Large Eddy Simulation of Lean Mixed-Mode Combustion Assisted by Partial Fuel Stratification in a Spark-Ignition Engine

With the increasing demand in global energy supplies and stringent emission regulations, cleaner and more efficient engine combustion technologies remain to be developed [1]. Lean operation can provide high thermal efficiency compared with conventional stoichiometric operation [2]. However, operation with early injections creates a near-homogeneous lean mixture which is prone to excessive cycle-to-cycle variations (CCVs) due to a reduced flame propagation speed and hence an increased importance of flame-flow interactions. Partial fuel stratification (PFS) is a promising strategy to improve the stability for lean combustion by stabilizing the spark-initiated deflagration [3]. In PFS, in addition to the early injections during the intake stroke, a fraction of fuel, referred to as pilot injection, is injected concurrently with the spark-ignition, thus creating a partially stratified local mixture which enables stable deflagrative combustion. However, relying only on deflagration is insufficient for ultra-lean operation because of an excessively long burn duration. To overcome this challenge, mixed-mode combustion or spark-assisted compression ignition (SACI) can be further utilized to speed up the overall combustion whereby a portion of the charge is consumed via normal deflagration and the remainder via autoignition of the end-gas. Therefore, PFS-assisted lean SACI combustion is an emerging advanced engine combustion technology featuring high efficiency, low emission, and stable operation at the same time, and a similar technology, namely Spark Plug Controlled Compression Ignition (SPCCI), has been demonstrated in the Mazda’s Skyactive-X Engine [4].

The performance and operability of the PFS-assisted lean SACI engine operation depend on a variety of in-cylinder processes, including pilot injection, flame kernel initiation and development, and end-gas autoignition. For example, the local mixture stratification near the spark has a strong implication on ignition reliability, the initial speed of flame kernel development, as well as soot emission [3]. The rate of flame propagation determines the onset of end-gas autoignition, while the heat release rate due to autoignition is largely affected by temperature and mixture stratification in the end-gas [5,6]. All of these processes actively interact with, or may even be controlled by, the complex flow patterns in a turbulent environment. Provided with such tight couplings of spray and spark, and of deflagration and autoignition, understanding the detailed physics involved in these processes is key to improving combustion control for PFS-assisted lean SACI operation.

On the other hand, the performance of an spark-ignition (SI) engine can strongly depend on the fuel being used, or more particularly, the physical and chemical properties of the fuels. For conventional stoichiometric boosted SI engine operation, impacts of different fuel properties have been systematically investigated in the literature [7–10]. A recent report from U.S. Department of Energy presented an engine efficiency merit function correlated with fuel properties, based on the experimental data from various sources [11]. The merit function identified positive correlations between the efficiency gain and key fuel properties, such as the Research Octane Number (RON), octane sensitivity S (S = RON − MON, MON is Motor Octane Number), heat of vaporization (HoV) and laminar burning speed (S_L), either through knock mitigation or higher exhaust gas recirculation (EGR)-dilution potential. These observed fuel property impacts, however, are limited to the stoichiometric EGR-dilute boosted-SI operation, and the extension to the SACI operation is not straightforward.

High-fidelity computational fluid dynamics (CFD) simulations provide a powerful tool to enable full three-dimensional (3D) access to all physical and chemical processes in a time-resolved fashion, and to allow isolated investigations of fuel property impacts, both of which are challenging using experiments. For example, CFD simulations have been used in previous studies to investigate SACI combustion using Reynolds-averaged Navier–Stokes (RANS) [12–15] or large eddy simulation (LES) [16,17] based models. In these previous studies, RANS-based models, in general, were able to capture the mean combustion characteristics, while LES-based models further showed the capability of predicting CCV [14]. However, very few numerical studies have been focused on the PFS-assisted lean SACI operation.

In this study, an LES-based CFD model for multi-cycle engine simulations of PFS-assisted lean SACI is presented. The previously developed RANS model targeting the well-mixed charge operation [15] is extended to account for pilot injection and to couple LES for more accurate spray and flame characterizations. The model will be validated against experimental data on crank-angle (CA) resolved pressure traces as well as in-cylinder spray and flame imaging. The multi-cycle LES model will also be compared with the widely used RANS model. The validated LES model will then be used to characterize spray and flame structures as well as mixture inhomogeneity. The model will be further employed to investigate impacts of fuel properties on combustion characteristics of PFS-assisted SACI, targeting future optimization of fuels and engines. To the authors’ best knowledge, this is the first multi-cycle LES study for the PFS-assisted lean SACI engine operation.

The experiments were performed in a single-cylinder, four-valve, direct-injection spark-ignition (DISI) research engine operated at Sandia National Laboratories. Figure 1 shows a cross-sectional view of the combustion chamber at top dead center (TDC). One of the intake valves was deactivated to enhance the in-cylinder swirl level and thereby the overall mixing process. The engine can be setup in two configurations, namely, an optical configuration and an all-metal configuration. The two configurations are nearly identical except that the optical engine has quartz windows in the pent-roof and piston to provide optical access to in-cylinder spray and combustion processes, while all-metal piston and metal blank in the pent-roof were used to enable continuously fired all-metal engine experiments. In this study, the former engine configuration was employed to provide spray and flame imaging for model validation and enhancement.

A certification gasoline blended with 30% ethanol by volume (referred to as E30 hereinafter) was used in the experiments. The E30 fuel has RON and MON of 105 and 91, respectively. Fuel can be directly injected into the combustion chamber during the intake stroke to generate a well-mixed charge or during the compression stroke to generate a stratified charge. In this study, a combined injection strategy, i.e., PFS, was employed: triple early injections to provide a near-homogeneous lean mixture in the background with an equivalence ratio (ϕ) of 0.48, and a late pilot injection to create localized fuel stratification near spark timing resulting in a global ϕ of 0.5. Compared with the well-mixed charge engine operation, PFS has proven to have supreme controls on the start of combustion (SoC) [3] and to maintain a low level of CCV. Engine specifications and key parameters for the PFS engine operation are provided in Table 1. For more details on engine operation strategies, readers are referred to Ref. [3].

The simulations are performed using the converge code v2.4 [18] on a full-scale engine geometry shown in Fig. 2. The code features second-order accuracy for spatial discretization and first-order accuracy for time integration. The computational domain contains intake and exhaust ports and valves, cylinder, spark plug, and fuel injector. Crank-angle resolve pressure measurements at the intake port inlet and the exhaust port outlet provide the pressure boundary conditions for simulations. The liquid spray and gas-phase combustion are solved with the Eulerian–Lagrangian approach. Droplet breakup, evaporation, and drag are modeled with the Kelvin–Helmholtz and Rayleigh–Taylor model [19], the Frossling correlation [20], and dynamic drag model [21], respectively. Wall heat transfer is modeled with a temperature wall function [22]. Temperature of the cylinder walls is specified as 435 K. The dynamic structure LES model [23] is employed in this study and its performance is further compared with RANS simulations based on the re-normalized group (RNG) k − ε model. Note that one focus of this study is to demonstrate the improved fidelity of the multi-cycle LES model compared with the conventionally used RANS model.

Non-uniform mesh with a base grid size of 4 mm is adopted. A fixed grid embedding of 0.25–0.5 mm is applied during spray injection and near the walls. A fine resolution of 0.125 mm is applied in a spherical region enclosing the spark gap for capturing early flame development. Adaptive mesh refinement with a minimum grid size of 0.5 mm is adopted to better resolve the temperature and the velocity gradients. The resultant mesh consists of a peak cell count of approximately 1.6 million in LES and approximately 1.2 million in RANS.

Using the realistic spray cone angle and included angle in Lagrangian spray models, the CFD model is then applied to simulate fired engine operation and validated against experimental data. Figures 5 and 6 show the predicted pressure and apparent heat release rate (AHRR) traces for ten consecutive simulation cycles obtained from RANS and LES, respectively, in comparison with 32 skip-fired experimental cycles. Here, the ten simulation cycles are used to obtain statistically meaningful ensemble average of pressure and AHRR profiles. Results from the RANS simulation show a reasonable agreement with experiments in terms of the mean pressure and AHRR traces. Both combustion modes, namely early deflagration and late autoignition, are reasonably captured in RANS, while visible discrepancies are observed in the timings at which peak pressures and peak AHRRs occur. In addition, a negligible level of CCV for deflagration and a moderate level of CCV for autoignition are predicted by RANS, which is expected since RANS-based models, by design, are intended to capture the mean rather than the fluctuation of a given quantity. In contrast, as seen in Fig. 6, LES significantly improves the prediction of mean and CCV for both pressure and AHRR traces, and especially the prediction of combustion phasing. Moreover, larger CCV is observed for the peak pressure and the peak AHRR associated with autoignition (the second peaks) than those associated with deflagration (the first peaks). This can be explained as follows. First, the temperature of the end-gas is directly affected by the combustion phasing of the deflagration-based combustion. Second, in addition to temperature, the autoignition process is also sensitive to the radicals in the end-gas that can be either from the trapped residual gas or newly formed in the partially reacted fresh mixture. Such temperature and radical dependencies along with the CCV of the flow field result in larger cyclic variability of the second AHRR peak.

Table 2 quantitatively compares several key engine performance metrics calculated from simulations and experiments, including mean and standard deviation of the peak cylinder pressure (P_max), mean and coefficient of variation (CoV) of the net indicated mean effective pressure (IMEP_n), and combustion phasing corresponding to 10% (CA10), 50% (CA50), and 90% (CA90) of the total heat release. Both RANS and LES show good agreement with the experiment in the mean peak pressure with relative errors of 2% and 3%, respectively. IMEP_n is slightly over-predicted (by 7%) using both RANS and LES, which can be attributed to the uncertainty in the thermal wall boundary condition given that detailed wall temperature measurement is not available in the experiment. Such discrepancy may also be attributed to insufficient description of flame-wall interaction in the current G-equation model, which merits further investigation. Both RANS and LES simulations predict retarded CA10 and CA50 compared with the experiment and this is consistent with the slight delay in SoC observed in Figs. 5 and 6. This mismatch may result from the slight under-prediction of liquid evaporation, which will be further detailed in Sec. 3.3. Clearly, LES shows an advantage over RANS on capturing the cyclic variations of the peak pressure and IMEP_n.

The improved accuracy of the LES-based model is further demonstrated in Fig. 7, where the AHRR profiles are plotted in the burned mass fraction (ξ) space. The burned mass fraction is calculated based on the accumulative AHRR normalized by its peak value at the end of combustion (EoC). The experimental AHRR peaks at ξ = 0.39 and 0.77, respectively, signifying the deflagration and autoignition modes, with its transition (local minima) occurring at ξ = 0.6. The LES-based model accurately captures the phasing associated with the two modes (ξ = 0.375 and 0.745, respectively) and the transition point (ξ = 0.59). However, the RANS-based model fails to capture the phasing of peak AHRRs, especially for the deflagration mode, resulting in an error of 33% compared with a 4% error using LES. The observed model deficiency of RANS suggests the need of further improvement of the G-equation model in the RANS context, e.g., by better accounting for the strain effect on flame propagation which has been conventionally neglected [25].

Individual in-cylinder processes in PFS-assisted SACI combustion, including pilot spray injection, mixing, flame propagation, and end-gas autoignition are further analyzed in detail based on 3D simulation data. Representative experimental and simulation cycles are selected such that they have a comparable peak AHRR with the ensemble mean of all experimental cycles. The pressure and AHRR profiles for the two selected simulation cycles from RANS and LES, respectively, are shown in Fig. 8.

Figure 9 shows the spray patterns for the two cycles in Fig. 8 that are predicted by RANS and LES models, respectively. Spray patterns from optical imaging enabled by light-emitting diode illumination are also shown for comparison. RANS and LES predict similar spray structure and penetration length, indicating the negligible impact of the flow field on spray characteristics, due to the large injection momentum of the sprays. Both RANS and LES simulations show limited plume-to-plume interaction, which is consistent with the experimental observations. The diminished experimental spray signal at −26.1 CA, probably indicating strong evaporation, is also captured in simulations with a small shift in the timing. Both RANS and LES simulations, however, show a smaller spray penetration length than the experiment. This can be probably explained by the differences in injection duration (210 versus 325 μs) and intake temperature (100 versus 30 °C) between the fired and motored cases (Sec. 3.1). The pilot-injection rate of injection (ROI) profile (which works well for the motored simulations) may not be consistent with that in experiments for the fired case due to the increasingly more relevant opening and closing transients with shorter pilot injection. This is a common limitation of the Lagrangian spray models in predicting highly transient near-nozzle behaviors. Further improvement is possible by incorporating more detailed information of ROI from experiments or through one-way coupling with internal nozzle flows. This, however, requires further investigation. Moreover, the under-prediction of liquid penetration is probably the cause of the delay in SoC (Figs. 5 and 6) because a smaller liquid penetration can delay the spray-driven spread of the flame from the spark plug gap into the combustion chamber.

Flame dynamics of the two representative cycles in Fig. 8 predicted from RANS and LES simulations are compared with measured flame structure based on natural luminescence in Fig. 10. The bright region in the experiment represents the high-sooting diffusion flame or the rich premixed flame, while the blue region indicates the lean premixed flame. To enable direct comparison between simulations and experiments, false color images are constructed from line-of-sight projection of 3D simulation data. In particular, since soot is not modeled in the current study, the bright color is mimicked by C₂H₂ which is a widely used soot precursor (e.g., in the two-equation soot model [29]). The blue color is typically due to emission of highly energetic radicals such as CH* which is only present at the flame fronts. Since CH* is not readily available from the current chemical mechanism, [OH] × [CH₂O] ([] denotes the molar concentration), a premixed flame indicator that is commonly used experimental diagnostics [30], is used to indicate the lean premixed flames. The transition from a sooting flame to a non-sooting lean premixed flame is well captured by both RANS and LES models. From instantaneous flame typology point of view, RANS simulation shows a near-spherical flame shape, while LES predicts more winkles along the premixed flame fronts which exhibit a preferential orientation similar to the experimental measurement at −19.5 deg CA. This preferential orientation is probably determined by the local flow condition and is not observed in the RANS simulation. Corresponding 3D flame structure is also shown in the last row. Deflagration flame fronts are marked by $G~=0$⁠, and the stoichiometric isosurface is represented by ϕ = 1. The equivalence ratio distribution is shown on a two-dimensional (2D) plane cutting through the spark plug gap. Clearly, as early combustion transitions from sooting to non-sooting flames, rich mixtures are quickly consumed, and flames propagate into lean mixtures.

To further depict the combustion modes associated with the early-stage sooting flame, Fig. 11 shows the contour of local heat release rate on a 2D plane cutting through the spark plug gap for the LES cycle. Isolines of $G~=0$ and ϕ = 1 are superimposed to delineate the local flame structure. It is clear that early-stage flames (−17.1 and −21.9 deg CA) are a combination of lean and rich premixed flames along with a stoichiometric diffusion flame that forms a canonical “triple flame” structure. As rich mixtures are consumed by the premixed flame front (−17.1 deg CA), the entire ϕ = 1 isoline is enclosed by the $G~=0$ isoline, indicating that fresh mixtures start to burn purely in the lean premixed mode. It is worth noting that successful prediction of such transition is enabled by the combined usage of G-equation for flame propagation and finite rate chemistry for the post-flame zone.

As flame propagates toward the cylinder wall, the unburned fresh mixture between the flame and the wall is commonly referred to as the “end-gas” (identified by $G~<0$ when the G-equation model is activated), which has the potential to auto-ignite when the temperature and pressure become sufficiently high. Figure 12 shows the stoichiometric isosurfaces, deflagration flame fronts, the autoignition pockets and the cross-sectional equivalence ratio distribution for the same LES cycle. Here, the deflagration flame fronts are identified by $G~=0$⁠, while the autoignition pockets are identified by large concentration of the major combustion product H₂O, i.e., $YH2O/YH2Oeq=70$% in the region where $G~<0$⁠. It is clearly seen that the pilot injection creates a locally stratified mixture ranging from rich to lean conditions. This stratified mixture lasts for approximately ten crank angles until either quickly consumed by flame or diluted by the lean background mixture. The presence of the near-stoichiometric local mixture creates a stable mechanism for fast initial flame propagation, thus stabilizing the overall combustion process. As the stratified fresh mixture is completely consumed, the flame continues to propagate in the lean background until autoignition takes place in the end-gas as the in-cylinder temperature and pressure become sufficiently high. The autoignition event evolves in a much shorter duration than deflagration and thus avoids retarded combustion phasing.

The nature of sequential autoignition in the end-gas is similar to that in homogeneous charge compression ignition (HCCI), the intensity of which is closely related to the available stratification levels of temperature and equivalence ratio [5,31]. It is therefore worthwhile to investigate temperature and mixture distributions in the end-gas to understand the knock mitigation mechanisms. Figure 13 shows the probability density functions (PDF) of the temperature and the normalized equivalence ratio (ϕ_n = ϕ/(ϕ + 1)) in the end-gas at various crank angles, ranging from spark timing, to transition of flame mode, and to the timing just prior to autoignition. Here, ϕ_n instead of ϕ is used so that contributions from extremely rich mixtures can be more conveniently quantified and visualized in the ϕ_n space. As time evolves, the mean temperature monotonically increases due to the compression of the end-gas by piston motion and flame propagation. The stratification level in temperature also increases due to continuous heat loss to the wall. The mean end-gas equivalence ratio remains largely unchanged, while the stratification level first increases due to spray evaporation (from −27 to −22 deg CA) and then decreases due to the consumption of the stratified fuel and the persistent mixing process. The presence of non-negligible stratification in temperature and equivalence ratio may play an important role in mitigating excessive engine knock for the SACI operation, which will be a topic for future investigation.

With the demonstrated capability on capturing the in-cylinder mixing and combustion processes in PFS-assisted SACI, the CFD model is employed to systematically investigate impacts of fuel properties. Different from previous studies [10,32] which used the lower-fidelity RANS model, the higher-fidelity LES model is employed for the sensitivity analysis in this study. In particular, heat of vaporization (HoV) and laminar burning speed (S_L) are of interest, and perturbations are applied to baseline HoV and S_L values with a fixed spark timing (−27 deg CA). Here, the laminar burning speed is assumed as a fuel property that is independent of the autoignition propensity for a given fuel. Therefore, when perturbation is applied to S_L, the chemical kinetic parameters for the finite rate chemistry modeling remain unchanged. While an alternative way to perform sensitivity analysis is to perturb the chemical kinetic parameters for both laminar burning speed calculation and finite rate chemistry modeling, the method used in the current study allows isolated investigation of individual fuel properties (which is nevertheless challenging in experiments), thus providing physical insights to the search of new fuels that might bring engine efficiency benefits with tailor-made fuel properties. For each perturbation, ten consecutive cycles are simulated to achieve the statistically steady state.

Figure 14 shows the AHRR traces in both the crank angle space and in the mass burned space, for HoV perturbed by $±30%$ relative to its baseline value. A decrease in HoV is seen to slightly advance deflagration combustion phasing while significantly accelerating autoignition and advancing the phasing. Interestingly, although the peak autoignition AHRR drops dramatically, the combustion phasing in the mass burned space remains unaffected for both deflagration and autoignition. The observed HoV sensitivity is mainly due to the charge cooling effect in the cylinder especially for direct injection engines, where a lower HoV normally leads to a higher fresh mixture temperature and thus higher laminar burning speed and shorter ignition delay time. This is confirmed by Fig. 15 where CA50 and intake valve closing (IVC) temperature for the PFS operation are plotted as functions of the HoV perturbation and compared with those for the well-mixed operation reported in a previous study [15]. Similar sensitivities of CA50 and IVC temperature are observed for both PFS and well-mixed operations, except that the well-mixed operation exhibits a non-monotonic trend in CA50 due to the excessive level of CCV.

Figure 16 shows the pressure traces, and the AHRR traces in both the crank angle coordinate and the mass burned space, for S_L perturbed by $±10%$⁠, $±30%$⁠, and $±50%$ relative to its baseline value, respectively. The relatively large perturbations of $±50%$ are included to mimic the fuels with extremely low (e.g., ammonia) and high (e.g., hydrogen) laminar burning speeds, respectively. The value of S_L is found directly associated with the initial ramp-up rates and the peak values of both pressure and AHRR during deflagration in the crank angle coordinate. The enhanced deflagration by a larger S_L further promotes autoignition by raising pressure and temperature in the end-gas, thus simultaneously advancing the combustion phasing and increasing the peak pressure and AHRR during autoignition in the crank angle coordinate. Similar to HoV, the change in S_L does not modify the combustion phasing in the mass burned space. In Fig. 17, the S_L sensitivities of CA50 and peak AHRR for the PFS operation are further compared with those for the well-mixed charge operation described in Ref. [15]. Despite the similar trends of increasing peak AHRR and advanced CA50 against S_L, the PFS operation shows a smaller sensitivity to S_L compared with the well-mixed operation, which can be attributed to the improved combustion phasing control capability of PFS. For lean well-mixed operation, there is a relatively long induction period from the spark until the main combustion event, resulting in a high sensitivity of the combustion phasing to the change of laminar burning speed. For PFS operation, Fig. 17 reveals a change in the slopes of CA50 and peak AHRR at S_L perturbation of $−10%$⁠, coinciding with the transition in the dominant combustion mode, i.e., from the deflagration-only mode to the mixed deflagration-autoignition mode (evident in Fig. 16(c)).

The sensitivities of the two peak AHRRs (associated with deflagration and autoignition, respectively) to HoV and S_L are further quantified in Fig. 18. Clearly, the autoignition peak AHRR shows a non-linear response to S_L, in contrast to the near-linear response of the deflagration peak AHRR. In addition, the peak AHRR associated with autoignition is much more sensitive to either HoV or S_L than that associated with deflagration. Therefore, to avoid excessive combustion noise and engine knock, controlling the autoignition-based peak AHRR is the primary concern when fuels of different laminar burning speeds are used. In a practical implementation, this can be achieved by adjusting the spark timing, preferably via closed-loop feedback using a cylinder pressure sensor.

A multi-cycle LES-based CFD model was developed in this study for lean SACI combustion assisted by the PFS fueling strategy in a DISI engine. The spray model was first improved to accurately represent the spray morphology during pilot injection. The improved spray model showed good agreement in liquid and vapor penetrations with experimental measurements. PFS-assisted SACI was modeled by a hybrid combustion modeling approach coupling the level-set G-equation model for flame propagation and the well-stirred reactor model for end-gas autoignition and diffusion flames in the post-flame region. Good agreement was observed between multi-cycle LES simulations and experimental data, in terms of mean and variations in pressure and heat release rate traces. LES also well predicted mean and fluctuations of global quantities such as peak pressure, IMEP, and combustion phasing. In comparison, multi-cycle RANS simulations failed to capture CCV and showed larger deviations in the combustion phasing, especially for the peak heat release rate associated with deflagration, while it captured the mean pressure and IMEP reasonably well. Pilot injection, combustion, and mixing processes were analyzed in detail. LES and RANS showed similar predictions in spray dynamics. LES captured flame wrinkling and a preferential flame orientation similar to experimental observations. In contrast, RANS predicted a smoother, spherical flame topology. With LES, the canonical triple flame structure was identified at the early stage of flame propagation before transitioning into lean-premixed flames. Non-negligible stratification in the end-gas was observed for both temperature and equivalence ratio, which may play an important role in smoothing out the peak heat release rate during sequential autoignition.

The LES-based CFD model was then employed to investigate fuel property effects by perturbing the HoV and S_L of the fuel. HoV was found to play an important role by modifying the unburned gas temperature and thus combustion phasing, and its sensitivity for the PFS operation is comparable to that for the well-mixed charge operation. A perturbation of S_L is found to directly modify the initial ramp-up rate and peak values of the deflagration AHRR and subsequently affect end-gas autoignition through changes in temperature and pressure. Compared with the well-mixed charge operation, PFS operation shows a smaller S_L sensitivity. It is also found that either a smaller HoV or a larger S_L would advance CA50 and increase the peak AHRR, and the sensitivity of autoignition peak AHRR is much larger than that of deflagration peak AHRR. The identified fuel property sensitivities through local perturbations shed light on future research of quantifying fuel property effects on knock limit and on optimal engine efficiency toward co-optimization of fuels and PFS-assisted lean SACI engine operation.

UChicago Argonne, LLC, operator of Argonne National Laboratory (“Argonne”), a U.S. Department of Energy (DOE) Office of Science Laboratory, is operated under Contract No. DE-AC02- 06CH11357. This research was partially funded by DOE’s Office of Vehicle Technologies, Office of Energy Efficiency and Renewable Energy under Contract No. DE-AC02-06CH11357. The authors wish to thank Gurpreet Singh, Michael Weismiller, and Kevin Stork, program managers at DOE, for their support. This research was conducted as part of the Co-Optimization of Fuels & Engines (Co-Optima) project sponsored by the U.S. DOE’s Office of Energy Efficiency and Renewable Energy (EERE), Bioenergy Technologies and Vehicle Technologies Offices. We gratefully acknowledge the computing resources provided on Bebop, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. The engine experiments were performed at the Combustion Research Facility, Sandia National Laboratories, Livermore, CA. Cinzia Tornatore of Istituto Motori is acknowledged for contributions to the spray and flame imaging presented in this study. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

There are no conflicts of interest.

p =

pressure

u′ =

turbulent velocity

$G~$ =

mean signed distance to the mean turbulent flame

D_T =

turbulent diffusivity

P_max =

peak cylinder pressure

S_L =

laminar burning speed

S_T =

turbulent burning speed

T₀ =

unburned temperature

Da =

Damköhler number

$G″~$ =

mean variance of signed distance to turbulent flame

μ =

molecular viscosity

μ_T =

turbulent viscosity

ρ =

density

ϕ =

equivalence ratio

γ =

residual gas fraction

ξ =

burned mass fraction