1 Introduction

Modern engine operating strategies aim to achieve high efficiency. Downsizing has been identified as a promising technology for accomplishing this goal. However, a higher tendency towards knocking combustion has been observed (Heywood 1988; Wang et al. 2017). In particular, a deeper understanding of the cycle-to-cycle variations (CCV) at the knock limit is important for further optimization of the combustion process (Ozdor et al. 1994).

In knocking combustion, auto-ignition occurs in the unburned mixture ahead of the flame front (Spicher et al. 1991). The different types of propagation that follow auto-ignition determine the intensity of knock (König et al. 1990; Bradley et al. 2002; Bradley and Kalghatgi 2009). At the knock limit, no detonations occur since the reaction front evolving from the auto-ignition is immediately separated from the respective pressure wave, and the knock intensity depends on the auto-ignited mass (Robert et al. 2015b). The auto-ignition process itself depends on the fuel under investigation. For example, an increase of the ignition delay times with rising temperature can be observed for long-chained hydrocarbon fuels inside the so-called negative temperature coefficient (NTC) regime (Liu et al. 2004; Ju et al. 2011). For the thermodynamic conditions in this regime, two-stage auto-ignition occurs (Law and Zhao 2012; Pan et al. 2016). This can become relevant for SI engines, since gasoline is a mixture of several large hydrocarbons. This multicomponent mixture is usually modeled by a reduced number of components in numerical simulations. Pera and Knop (2012) showed that ternary mixtures, e.g. Toluene Reference Fuels (TRF), are needed to formulate a surrogate fuel that captures the main gasoline properties. These TRFs were extended by ethanol to account for the composition of modern gasoline fuels (Kim et al. 2019).

The common approaches of knock modeling in the context of CFD can be divided into the direct modeling of the auto-ignition process, on one hand, and the determination of auto-ignition reaction progress by a precursor, on the other. Directly modeling the auto-ignition process requires detailed chemical kinetics. The associated high computational costs can be reduced by employing simplified kinetic approaches (Eckert et al. 2003; Liberman et al. 2005). A strategy based on the G-equation model for flame propagation and finite-rate chemistry for the unburned gas was employed in several RANS and one LES study for different surrogate fuels with reduced or skeletal mechanisms of varying complexity (Liang et al. 2007; Wang et al. 2012; Pal et al. 2018a, b; Chen et al. 2020). The pre-tabulation of chemistry is used in the tabulated kinetics of ignition (TKI) model (da Cruz 2004; Colin et al. 2005), which was applied to RANS studies investigating homogeneous charge compression ignition (HCCI) and Controlled Auto-Ignition (CAI) engines (Knop et al. 2009a, b) and modified to analyze auto-ignition in a spark ignition (SI) engine (Knop et al. 2011). With modifications regarding coupling and modeling, a similar approach was used for LES investigations of knocking combustion by Robert et al. (2015a, 2015b, 2019). In a precursor model, the evolution of a real (e.g. formaldehyde \({\text {CH}}_{2{\mathrm{O}}}\)) or pseudo species is expected to capture the auto-ignition process. In this regard, Livengood and Wu (1955) proposed the knock integral method (KIM). For simplicity, a linear relation between the auto-ignition time and the precursor mass fraction is assumed. In particular, the chemical source term of the precursor is expressed as the reciprocal of the ignition delay time. For the application in CFD simulations, empirical correlations (Lafossas et al. 2002; Kleemann et al. 2003) or reaction mechanisms of varying complexity (Teraji et al. 2005; Jaworski et al. 2010) were used to calculate and tabulate the ignition delay times, respectively. Linse et al. (2014) developed the generalized knock integral method (gKIM), which incorporates transport effects and the effect of turbulent fluctuations. Further adaption and application to a broader range of operating conditions demonstrated the capabilities of this approach (Kircher et al. 2021). These RANS studies were based on complex mechanisms for iso-octane.

Recent LES studies looked at knock of iso-octane (Robert et al. 2015a, b) and TRF surrogate fuels (Fontanesi et al. 2013; Robert et al. 2019; Chen et al. 2020). All of these studies demonstrated the benefits of LES over RANS. In particular, by resolving cycle-to-cycle variations, the influence of single cycles on the mean knock behavior can be investigated. However, the operating conditions were outside the NTC regime of the investigated fuels or the effects of complex chemistry associated with NTC behavior on the auto-ignition process have not been addressed.

With that, the influence of NTC behavior on the local auto-ignition process in engine operation under knocking conditions has not yet been investigated. In particular, the question remains how temperature stratification and flame propagation affect the local auto-ignition process under thermodynamic conditions in the NTC regime. Furthermore, it is of interest how cycle-to-cycle variations influence the auto-ignition process under these conditions.

To address these aspects, combustion at knocking conditions of a four-component Toluene Reference Fuel with ethanol addition matching the RON and MON of a real gasoline fuel is investigated for a single-cylinder research engine in a joint experimental-numerical study. The experiments conducted on a single-cylinder research engine are the base for subsequent numerical analyses. First, the operating conditions are characterized and the homogeneous auto-ignition delay times of the surrogate fuel at the engine-relevant thermodynamic conditions are analyzed. This is followed by the investigation of the auto-ignition process under homogeneous conditions in 0-D analyses. Subsequently, a multi-cycle engine LES study is conducted. Here, the auto-ignition process is accounted for by an extended version of the gKIM model (Linse et al. 2014; Kircher et al. 2021). The local combustion and auto-ignition process is investigated under stratified conditions and the influence of the NTC behavior is discussed. The investigations on cycle-to-cycle variation is subsequently performed based on a comparison of the model predictions to experimental knock trends. In addition, analyses of the auto-igniting mass and its relation to mean thermodynamic conditions for LES realizations that differ in mass burn rate are conducted.

The remainder of this work is structured as follows: In Sect. 2, the experimental configuration is presented. This is followed by the description of the numerical setup and the applied modeling approaches in Sect. 3. After that, the results are discussed in Sect. 4. The work closes with a summary of the main findings in Sect. 5.

2 Experimental Configuration

Experiments are conducted on a 4-valve single-cylinder research engine operated with port fuel injection (PFI) under homogeneous stoichiometric (\(\lambda =1\)) conditions. The valve timing results in an internal exhaust gas recirculation (EGR) of around 4%. The engine is operated close to full load with a spark timing close to the knock limit.

The engine specification and operating conditions are summarized in Table 1. For 1000 consecutive cycles, the in-cylinder pressure is measured using flush-mounted KistlerA6061B piezoelectric pressure transducers. A weighted moving average low-pass filter is used to separate high-frequency pressure oscillations induced by auto-ignition from the mean pressure trace.

Table 1 Engine specifications and operating conditions
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The experimental investigations are conducted for a Toluene Reference Fuel (TRF) consisting of iso-octane, n-heptane, toluene and ethanol with the respective mass fraction of 66.0%, 11.3%, 17.4% and 5.3%, based on a study by Kim et al. (2019). The RON and MON are 95.3 and 92.6, respectively. The NTC behavior of this specific TRF is discussed in Sect. 4.1.1.

3 Numerical Setup and Modeling

In this section, the numerical setup of the engine LES is presented (see Sect. 3.1), followed by the discussion of modeling of combustion (see Sect. 3.2) and auto-ignition (see Sect. 3.3).

3.1 LES Setup

The LES are performed using an OpenFOAM framework developed for engine simulations (Pati 2022). The meshes are hexa-dominant structured grids, with a base cell size of 0.5 mm in the cylinder. This is comparable to other LES studies presented in literature, e.g., by Robert et al. (2015a, 2015b). To capture geometric features, a local refinement down to 0.125 mm is applied. The maximum number of cells is approximately 5.9 million at bottom dead center. The sub-grid scale turbulence is modeled using the \(\sigma\) model (Nicoud et al. 2011).

A combustion LES matching the experimentally obtained mean pressure trace (cf. Fig. 1) is used to initialize a consecutive full-cycle simulation. Those results serve as initial conditions for the multi-cycle LES realizations discussed in Sect. 4.2. A detailed gasoline surrogate mechanism with 485 species and 2081 reactions (Cai et al. 2019) is used for the combustion and auto-ignition modeling approaches, which are described in the next sections.

3.2 Combustion Modeling

Modeling of the turbulent flame propagation is based on an algebraic flame surface density (FSD) model by Muppala et al. (2005) coupled to a flamelet-generated manifold (FGM) approach (van Oijen and de Goey 2000). For the construction of the FGM, adiabatic 1-D laminar freely propagating flames are computed with an in-house solver (Zschutschke et al. 2017). The manifold is generated by varying the pressure p, the unburned temperature \(T_\mathrm {u}\) and the initial composition - taking into account different exhaust gas recirculation (EGR) mass fractions. The variation in \(T_\mathrm {u}\) spans different enthalpy levels h, which are then mapped to a normalized enthalpy \(h_\mathrm {norm}\):

$$\begin{aligned} h_\mathrm {norm}(Z,p,\mathrm {EGR}) = \frac{h - h_{\min }(Z,p,\mathrm {EGR}) }{h_{\max }(Z,p,\mathrm {EGR}) - h_{\min }(Z,p,\mathrm {EGR})}\, . \end{aligned}$$
(1)

Here, \(h_{\max }(Z,p,\mathrm {EGR})\) and \(h_{\min }(Z,p,\mathrm {EGR})\) are the maximum and minimum enthalpies for a given level of Z, p and EGR. The reaction progress variable \(Y_\mathrm {c}\) is defined by the sum of mass fractions of carbon dioxide \(Y_\mathrm {CO2}\) and carbon oxide \(Y_\mathrm {CO}\). The normalized reaction progress variable c is given by

$$\begin{aligned} \begin{aligned}&{c}(Z,p,h_\mathrm {norm},\mathrm {EGR}) =&\frac{{Y}_\mathrm {c} - {Y}_\mathrm {c, min}(Z,p,h_\mathrm {norm},\mathrm {EGR})}{{Y}_{\mathrm {c,max}}(Z,p, h_\mathrm {norm},\mathrm {EGR}) - {Y}_\mathrm {c, min}(Z,p,h_\mathrm {norm},\mathrm {EGR})} \, , \end{aligned} \end{aligned}$$
(2)

where \({Y}_{\mathrm {c,min}}(Z, p, h_\mathrm {norm},\mathrm {EGR})\) is the respective value in the unburned and \({Y}_{\mathrm {c,max}}(Z,p,h_\mathrm {norm},\mathrm {EGR})\) that in the burned mixture.

The parameterization of the thermo-chemical state \(\varvec{\varphi }\) is defined by the mixture fraction Z, normalized reaction progress variable c, pressure p, normalized enthalpy \(h_\mathrm {norm}\) and exhaust gas recirculation EGR, i.e. \(\varvec{\varphi } = \varvec{\varphi } (Z, c, p, h_\mathrm {norm}, \mathrm {EGR})\). In the LES, a transport equation is solved for the normalized reaction progress variable \({\widetilde{c}}\) (Janas 2017):

$$\begin{aligned} \frac{\partial (\bar{\rho } {\widetilde{c}} )}{\partial t} + \nabla \cdot (\bar{\rho } \widetilde{\varvec{u}} {\widetilde{c}})= \nabla \cdot \left[ \left( \bar{\rho } D + \frac{\mu _\mathrm {t}}{\mathrm {Sc_{t}}}\right) \nabla {\widetilde{c}}\right] + \rho _\mathrm {u} s_\mathrm {L} \Sigma \, . \end{aligned}$$
(3)

Here, \(\varvec{u}, \rho , \rho _\mathrm {u}, D, \mu _\mathrm {t}, s_\mathrm {L}\) and \(\Sigma\) denote the velocity vector, density, density conditional upon the unburned state, diffusion coefficient, eddy viscosity, laminar burning velocity and FSD, respectively. For diffusion modeling, a unity Lewis number is assumed. A gradient assumption is used for turbulent flux closure, with the turbulent Schmidt number set to \(\mathrm {Sc_t} = 0.4\). The combustion process is initiated by imposing a sphere of burned state (\({\widetilde{c}} = 1,r=1.5\) mm) in the spark plug gap. Different mass burning rates result from a variation in the timing of the start of combustion.

The passive scalar \(\mathrm {EGR}\) is accounting for the burned gas mass fraction of the previous cycle and a respective transport equation is solved. Prior to exhaust valve opening (EVO), \({\widetilde{c}}\) is mapped to EGR and reset to zero (Janas 2017). During the simulation, the state \(\varvec{\varphi }\) is used to access the tabulated manifold. Transport and thermodynamic data are determined by the extracted species composition. Further, the tabulated unburned density and the laminar burning velocity are used in Eq. (3) and for determining \(\Sigma\) in the FSD model (Muppala et al. 2005):

$$\begin{aligned} \Sigma =\left[ 1+0.46Re_{\Delta }^{0.25}\left( \frac{u_{\Delta }^{\prime }}{s_{L}}\right) ^{0.3}\left( \frac{p}{p_{0}}\right) ^{0.2}\right] \vert \nabla {\widetilde{c}} \vert \, , \end{aligned}$$
(4)

where \(Re_\Delta\) and \(u_{\Delta }^{\prime }\) are turbulent Reynolds number and unresolved turbulent velocity fluctuation, respectively. The reference pressure \(p_0\) is 0.1 MPa.

For the analyses described in Sect. 4.2, the combustion progress during the engine cycle is expressed in terms of the mass fraction burned (MFB):

$$\begin{aligned} \mathrm {MFB} = \frac{\int \bar{\rho } {\widetilde{c}}\mathrm {d}V}{\int \bar{\rho } \mathrm {d}V} \, . \end{aligned}$$
(5)

The unburned mass is calculated from the initial mass, \(m_\mathrm {u,0}\), and the MFB:

$$\begin{aligned} m_\mathrm {u} = m_\mathrm {u,0}\cdot (1-\mathrm {MFB}) \, . \end{aligned}$$
(6)

3.3 Auto-ignition Modeling

The modeling of the auto-ignition is based on the generalized knock integral method (gKIM) proposed by Linse et al. (2014). The fuel consumption is used as a pseudo precursor species and determines the auto-ignition progress variable conditioned on the unburned state \(Y_{{\mathrm{c}_{\mathrm{I},\mathrm{u}}}}\). This is used to track the auto-ignition process in the unburned mixture. With the parameterization of the thermo-chemical state \(\varvec{\varphi }_\mathrm {I} = \varvec{\varphi }_\mathrm {I} (Z, Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}, p, h_\mathrm {norm}, \mathrm {EGR})\), an auto-ignition manifold is constructed based on homogeneous reactor calculations with the aforementioned in-house solver (Zschutschke et al. 2017), where again the pressure p, the unburned temperature \(T_\mathrm {u}\) and the EGR level are varied.

The source term \({\dot{\omega }}_{Y_{{\mathrm{c}}_{\mathrm{I,u}}}}\) is given by the temporal evolution of \(Y_{{\mathrm{c}}_{\mathrm{I,u}}}\). In previous works (Linse et al. 2014; Kircher et al. 2021), a linearization was used for source term modeling. Following Livengood and Wu (1955), it was assumed that the auto-ignition progress variable increases linearly until auto-ignition. With that, the source term is given by the reciprocal of the auto-ignition delay time \(\tau\):

$$\begin{aligned} {\dot{\omega }}_{Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}, \mathrm{lin}}}} = \frac{1}{\tau } \, . \end{aligned}$$
(7)

This model is referred to as linear gKIM in the following. In this work, the detailed non-linear auto-ignition evolution is captured in the source term. Similar to existing modeling approaches, e.g., the TKI model (Knop et al. 2011), the source term is defined by the instantaneous reaction rates of the auto-ignition progress variable:

$$\begin{aligned} {\dot{\omega }}_{Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}, \mathrm{det}}}} = \frac{\partial {Y}_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}}{\partial t} \, . \end{aligned}$$
(8)

This model is referred to as detailed gKIM in the following. A transport equation for \(Y_{{\mathrm{c}}_{\mathrm{I,u}}}\) is solved (Kircher et al. 2021):

$$\begin{aligned} \frac{\partial (\bar{\rho } {\widetilde{Y}}_{\mathrm{c}_{\mathrm{I},\mathrm{u}}})}{\partial t}+\nabla \cdot (\bar{\rho } \varvec{{\widetilde{u}}} {\widetilde{Y}}_{\mathrm{c}_{\mathrm{I},\mathrm{u}}})= \nabla \cdot \left[ \left( \bar{\rho } D +\frac{\mu _\mathrm {t}}{{\mathrm{Sc}_{\mathrm{t}}}}\right) \nabla {\widetilde{Y}}_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}\right] + \bar{\rho }\widetilde{\dot{\omega }}_{Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}} \, . \end{aligned}$$
(9)

Again, a gradient assumption is used for turbulent flux closure using a turbulent Schmidt number of \(\mathrm {Sc_t}=0.4\) and diffusion modeling assumes the Lewis number being unity.

In the following, the conditional auto-ignition progress variable is normalized to \({c}_{\mathrm{I},\mathrm{u}} = Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}/Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}, {\max }}}\), where \(Y_{c_{\mathrm{I},\mathrm{u}, \max }}\) is the value after auto-ignition. For further data processing (cf. Sect. 4.2), the auto-ignition progress variable is retrieved from \({c}_\mathrm {I} = (1 - {c}) \cdot {c}_{\mathrm{I},\mathrm{u}}\), assuming that no auto-ignition can occur in the burned gas (Linse et al. 2014). The set \(\Omega _\mathrm {crit}\), representing all computational cells that exhibit a critical auto-ignition state, is defined by \(c_\mathrm {I} \ge \varepsilon = 0.95\) in this work. In accordance with Kircher et al. (2021), the mass represented by these cells is referred to as critical mass \(m _\mathrm {crit}\) hereafter.

4 Results

First, the auto-ignition process is investigated under homogeneous engine-relevant conditions (see Sect. 4.1). Subsequently, stratification and flame propagation is accounted for by performing a multi-cycle engine LES (see Sect. 4.2).

4.1 Auto-ignition Characterization Under Homogeneous Conditions

In this section, the engine operating conditions are discussed and the auto-ignition delay times of the investigated fuel are analyzed (see Sect. 4.1.1). For further characterization, the auto-ignition process of the surrogate fuel is investigated under homogeneous conditions in 0-D analyses. This is performed under engine operating conditions first (see Sect. 4.1.2) and for the variable unburned temperature and pressure of the mean engine cycle thereafter (see Sect. 4.1.3).

4.1.1 Investigation of Auto-ignition Delay Times Under Engine Operating Conditions

The engine operating conditions and the global combustion progress are investigated by means of a pressure trace analysis based on the average of the 1000 measured cycles with the Three Pressure Analysis (TPA) incorporated in the engine simulation tool GT-Power. In Fig. 1 the evolution of the respective unburned temperature \(T_\mathrm {u}\), pressure p and the mass fraction burned (MFB) are shown. The start of combustion, expressed as 1% of the mass fraction burned (MFB1), and end of combustion (MFB99) are marked for orientation. After the onset of combustion at approximately 5\(^\circ\)CA, the pressure and unburned temperature increase to about 840 K and 45 bar (around 20\(^\circ\)CA) and then decrease until the end of combustion at about 33\(^\circ\)CA and beyond.

Fig. 1
figure 1

Evolution of unburned temperature \(T_\mathrm {u}\), pressure p and the mass fraction burned (MFB) for the mean experimental cycle. MFB1 and MFB99 are marked for orientation

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In the range of these thermodynamic conditions, the auto-ignition delay times \(\tau\) of the TRF are investigated. These are shown in Fig. 2 as a function of \(T_\mathrm {u}\) and p. For unburned temperatures between approximately 750 K and 850 K (depending on the pressure) auto-ignition delay times increase with higher temperatures. Thus, the TRF investigated in this study shows a distinctive negative temperature coefficient (NTC) behavior. The NTC regime is marked with dashed lines in Fig. 2.

Fig. 2
figure 2

Auto-ignition delay times \(\tau\) as a function of unburned temperature \(T_\mathrm {u}\) and pressure p. The NTC limits are shown as dashed lines. A trajectory of the conditions in the fresh gas during the combustion progress is overlaid as a black curve with MFB50 marked for reference

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In order to prove the relevance of the NTC regime for the current operating conditions, Fig. 2 additionally shows the trajectory of the auto-ignition delay times, determined based on \(T_\mathrm {u}\) and p of the mean experimental cycle in the interval between MFB1 and MFB99 (cf. Fig. 1), MFB50 is denoted as reference. It can be clearly observed that most of the cycle happens in the NTC regime.

As shown by Law and Zhao (2012), a two-stage auto-ignition process can be observed in the NTC regime. The presence of two-stage auto-ignition for the TRF used in this study is investigated in the next section.

4.1.2 Auto-ignition Process Under Engine Operating Conditions

The auto-ignition process under representative thermodynamic conditions is further analyzed with respect to the precursor modeling. Therefore, a homogeneous reactor simulation using the detailed mechanism (Cai et al. 2019) is initialized with stoichiometric conditions at unburned temperature and pressure corresponding to MFB50 of the mean cycle (cf. Fig. 2). The obtained temperature evolution during the auto-ignition process is depicted in Fig. 3. The upper plot shows the evolution of the auto-ignition progress variable \(Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}\), and the lower plot shows its source term \(\dot{\omega }_{Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}}\).

Fig. 3
figure 3

Auto-ignition process for thermodynamic conditions of MFB50. Development of temperature, auto-ignition progress variable (top) and source term of auto-ignition progress variable (bottom). Results for a coupled simulation applying the auto-ignition modeling are shown in blue; the results of the detailed gKIM are shown as dashed-dotted lines, and those of the linear gKIM as dashed lines

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The temperature evolution exhibits the expected two-stage auto-ignition behavior. The first-stage ignition results in a temperature rise of approximately 45 K. During the second-stage ignition, the temperature increases significantly by 1600 K. The two-stage behavior is reflected in the evolution of the auto-ignition progress variable, which is shown as a solid black line. Both, the first-stage and the second-stage auto-ignition are characterized by the increased formation of \(Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}\) and a corresponding peak in the source term. This source term is used in the detailed gKIM (see Sect. 3.3). When applying the auto-ignition modeling in a coupled simulation to a homogeneous reactor under the same thermodynamic conditions, the detailed gKIM is able to reproduce the evolution of the auto-ignition progress variable \(Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}\) with very high accuracy, as shown by the dash-dotted blue line. The simplified source term assumption of the linear gKIM leads to a linear increase of the auto-ignition progress variable \(Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}\) until auto-ignition (cf. dashed blue line). While the auto-ignition delay time is correctly reproduced, the two-stage auto-ignition is obviously not captured by this model.

The analysis confirms, that for the representative thermodynamic conditions inside the NTC regime, a two-stage auto-ignition occurs under homogeneous conditions for the investigated surrogate fuel. This is consistent with the detailed analysis for n-heptane performed by Law and Zhao (2012). They showed, in particular, that inside the NTC regime, the first-stage auto-ignition determines the total auto-ignition delay time by changing the initial conditions of the second-stage auto-ignition. Adopting the methodology used by Law and Zhao (2012), the auto-ignition delay times of the two stages are analyzed for the TRF at the highest and lowest pressure investigated before, see Fig. 4. The total and first-stage auto-ignition delay times are shown in black and orange, respectively. The second-stage auto-ignition delay time is defined by the difference of the total and the first-stage auto-ignition delay time and shown in blue. Squares refer to 20 bar and circles to 60 bar pressure. In accordance with the findings of Law and Zhao (2012), the first-stage auto-ignition delay times decrease with increasing temperature, while the second-stage auto-ignition delay times increase. This results in increasing total auto-ignition delay times in the NTC regime. This behavior is also observed for higher pressures but with lower first-stage auto-ignition delay times and the NTC regime shifted to higher temperatures.

Fig. 4
figure 4

Analysis of total (black), first-stage (orange) and second-stage auto-ignition delay times for 20 (squares) and 60 bar (circles)

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This analysis demonstrates that two-stage auto-ignition behavior determines the NTC behavior for the TRF investigated. The relevance of two-stage auto-ignition for the variable thermodynamic conditions of the mean engine cycle is investigated in the next section.

4.1.3 Auto-ignition Process During the Mean Engine Cycle

The auto-ignition process is further investigated under variable engine-relevant thermodynamic conditions in a 0-D configuration. Based on the pressure and unburned temperature of the mean experimental cycle (cf. Fig. 1) and stoichiometric mixture, the source term for the precursor modeling is extracted from the auto-ignition manifold. With that, Eqs. (7) and (8) are solved to obtain the evolution of the auto-ignition progress variable \(Y_{{\mathrm{c}}_{\mathrm{I,u}}} \), which is then normalized (cf. Sect. 3.3). The methodology is illustrated in Fig. 5.

Fig. 5
figure 5

Methodology of 0-D analysis: the thermodynamic state of the mean experimental cycle \(\varvec{\varphi _\mathrm {I,exp}}\) is used to access the auto-ignition manifold. The extracted source term is integrated over time to retrieve the auto-ignition progress variable \(Y_{\mathrm{c}_{\mathrm{I},\mathrm{u}}}\)

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The upper plot of Fig. 6 shows the evolution of the normalized auto-ignition progress variable, and the lower plot the respective source term. Additionally, the imposed pressure is depicted for orientation.

Fig. 6
figure 6

0-D analysis of auto-ignition modeling based on the thermodynamic conditions of the average experimental cycle. Imposed pressure, normalized auto-ignition progress variable (top) and source term of auto-ignition progress variable (bottom). The detailed gKIM is shown as solid lines, the linear gKIM as dashed lines

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The importance of non-linear auto-ignition chemistry for the variable thermodynamic conditions can be illustrated by comparing the two different approaches for the source term of the precursor models (cf. Sect. 3.3). Without the incorporation of the two-stage auto-ignition characteristics in the linear gKIM (cf. Fig. 3), the final value of the progress variable is below unity and hence no auto-ignition is predicted. Given the knocking operating conditions, this is not reasonable and the linear gKIM is not considered in the further analysis. The prediction of the detailed gKIM, incorporating complex auto-ignition chemistry, is shown as solid lines. A two-stage ignition behavior can be observed. Around top dead center (TDC, 0.0\(^\circ\)CA), first-stage ignition is predicted. At this time, the normalized auto-ignition progress variable increases to around 0.5 since the source term shows a respective local maximum. At peak pressure, the source term has a second peak and \(c_\mathrm {I,u}\) reaches unity, predicting second-stage ignition.

The investigations conducted in this section show, that the two-stage auto-ignition behavior observed in a representative homogeneous reactor simulation (cf. Sect. 4.1.2), also occurs under the engine-relevant variable thermodynamic conditions. However, those reactor simulations cannot describe spatial stratification and the influence of turbulent flame propagation. This is incorporated in the engine LES conducted in the following.

4.2 Engine LES

The previous analyses confirmed, that engine combustion takes place under NTC-relevant conditions and a two-stage auto-ignition process is likely to occur. In this section, a multi-cycle engine LES is performed to investigate the influence of turbulent flame propagation and temperature stratification on the local auto-ignition process under the NTC related thermodynamic conditions investigated. After validating the engine LES combustion methodology (see Sect. 4.2.1), the process of flame propagation and auto-ignition is investigated for a selected LES cycle (see Sect. 4.2.2). Subsequently, the model predictions in terms of the onset of auto-ignition and the mass affected by it (critical mass) are compared to experimentally recorded pressure traces including cycle-to-cycle variations (see Sect. 4.2.3). Further investigations on cycle-to-cycle variations are conducted to analyze the spatial distribution and temporal evolution of the critical mass (see Sect. 4.2.4) and the influence of combustion progress on the unburned mass at knock onset (see Sect. 4.2.5).

4.2.1 Combustion Process

At first, experimental and LES pressure traces are depicted in Fig. 7. The pressure traces of the 1000 single cycles recorded in the experiment are shown in gray color. After spark ignition, the pressure traces show large cycle-to-cycle variations (CCV). Faster combustion cycles lead to higher peak pressures, with differences in the peak pressure of around 20 bar compared to the slowest cycles, 17 LES realizations are shown in blue color. They properly reproduce both the fast and the slow cycles, and almost the entire experimental envelope is captured by the LES.

Fig. 7
figure 7

Pressure traces of experiments and LES realizations

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4.2.2 Local Auto-ignition Process and Flame Propagation

The local auto-ignition process and the influence of flame propagation is investigated in the following. In particular, the fastest burning LES cycle is considered due to its relevance for auto-ignition, since experimental cycles with a comparable pressure trace show knock. The analysis depicted in Fig. 8 is based on a horizontal slice through the cylinder with a vertical position of 2.5 mm above the TDC of the piston, cutting the lower electrode of the spark plug. A perspective from above is chosen, with the engine geometry shown transparently for orientation. The times −9.0\(^\circ\)CA, 0.0\(^\circ\)CA, 9.0\(^\circ\)CA and 18\(^\circ\)CA are considered, covering the relevant time span of flame propagation and the auto-ignition process. The earliest time is at the top, the latest at the bottom. The left column shows the unburned temperature distribution, and the right column depicts the results of the precursor model. The flame front defined by \({\widetilde{c}}=0.5\) is shown as a white iso-line to illustrate the influence of flame propagation on auto-ignition in the unburned gas. The mean cylinder pressure \(\langle {\overline{p}} \rangle _\mathrm {cyl}\) is given for reference.

Fig. 8
figure 8

Horizontal slice at 2.5 mm above TDC position of the piston; perspective from above, with engine geometry shown transparently. Analysis of local auto-ignition process for four different instants in time: prior to spark ignition, shortly after spark ignition, early combustion phase and close to peak pressure. The earliest time is at the top, the latest at the bottom. The plots show unburned temperature, \({\widetilde{T}}_\mathrm {u}\), on the left side and auto-ignition progress, \({\widetilde{c}}_\mathrm {I}\), on the right. The flame front is visualized based on \({\widetilde{c}}=0.5\) as a white iso-line. The mean cylinder pressure, \(\langle {\overline{p}} \rangle _\mathrm {cyl}\), is given for reference

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Already prior to spark ignition (−9.0\(^\circ\)CA), the \({\widetilde{c}}_\mathrm {I}\)-field shows the onset of auto-ignition. The influence of unburned temperature on the first-stage auto-ignition delay time in the NTC regime, observed under homogeneous conditions in Sect. 4.1.2, is also visible for the auto-ignition process in the engine under temperature stratification. In particular, the first-stage auto-ignition progress is highest (\({\widetilde{c}}_\mathrm {I} \approx 0.15\)) in the regions of the highest unburned temperatures (\({\widetilde{T}}_\mathrm {u}\approx 780\) K), due to the lowest first-stage auto-ignition delay times (cf. Fig. 4).

At TDC, shortly after spark ignition (0.0\(^\circ\)CA), a widespread first-stage auto-ignition is observed. The increased mean unburned temperature and pressure are associated with lower first-stage auto-ignition delay times (cf. Fig. 4). Thus, regions where auto-ignition had not yet progressed, undergo a more pronounced first-stage auto-ignition. For regions that already underwent first-stage auto-ignition, the further progress is related to the second-stage auto-ignition delay time. Since these increase towards the upper NTC limit at approximately 850 K, the auto-ignition process is decelerated in regions with higher unburned temperatures. Thus, as a result of the NTC behavior, the highest values of \({\widetilde{c}}_\mathrm {I}\) are observed in regions of intermediate temperatures of approximately 750 K.

With further combustion progress at 9.0\(^\circ\)CA, the previously not-ignited areas of the unburned mixture now reached unburned temperatures high enough (\({\widetilde{T}}_\mathrm {u}\approx 750\) K) to undergo first-stage auto-ignition. Again it is more pronounced due to the increased pressure, while at higher unburned temperatures the auto-ignition process is prolonged. Now almost the entire unburned mixture is in an auto-igniting state. However, the propagating flame reduces the unburned mass and terminates the auto-ignition process in the burned mixture (\({\widetilde{c}}_\mathrm {I}\)=0).

At a time close to peak pressure (18.0\(^\circ\)CA), the majority of the mixture is consumed by the flame, leaving only pockets of unburned gas. Within those, the value of \({\widetilde{c}}_\mathrm {I}\) is almost uniform, due to the previously widespread first-stage auto-ignition. In particular, the further pressure increase lowered the second-stage auto-ignition delay times and hence a relevant part of the remaining unburned mass is predicted to undergo second-stage auto-ignition. This is indicated by values of \({\widetilde{c}}_\mathrm {I} > \varepsilon\) and contributes to the critical mass (see Sect. 3.3), which can be related to knock intensity as discussed in the next section.

4.2.3 Global Knock Characteristics

For operation at the knock limit, the knock intensity is directly proportional to the mass consumed during auto-ignition (Robert et al. 2015a). For the precursor modeling utilized in this work, this mass is given by the critical mass and the location of the peak of its temporal evolution indicates the crank angle of knock onset, as shown in a previous work (Kircher et al. 2021).

The experimental unfiltered pressure signal p and the respective knock amplitude \(p^\prime\) are depicted on the left-hand side of Fig. 9 for representative fast, medium and slow cycles, with three individual cycles each. For faster cycles, the peak pressure appears closer to TDC (0.0\(^\circ\)CA) and the knock amplitudes increase. The highest amplitudes, associated with the mean knock onset, appear in the vicinity of the crank angle of peak pressure. The overall knock intensity is low.

Fig. 9
figure 9

Experimental unfiltered pressure signal and knock amplitude for three levels of peak pressure with three cycles each (left). Pressure traces and development of critical mass for three LES realizations (right)

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The pressure traces of three corresponding LES cycles are shown on the right-hand side of Fig. 9. Additionally, the development of the critical mass over the engine cycle is shown. For all three cycles, the peak of the critical mass is close to the crank angle of peak pressure. Furthermore, it is located closer to TDC for the fast cycle. With that, location and sensitivity are in good agreement with the experimental knock onset. The maximum critical mass increases with faster combustion, meaning a higher amount of unburned mixture is predicted to undergo auto-ignition. The trend of increasing critical mass agrees well with the increasing knock amplitudes in the experiment.

Thus, the amount of critical mass and the time at which its maximum is observed serve as indicator of the knock amplitude and its onset for the fuel under investigation with operating conditions in the NTC regime. Furthermore, the amount of critical mass and its temporal evolution are shown to be affected by cycle-to-cycle variations. Both aspects are further considered in the next section, based on the local distribution of the critical mass.

4.2.4 Local Distribution of Critical Mass

The further investigation of the local evolution of the critical mass and the influence of the propagating flame is based on a time-series, where the crank angle of the maximum critical mass, defining the knock onset, is considered as the reference time. For that, the set of critical cells, \(\Omega _\mathrm {crit}\) (see Sect. 3.3), is extracted for the three representative LES realizations shown in the last section (cf. Fig. 9).

Fig. 10
figure 10

Isometric view of fast (left), medium (middle) and slow cycle (right). Spatial distribution of critical cells shown in red color. Flame front (\({\widetilde{c}}=0.5\)) visualized in gray. Time series for 4 crank angles each. Reference time of peak of critical mass in third row, 2.0\(^\circ\)CA and 1.0\(^\circ\)CA before and 1.0\(^\circ\)CA after reference time in first, second and last row, respectively

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In Fig. 10, the critical cells are highlighted in red color in an isometric view of the cylinder for the fast (left), medium (middle) and slow cycle (right). The flame front (\({\widetilde{c}}=0.5\)) is visualized as a gray iso-contour. The reference time is shown in the third row, while the first, second and last row correspond to 2.0\(^\circ\)CA and 1.0\(^\circ\)CA before and 1.0\(^\circ\)CA after this reference time, respectively. The temporal development of critical cells is comparable for all three LES realizations, based on the common reference time. At 2.0\(^\circ\)CA before knock onset (top row), the first critical cells appear in the unburned mixture ahead of the flame front. Within the time of 1.0\(^\circ\)CA (second row), a relevant part of the volume enclosed by the cylinder walls and the wrinkled flame front reaches a critical state (\({\widetilde{c}}_\mathrm {I} > \varepsilon\)) due to the NTC related widespread second-stage auto-ignition (cf. Sect. 4.2.2). Thus, the turbulent flame propagation, which defines the regions of unburned mixture, is a main factor determining the auto-ignition locations. At the time of knock onset (third row), the distance between flame front and auto-ignition spots is small. Therefore, the propagating flame starts to consume the auto-ignited mass within the next 1\(^\circ\)CA (cf. bottom row) and the critical mass reduces (cf. Fig. 9). With the short time available for a potential reaction front to develop from the auto-ignition locations before being consumed by the flame front, detonation waves are unlikely to occur. This explains the direct relation between knock intensity observed in the experiments and the amount of critical mass shown in the previous section.

4.2.5 Unburned Mass at Knock Onset

Beside these similarities in the temporal evolution of the critical mass, cycle-to-cycle variations in the flame front propagation are suggested in Fig. 10. At the reference time of knock onset (third row), the largest distance of the flame front to the walls is observed for the fast cycle, which indicates less combustion progress at knock onset. To further investigate this observation, the global combustion and auto-ignition process is shown in Fig. 11. The upper plot depicts the MFB for the three LES cycles discussed in the last section in solid lines, and the respective MFB at knock onset (MFBKO) is indicated by a dot. In addition, the mean value of the auto-ignition progress variable in the unburned mixture is depicted as a dashed line. The lower plot shows the pressure trace (solid) and the mean unburned temperature of the unburned mixture (dashed).

Fig. 11
figure 11

Evolution of combustion (solid lines) and auto-ignition progress (dashed lines) (top), as well as pressure (solid lines) and unburned temperature conditioned on the unburned mixture (dashed lines) (bottom) for three LES realizations. The dots in the upper plot indicate CA and MFB at knock onset

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Prior to spark ignition, the (first-stage) auto-ignition process is solely controlled by the increasing pressure and unburned temperature resulting from the compression caused by the piston movement. This leads to similar mean auto-ignition progress for all three cycles in this time span. However, the variation in the onset of combustion results in differences in the subsequent auto-ignition progress. With decreasing crank angle of combustion onset, higher mass burn rates are observed, leading to stronger compression heating. As a result, pressure and temperature increase earlier and more rapidly. This in turn increases the reactivity in the unburned mixture, leading to earlier second-stage auto-ignition and thus knock onset. Thus, despite the higher mass burn rate, the earlier knock onset results in a higher fraction of unburned mixture at knock onset. With that, the mass fraction burned at knock onset (MFBKO) is lowest for the fast cycle (74.7%), resulting in the highest unburned mass at knock onset \(m_\mathrm {u,KO}\) (cf. Fig. 11 and Table 2). Since a relevant part of the unburned mass auto-ignites, the critical mass at knock onset \(m_\mathrm {crit,KO}\) correlates with \(m_\mathrm {u,KO}\). With that, the highest peak of critical mass of the fast cycle (cf. Fig. 9) can be related to the lowest MFBKO.

Table 2 Summary of combustion progress (MFBKO), unburned mass (\(m_\mathrm {u,\mathrm {KO}}\)) and critical mass (\(m_\mathrm {crit,KO}\)) at knock onset
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Hence, the NTC behavior affects the local auto-ignition process in the unburned mixture, in particular leading to a widespread two-stage auto-ignition process. Here, the second-stage auto-ignition is mainly influenced by differences in the mass burn rate. Thus, for the operating conditions at the knock limit under consideration, the flame propagation determines the location and amount of the auto-igniting mass, and thus the knock intensity.

5 Conclusions

In this work, combustion and auto-ignition of a four-component Toluene Reference Fuel (TRF) under engine operating conditions at the knock limit were investigated. In particular, auto-ignition takes place inside the negative temperature coefficient (NTC) regime and is captured by a precursor model.

Investigations of the auto-ignition process were conducted under homogeneous conditions in 0-D configurations based on the experimental data. Subsequently, a multi-cycle engine LES study was performed, investigating the influence of turbulent flame propagation and temperature stratification on the local auto-ignition process under the NTC related thermodynamic conditions, as well as the influence of cycle-to-cycle variations on auto-ignition and knock behavior.

The main findings of the 0-D analyses can be summarized as follows:

  • For the engine-relevant thermodynamic conditions inside the NTC regime, a two-stage auto-ignition process occurs.

  • A non-linear source term formulation is needed to capture this.

  • While the second-stage auto-ignition delay time increases with temperature in the NTC regime, the first-stage auto-ignition delay time decreases.

The following was observed in the engine LES:

  • The NTC behavior affects the local auto-ignition process, leading to a widespread first-stage auto-ignition.

  • A relevant part of the unburned mass undergoes second-stage auto-ignition.

  • For the fastest cycle, the highest amount of mass is consumed by auto-ignition and the earliest knock onset is predicted.

  • The knock intensity observed in the experiments is directly related to the mass consumed by auto-ignition.

  • Detonation waves are unlikely to occur due to the fast consumption of the auto-ignited mass by the flame front.

  • Higher mass burn rates lead to higher reactivity of the unburned mixture and thus to earlier knock onset.

  • The auto-ignited mass correlates with the combustion progress at knock onset.

The results clearly show that the NTC behavior affects the local auto-ignition process in the unburned mixture. However, the flame propagation determines the knock intensity for the operating conditions at the knock limit. In this context, the inclusion of a wider range of operating conditions represents an interesting direction for further studies on the influence of NTC behavior on knocking combustion.