at $t^{\ast }=18$ (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 $t^{\ast }=9$ (near the peak turbulent intensity) until the end of the simulation. The extent of the joint PDF decreases rapidly following the HTC ignition at $t^{\ast }=19.8$ due to the increase in viscosity relative to inertial forces. Conditioning the joint PDF on $\unicode[STIX]{x1D709}$ (also not presented here), shows that the extent of the joint PDF increases with increasing $\unicode[STIX]{x1D709}$ , because low $\unicode[STIX]{x1D709}$ values reside mostly at the jet periphery while high $\unicode[STIX]{x1D709}$ values exist within the shear layer and turbulent jet core.

Figure 18. Cartoon illustrations of the flow topologies identified in this study. Reproduced from Cifuentes et al. (Reference Cifuentes, Dopazo, M. and Jimenez2014) (doi:http://dx.doi.org/10.1063/1.4884555), with the permission of AIP Publishing.

Figure 19. Statistics at $t^{\ast }=18$ , just before HTC ignition in the $QR$ plane: (a) joint PDF for $\unicode[STIX]{x1D709}>0.005$ (coloured on a log scale) where the white dashed line bounds the joint PDF conditioned on $0.1<\unicode[STIX]{x1D709}<0.3$ ; (b) mean $\dot{\unicode[STIX]{x1D714}_{P}}$ ; and (c) mean $\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C}$ corresponding to $0.1<\unicode[STIX]{x1D709}<0.3$ (the region delineated by the dashed line in (a)).

The white dashed isocontour in figure 19(a) bounds the joint PDF conditioned on $0.1<\unicode[STIX]{x1D709}<0.3$ , where all of the kernels form. The doubly conditioned means in $QR$ space for $\dot{\unicode[STIX]{x1D714}_{P}}$ and $\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C}$ , further conditioned on $0.1<\unicode[STIX]{x1D709}<0.3$ are also presented in figure 19 (b and c, respectively). $\dot{\unicode[STIX]{x1D714}_{P}}$ increases away from the origin and is generally higher for $Q\geqslant 0$ , but peaks near $Q\approx 0$ for positive $R$ . $\unicode[STIX]{x1D712}$ increased to the upper left and lower right of the joint PDF and is lower near the peak in $\dot{\unicode[STIX]{x1D714}_{P}}$ , revealing that the anti-correlation of $\unicode[STIX]{x1D712}$ and $\dot{\unicode[STIX]{x1D714}_{P}}$ is also apparent in $QR$ space. For samples with $P\approx 0$ (dilation free), the region of peak $\dot{\unicode[STIX]{x1D714}_{P}}$ and low $\unicode[STIX]{x1D712}$ correspond to classification 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 $\dot{\unicode[STIX]{x1D714}_{P}}$ . (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 $\unicode[STIX]{x1D709}$ 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 $\unicode[STIX]{x1D709}$ space ( $y$ -axis).

At $t^{\ast }=4.5$ , the most reactive samples are concentrating in the strain dominated N/S/S steady and unsteady topologies. By $t^{\ast }=9$ , 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 $\unicode[STIX]{x1D709}$ PDFs of the most reactive samples prior to ignition are located in rich $\unicode[STIX]{x1D709}$ values, where the LTC is concentrated. The grey shading between $t^{\ast }=19.8$ and $t^{\ast }=31.5$ 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 $0.05<\unicode[STIX]{x1D709}<0.3$ , which includes regions of both edge flames and expanding ignition kernels. The most reactive regions correspond to nodal topologies for the remainder of the simulation, with the largest contributions from Node/Saddle/Saddle unstable classification.

The value of $\dot{\unicode[STIX]{x1D714}_{P}}$ also depends on $\unicode[STIX]{x1D709}$ , and so figure 21 presents the same analysis for the following ranges of $\unicode[STIX]{x1D709}$ : $0.0<\unicode[STIX]{x1D709}<0.1$ , which experiences little chemistry until for formation of edge flames; $0.1<\unicode[STIX]{x1D709}<0.3$ , which experiences LTC before $t^{\ast }=19.8$ followed by ignition kernels and then rich premixed branches of edge flames; and $0.3<\unicode[STIX]{x1D709}<0.7$ where the LTC is most prominent and HTC develops gradually from $t^{\ast }\approx 27$ onwards. Figure 20 suggests that the HTC ignition develops from vortical regions (as judged by the result just before ignition at $t^{\ast }=18$ ) as has been previously suggested by Sreedhara & Lakshmisha (Reference Sreedhara and Lakshmisha2002) and observed for non-premixed, vortex-mixing layer autoignition at atmospheric pressure by Thévenin & Candel (Reference Thévenin and Candel1995). It is also seen that the most reactive regions are strain dominated following ignition (for $\unicode[STIX]{x1D709}<0.3$ ). This is consistent with prior DNS Grout et al. (Reference Grout, Gruber, Yoo and Chen2011), Cifuentes et al. (Reference Cifuentes, Dopazo, M. and Jimenez2014) that also showed flames to exist predominately within Node/Saddle/Saddle unstable type topologies.

Figure 20. (a) Evolution of the classification of the most reactive locations (top 0.01 % of $\dot{\unicode[STIX]{x1D714}_{P}}$ values) in the domain (in increments of $t^{\ast }=4.5$ ); blue bars represent the nodal type (strain dominated) modes and red bars represent foci type (vortically dominated) modes. (b) PDFs for $\unicode[STIX]{x1D709}$ 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 $\unicode[STIX]{x1D709}$ space ( $y$ -axis). The dashed line marks $\unicode[STIX]{x1D709}_{MR}$ and the solid line marks $\unicode[STIX]{x1D709}_{ST}$ . The grey shaded region in each panel marks the duration between the first and last kernel formation.

Figure 21. Evolution of the classification of the most reactive locations in the domain, conditioned on ranges of $\unicode[STIX]{x1D709}$ : $0.0<\unicode[STIX]{x1D709}<0.1$ (a), $0.1<\unicode[STIX]{x1D709}<0.3$ (b), $0.3<\unicode[STIX]{x1D709}<0.7$ (c). Colour bars and grey shading correspond to those presented in figure 20.

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, $\boldsymbol{k}$ , over time. For each kernel, the classification is plotted between the time when mixing starts (as judged by $\unicode[STIX]{x1D712}$ ) until the time when HTC ignition occurs (when $T_{OX}$ exceeds 1400 K). The results confirm that most kernels spend most of their history from mixing to ignition in vortical (foci) topologies.

Figure 22. 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.

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 conditions have identified $\unicode[STIX]{x1D712}$ 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, $S_{e}$ , are negatively correlated with $\unicode[STIX]{x1D712}$ during extinction processes (Pantano Reference Pantano2004; Hawkes, Sankaran & Chen Reference Hawkes, Sankaran and Chen2008; Karami et al. Reference Karami, Hawkes, Talei and Chen2016) but may have a non-monotonic correlation during reignition processes under some conditions (Hawkes et al. Reference Hawkes, Sankaran and Chen2008). A recent DNS study of edge-flame statistics at non-autoignitive conditions with simple chemistry (Karami et al. Reference Karami, Talei, Hawkes and Chen2017) 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 (Reference Chakraborty and Mastorakos2008) showed that $\unicode[STIX]{x1D712}$ influenced the reactivity of the expanding ignition kernels, which was also dependent on the forcing location in $\unicode[STIX]{x1D709}$ -space. Krisman et al. (Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2016) 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 $\unicode[STIX]{x1D712}$ reduced the displacement speed while fluctuations in the upstream mixture composition (due to varying upstream progress in autoignition) were not found to be strongly correlated. Minamoto & Chen (Reference Minamoto and Chen2016) 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 (Reference Choi and Chung2013), Al-Noman, Choi & Chung (Reference Al-Noman, Choi and Chung2015) 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 $\unicode[STIX]{x1D712}$ and $S_{e}$ and its components, and on the non-dimensional values of $s_{e}/s_{L}$ . 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 $\unicode[STIX]{x1D709}_{ST}$ surface with a surface of product mass fraction $Y_{P}=0.19$ , 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 $Y_{P}$ and $\unicode[STIX]{x1D709}_{ST}$ surfaces (a singular point in figure 23) defines the location of the edge flame. The edge-flame speed ( $S_{e}$ ) is defined in terms of the displacement speeds of the $Y_{P}$ isosurface ( $S_{Y_{P}}$ ) and the $\unicode[STIX]{x1D709}_{ST}$ isosurface ( $S_{\unicode[STIX]{x1D709}}$ ) and inner product of their respective surface-normal vectors ( $\boldsymbol{k}$ ), where $\boldsymbol{k}=1$ corresponds to parallel surfaces and $\boldsymbol{k}=0$ corresponds to orthogonal surfaces. The edge-flame speed is given by $S_{e}=(S_{Y_{P}}-\boldsymbol{k}S_{\unicode[STIX]{x1D709}}/\sqrt{1-\boldsymbol{ k}^{2}})$ , which may be derived using the same methodology as presented by Karami et al. (Reference Karami, Hawkes, Talei and Chen2015, Reference Karami, Hawkes, Talei and Chen2016). In this paper the edge-flame speed and its components are presented normalised by the laminar flame speed $s_{L}=1.18~\text{m}~\text{s}^{-1}$ calculated in § 2.3. It is important to note that while the intersection of $Y_{P}$ and $\unicode[STIX]{x1D709}_{ST}$ 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.

Figure 23. Cartoon of edge-flame cross-section. The solid black line represents the $Y_{P}=0.19$ isosurface and the dashed red line shows the $\unicode[STIX]{x1D709}_{ST}$ isosurface.

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. (Reference Sripakagorn, Mitarai, Kosály and Pitsch2004) identified three distinct reignition modes. These modes were: edge-flame propagation, which involves positive $S_{e}$ 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 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 $S_{e}$ and $\unicode[STIX]{x1D712}$ (Pantano Reference Pantano2004; Hawkes et al. Reference Hawkes, Sankaran and Chen2008; Karami et al. Reference Karami, Hawkes, Talei and Chen2016) and occurs at low values of $\boldsymbol{k}$ (Hawkes et al. Reference Hawkes, Sankaran and Chen2008); flame folding occurs at relatively higher $\unicode[STIX]{x1D712}$ values and high values of $\boldsymbol{k}$ (Hawkes et al. Reference Hawkes, Sankaran and Chen2008); and that independent flamelet reignition is a very small contribution to the reignition process (Sripakagorn et al. Reference Sripakagorn, Mitarai, Kosály and Pitsch2004) 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 $S_{e}$ with respect to $\unicode[STIX]{x1D712}$ and $\boldsymbol{k}$ (Hawkes et al. Reference Hawkes, Sankaran and Chen2008). A key finding was that during the reignition phase a non-monotonic relationship between $S_{e}$ and $\unicode[STIX]{x1D712}$ exists, such that peak $S_{e}$ values occurred at relatively high $\unicode[STIX]{x1D712}$ values. The joint PDF of $S_{e}$ and $\unicode[STIX]{x1D712}$ , combined with $k$ values doubly conditioned upon $S_{e}$ and $\unicode[STIX]{x1D712}$ , 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 $\unicode[STIX]{x1D709}_{ST}$ 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 $\unicode[STIX]{x1D709}_{ST}$ surface, and with high values of $\unicode[STIX]{x1D712}$ , due to the peak in $\unicode[STIX]{x1D712}$ observed at the leading edge of an expanding ignition kernel (as also observed by Mukhopadhyay & Abraham (Reference Mukhopadhyay and Abraham2012a ) and Krisman et al. (Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2016)).

Statistics gathered on the edge flames, over the entire simulation are presented in order to first consider the effects of $\unicode[STIX]{x1D712}$ on $S_{e}$ and its components. Here, all $\unicode[STIX]{x1D712}$ values are normalised by the critical scalar dissipation rate, $\unicode[STIX]{x1D712}_{C}=785~\text{s}^{-1}$ , as calculated in § 2.3.

Figure 24 presents the joint PDFs of the edge-flame speed, its components and $\boldsymbol{k}$ , with the logarithm of the normalised scalar dissipation rate, $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})$ . The edge-flame speed distribution is monomodal, centred at $s_{e}/s_{L}\approx 2.4$ (2.8  $\text{m}~\text{s}^{-1}$ ), with most values lying between 1.8 and 3.5. In terms of $\unicode[STIX]{x1D712}$ , almost all samples are below $\unicode[STIX]{x1D712}_{C}$ and the joint PDF mode is located at $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})\approx -1.5$ . The low values of $\unicode[STIX]{x1D712}$ on the $\unicode[STIX]{x1D709}_{ST}$ surface are due to the small value of $\unicode[STIX]{x1D709}_{ST}$ and the initial centreline $\unicode[STIX]{x1D709}$ value of unity, causing the conditional $\unicode[STIX]{x1D709}$ profile to peak in very rich mixtures and become small near $\unicode[STIX]{x1D709}_{ST}$ (see figure 9). It is possible, that for the same nominal turbulent and chemical conditions but with a conditional $\unicode[STIX]{x1D712}$ profile that peaks near $\unicode[STIX]{x1D709}_{ST}$ , 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.

Figure 24. Joint PDF of the logarithm of dissipation rates with $S_{e}/s_{L}$ , its components $(S_{Y_{P}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ and $(-\boldsymbol{k}S_{\unicode[STIX]{x1D709}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ and $\boldsymbol{k}$ . The colour map shows the joint PDF information and the grey dashed line indicates the mean ordinate value, conditioned upon the abscissa value, $\ln (\unicode[STIX]{x1D712})$ .

Regarding the correlations, for $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})$ between $-5$ and $-3.5$ , there is a positive correlation with $S_{e}$ , but for $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})$ between $-3.5$ and 0.5 (the most probable range of values), $S_{e}$ is negatively correlated. At very high $\unicode[STIX]{x1D712}$ values a slight increase in $S_{e}$ is observed. The conditional mean of $S_{e}$ peaks at approximately $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})=-3.5$ and the marginal PDF in $S_{e}$ space at this point is broad and skewed towards high values of $S_{e}$ . The first component of the edge-flame speed, $(S_{Y_{P}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ , is positive and higher in magnitude than $S_{e}$ . At high levels of $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})$ , there is a strong, positive correlation with increasing $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})$ . The second component of $S_{e}$ , $(-\boldsymbol{k}S_{\unicode[STIX]{x1D709}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ , has a more narrow joint distribution. At low levels of dissipation, the magnitude of $(-\boldsymbol{k}S_{\unicode[STIX]{x1D709}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ is small and the most likely values are negative. At high levels of dissipation the speed decreases rapidly, balancing the rapid increase in the $(S_{Y_{P}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ component. The results for $k$ reveal a positive correlation between $\unicode[STIX]{x1D712}$ and $k$ . The correlation is most pronounced for high values of $\unicode[STIX]{x1D712}$ 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 $\unicode[STIX]{x1D712}$ values for the components of $S_{e}$ . At lower $\unicode[STIX]{x1D712}$ values the $S_{e}/s_{L}$ statistics closely resemble those of the $(S_{Y_{P}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ component, with a near-uniform reduction due to the negative $(-\boldsymbol{k}S_{\unicode[STIX]{x1D709}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ term.

The results indicate that the scalar dissipation rate has a non-monotonic impact on the edge-flame speed. For low values of $\unicode[STIX]{x1D712}$ , the edge-flame speed is slightly promoted by increasing $\unicode[STIX]{x1D712}$ . This can be attributed to the increase in the $(S_{Y_{P}}/s_{L})/\sqrt{1-\boldsymbol{ k}^{2}}$ component. At intermediate values of $\unicode[STIX]{x1D712}$ , both components of $S_{e}$ are reduced with increasing $\unicode[STIX]{x1D712}$ , causing a modest attenuation in $S_{e}$ . At very high $\unicode[STIX]{x1D712}$ rates, the $Y_{P}$ and $\unicode[STIX]{x1D709}_{ST}$ surfaces become aligned, causing the components of $S_{e}$ to increase in magnitude rapidly and with opposite sign. Overall, the effect on $S_{e}$ is a slight increase at very high $\unicode[STIX]{x1D712}$ 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 $s_{e}/s_{L}$ . Firstly, the flow velocities near the stabilisation location in diesel flames are ${\approx}10~\text{m}~\text{s}^{-1}$  (Pei et al. Reference Pei, Hawkes, Bolla, Kook, Goldin, Yang, Pope and Som2016). This is only 2–3 times the observed (dimensional) local $s_{e}$ 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 the edge-flame structure, similar to how in premixed turbulent flames the net burning velocity is much larger than $s_{L}$ .

Figure 25. Left $y$ -axis: non-dimensional speeds for $s_{L}/s_{L,0}$ (circles), $\sqrt{\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{b}}s_{L}/s_{L,0}$ (pluses) and $\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{f}s_{L}/s_{L,0}$ (triangles) with respect to the ignition progress of the reactant mixture. Right $y$ -axis: PDF for the ignition progress measured $2\times \unicode[STIX]{x1D6FF}_{L}$ ahead of the edge flame on the $\unicode[STIX]{x1D709}_{ST}$ isosurface.

The second observation is that the non-dimensionalised speed $s_{e}/s_{L}$ exceeds unity. This is expected for both hydrodynamic reasons due to flow expansion (Ruetsch, Vervisch & Liñán Reference Ruetsch, Vervisch and Liñán1995; Im & Chen Reference Im and Chen1999) and thermochemical reasons due to enhanced flame propagation into a partially reacted mixture ahead of the flames (Minamoto & Chen Reference Minamoto and Chen2016). 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, $\unicode[STIX]{x1D70C}_{f}$ , to the unburnt density, $\unicode[STIX]{x1D70C}_{u}$ (Echekki & Chen Reference Echekki and Chen1998; Im & Chen Reference Im and Chen1999). Here, this equates to the expression $s_{e}/s_{L}=\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{f}=1.88$ . Alternatively, the deviation between $s_{L}$ and $s_{e}$ can be explained by the divergence of streamlines across the edge flame (Ruetsch et al. Reference Ruetsch, Vervisch and Liñán1995). 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 $s_{e}/s_{L}=\sqrt{\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{b}}=1.47$ , where $\unicode[STIX]{x1D70C}_{b}$ 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 $\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{f}=1.88$ . This factor is insufficient to account for the values of $s_{e}/s_{L}$ 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 (Reference Minamoto and Chen2016), the value of $s_{L}$ 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 $Y_{P}/Y_{P,b}$ , where $Y_{P,b}$ is the burnt value of $Y_{P}$ and the laminar flame speed at zero progress, $s_{L,0}$ , 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 $\unicode[STIX]{x1D709}_{ST}$ isosurface during ignition that are $2\times \unicode[STIX]{x1D6FF}_{L}$ (Sensitivity to this distance was assessed for values between 1.5 and $3\times \unicode[STIX]{x1D6FF}_{L}$ 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, $s_{L}/s_{L,0}$ increases with ignition progress but by itself does not account for the discrepancy with the observed values of $s_{e}/s_{L}$ . The correction factor $\sqrt{\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{b}}s_{L}/s_{L,0}$ is also lower than most observed values of $s_{e}/s_{L}$ . However, the correction factor due to dilatation at the flame location, $\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{f}s_{L}/s_{L,0}$ , is close to the range of likely values of $s_{e}/s_{L}$ as shown in figure 24. Even with this correction, there is still a slight discrepancy between $s_{e}/s_{L}$ (higher) and $\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{f}s_{L}/s_{L,0}$ (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 $s_{e}/s_{L}$ statistics, which do not distinguish between ignition modes; (ii) turbulent enhancement of $s_{e}/s_{L}$ , via wrinkling of the edge-flame front; (iii) the definition of $s_{L}$ 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 $\unicode[STIX]{x1D709}_{ST}$ isosurface is primarily explained by autoignition-assisted edge-flame propagation, as opposed to the expansion of independent autoignition kernels from rich mixtures engulfing the $\unicode[STIX]{x1D709}_{ST}$ 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 $\unicode[STIX]{x1D709}_{ST}$ isosurface over time. For example, figure 7 showed the evolution of the state of the $\unicode[STIX]{x1D709}_{MR}$ 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 $\unicode[STIX]{x1D709}_{ST}$ isosurface, but that it is edge-flame propagation that consumes the majority of the $\unicode[STIX]{x1D709}_{ST}$ isosurface.

Figure 26. Ensemble mean of alignment factor $k$ , conditioned upon $\unicode[STIX]{x1D712}$ and $S_{e}$ . Superimposed in white is the joint PDF of $\unicode[STIX]{x1D712}$ and $S_{e}$ and superimposed in black is the mean value of $S_{e}$ conditioned on $\unicode[STIX]{x1D712}$ , as presented in figure 24.

In order to provide further evidence for the relative contribution of ignition modes, the ensemble-averaged alignment factor $k$ , doubly conditioned upon $S_{\unicode[STIX]{x1D709}}$ and $\unicode[STIX]{x1D712}$ , is presented alongside the joint PDF of $S_{\unicode[STIX]{x1D709}}$ and $\unicode[STIX]{x1D712}$ in figure 26. For $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})<-0.5$ and $S_{e}/s_{L}<4.5$ , the alignment factor takes a low to intermediate value, consistent with an edge-flame propagation ignition mode (Hawkes et al. Reference Hawkes, Sankaran and Chen2008). This corresponds to the most probable region of the joint PDF where the $s_{e}/s_{L}$ values correspond to those predicted by the $\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{f}s_{L}/s_{L,0}$ 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 flames previously observed with detailed dimethyl-ether chemistry in two dimensions (Krisman et al. Reference Krisman, Hawkes, Talei, Bhagatwala and Chen2015) and three dimensions (Minamoto & Chen Reference Minamoto and Chen2016) DNS.

Two regions of high $\boldsymbol{k}$ value and low probability are also observed. The first region occurs for $\ln (\unicode[STIX]{x1D712}/\unicode[STIX]{x1D712}_{C})>-0.5$ . 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 $\unicode[STIX]{x1D712}$ . Flame folding requires a strong interaction between the turbulent field and the stoichiometric surface. However, due to the low value of $\unicode[STIX]{x1D709}_{ST}$ in the present case, the $\unicode[STIX]{x1D709}_{ST}$ 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 $\unicode[STIX]{x1D712}$ , high $\boldsymbol{k}$ , but low probability region of the joint PDF.

The second region occurs for $S_{e}/s_{L}>4$ . This region coincides with an extremely low probability region of the joint PDF at intermediate $\unicode[STIX]{x1D712}$ 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 $\unicode[STIX]{x1D709}_{ST}$ surface do. This causes the $Y_{P}$ surface to become aligned with the $\unicode[STIX]{x1D709}_{ST}$ surface and therefore results in a high value of $k$ , but not necessarily a larger value of $\unicode[STIX]{x1D712}$ . Visual inspection also suggests that the edge-flame collisions are associated with an acceleration of the ignition of the $\unicode[STIX]{x1D709}_{ST}$ 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 $S_{e}$ , high $k$ , but low probability region of the joint PDF. This ignition mode therefore introduces a positive bias to the mean values of $s_{e}/s_{L}$ reported here and may explain why some samples have $s_{e}/s_{L}$ larger than expected from the $\unicode[STIX]{x1D70C}_{u}/\unicode[STIX]{x1D70C}_{f}s_{L}/s_{L,0}$ scaling argument.