Effects of Mixture Distribution on Localized Forced Ignition of Stratified Mixtures: A Direct Numerical Simulation Study

ABSTRACT The influences of initial mixture distribution on localized forced ignition of globally stoichiometric stratified mixtures have been analyzed using three-dimensional compressible direct numerical simulations. The globally stoichiometric mixtures (i.e., ) for different root-mean-square (rms) values of equivalence ratio (i.e., = 0.2, 0.4, and 0.6) and the Taylor micro-scale of equivalence ratio variation (i.e., 2.1, 5.5, and 8.3 with being the Zel’dovich flame thickness of stoichiometric mixture) have been analyzed for different initial rms values of turbulent velocity . The equivalence ratio variation is initialized following both Gaussian and bi-modal distributions for a given set of values of and in order to analyze the effects of mixture distribution. The localized forced ignition is accounted for by considering a source term in the energy conservation equation that deposits energy for a stipulated time interval. It has been demonstrated that the initial equivalence ratio distribution has significant effects on the extent of burning of stratified mixtures following successful localized forced ignition. It has been found that an increase in ) has adverse effects on the burned gas mass, whereas the effects of on the extent of burning are non-monotonic and dependent on for initial bi-modal mixture distribution. The initial Gaussian mixture distribution exhibits an increase in burned gas mass with decreasing , but these cases are more prone to flame extinction for high values of than the corresponding bi-modal distribution cases. Detailed physical explanations have been provided for the observed mixture distribution, , , and dependences on the extent of burning following localized forced ignition of stratified mixtures.


Introduction
Premixed combustion offers an option of controlling flame temperature and reducing pollutant (e.g., NO x ) emission, but, in practice, perfect mixing is often difficult to achieve and thus combustion in many engineering applications takes place in turbulent stratified mixtures. A number of previous analyses concentrated on flame propagation in stratified mixtures based on experimental (Anselmo-Filho et al., 2009;Balusamy et al., 2014;Grune et al., 2013;Kang and Kyritsis, 2005;Mulla and Chakravarthy, 2014;Renou et al., 2004;Samson, 2002;Sweeney et al., 2013;Zhou et al., 1998Zhou et al., , 2013 and direct numerical simulations (DNS) (Cruz et al., 2000;Haworth et al., 2000;Hélie and Trouvé, 1998;Jiménez et al., 2002;Malkeson and Chakraborty, 2010;Patel and Chakraborty, 2014;Pera et al., 2013;Swaminathan et al., 2007) data. These studies demonstrated that the flame propagation statistics are strongly affected by the local gradient of equivalence ratio. It has been found that local variations of equivalence ratio have significant influences on the statistical behavior of surface density function (SDF) and flame curvature for small values of turbulent Reynolds number (Anselmo-Filho et al., 2009), while these effects seem to disappear when turbulence intensity increases . Recently, Pera et al. (2013) demonstrated that the mixture inhomogeneities typical of cycle-to-cycle variation in internal combustion (IC) engines can have significant influences on flame wrinkling based on mainly two-dimensional (2D) detailed chemistry DNS simulations. A similar qualitative conclusion was also drawn by Swaminathan et al. (2007) based on a preliminary analysis centered on simple chemistry 3D DNS simulations. Patel and Chakraborty (2014) have recently demonstrated, based on DNS simulations, that the root-mean-square (rms) value of equivalence ratio and the length scale of equivalence ratio fluctuations have profound influences on the extent of burning following successful forced ignition of stratified mixtures where the equivalence ratio fluctuation was initialized by a presumed bi-modal distribution.
Forced ignition of inhomogeneous mixtures arising from evaporation of droplets in an air stream has been experimentally analyzed in depth by Ballal and Lefebvre (1980) in relation to critical spark energy and optimum spark duration. Ballal and Lefebvre (1980) showed that an increase in turbulent velocity fluctuation has detrimental effects on successful ignition and early stages of combustion of inhomogeneous mixtures. Similar effects have been reported for premixed combustion based on both experimental (Ballal and Lefebvre, 1977a;Huang et al., 2007) and numerical (Chakraborty et al., 2007;Klein et al., 2008;Poinsot et al., 1995) analyses. A number of DNS-based analyses (Chakraborty et al., 2007Chakraborty and Mastorakos, 2008) also demonstrated that an increase in turbulent velocity fluctuation for a given value of integral length scale of turbulence has a detrimental effect on the success of localized forced ignition of single-phase gaseous inhomogeneous mixtures. Recently, a similar effect has been reported by Patel and Chakraborty (2014) for localized ignition of turbulent stratified mixtures with initial bi-modal distribution of equivalence ratio. The experimental data by Ahmed and Mastorakos (2006) demonstrated that an increase in mean velocity in jets also leads to a deterioration of ignition performance, which was quantified by a reduction in ignition probability. The detrimental effects of turbulent velocity fluctuation have also been reported for localized ignition of droplet-laden mixtures (Wandel, 2013(Wandel, , 2014Wandel et al., 2009).
Most existing analyses on localized forced ignition of inhomogeneous gaseous mixtures (Chakraborty et al., 2007Chakraborty and Mastorakos, 2008) have been carried out for a mixture distribution, which is characterized by a mean variation of equivalence ratio ϕ. By contrast, localized forced ignition of stratified mixtures characterized by a constant global mean value of equivalence ratio hϕi with non-zero rms values of equivalence ratio ϕ 0 has received limited attention (Patel and Chakraborty, 2014;Pera et al., 2013;Swaminathan et al., 2007) in spite of its practical importance in direct injection (DI) engines and lean premixed prevaporized (LPP) combustors. Patel and Chakraborty (2014) demonstrated that the rms value of equivalence ratio ϕ 0 and the length scale of equivalence ratio variation have significant influences on the early stages of combustion following localized forced ignition of stratified mixtures. Their analysis was limited to inhomogeneous mixtures with initial equivalence ratio distributions characterized by bi-modal probability density functions (pdfs). However, the effects of the nature of equivalence ratio distribution on localized forced ignition of stratified mixtures for a given set of global mean and rms values of equivalence ratio (i.e., hϕi and ϕ 0 ) and the associated length scale of mixture inhomogeneity are yet to be analyzed in detail. The current study aims to address the aforementioned gap in existing literature. In the current analysis, 3D compressible direct numerical simulations (DNS) have been carried out for localized forced ignition of globally stoichiometric stratified mixtures (i.e., hϕi ¼ 1:0) for a range of different rms turbulent velocity u 0 for both initial Gaussian and bi-modal distributions of equivalence ratio for a given set of values of ϕ 0 and the Taylor micro-scale l ϕ of equivalence ratio variation. The Taylor microscale l ϕ of the equivalence ratio variation is defined as (Eswaran and Pope, 1988): where the angle bracket indicates the global mean evaluated over the whole of computational domain. The main objectives of the present study are: (1) to demonstrate the influences of initial mixture inhomogeneity distribution on localized forced ignition and early stages of combustion for globally stoichiometric mixtures; and (2) to provide physical explanations for the observed mixture distribution, u 0 , ϕ 0 , and l ϕ dependences of the extent of burning following successful localized forced ignition of stratified mixtures.
The remainder of the article will be organized as follows. The information related to mathematical background and numerical implementation pertaining to the current analysis will be presented in the next section. This will be followed by the presentation of the results and subsequent discussion. The main findings will be summarized and conclusions will be drawn in the final section of this article.

Mathematical background and numerical implementation
It is extremely expensive to carry out detailed chemistry 3D DNS simulations for an extensive parametric analysis (Chen et al., 2009), as done in the current study. Thus, a modified single step chemical mechanism (Tarrazo et al., 2006) has been considered here: where s indicates the mass of oxidizer consumed per unit mass of fuel consumption under stoichiometric conditions and the fuel reaction rate is given by an Arrhenius-type expression (Chakraborty et al., 2007Chakraborty and Mastorakos, 2008;Mastorakos et al., 1997;Tarrazo et al., 2006;Wandel, 2013Wandel, , 2014Wandel et al., 2009): where Y F and Y O are the local fuel and oxidizer mass fractions respectively, ρ is the gas density and T ¼T À T 0 À Á = T ad ϕ¼1 ð Þ À T 0 À Á is the nondimensional temperature, withT, T 0 , and T ad ϕ¼1 ð Þ being the instantaneous dimensional temperature, unburned gas temperature, and the adiabatic flame temperature of the stoichiometric mixture, respectively.
(2) to mimic the realistic equivalence ratio dependence of the unstrained laminar burning velocity S b ϕ ð Þ (Tarrazo et al., 2006) obtained from experimental measurements (Egolfopoulos et al., 1989). The Lewis numbers of all species are taken to be unity for the current analysis. The combustion is assumed to take place in the gaseous phase where all the species are considered to be perfect gases. The mixture inhomogeneity in stratified mixtures is often characterized in terms of mixture fraction , which can be expressed in terms of both fuel and oxidizer mass fractions (i.e., Y F and Y O ) as: (Bilger, 1988) where Y F1 is the fuel mass fraction in the pure fuel stream and Y O1 is the oxidizer mass fraction in air. The equivalence ratio ϕ can be expressed in terms of mixture fraction and the stoichiometric mixture fraction st as: For the present analysis, s ¼ 4; Y F1 ¼ 1:0, and Y O1 ¼ 0:233 have been taken, which yields Y Fst ¼ 0:055 and st ¼ 0:055. These values represent methane-air mixtures. The extent of the completion of the chemical reaction can be characterized by a reaction progress variable c, which is defined as (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Hélie and Trouvé, 1998;Malkeson and Chakraborty, 2010;Patel and Chakraborty, 2014): According to this equation, c rises monotonically from 0 in the fully unburned reactants to 1.0 in the fully burned products. The heat addition by the ignitor is accounted for by a source term q 000 in the energy conservation equation (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Neophytou et al., 2010;Patel and Chakraborty, 2014): It is worth noting that in Eq. (6) the term @ @x i ρ the present analysis as the specific heats at constant pressure and volume are taken to be constant and the same for all of the species. In Eq. (6), h s;k is the specific enthalpy, P is the pressure, τ ki is the viscous shear stress, _ w T ¼ _ w F j jH ϕ is the source term originating from heat release due to combustion, and E ¼ ò is the specific stagnation internal energy.
The source term q 000 in Eq. (6) is assumed to follow a Gaussian distribution in the radial direction from the center of the ignitor (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Espi andLiñán, 2001, 2002;Neophytou et al., 2010;Patel and Chakraborty, 2014;Wandel, 2013Wandel, , 2014Wandel et al., 2009) and is expressed in the following manner: q 000 r ð Þ ¼ A q exp Àr 2 =2R 2 ð Þ , where r is the radial direction from the center of the ignitor and R is the characteristic width of energy deposition, which is taken to be ð Þ is the Zel'dovich flame thickness of the stoichiometric mixture with D 0 and S b ϕ¼1 ð Þ being the unburned gas diffusivity and the unstrained laminar burning velocity of the stoichiometric mixture, respectively. The constant A q is determined by the volume integration _ Q ¼ ò V q 000 dV, where _ Q is the ignition power, which is defined as: where a sp is a parameter that determines the total energy deposited by the ignitor and is taken to be a sp ¼ 3:6 in the current analysis following previous studies (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Patel and Chakraborty, 2014). The energy deposition duration t sp is expressed as: where b sp is the energy deposition duration parameter and t f is a characteristic chemical timescale given by: The parameter b sp for optimum spark duration varies between 0:2 < b sp < 0:4 (Ballal and Lefebvre, 1977b) and b sp ¼ 0:2 has been taken for the current analysis following previous studies (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Patel and Chakraborty, 2014;Wandel, 2013Wandel, , 2014Wandel et al., 2009). The details of spark formation (momentum modification contribution, plasma formation, and shock wave) are kept beyond the scope of the present analysis in order to keep this study computationally feasible. The pseudo-spectral methods proposed by Eswaran and Pope (1988) and Rogallo (1981) were used for generating initial ϕ fields following bi-modal and Gaussian distributions, respectively. The pseudo-spectral method proposed by Rogallo (1981) is also used to produce a homogeneous isotropic incompressible velocity field to initialize turbulent velocity fluctuations according to the Batchelor-Townsend spectrum (Batchelor and Townsend, 1948). For the current analysis, a globally stoichiometric (i.e., hϕi ¼ 1:0) mixture has been considered. The initial magnitudes of the normalized rms value of turbulent velocity u 0 =S b ϕ¼1 ð Þ , rms value of equivalence ratio ϕ 0 , and normalized Taylor micro-scale of equivalence ratio variation l ϕ =l f are listed in Table 1. The ratio of longitudinal integral length scale to flame thickness is considered to be L 11 =l f ¼ 3:36 for all turbulent cases following previous analyses (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Neophytou et al., 2010;Patel and Chakraborty, 2014;Wandel, 2013Wandel, , 2014Wandel et al., 2009). The case names are chosen in such a manner so that G and B stand for initial Gaussian and bi-modal equivalence ratio ϕ distributions; T0, T4, and T6 indicate increasing turbulence intensity u 0 =S b ϕ¼1 ð Þ ; X, Y, and Z indicate increasing values of ϕ 0 ; and D, E, and F denote increasing values of l ϕ =l f (e.g., GYT6F corresponds to a case with initial Gaussian ϕ distribution with initial values of ϕ 0 ¼ 0:4; u 0 =S b ϕ¼1 ð Þ ¼ 6:0; lϕ=l f ¼ 8:3). The initial mixture distributions for ϕ 0 ¼ 0:4 with different values of l ϕ =l f are shown in Figure 1, which indicates that the clouds of mixture inhomogeneities increase in size with increasing l ϕ =l f .
A compressible 3D DNS code SENGA (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Neophytou et al., 2010;Patel and Chakraborty, 2014, Wandel, 2013, 2014Wandel et al., 2009) was used to carry out the simulations in a domain of size 33l f Â 33l f Â 33l f , which ensures that about 10 integral eddies are retained on each side of the domain. The simulation domain is discretized by a Cartesian grid of size 200 Â 200 Â 200 with uniform grid spacing Δx. This grid spacing ensures 10 grid points within the thermal flame thickness δ th ϕ¼1 where η is the Kolmogorov length scale. The boundaries in Table 1. Initial values of the simulation parameters.

Gaussian
Gaussian Gaussian Bi-modal Bi-modal Bi-modal the x 1 -direction are taken to be partially nonreflecting and are specified using the Navier-Stokes characteristic boundary conditions (NSCBC) technique (Poinsot and Lele, 1992), whereas the boundaries in the other directions are considered to be periodic. A 10th-order central difference scheme is used for spatial differentiation for the internal grid points, and the order of differentiation gradually reduces to a one-sided 2nd-order scheme at the nonperiodic boundaries. The time advancement is carried out using a third order lowstorage Runge-Kutta scheme (Wray, 1990 is the initial eddy turnover time with k being the initial turbulent kinetic energy. Statistics for all cases are presented at t ¼ 8:40t sp ¼ 1:68t f , which corresponds to about 2t e and 3t e for the initial values of u 0 =S b ϕ¼1 ð Þ ¼ 4:0 and u 0 =S b ϕ¼1 ð Þ ¼ 6:0, respectively. The simulation time used in the current analysis is comparable to that used in several analyses, which contributed to the fundamental physical understanding of localized forced ignition in the past (Chakraborty et al., 2007;Chakraborty and Mastorakos, 2008;Neophytou et al., 2010;Patel and Chakraborty, 2014, Wandel, 2013, 2014Wandel et al., 2009;Yu and Bai, 2013).

Results and discussion
The temporal evolution of the nondimensional maximum temperature (i.e., T max 1 T max À T 0 À Á = T ad ϕ¼1 ð Þ À T 0 À Á and the normalized maximum fuel reaction rate magni- Þ ) for the initial Gaussian and bi-modal distributions are shown in Figures 2 and 3, respectively. It can be seen from Figures 2 and 3 that T max rises with time due to energy deposition during 0<t<t sp , which eventually gives rise to a thermal runaway once T max attains a value close to T c % 1 À 1=β ϕ¼1 , leading to rapid increases in T max and ð _ Ω F Þ max with time. The high thermal gradient between the hot gas kernel and the surrounding unburned gas leads to a high rate of heat transfer from the ignition kernel. This, in turn, leads to decreases in T max and ð _ Ω F Þ max with time once the ignitor is switched off, but T max ultimately settles to the nondimensional adiabatic flame temperature of the stoichiometric mixtures (i.e., T % 1:0) and ð _ Ω F Þ max settles to a much smaller value, which no longer changes appreciably with time for t ) t sp in the cases where self-sustained combustion is obtained following successful ignition.
A comparison between Figures 2 and 3 shows that self-sustained combustion has been obtained for all Gaussian distribution cases with u 0 =S b ϕ¼1 ð Þ ¼ 0.0 and 4.0, whereas some cases with initial bi-modal distribution (e.g., BT4YE and BT4ZE) fail to achieve selfsustained combustion for u 0 =S b ϕ¼1 ð Þ ¼ 4.0. It has been found that the bi-modal distribution cases with initial l ϕ =l f ¼ 5:5 are more prone to flame extinction at t > t sp than the initial l ϕ =l f ¼ 2:1 and 8.3 cases for initial ϕ 0 ¼ 0.4 and 0.6, but no such trend is observed for the corresponding cases with initial Gaussian equivalence ratio distribution. By contrast, some cases with initial bi-modal distribution show more resistance to flame extinction at t > t sp for initial u 0 =S b ϕ¼1 ð Þ ¼ 6.0 (e.g., the initial bi-modal distribution cases BT6XD, BT6XE, BT6XF, and BT6YF exhibit self-sustained combustion) than the corresponding cases with initial Gaussian distribution (all cases with initial Gaussian distribution fail to achieve self-sustained combustion for initial u 0 =S b ϕ¼1 ð Þ ¼ 6.0). In general, the probability of flame extinction at t > t sp (i.e., without any external addition of energy) increases with increasing u 0 =S b ϕ¼1 ð Þ for both initial bi-modal and Gaussian distribution cases. The observations made from Figures 2 and 3 indicate that the nature of initial mixture distribution, u 0 =S b ϕ¼1 ð Þ , ϕ 0 , and l ϕ =l f , have important influences on the possibility of obtaining self-sustained combustion following successful ignition in stratified mixtures.
The distributions of fuel mass fraction (i.e., Y F ), nondimensional temperature (i.e., T), normalized fuel reaction rate magnitude (i.e., _ Þ ), and equivalence ratio ϕ at t ¼ 1:05t sp and t ¼ 8:40t sp at the central x 1 À x 2 plane for the cases GT4YD and BT4YD are shown in Figure 4. Similar qualitative behavior has been observed for other cases but the burned gas volume has been found to decrease with increasing u 0 =S b ϕ¼1 ð Þ irrespective of the nature of the initial mixture distribution. The contours of T remain ð Þ for all cases with initial Gaussian mixture distributions from Table 1  approximately spherical during the period of energy deposition but they become increasingly wrinkled as time progresses for all turbulent cases. The evolution of T contours is principally determined by the diffusion of deposited energy during 0<t<t sp , whereas after ignition the evolution of isotherms depends on the magnitude of the reaction rate at the local mixture composition and the flame stretch induced by the background fluid motion. The stretch rate dependencies of local flame propagation are found to be qualitatively similar to those reported earlier by Malkeson and Chakraborty (2010) and thus are not repeated here for the sake of brevity. Additionally, it can be seen from Figure 4 that the level of nonuniformity of ϕ decreases as time progresses in all cases. The evolution of the mixing process can be illustrated by the temporal evolution of the pdfs of ϕ, which is shown in Figure 5. The cases with an initial Gaussian distribution exhibit higher probabilities of finding ϕ % hϕi ¼1:0 than the corresponding cases with an initial bi-modal distribution. The equivalence ratio pdfs for the initial Gaussian distribution cases show peak values at ϕ % hϕi ¼1:0, whereas the cases with initial bi-modal distribution show higher probabilities of finding ϕ < hϕi and ϕ > hϕi than the initial Gaussian distribution cases. Figure 5 shows that the pdf of ϕ for the initial bi-modal distribution cases approaches an approximate Gaussian distribution as time progresses due to mixing, ð Þ for all cases with initial bi-modal mixture distributions from Table 1  whereas the width of ϕ pdf decreases, and the peak value of ϕ pdf at ϕ % hϕi ¼1:0 increases with time in the initial Gaussian distribution cases. A comparison between the equivalence ratio pdfs between the initial bi-modal and Gaussian distribution cases reveals that the probability of finding ϕ % 1:0 (ϕÞ1:0) is greater for the cases with initial Gaussian (bi-modal) distribution than those with initial bi-modal (Gaussian) distribution for a given set of values of ϕ 0 and l ϕ =l f . It can further be seen from Figure 5 that the effects of mixing are stronger for smaller values of l ϕ =l f for a given value of u 0 =S b ϕ¼1 ð Þ , as the mean scalar dissipation rate N ¼ DÑ Á Ñ of mixture fraction scales as hN i,D 0 2 =l 2 ϕ , Gaussian distribution case GT4YD Bimodal distribution case BT4YD Figure 4. Distribution of mass fraction (i.e., Y F -1st row), nondimensional temperature (i.e., T-2nd row), normalized fuel reaction rate magnitude (i.e., _ ð Þ -3rd row), and equivalence ratio (i.e., ϕ-4th row) at t ¼ 1:05t sp and t ¼ 8:40t sp at the central x 1 À x 2 plane for the cases GT4YD (columns 1 and 2) and BT4YD (columns 3 and 4). The white broken line shows the stoichiometric mixture fraction ¼ st .
where 0 is the rms value of mixture fraction. This suggests that N is likely to assume high magnitudes for small values of l ϕ =l f for a given value of 0 . This behavior can be confirmed from the temporal evolution of ϕ 0 shown in Figure 5, which demonstrates that the high rate of mixing (i.e., rapid decay rate of ϕ 0 ) is associated with small values of l ϕ =l f and vice versa. Furthermore, the probability of finding high values of N increases with increasing ð Þ for cases with initial ϕ 0 ¼0:2 and l ϕ =l f ¼ 2:1. u 0 =S b ϕ¼1 ð Þ , as turbulent straining acts to generate scalar gradient (Vedula et al., 2001), which in turn increases the rate of micro-mixing. This is consistent with high decay rate of ϕ 0 for high values of u 0 =S b ϕ¼1 ð Þ (see Figure 5). It is vital to understand the flame structure originating from localized forced ignition to explain the observed burning behavior for different mixture distributions. The mode of combustion can be characterized by the flame index I c ¼ ÑY F Á ÑY O =ð ÑY F j j ÑY O j jÞ, which assumes positive (negative) values for premixed (non-premixed) mode of combustion (Hélie and Trouvé, 1998;Yamashita et al., 1996). The percentage of overall heat release arising from premixed (i.e., I c > 0) and non-premixed (i.e., I c < 0) modes of combustion at t ¼ 8:40t sp are shown in Figure 6 for a selection of cases. It is evident from Figure 6 that the chemical reaction takes place predominantly in premixed mode but some pockets of non-premixed combustion can also be found. Moreover, Figure 6 shows that the percentage of heat release from the premixed mode of combustion is greater in the initial Gaussian mixture distribution cases in comparison to the initial bi-modal distribution cases. This is consistent with higher probability of finding ϕ % hϕi in the initial Gaussian mixture distribution cases than in the cases with initial bi-modal distribution, as shown in Figure 5. The percentage of heat release arising from the non-premixed mode of combustion increases with increasing ϕ 0 for both types of initial mixture distributions, but this effect is more prominent in the initial bi-modal distribution cases in comparison to the initial Gaussian distribution cases. Furthermore, the percentage of heat release arising from non-premixed combustion (i.e., I c < 0) decreases with decreasing (increasing) values of l ϕ =l f (u 0 ) for a given value of ϕ 0 , as a result of improved mixing due to high magnitudes of N , and this effect is more prominent in the initial bi-modal distribution cases than in the initial Gaussian distribution cases (see Figure 6c). It is important to understand the reaction zone structure of the flames initiated by the localized forced ignition, in order to explain the effects of stratification on the extent of burning following successful ignition. The scatter of _ ð Þ with reaction progress variable c is presented in Figure 7a at t ¼ 8:40t sp for the case GT4YF and a similar qualitative behavior has been observed for all cases, including the initial bi-modal distribution cases (Patel and Chakraborty, 2014), where the self-sustained combustion has been obtained. Figure 7a shows that the high values of _ Ω F are obtained close to c ¼ 0:8, which is consistent with reaction rate profiles obtained from previous analyses (Chakraborty et al., 2007Chakraborty and Mastorakos, 2008;Patel and Chakraborty, 2014). The scatter of _ Ω F with mixture fraction for the case GT4YF is presented in Figure 7b, which shows that a considerable amount of scatter, and the same qualitative behavior is observed for other cases including the initial bi-modal distribution cases (Patel and Chakraborty, 2014). The large variation of nondimensional temperature T on a given isosurface (due to both unburned and burned contributions) ð Þ with reaction progress variable c, (b) scatter of _ Ω F with mixture fraction , (c) variation of the mean values of _ Ω F conditional on mixture fraction for reaction progress variable range 0:5 c 0:9, (d) scatter of _ Ω F with normalized scalar dissipation rate Λ ¼ N t f = 2 st 1 À st ð Þ 2 , (e) variation of the mean values of _ Ω F conditional on normalized scalar dissipation rate Λ for the progress variable range 0:7 c 0:9. All variation are shown for the case GT4YF. leads to a large extent of scatter of _ Ω F . Furthermore, Figure 7b shows that the high values of _ Ω F are obtained towards the slightly rich side (i.e., % 0:06), which corresponds to ϕ % 1:10, where the unstrained planar laminar burning velocity attains its maximum value (Egolfopoulos et al., 1989;Tarrazo et al., 2006). The temporal evolution of the mean values of _ Ω F conditional of mixture fraction in the region corresponding to 0:5 c 0:9 (i.e., where the high values of reaction rate magnitudes are obtained; see Figure 7a) for the case GT4YF is shown in Figure 7c for different time instants. Figure 7c demonstrates that the temperature distribution remains qualitatively similar following successful ignition and the same qualitative behavior has been observed for other cases. In stratified mixture combustion, the scalar dissipation rate of mixture fraction N ξ = D ▽ ξ• ▽ ξ also affects the reaction zone structure (Chakraborty et al., 2007Chakraborty and Mastorakos, 2008;Im et al., 1998;Mastorakos et al., 1997;Patel and Chakraborty, 2014). The scatter of _ Ω F with varying normalized scalar dissipation rate Λ ¼ N t f = 2 st 1 À st ð Þ 2 at t ¼ 8:40t sp for the case GT4YF is shown in Figure 7d. Additionally, the mean values of _ Ω F conditional on the values for Λ in the most reactive region (i.e., 0:7 c 0:9) is shown in Figure 7e for the case GT4YF. Figure 7e shows a predominant negative correlation between _ Ω F and Λ , which further indicates that _ Ω F attains relatively high values where combustion takes place predominantly in the premixed mode and thus high values of _ Ω F are associated with small values of Λ . A similar behavior is observed for all other initial Gaussian and bi-modal distribution cases, which is also consistent with previous findings (Chakraborty et al., 2007Chakraborty and Mastorakos, 2008;Im et al., 1998;Mastorakos et al., 1997;Patel and Chakraborty, 2014) in the context of localized ignition.
The extent of burning can be characterized by the burned gas mass m b with c ! 0:9 (Chakraborty et al., 2007Chakraborty and Mastorakos, 2008;Patel and Chakraborty, 2014;Wandel et al., 2009). The temporal evolution of the mean and standard deviations of burned gas mass normalized by the mass of an unburned gas sphere with a radius equal to l f (i.e., 4=3 ½ πρ 0 l 3 f ) for all cases with initial Gaussian and bimodal distributions of equivalence ratio are shown in Figures 8 and 9, respectively. It is evident from Figures 8 and 9 that the mean value of m b c ! 0:9 ð Þdecreases with increasing u 0 =S b ϕ¼1 ð Þ in both types of initial mixture distributions. An increase in u 0 leads to an increase in eddy diffusivity D t ,u 0 L 11 for a given value of L 11 , which leads to a greater amount of heat loss from the hot gas kernel for high values of u 0 . The heat release due to combustion must overcome the heat loss in order to sustain combustion following successful ignition. The probability of findings high values of c decreases with increasing u 0 =S b ϕ¼1 ð Þ due to enhanced heat transfer rate from hot gas kernel for both types of initial mixture distributions. The smaller extent of burning rate for higher values of u 0 =S b ϕ¼1 ð Þ and the adverse effect of u 0 =S b ϕ¼1 ð Þ on the extent of burning are consistent with previous findings (Ballal and Lefebvre, 1977a;Chakraborty et al., 2007Chakraborty and Mastorakos, 2008;Huang et al., 2007;Klein et al., 2008;Patel and Chakraborty, 2014;Poinsot et al., 1995). Furthermore, Figures 8 and 9 show that an increase in ϕ 0 leads to a decrease of the mean value of m b c ! 0:9 ð Þirrespective of the nature of initial mixture distribution. The burning velocity of mixtures with ϕ < 1:0 and ϕ > 1:10 is smaller than that in the mixtures with 1:0 ϕ 1:10. The probability of finding 1:0 ϕ 1:10 decreases with increasing ϕ 0 and this gives rise to a reduction in burning rate for high values of ϕ 0 . Additionally, Figures 8 and 9 indicate that increasing ϕ 0 for stratified cases increases the variability of burning irrespective of the nature of initial mixture distribution. The reduction in burning rate due to mixture stratification in globally stoichiometric mixtures is consistent with several previous analyses (Haworth et al., 2000;Jiménez et al., 2002;Malkeson and Chakraborty, 2010;Patel and Chakraborty, 2014;Renou et al., 2004;Samson, 2002;Zhou et al., 1998).  It is evident from Figures 8 and 9 that the mean values of m b c ! 0:9 ð Þ remain comparable for the quiescent cases with initial ϕ 0 ¼ 0:2 for all values of l ϕ =l f in both initial bi-modal and Gaussian distributions. A comparison between Figures 8 and 9 indicates that the initial Gaussian mixture distribution cases with initial u 0 =S b ϕ¼1 ð Þ ¼ 6:0 fail to sustain combustion once the ignitor is switched off irrespective of the initial values of ϕ 0 and l ϕ =l f , whereas the initial bi-modal distribution cases with initial ϕ 0 ¼ 0:2 and l ϕ =l f ¼ 5:5 and 8.3 exhibit self-sustained combustion for initial u 0 =S b ϕ¼1 ð Þ ¼ 6:0. The bimodal distribution cases show higher probability of finding 1:0 ϕ 1:10 than in the Gaussian distribution cases for l ϕ =l f > 2:1 (see Figure 5). Thus, the heat release due to combustion for the initial bi-modal distribution cases with initial ϕ 0 ¼ 0:2 and l ϕ =l f ¼ 5:5 and 8.3 is greater than that in the corresponding Gaussian distribution cases and thus these initial bi-modal distribution cases show self-sustained combustion whereas flame extinction takes place for the initial Gaussian distribution cases at t > t sp .
Among the bi-modal cases with initial ϕ 0 = 0.2, the mean value of burned gas mass m b c ! 0:9 ð Þassumes the highest (lowest) magnitude for the cases with initial l ϕ =l f = 5.5 (l ϕ =l f = 2.1). By contrast, the mean value of burned gas mass m b c ! 0:9 ð Þassumes the highest (lowest) magnitude for the cases with initial l ϕ =l f = 8.3 (l ϕ =l f = 5.5) for initial ϕ 0 = 0.4 and 0.6 in the initial bi-modal distribution cases. By contrast, the initial Gaussian distribution cases show an increase in the mean value of burned gas mass m b c ! 0:9 ð Þwith decreasing l ϕ =l f irrespective of initial ϕ 0 . The probability of finding highly reactive 1:0 ϕ 1:10 mixtures increases with decreasing l ϕ =l f , due to more efficient mixing for small values of l ϕ for initial ϕ 0 = 0.4 and 0.6 cases (see Figure 5). The probability of finding highly reactive mixture corresponding to 1:10 > ϕ > 1:0 is greater in the cases with initial values of l ϕ =l f = 5.5 and 8.3 (the probability of finding 1:10 > ϕ > 1:0 is greater in the initial l ϕ =l f = 5.5 cases than in the initial l ϕ =l f = 8.3 cases) for initial ϕ 0 = 0.2 than in the initial l ϕ =l f = 2.1 cases. This gives rise to a greater rate of burning in the bi-modal distribution cases with initial values of l ϕ =l f = 5.5 and 8.3 than in the initial l ϕ =l f = 2.1 cases for initial ϕ 0 = 0.2. The probability of finding highly reactive 1:0 ϕ 1:10 mixture is smaller for initial l ϕ =l f ¼ 5:5 cases than the initial l ϕ =l f ¼ 2:1 cases for bi-modal distribution with initial values of ϕ 0 ¼ 0:4 and 0.6. This gives rise to smaller mean value of burned gas mass m b c ! 0:9 ð Þfor initial l ϕ =l f ¼ 5:5 cases than the initial l ϕ =l f ¼ 2:1 cases for bi-modal distribution. For cases with initial l ϕ =l f = 8.3 the clouds of mixture inhomogeneities are relatively big (see Figure 1), and as a result, there is a high probability of obtaining a large region of almost homogeneous mixture in the vicinity of the center of the ignitor, which leads to a large variation of m b c ! 0:9 ð Þbetween different realizations. If the ignitor center is located in the vicinity of a large cloud of 1:0 ϕ 1:1, the slow burning rate in the pockets with 1:0 < ϕ and ϕ > 1:1 encountered during the expansion of hot gas kernel is mostly compensated by the high burning rate in the 1:0 ϕ 1:1 mixture and this leads to greater burned gas mass in the cases with initial l ϕ =l f = 8.3 than in the cases with initial l ϕ =l f = 2.1 for a given realization when the initial ϕ 0 is 0.4 and 0.6 for the bi-modal distribution. Furthermore, it has been demonstrated in Figure 7 that the burning rate assumes small values in the regions, which are associated with high values of N , and the probability of finding high values of N is greater in the cases with initial l ϕ =l f = 2.1 than in the cases with initial l ϕ =l f = 8.3. This also acts to increase the mean value of the burned gas mass m b c ! 0:9 ð Þin the bi-modal distribution cases with initial l ϕ =l f = 8.3 in comparison to the corresponding initial l ϕ =l f = 2.1 cases for initial ϕ 0 ¼ 0:4 and 0.6. The summary of the effects of initial mixture distribution for different values of u 0 , ϕ 0 , and l ϕ =l f on the outcome of localized forced ignition in terms of self-sustained combustion is presented in Table 2. It can be seen from Table 2 that the initial bi-modal mixture distribution cases offer higher probability of obtaining self-sustained combustion than in the cases with initial Gaussian distribution for initial values of ϕ 0 ¼ 0:2 and u 0 =S b ϕ¼1 ð Þ ¼ 6:0. The temporal evolution of the mean and standard deviation of burned gas mass for all cases listed in Table 1 are re-plotted in Figure 10 to demonstrate the effects of different distribution of stratified mixtures in comparison to the homogeneous mixture (i.e., premixed) cases. It is evident from Figure 10 that both initial Gaussian and bi-modal distribution cases with small initial values of ϕ 0 and l ϕ =l f behave similar to homogeneous mixtures due to rapid mixing (i.e., sharp decay of ϕ 0 ). Furthermore, Figure 10 shows that it is possible to achieve a higher value of m b c ! 0:9 ð Þthan the corresponding homogeneous mixture case for a given realization for both Gaussian and bimodal distribution cases but this propensity is higher in the initial bi-modal distribution cases than in the initial Gaussian distribution cases (see also Table 2). The variation of burned gas mass m b c ! 0:9 ð Þbetween different realizations increases with increasing l ϕ =l f and ϕ 0 for both bimodal and Gaussian distributions of ϕ but the degree of variability is smaller for initial Gaussian distribution cases than in the initial bi-modal distribution cases. This behavior originates due to the higher probability of finding ϕ % 1:0 (ϕ Þ 1:0) in the initial Gaussian (bi-modal) distribution case than those with initial bi-modal (Gaussian) distribution cases.
The variability of burned gas mass is routinely obtained in the cylinder of IC engines due to cycle-to-cycle variations (Pera et al., 2013). The findings from Figures 8-10 reveal that the degree of variability of burning depends not only on ϕ 0 and l ϕ but also on the nature of mixture distribution, which can be manipulated by careful design of the nozzle and fuel injection systems in IC engines. Moreover, in-cylinder turbulence together with injection characteristics can influence the values of u 0 , ϕ 0 , and l ϕ , and thus the effective control of mixing characteristics in IC engine combustion chamber can potentially play a pivotal role in ensuring successful ignition and reducing the variability associated with the ignition event. Furthermore, the minimum ignition energy for ensuring self-sustained combustion subsequent to successful ignition in turbulent stratified mixtures is not only dependent on u 0 , ϕ 0 , and l ϕ , but also on the nature of mixture distribution. This suggests that the success or failure of the ignition of stratified mixtures Table 2. Summary of outcomes for the self-sustained combustion for all cases (✓ and ✕ marks indicate the cases with successful self-sustained combustion following ignition and failed self-sustained combustion subsequent to forced ignition, respectively). l ϕ =l f = 2.1 (1st row); 5.5 (2nd row); and 8.3 (3rd row).
is a highly random event and that a small change in mixture distribution has the potential to alter the outcome. A limited number of simulations has also been carried out for globally fuel-lean (i.e., hϕi ¼ 0:8) and the effects of u 0 , ϕ 0 , l ϕ , and mixture distribution on localized forced ignition of hϕi ¼ 0:8 mixtures have been found to be qualitatively similar to those in case of globally stoichiometric (i.e., hϕi ¼ 1:0) mixtures but hϕi ¼ 0:8 mixtures need a higher amount of energy than hϕi ¼ 1:0 mixtures to successfully ignite and exhibit self-sustained combustion.

Conclusions
The effects of l ϕ =l f , ϕ 0 , and u 0 =S b ϕ¼1 ð Þ on early stages of combustion following localized forced ignition of globally stoichiometric (i.e., hϕi ¼ 1:0) stratified mixtures have been numerically investigated using 3D DNS simulations for both initial Gaussian and bi-modal distributions of equivalence ratio. The flame resulting from localized forced ignition shows a predominantly premixed mode of combustion although some pockets of the non-premixed mode of combustion have also been observed. The probability of finding non-premixed pockets is relatively greater in the case of initial bi-modal distribution than in the initial Gaussian distribution cases because the cases with initial Gaussian mixture distribution show greater probabilities of finding ϕ % hϕi ¼ 1:0 than the corresponding initial bi-modal distribution cases. The extent of nonpremixed combustion decreases with decreasing (increasing) ϕ 0 and l ϕ =l f (u 0 ). For a given value of l ϕ =l f , an increase in ϕ 0 leads to a reduction of burned gas mass for both Gaussian and bi-modal distributions, whereas the influence of l ϕ =l f on the extent of burning has been found to be nonmonotonic and dependent on ϕ 0 for initial bi-modal mixture distributions, whereas the cases with initial Gaussian mixture distribution show an increase in burned gas mass with decreasing values of l ϕ =l f for all initial values of ϕ 0 considered here. The increase in heat transfer rate from hot gas kernel with increasing u 0 =S b ϕ¼1 ð Þ leads to a decrease in the extent of burning irrespective of the values of l ϕ =l f and ϕ 0 for both mixture distributions. However, some cases with initial bimodal distribution show more resistance to flame extinction than the cases with initial Gaussian distribution for high values of u 0 =S b ϕ¼1 ð Þ . The above findings demonstrate that the effects of l ϕ =l f , ϕ 0 , and u 0 =S b ϕ¼1 ð Þ on the early stages of combustion following successful forced ignition of globally stoichiometric stratified mixtures are dependent on the nature of initial equivalence ratio distribution. Thus, it is not sufficient to characterize the minimum ignition energy of turbulent stratified mixtures in terms of l ϕ =l f , ϕ 0 , and u 0 =S b ϕ¼1 ð Þ , because it is possible to have successful self-sustained combustion for one type of mixture distribution but the same might not be true for a different type of mixture distribution with same set of values of l ϕ =l f , ϕ 0 , and u 0 =S b ϕ¼1 ð Þ . The qualitative nature of the present findings are not likely to change in the presence of detailed chemical kinetics but further analyses based on detailed chemistry and transport will be necessary for more comprehensive physical understanding and quantitative predictions.