Due to diesel-like thermal efficiency and low emissions, advanced compression ignition (ACI) engines operating in the low temperature combustion regime have recently attracted extensive research interest (Dec, 2009; Dempsey et al., 2016; Cung et al., 2017; Pachiannan et al., 2019). Such engine combustion concepts include the homogeneous charge compression ignition (HCCI), HCCI with partial fuel stratification (PFS), partially premixed compression ignition (PPCI), reactivity-controlled compression ignition (RCCI), etc. Typically, combustion in these engines is globally fuel lean and highly diluted. In addition, another essential characteristic of ACI combustion is the omnipresent temperature and concentration stratification induced by fuel injection as well as in-cylinder flow and conjugate heat transfer. Controlled stratifications can offer high flexibility to modulate heat release rate and adjust engine combustion phasing (Dempsey et al., 2016), yet result in substantial challenges in the understanding of ignition and subsequent combustion evolution in ACI conditions.

Recently, a framework by overlapping the engine thermodynamic pressure-temperature trajectory and the fuel ignition delay iso-contour has been developed, aiming to provide general insights on the knock in spark-ignition (SI) engines and bulk combustion phasing in ACI engines (Szybist and Splitter, 2017; Szybist et al., 2017; Tao et al., 2019). Manifested by the changes in fuel ignition delay iso-contour and engine thermodynamic trajectory, this framework can qualitatively show the effects of conventional fuel reactivity metrics such as research and motor octane numbers (RON and MON) and octane sensitivity (OS), as well as the effects of engine operating condition, e.g., intake conditions, global equivalence ratio and level of exhaust gas recirculation (EGR). This approach is fundamentally an extension of the Livengood-Wu integration (Livengood and Wu, 1955) facilitated by the recent developments of fuel surrogate and kinetic models, and is able to explain features of engine auto-ignition as observed from experiments, for example, the reduced effectiveness of EGR to mitigate knock in boosted SI engines (Szybist et al., 2017). Useful as it is, a key dimension that this P-T framework is missing is the mixture thermal and concentration stratification in ACI engines, in that the ignition of a gas pocket occurs in the existence of temperature and equivalence ratio gradient (Dec, 2009; Dempsey et al., 2016).

With challenges to conduct well-controlled experiment and fast progress in computational combustion and chemical kinetics, extensive computational investigations have recently been conducted on the flame propagation and ignition process in mixtures with temperature and fuel stratifications. Among these studies, a representative approach is to adopt the counterflow and diffusion mixing layer configuration (Sankaran and Im, 2005; Mukhopadhyay and Abraham, 2011). For example, Sankaran and Im (2005) studies the auto-ignition characteristics of lean premixed iso-octane/air stream impinging onto a hot stream of exhaust gas. The ignition delay and front propagation speed across the mixing layer were determined as a function of a local mixture fraction variable as well as the scalar dissipation rate. Another widely-adopted approach is to perform transient computations in the 1-D planar domain with initially non-uniform temperature or fuel concentration profiles and the emphasis is placed on the subsequent reaction front development and propagation (Sun et al., 2015; Dai et al., 2017; Qi et al., 2017; Shi et al., 2017). These studies include laminar flame propagation in a stratified mixture, detonation development with temperature or concentration gradient, ignition to flame transition in stratified charge. Beyond the theoretical framework of a premixed flame, a widely adopted theory to describe reaction wave propagation in a mixture with non-uniform temperature or concentration is the reactivity gradient theory of Zeldovich (Zeldovich, 1980), which introduces the spontaneous propagation (SP) combustion waves driven by the sequential auto-ignition of adjacent particles. Such a reaction front is completely kinetics-driven and fundamentally different from the classical flame through the coupling between convective-diffusive transport and reaction.

More recently, rapid compression machine (RCM) experiment has shown that three distinct stages of oxidation exhibit in lean n-heptane/air ignition (Sarathy et al., 2019), controlled by cool flame chemistry under low temperature (Zhao et al., 2016), H₂O₂ chain branching under intermediate temperature, and CO oxidation during high temperature. It is therefore intriguing to investigate the relevance of such kinetic behaviors in the ignition of stratified mixtures under ACI conditions, and to see if these distinct stages of oxidation can form novel reaction front and provide further insights into combustion evolution and heat release profile in lean ACI combustion. Next, we shall introduce the computational platform, configuration, case setup and the concept of three-stage ignition.

The numerical simulation of ignition in mixture with thermal and concentration stratification is conducted using a novel in-house reacting flow CFD code reactingFOAM-SCT based on OpenFOAM 6.0 (Yang et al., 2019). The new features of this open-source platform include an advanced accurate mid-point Splitting scheme (Lu et al., 2017), a fast and accurate stiff Chemistry solver (Imren and Haworth, 2016) and a detailed transport property evaluation package (Dasgupta et al., 2019). The performance of these advanced features has been extensively validated against either DNS results or analytical solutions in different configurations, including those from a diffusion-convection problem, homogeneous ignition delay, laminar premixed flame, shock wave, 1D spherical flame initiation and propagation, 2D premixed flame instabilities, 1D non-premixed counterflow flame and 2D non-premixed co-flow flame. More details of the governing equation, numerical scheme, spatial and temporal accuracy, and validation results are available in Yang et al. (2019). More recently, this code has been utilized to demonstrate the minimum ignition energy, regimes, and propagation dynamics of direct cool flame initiation (Yang and Zhao, 2020).

In the current work, first-order implicit Euler scheme is utilized to discretize the unsteady term, a second-order Gauss linear scheme is used for spatial terms, including both convection and diffusion terms. CFL number is set as 0.4. The upgraded algorithm named PIMPLE which combines PISO and SIMPLE is used for iteration. The current computational configuration is a 1D planar domain of 8 cm in length including a stratified segment from the left boundary x = 0 to x = x₁ and a uniform segment from x = x₁ to x = 8 cm, as shown in Figure 1. This domain is under adiabatic conditions with initial pressure 40 atm. At x = 0, zero gradient boundary condition is applied for temperature, pressure, and species mass fraction; at the other boundary of x = 8 cm, zero gradient is applied for temperature and species mass fraction, while wave-transmissive boundary condition is applied for pressure. As such, unlike (Dai and Chen, 2015) where substantial effects of pressure wave and shock are observed with a reflective boundary condition, there is no reflection of pressure waves from the right boundary in the current study, and the pressure in the domain eventually will drop to 40 atm. The fuel involved in the computation is n-heptane, one of the primary reference fuels representing long-chain n-alkane with low-temperature chemistry. The mechanism we adopt is from Andrae (2013), which includes 137 species and 635 reactions, and has been extensively validated and applied in engine combustion.

The computation cases we select in the current work are summarized in Table 1. The range of equivalence ratio covers 0.3 to 2.0 and the temperature covers 800–1,000 K, adopted from the operating conditions of typical low temperature combustion conditions (Kokjohn et al., 2012, 2015; Wang et al., 2014; Luong et al., 2017; Naser et al., 2017; Liu et al., 2019). For example, as in Liu et al. (2019), thermal stratifications up to 30 K/cm is observed near wall region. In our simulation, the temperature gradient is set at 16, 33, and 66 K/cm. A large gradient of 66 K/cm is adopted to showcase the effect of large temperature gradient on the combustion modes. According to Kokjohn et al. (2012), in an RCCI engine, non-uniform equivalence ratio distribution is observed at a distance from 30 to 65 mm away from the center of bore, and the gradient of equivalence ratio varies approximately from 0.1 to 1.1 per centimeter. In our simulation, the equivalence ratio gradient is set from 0.1 to 0.6 cm⁻¹. Among these cases, Cases 1, 2, and 3 share the same domain length of temperature stratification at 3 cm, while their temperature gradients are different as induced by different temperature ranges. Cases 6, 7, and 8 share the same domain length of equivalence ratio stratification at 3 cm, while their ϕ gradients are different as induced by different ϕ ranges. Cases 1, 4 and 5 share the same temperature range in the segment of stratification, but with different temperature gradients due to different length of stratification segment. Similarly, Cases 6, 9, and 10 share the same ϕ range in the segment of stratification, but with different ϕ gradients due to different length of stratification segment. Beyond the stratification segment, temperature and equivalence ratio are set as uniform at 800 K and 0.3, respectively. It is shown later that due to the different initial stratifications, reaction fronts propagating into the same final uniform segment can experience different thermal-chemical states and exhibit very different structure and propagation behaviors.

Recent RCM experiment has suggested three distinct oxidation stages for lean n-heptane/air at engine relevant conditions (Sarathy et al., 2019). As shown in Figure 2, a comparison has been made between the RCM ignition data and the simulation results for n-heptane/air at equivalence ratio 0.3, under 780 K and 21.6 atm. Quantitative difference has been observed in the pressure and CO evolution from the comparison, yet both simulation and experiment evidently exhibit three-stage oxidation phenomena. It is observed that while pressure keeps increasing throughout this process, the mole fraction of CO increases at the first and second stage ignition, while decreases at the third stage oxidation.

To further show the detailed kinetic behavior of this three-stage oxidation phenomena, simulated temperature, heat release rate (HRR) and major species profiles are shown in Figure 3, for constant pressure adiabatic ignition for n-heptane/air at ϕ = 0.3, P = 40 atm and T = 800 K. Here the delay time for the first, second and total ignition stages are defined at the instant with local maximum temperature rise rate, as τ₁, τ₂ and τ_T respectively, as shall be adopted in the rest of this study. Three distinct peaks appear at each stage with similar orders of magnitude. Under this specific condition, the first-stage ignition occurs at lowest temperature (~850 K), which is the well-known “cool flame” stage. The dominant kinetics is the low temperature chemistry, leading to mild temperature rise and formation of H₂O₂, CO, and CH₂O, etc. The second-stage ignition occurs at intermediate temperature (~1,100 K), featured by depletion of H₂O₂, CH₂O and formation of CO. This stage is driven by the H₂O₂ chain branching via reaction H₂O₂+M=OH+OH+M. Due to the relatively low temperature at the second-stage, CO is not immediately oxidized to CO₂, instead, its oxidation occurs at a much higher temperature around 1,400 K, via a major exothermic reaction CO+OH=CO₂+H. For a near stoichiometric mixture, the difference between τ₂ and τ_T is usually not discernible due to the higher temperature rise rate at the second-stage, which immediately triggers the third-stage. As such, two-stage ignition is more commonly observed and discussed in the literature. Due to the absence of non-reactive pressure trace from experiment and additional kinetic targets, it is difficult to evaluate the heat loss and hence the effective volume evolution during the actual RCM experiment, for a more quantitative comparison. The qualitative agreement between the simulation and experiment nevertheless substantiates the three-stage oxidation during lean auto-ignition and provides necessary justification of the current work. Also, beyond the goal of the current work, more detailed combustion target measurement should be conducted under these engine relevant conditions, along with other novel combustion regimes and configurations (Zhao et al., 2016; Reuter et al., 2017) for the sake of kinetic model development.

The temperature dependence of the delay time τ₁, τ₂ and τ_T for lean n-heptane/air with ϕ = 0.3 and P = 40 atm is shown in Figure 4A. All three stages exhibit the phenomena of negative temperature coefficient (NTC). As temperature becomes either lower or higher outside of the NTC regime, the difference between τ₂ and τ_T becomes less discernible and eventually the three-stage oxidation transitions into two-stage ignition. Figure 4B shows the equivalence ratio dependence of τ₁, τ₂ and τ_T for n-heptane/air with P = 40 atm and T = 800 K. As ϕ increases from lean to rich, all three stages advance. For mixtures with ϕ between 0.45 and 0.85, second and third stages merge and exhibit two-stage ignition only. The three-stage ignition is most obvious under lean conditions, yet can still be identified for richer mixtures. In addition, the first-stage ignition delay τ₁ shows much lower ϕ-sensitivity compared with τ₂ and τ_T. In the following, we shall demonstrate how these ignition delay information and sensitivity can be used to explain the ignition evolution and reaction wave propagation.

In this study, auto-ignition and subsequent reaction front evolution and wave dynamics in mixture with thermal and concentration stratification is computationally investigated using a recently developed computational platform with detailed chemistry and transport. The computational configuration is one-dimensional with an initially stratified segment followed by a uniform segment. Insights into stratified combustion are obtained through a combined analysis using the three-stage oxidation and the classical Zeldovich reactivity gradient theory:

For all the auto-ignition induced reaction fronts, it is found that diffusion plays a minor role as justified using budget term analysis.

In the future, more complexities will be included to study the combined effects of thermal and equivalence ratio gradient on charge auto-ignition and subsequent wave dynamics, for example, with either parallel or opposite temperature and concentration gradients.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

MT: validation, investigation, data processing, visualization, and writing. QY: software and tool development. PL: conceptualization, review, and editing. PZ: conceptualization, methodology, analysis, writing, and supervision. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.