Direct Numerical Simulation of Preignition and Knock in Engine Conditions

Abstract

Theoretical framework and high-fidelity direct numerical simulations (DNS) on preignition and knock encountered in internal combustion (IC) engines are briefly reviewed in this chapter. The theoretical framework is presented first, and systematically followed by one-dimensional and multi-dimensional simulations. A number of high-fidelity simulations under realistic IC-engine conditions were used to demonstrate the super-knock mechanism in the presence of the complex chemistry-turbulence interaction. The ratio of ignition delay time to eddy-turnover time, $\tau _\mathrm{ig}/\tau _t$, and the ratio of the most energetic length scale of turbulence and temperature fluctuations, $l_t/l_T$, affect the knock intensities are discussed. Different statistical metrics extracted from the multi-dimensional simulations to predict knock intensity are also presented. The quantitative determination of strong and weak ignition modes using the predicted Sa-based metrics is also cover in this chapter.

1 Introduction

Modern downsized and boosted internal combustion (IC) engines provide higher thermodynamic efficiencies and cleaner combustion. However, these engines operated under extreme conditions of high load suffer from a higher possibility of abnormal preignition process, a premature auto-ignition induced by hot spots leading to ignition front propagation [1,2,3,4,5,6,7]. The mechanism and the source of such irregular preignition events are still not well-understood [8]. The early flames developed from preignition kernels increase the temperature and pressure of the end-gas that may induce a stronger end-gas auto-ignition, leading to extremely high-knock intensities, and even super-knock under high-load and/or high-temperature conditions. As representatively shown in Fig. 1 [6], super-knock cycle (S2) is characterized by high-pressure amplitudes and intensities due to localized detonation development that may lead to severe structural damage to the engine components. Therefore, a reliable criterion to predict irregular combustion phenomena is of critical importance [1, 3, 4, 6, 9,10,11,12].

Fig. 1

(taken from Refs. [6, 7])

Examples of in-cylinder pressure traces show preignition and superknock in a turbocharged SI engine: super-knock cycle (S2) with preignition at 19 CAD BTD and KI $\sim$ 14.5 MPa, moderate knock cycle (K2) with preignition at 6 CAD BTDC and KI $\sim$ 2.4 MPa, and normal (N2) cycle without preignition

A large spectrum of turbulence length and time scales combined with extremely short time scales of complex combustion process in detonation simulations pose a significant challenge in reproducing the phenomena using computational simulations, which require extremely fine spatial and temporal resolutions to capture the highly intermittent localized characteristics. This chapter reviews our recent efforts to investigate the fundamental issues of the knock events using high-fidelity direct numerical simulations (DNS) with the capability of fully resolving all temporal and spatial scales and the complex interaction of thermochemistry and turbulence [3, 4, 13, 14]. First-principle DNS is capable of unraveling the complex interplay between turbulence and chemical reactions to provide a better understanding of the mechanism of detonation development encountered in modern combustion devices under extreme high-load operating conditions, and to develop a reliable predictive model for real-world industrial applications [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

In the following, a theoretical framework on super-knock development encountered in internal combustion (IC) engines is briefly reviewed, followed by high-fidelity DNSs in the subsequent sections. A systematically parametric set of two-dimensional (2-D) and three-dimensional (3D) DNS simulations are conducted at engine-relevant conditions [1,2,3,4, 32]. Effects of various parameters such as length and time scales of turbulence and scalar fluctuations on the occurrence and intensity of detonation are examined and a scaling relation for knock prediction is proposed [33,34,35,36,37,38,39,40,41,42,43].

2 Developing Detonation Regime Diagram

2.1 Theoretical Background

2.2 Zeldovich’s Theory on Reactivity Gradient

According to Zeldovich’s theory [44], a spontaneous ignition front is defined as a successive ignition events in the presence of spatial reactivity gradients. Its propagation speed, $S_\mathrm{sp}\mathrm{{}}$, is determined as

$$\begin{aligned} {{S_\mathrm{sp}}} = |\nabla \tau _{\mathrm{ig}}|^{-1} \,, \end{aligned}$$

(1)

where $\tau _\mathrm{ig}$ is the ignition delay time of the local reactant mixture as schematically shown in Fig. 2, defined as the time to reach the maximum heat release as illustrated in Fig. 3.

Based on the relative ratio of $S_\mathrm{sp}$ to the laminar flame speed, $S_\mathrm{sp}/S_L$, and to the sound speed, $S_\mathrm{sp}/a$, of the local mixture as illustrated in Figs. 2 and 3, Zeldovich [44] classified four different ignition regimes of a reaction front propagation in a combustible gaseous mixture:

• 1. weak ignition: In the presence of large spatial variations in $\tau _\mathrm{ig}$, $S_\mathrm{sp} \sim S_L$, the local ignition establishes a slow-speed thin-front deflagration (see Figs. 2c and 3).

• 2. spontaneous ignition: For a mixture with a smaller level of $\tau _\mathrm{ig}$ variations, $S_\mathrm{sp} \gg a$, the mixture ignites spontaneously and $S_\mathrm{sp}\mathrm{{}}$ exhibits a rapid ignition front propagation (see Figs. 2a and 3).

• 3. detonation: If $S_\mathrm{sp}\mathrm{{}}$ is sufficiently large and comparable to the sound speed, $S_\mathrm{sp} \sim a$ as visually shown in Fig. 2b, the combustion wave couple with the acoustic wave may lead to detonation development characterized by extremely high-peak amplitudes of pressure oscillations.

• 4. homogeneous ignition: In the asymptotic limit of zero gradient, $S_\mathrm{sp}\mathrm{{}}$ becomes infinity, resulting in the purely homogeneous ignition mode.

Fig. 2

Schematic of 1-D spatially monotonic distribution of ignition delay times, $\tau ^1_\mathrm{ig}< \tau ^2_\mathrm{ig} \dots < \tau ^6_\mathrm{ig}$, (first row), and the relative speed between the reaction front and the acoustic wave (second row) with a spontaneous ignition, b detonation with a perfect coupling between the reaction front and acoustic wave, and c deflagration

Fig. 3

Representative 2D contours of HRR show deflagrations (first row) and spontaneous ignition fronts (second row) and their temporal evolutions of HRR (right column). The definition of ignition delay time, $\tau _\mathrm{ig}$, and excitation time, $\tau _e$ (the time duration from 5% of the peak HRR to the peak HRR), are shown in the right column

2.3 $\varepsilon \!-\!\xi$ Regime Diagram

Gu and Bradley [45] further mapped different propagation modes into a regime diagram characterized by two key non-dimensional parameters, $\xi$ and $\varepsilon$, which are defined as:

$$\begin{aligned} \xi \!=\! a/S_\mathrm{sp} \,, \end{aligned}$$

(2)

$$\begin{aligned} \varepsilon \!=\! (r_\mathrm{hs}/a)/\tau _e \,, \end{aligned}$$

(3)

where $\varepsilon$ is defined as the ratio of the residence time, $r_\mathrm{hs}/a$, of the acoustic wave within the hot spot with a radius, $r_\mathrm{hs}$, to the excitation time, $\tau _e$, in which most of the chemical energy is released. $\tau _e$ is typically defined as the time interval between 5% of the peak heat release rate (HRR) and the peak HRR as schematically illustrated in Fig. 3.

Based on the range of $\xi$, five different modes of ignition front propagation within a hot spot were identified [45, 46]:

• 1. $\xi = 0$ for instant thermal explosion,

• 2. $0< \xi < \xi _l$ for spontaneous (supersonic) ignition where the autoignitive wave is ahead of acoustic wave,

• 3. $\xi _l< \xi < \xi _u$ developing and developed detonation, through the thermoacoustic coupling process referred to as the shock wave amplification by coherent energy release (SWACER) mechanism [47, 48]),

• 4. $\xi _l \ge \xi < a/S_L$ subsonic autoignitive deflagration where $S_L$ is the laminar burning velocity,

• 5. $\xi _l \ge \xi < a/S_L$ laminar burning deflagration propagating at the speed comparable to the laminar flame speed.

Figure 4 shows the $\xi \!-\! \varepsilon$ peninsula with the upper and lower limit, $\xi _u$ and $\xi _l$, respectively, between which the developing detonation regime is identified for a stoichiometric 50%H$_2$–50%CO/air mixture at 5 atm [45, 46]. Note that the upper and lower $\xi$ limits of the C-shaped curve strongly depend on the initial conditions of $T_0$, $P_0$, and $\phi _0$, fuel types, and the kinetic model used in simulations [49, 50]. In general, the developing detonation regime is expected when $\xi$ ranges between 1 and O(10) quantity [49, 50]. As representatively shown in Fig. 5, due to the NTC behavior of two-stage ignition fuels (e.g., DME), they are much more prone to detonation development (gray region with roughly $1< \xi < 10$) than single-stage ignition fuels (e.g., ethanol) for the same initial conditions of temperature, pressure, equivalence ratio, temperature gradient.

Fig. 4

(taken from Refs. [6, 7])

Conditions for the occurrence of developing detonations in terms of Xi and epsilon for a stoichiometric 50%H$_2$–50%CO/air mixture at 5 atm [45, 46]. Three points (S2, K2, and N2) in the red line correspond to the three pressure traces shown in Fig. 1

Fig. 5

The $\xi$ values as a function of temperature for ethanol/air mixtures (red) and DME/air mixtures (blue) at $\phi$ of 0.5 and 1.0, and pressure of 35 atm, and ${\partial T} /{\partial r} = -2$ K/mm. The shaded region of $1 \le \xi \le 10$ is prone to the developing detonation for a small hot spot, i.e., $\varepsilon < 10$

2.4 Quantitative Analysis

Luong et al. [3] proposed three a posteriori statistical metrics to quantitatively assess the knock intensity, defined as follows:

$$\begin{aligned} F_{H} =\frac{\sum {\left\langle {\left. {\dot{q}} \right| {P} > {1.1P_e}} \right\rangle } }{\sum {\dot{q}} } \,, \end{aligned}$$

(4)

$$\begin{aligned} F_\mathrm{CJ} =\frac{\sum {\left\langle {\left. {\dot{q}} \right| {P} > {P_\mathrm{CJ}}} \right\rangle } }{\sum {\dot{q}} }, \end{aligned}$$

(5)

$$\begin{aligned} F_\mathrm{VN} =\frac{\sum {\left\langle {\left. {\dot{q}} \right| {P} > {P_\mathrm{VN}}} \right\rangle } }{\sum {\dot{q}} } \,, \end{aligned}$$

(6)

where $\dot{q}$ is the HRR; $P_e$, $P_\mathrm{CJ}$ and $P_\mathrm{VN}$ correspond to the homogeneous constant-volume equilibrium pressure, the Chapman–Jouguet pressure, and the von Neumann pressure, respectively. The summation is operated over all the cells of the computational domain and integrated over time. As such, $F_i$ ($i = {e, CJ, VN}$) measures the fraction of HRR associated with higher pressures exceeding the corresponding $P_i$, evaluated at the initial conditions of $T_0$, $P_0$, and $\phi _0$.

According to Fig. 4 [45], the developing detonation regime is bounded by the upper and lower $\xi$ limits of the C-shaped curve, $\xi _u$ and $\xi _l$, respectively, suggesting that the detonation intensity depends strongly on the statistical quantities extracted from the spatial distribution of $\xi$ such as its volume-average $\mathrm {\overline{{\xi }}}$, probability density function. To this end, $F_{D}$, defined as the volume fraction of $\xi _u< \xi < \xi _l$, is proposed to predict knock intensity:

$$\begin{aligned} F_{D} = \int \limits _{\xi _l}^{\xi _u} {P({\xi })\text {d}{\xi }}, \end{aligned}$$

(7)

which serves as an a priori predictive indicator of the subsequent detonation development while $F_H$ is an a posteriori statistical assessment of knock intensity. Note that both $F_D$ and $F_H$ range from zero to unity. $F_D$ was found to have a strong correlation with the fraction of the total heat release associated with detonation extracted from the full DNS data, $F_H$ [3]. The mixture is found to be more prone to strong detonation as $F_D$ approaches unity if $\varepsilon _p \gtrapprox 1$. For each initial condition, $F_D$ together with the volume-average, $\mathrm {\overline{{\xi }}}$, and probability density function, $P(\xi )$, are computed to quantify the knock intensity [3].

In the next sections, the above predictive criteria are discussed using 1D simulations first, followed by selected 2D and 3D simulations with and without turbulence in order to characterize the spontaneous ignition and detonation development.

2.5 One-Dimensional Detonation Development

The developing detonation $\varepsilon \!-\!\xi$ diagram [45, 46] has been used in many 1-D studies to investigate the interaction between flame propagation and end-gas auto-ignition, and the effects of boundary conditions, domain size, hot-spot location, hot-spot interaction, and cool flames on subsequent detonation formulation [6, 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71]. The effects of the bulk-mixture gradient on developing detonation were also studied by Sow et al. [60] and Nogawa and Terashima [72]. In the following, two representative 1-D cases with distinct detonation development characteristics are discussed.

The first case is shown in Fig. 6a features a deflagration to detonation transition (DDT) event where a hot spot is developed into deflagration, gradually consuming the unburned mixture, and finally accelerating to make a transition to detonation. Specifically, the initial temperature profile is designed such that $\xi$ ranges from 0 to 1000 within the hot spot (see Fig. 6). Figure 7 shows the temporal evolution of the temperature and pressure profiles. For such a strong ignition source, autoignition immediately forms a detonation front within the hot spot length, which is subsequently attenuated quickly upon leaving the hot spot (lines 1–4 in Fig. 7). Due to the low reactivity of the unburned mixture at $T_0=$ 1000 K, the heat release rate within the reaction zone is not fast enough to sustain the detonation front such that it decays quickly and transits back into deflagration at $x \sim 10$ mm. As more fuel/air mixture is gradually consumed by deflagration, the temperature and pressure of the remaining unburned mixture ahead of the flame are increased due to the compression heating effect (lines 5–7 in Fig. 7). Due to the increased reactivity of the unburned mixture together with the flame acceleration, the deflagration front transits into detonation at $x \sim 20$ mm and rapidly consumes the rest of the unburned mixture (lines 8–12 in Fig. 7). Much higher pressure peaks are observed during this period. These processes are typically referred as to deflagration to detonation transition (DDT).

Fig. 6

Two representative initial conditions of a hot spot with a 1-D linear-temperature profile: a $T_0 = 1000$ K and $T_\mathrm{max} = 2000$ K featuring deflagration to detonation transition (DDT) (see Fig. 7), and b $T_0 = 1200$ K and $T_\mathrm{max} = 1264$ K featuring direct detonation development (perfect coupling between the acoustic wave and ignition front, see Fig. 8). The speed of sound, a, and $\xi$ are evaluated at the center of hot spots (marked as dots). The spatial distribution of $\xi$ within the hot spots is also shown. Note that $\tau _\mathrm{ig}$ and $\tau _e$ are computed by using a 0-D homogeneous adiabatic constant-volume reactor using the localized temperature and pressure

Fig. 7

The temporal evolutions of a temperature and b pressure at different time. The initial conditions of $T_0$ of 1000 K and $P_0$ of 35 atm (see Fig. 6), a stoichiometric ethanol/air mixture

The second case (i.e., $T_0$ of 1200 K with $T_\mathrm{peak}$ of 1264 K in Fig. 6) features a direct detonation initiation in which the reaction front and the pressure wave are perfectly coupled to form an extremely high shock wave. The initial hot spot is designed for $\xi$ to range between 3 and 6 (see Fig. 6). Due to the high reactivity of the unburned mixture with $T_0$ of 1200 K, the hot spot immediately gives rise to a detonation front within the hot spot length, thereby rapidly consuming the unburned mixture (see Fig. 8). Once the detonation is established, the rapid heat release rate in the reaction zone is able to sustain the steady detonation front that propagates and consumes the rest of the unburned mixture.

Fig. 8

The temporal evolutions of a temperature and b pressure at different time.The initial conditions of $T_0$ of 1000 K and $P_0$ of 35 atm (see Fig. 6), a stoichiometric ethanol/air mixture

Fig. 9

(taken from Luong et al. [3])

a The normalized maximum pressure as a function of $\xi$, and b the $\xi -\varepsilon$ diagram for different hot spot sizes

The above cases show that the two parameters, $\xi$ and $\varepsilon$, within the hot spot and the bulk mixture determines the nature of the early detonation development. Figure 9 shows the compiled results of a series of 1-D simulations for various combinations of these parameters. For a fixed $T_0$ of 1200 K and $\phi$ of 1, the 1-D simulations were performed by varying $T_\mathrm{hs}$ and $r_\mathrm{hs}$ such that both $\xi$ and $\varepsilon$ are varied accordingly. As seen in Fig. 9, the maximum pressure above the equilibrium pressure (i.e., > 1.1$P_e$) for $0.2< \xi < 20$. The peak $P_{\max }$ occurs at $\xi \sim 6$ where the perfect coupling between the reaction front and the acoustic wave evolving into a stable detonation shock wave. The range of $\xi$ was used to determine the volume fraction of $\xi$ and $F_D$ to predict the amplitude of knock intensity in multi-dimensional cases.

2.6 2-D and 3-D Simulations of Super-Knock Under Realistic IC-engine Conditions

The basic characterization in 1D problem is now extended to multi-dimensional conditions in the presence of spatial fluctuations in temperature and velocity fields. Parametric 2-D and 3-D simulations were systematically performed to validate the criteria proposed in Fig. 2.4. The distribution of $\xi$ and $\varepsilon$ were varied by changing $T'$ over a wide range from 0.5 to 20 K, and $l_T$ of 1, 2, 5, and 10 mm. For each initial condition, the full detailed information of spatial distribution of $\xi$ and the corresponding statistical quantities of $\xi$ can be computed. Figure 10 shows the spatial distribution of $\xi$ of four cases with different $T'$ and $l_T$ that result in no-knock, mild-knock and super-knock [3].

Fig. 10

Representative 2-D contours of $\xi$ distribution computed from four different initial conditions for three different $l_T$ of 1 mm, 2 mm, 5 mm, with $T'$ of 15 K (a–c) and 0.6 K (d)

Figure 11 shows a representative 2-D contour of temperature field with $T_0$ of 1200 K, $T'$ of 15 K, and $l_T$ of 5 mm. According to Peters et al. [53, 73], the temperature field is subdivided into the dissipation elements over which the temperature profile is monotonic, ranging from the peak to the trough of each dissipation element. For isotropic temperature field fluctuations, Peters et al. [53] found $r_\mathrm{hs}$ comparable with the mean distance of the dissipation elements, $l_\mathrm{DE}$, and approximated as $l_\mathrm{DE} \!\approx \! 2\lambda _T$, where $\lambda _T$ is the Taylor mixing scale [53]. In the multi-dimensional problems, $l_\mathrm{DE}$ is believed to be a minimum size of a hot spot of unburned reactants required so that it allows a sufficient run-up distance of detonation development [12, 53].

Fig. 11

Representative 2-D contour of temperature field with $T_0$ of 1200 K, $T'$ of 15 K, and $l_T$ of 5 mm overlaid by the dissipation elements. The white lines mark the boundary between the dissipation elements

Based on the Taylor mixing scale, Towery et al. [12] proposed a statistical model to evaluate the predictive $\varepsilon _p$ for highly compressible homogeneous isotropic turbulence fluctuations. $\varepsilon _p$ is defined as

$$\begin{aligned} \varepsilon _p\!=\!2\lambda _T/(a_\mathrm{{rms}} \tau _\mathrm{{exo}}) \end{aligned}$$

(8)

where a and $\tau _\mathrm{{exo}}$ are the mass-weighted sound speed and exothermicity time, respectively (see Ref. [12] for details). In Refs. [3, 4], at the the initial conditions of $T_0 = 1200$ K, $P_0 = 35$ K, and $\phi _0 = 1$, $\varepsilon _p = 1$ corresponds to the initial temperature field of $l_T = 3.3$ mm. The cases with $\varepsilon _p <1$ (i.e., $l_T$ of 1 and 2 mm with $\varepsilon _p$ of 0.3 and 0.6) are less likely to develop detonation while the cases with $\varepsilon _p$ of 1.2, 1.4, and 3.8 for $l_T$ of 4 mm, 5 mm, and 10 mm, respectively, are more likely to develop strong detonation if $1< \xi < 10$.

For a uniform mixture with only thermal fluctuations, the predictive $\xi _p$ is scaled as [11, 12]

$$\begin{aligned} \xi _p = a|\frac{\text {d}\tau _\mathrm{ig}\mathrm{{}}}{\text {d}T} \nabla T| \cong a\left| \frac{\text {d} \tau _\mathrm{ig}}{\text {d} T}\right| |\widetilde{\nabla T}| \end{aligned}$$

(9)

where the statistical mean temperature gradient is further approximated by $|\widetilde{\nabla T}| \cong T'/ \lambda _T$, with the Taylor mixing length scale, $\lambda _T \cong (l_T \alpha /T')^{0.5}$ [11]. These predictive parameters, $\xi _p$, and $\varepsilon _p$, will be validated in the subsequent sections.

2.6.1 Baseline Cases with No Turbulence

In this subsection, three baseline (no turbulence) cases with $l_T$ of 1 mm, 2 mm, and 5 mm denoted as BL1, BL2, and BL5, respectively, are chosen for discussion. The probability density functions (PDF) of $\xi$, the temporal evolution of the normalized $P_{\max }$, and the contours of pressure at the time of the peak of $F_H$ for three baseline cases are shown in Figs. 12 and 13. According to the scaling of $\xi _p$ above, with a fixed $T'$ of 15 K, increasing $l_T$ from 1 to 5 mm decreases $\xi _p$ which is consistent with the PDF of $\xi$ shifting toward the developing detonation regime (see Fig. 12). Consequently, BL5 is predicted to be more prone to developing detonation as compared to BL1 and BL2.

Consistent with the PDFs of $\xi$ and $\varepsilon _p = 0.3$, $P_{\max }\le P_\mathrm{VN}$ and the isocontours of pressure in Figs. 12 and 13, respectively, confirm that BL1 features a spontaneous ignition process. BL2 with $F_\mathrm{VN} \approx 0.07$ is characterized as a mild detonation process as also quantitatively predicted by $\varepsilon _p = 0.6$. Unlike BL1 and BL2, a strong detonation process is observed for BL5, quantified by $F_\mathrm{VN} \approx 0.2$ and $P_{\max } \gg P_\mathrm{VN}$, which is accurately predicted by $\varepsilon _p = 1.6$ for BL5.

Fig. 12

(adapted from Luong et al. [3, 4])

a The probability density functions (PDF) of $\xi$ for three baseline cases (without turbulence) with three different most energetic length scale of temperature, $l_T$ of 1 mm, 2 mm, and 5 mm, featuring no knock, mild knock, and super-knock, respectively, and b their temporal evolution of the normalized maximum pressure, $P_\mathrm{max}/P_\mathrm{VN}$,  [4]. The gray area in the PDF figure depicts the region of $1< \xi < 10$ determined by a series of 1-D simulations under the same initial conditions, which is prone to developing detonation (see Fig. 9a)

Fig. 13

(adapted from Luong et al. [3])

Representative 2-D contours of pressure for three cases with the same $T'$ of 15 K, and $l_T$ of 1 mm, 2 mm, and 5 mm with no knock, mild knock, and super-knock, from left to right, respectively

Fig. 14

(taken from Luong et al. [3])

a $F_H$ (color symbols) and $F_D$ (black symbols) as a function of $\mathrm {\overline{{\xi }}}$, with $T'$ ranging from 0.5 K to 20 K for different $l_T$ ranging from 1 mm to 10 mm, b the correlation between $F_H$ and $F_D$. Filled (open) symbols are for 3-D (2-D) cases

The statistical quantities, $F_H$, $F_D$ and $\mathrm {\overline{{\xi }}}$, of all the baseline cases are plotted in Fig. 14. Three main points are noted from Fig. 14. First, $F_D$ has a better agreement with $F_H$ for $\varepsilon _p > 1$ (i.e., cases with $l_T > 4$ mm in Fig. 14b). Second, the magnitude of $F_H$ for 3-D cases is comparable with the corresponding 2-D cases as seen in Fig. 14a despite the peak $F_\mathrm{VN}$ of the 3-D cases slightly lower than that of the 2-D cases [3]. Third, Fig. 14 also reveals a good correlation between $\mathrm {\overline{{\xi }}}$ and the knock intensity characterized by the $F_H$ magnitude. The correlation indicates that the conditions with $\mathrm {\overline{{\xi }}}$ ranging between 0.5 and 30 experience developing detonation and the strongest detonation intensity ($F_H$ approaches unity) occurs at $\mathrm {\overline{{\xi }}}$ $\sim$ 6 with $\varepsilon _p > 1$. Specifically, the maximum values of $F_H$ at $\mathrm {\overline{{\xi }}}$ $\sim$ 6 are approximately 0.18, 0.22, 0.55, and 0.75 for $l_T$ of 1 mm, 2 mm, 5 mm, and 10 mm, respectively. These results verify that the prediction of $\varepsilon _p$ for knock propensity is consistent with $F_H$, the detonation intensity is greatly alleviated with decreasing $l_T$ (mild-to-no detonation for $l_T < 3.3$ mm with $\varepsilon _p < 1$).

2.6.2 Effect of Turbulence

The previous subsection demonstrated that no detonation was observed for BL1 while BL5 exhibited a strong detonation process because a minimum run-up distance is required for developing detonation as accurately predicted by $\varepsilon _t$. The results suggest that decreasing (increasing) the length scale of temperature fluctuations (hence the dissipation element length) is a key parameter to alleviate (promote) knock intensity. In this subsection, the additional effect of turbulent velocity fields on knock intensity is also investigated [4].

The isocontours of pressure and HRR of two representative 2-D simulations and a 3-D simulation in Figs. 15, 16 and 17 are chosen for discussion. With the same turbulent velocity fluctuation, $u'$ of 83.3 m/s, and root-mean-square temperature fluctuation, $T'$ of 15 K, $l_T/l_e$ and $\tau _{ig}/\tau _t$ were found to be two key parameters to determine knock propensity.

First, increasing turbulent intensity by increasing $u'$ and fixing $l_e = 5$ mm is not effective in reducing knock intensity since with $\tau _\mathrm{ig}/\tau _t$ of 1.0, turbulence is not fast enough to modify the autoignition and the subsequent detonation process as visually seen in Fig. 16. Rather, the main effect of turbulent eddies is to weakly wrinkle and distort the detonation fronts. Second, a finer turbulence structure, $l_e$ of 1 mm $\ll l_T = 5$ mm, and $\tau _\mathrm{ig}/\tau _t$ of 5.0, effectively attenuates the detonation intensity as visually seen in Figs. 16 and 17. High turbulence intensity together with a short mixing-time scale significantly increases the amount of reactants burned by corrugating turbulent flames rather than by the developing detonation phase. Fast and small turbulence structures also induce smaller and finer localized detonation waves during the developing detonation process. Third, for the 3-D cases with a slightly lower $F_{H}$, a consistent agreement between the 2-D and 3-D cases was observed in terms of $P_{\max }$, the duration of detonation process, and the level of pressure oscillations, $P'$ [4].

Contrary to the case with $l_e =1$ mm $\ll l_T = 5$ mm being able to attenuate detonation intensity, a fast turbulent mixing, $\tau _\mathrm{ig}/\tau _t$ of 5, together with $l_e = 5$ mm $\gg l_T =1$ mm, promotes detonation development as seen in Fig. 18. Turbulence with a larger eddy size effectively spreads the temperature field over a wider area, stretching out the run-up distance, and thus facilitates the developing detonation [4].

Fig. 15

(taken from Desai et al. [14])

Two-dimensional contours of pressure and heat release rate (HRR) at the times of onset of detonation development (left) and the peak detonation intensity (right) for a 2-D case with the most energetic length scale of temperature and turbulent field, $l_T$ of 5 mm and $l_e$ of 5 mm, respectively, the turbulent velocity fluctuation, $u'$ of 83.3 m/s, and the ratio of the ignition delay time to turbulent time scale $\tau _\mathrm{ig}/\tau _t$ of 1.0 [4]

Fig. 16

(taken from Desai et al. [14])

Two-dimensional contours of pressure and heat release rate (HRR) at the times of onset of detonation development (left) and the peak detonation intensity (right) for a 2-D case with the most energetic length scale of temperature and turbulent field, $l_T$ of 5 mm and $l_e$ of 1 mm, respectively, the turbulent velocity fluctuation, $u'$ of 83.3 m/s, and the ratio of the ignition delay time to turbulent time scale $\tau _\mathrm{ig}/\tau _t$ of 5.0 [4]

Fig. 17

(taken from Desai et al. [14])

The volume rendering of HRR and the iso-surface of pressure (green) with $P = P_\mathrm{CJ}$ at successive times prior to the onset of the detonation formation (top row) until the detonation intensity reaches its peak (bottom row) for a 3-D case with the most energetic length scale of temperature and turbulent field, $l_T$ of 5 mm and $l_e$ of 1 mm, respectively, the turbulent velocity fluctuation, $u'$ of 83.3 m/s, and the ratio of the ignition delay time to turbulent time scale $\tau _\mathrm{ig}/\tau _t$ of 5.0 [4]

Fig. 18

(adapted from Luong et al. [4])

Temporal evolution of the normalized maximum pressure, $P_\mathrm{max}/P_\mathrm{VN}$ for the cases with $l_T$ of 1 mm with different $l_e$ and $u'$

3 Weak and Strong Ignition Modes

3.1 Theoretical Background

Based on Zeldovich’s theory and validated by high-fidelity DNS data, Sankaran et al. [15] proposed a criterion to delineate the weak or strong ignition modes, referred to as the Sankaran number (Sa) herein, which is based on the ratio of laminar flame speed, $S_L$, to the ignition front speed, ${{S_\mathrm{sp}}} = |\nabla \tau _{\mathrm{ig}}|^{-1}$ [44].

$$\begin{aligned} \mathrm{{S{a}}} = \beta \frac{S_L\mathrm{{}}}{S_\mathrm{sp}\mathrm{}}, \end{aligned}$$

(10)

where $\beta$ is a weighting factor of 0.5 to account for some delay in developing a deflagration from the ignition kernel due to the interactions of thermal and radial diffusion between the neighboring regions. Sa = 1 serves as the boundary between the strong (Sa < 1) and the weak (Sa > 1) ignition.

In practical combustion devices, the detailed information of the spatially resolved scalar fluctuations is not available, and thus the prediction for in a real engine should be based on the statistical quantities such as the mean and variance of fluctuations in RANS or LES simulations. Im et al. [11] conducted a theoretical scaling analysis by incorporating the turbulent effect into the original Sankaran criterion to derive the turbulent ignition regime diagram as shown in Fig. 19.

Fig. 19

Regime diagram proposed by Im et al. [11] for strong and weak ignition modes for nearly homogeneous reactant mixture with velocity and temperature fluctuations

For compositionally homogeneous reactant mixtures with turbulent velocity and temperature fluctuations, the diagram classifies ignition modes into three categories: weak exhibiting a dominant slow-speed deflagration mode of fuel consumption, reaction-dominant strong, and mixing-dominant strong exhibiting a dominant volumetric-ignition mode of fuel consumption. Pal et al. [74] and Luong et al. [1] conducted a series of DNS simulations and found consistent results with the predictions by the turbulent ignition regime diagram. The ignition criterion [11] was further extended to accommodate both temperature and concentration fluctuations and validated for the mixtures with NTC and non-NTC behaviors [1].

The gradient of ignition delay time, $\nabla \tau _{\mathrm{ig}}$, for a mixture with a uniform composition and pressure is solely due to temperature variations expanded by the chain rule as [11]:

$$\begin{aligned} |\nabla \tau _\mathrm{ig}\mathrm{}| = \biggr \vert \frac{\text {d}\tau _\mathrm{ig}\mathrm{{}}}{\text {d}T} \nabla T \biggr \vert \,, \end{aligned}$$

(11)

where $\text {d}\tau _{\mathrm{ig}}/\text {d}T$ represents the ignition delay sensitivity to temperature as representatively shown in Fig. 20b, and $\nabla T$ is the temperature gradient of the bulk mixture [11, 74]. In turbulent conditions, the temperature gradient is approximated by the statistical mean temperature gradient, $\widetilde{|\nabla T|}$, which is estimated based on the variance of the temperature, $T'$, fluctuations and the Taylor mixing scale, $\lambda _T$, as

$$\begin{aligned} \widetilde{|\nabla T|} \approx \frac{T'}{\lambda _T} \,, \end{aligned}$$

(12)

where the Taylor mixing scale is approximated as

$$\begin{aligned} \lambda _T = l\mathrm{{Re}}_{l}^{-0.5} \,, \end{aligned}$$

(13)

with Re$_l = u'l/\nu$ being the turbulent Reynolds number, $\nu$ the kinematic viscosity of the bulk mixture gas, $u'$ the root mean square (RMS) turbulent velocity fluctuation, and l the integral scale of velocity field.

Further scaling analysis yields the expression for the predicted Sankaran number, Sa$_p$, [11] as follows:

$$\begin{aligned} \mathrm{{Sa}}_p = K\mathrm{{Da}}_l^{-1/2}, K = \beta \frac{1}{{(\tau _f}{\tau _\mathrm{ig}})^{1/2}} \left| \frac{\text {d}\tau _\mathrm{ig}\mathrm{{}}}{\text {d}T} T'\right| \,, \end{aligned}$$

(14)

where $\mathrm{{Da}}_l$ = $\tau _{t}\mathrm{}$/$\tau _{ig}\mathrm{}$ is the ignition Damköhler number, defined as the ratio of the turbulence integral time scale, $l/u'$, to the ignition delay time, and $\tau _f\mathrm{{}}$ = $\alpha /{S^2_L}$ is the characteristic flame time, where $\alpha$ is the thermal diffusivity of the initial bulk mixture. K is referred to as the normalized thermal ignition sensitivity.

In the presence of both temperature and equivalence ratio (concentration) fluctuations, $\tau _\mathrm{ig}$ is a function of T and $\phi$ such that $|\nabla \tau _\mathrm{{ig}}|$ is expanded by using the chain rule as flows:

$$\begin{aligned} |\nabla \tau _\mathrm{{ig}}| = \biggr \vert {{{\partial {\tau _\mathrm{ig}}} \over {\partial T}}\nabla T + {{\partial {\tau _\mathrm{ig}}} \over {\partial \phi }}\nabla \phi } \biggr \vert , \end{aligned}$$

(15)

where $\nabla \phi$ and $\partial {\tau _\mathrm{ig}} / {\partial \phi }$ are the equivalence ratio gradients, and the ignition delay sensitivity to equivalence ratio, respectively [1]. Assuming that the temperature and equivalence ratio fields are correlated with the turbulent flow field, it results in a comparable integral length scale, $l_T \sim l_{\phi } \sim l_e$, and $\widetilde{\nabla \phi } \sim \phi '/\lambda _T$ as approximated in the temperature case.

As such K in Fig. 14 is extended as:

$$\begin{aligned} K = \beta \frac{1}{{(\tau _f}{\tau _\mathrm{ig}})^{1/2}} \Bigg ( \biggr \vert {{{\partial {\tau _\mathrm{ig}}} \over {\partial T}} T'\biggr \vert + s\biggr \vert {{\partial {\tau _\mathrm{ig}}} \over {\partial \phi }} \phi ' } \biggr \vert \Bigg ) \,, \end{aligned}$$

(16)

where the sign of s is to account for the NTC behavior and the correlation of T-$\phi$ distribution. s is positive if T and $\phi$ fluctuations are uncorrelated, thereby being additive to $|\nabla \tau _\mathrm{{ig}}|$ that synergistically promotes the deflagration mode regardless of the NTC regime. s adopts a negative (positive) sign for a negative $T - \phi$ correlation if the product of ${\partial {\tau _\mathrm{ig}}} / {\partial T}$ and ${\partial {\tau _\mathrm{ig}}} / {\partial \phi }$ is positive (negative) that the combined effects of T and $\phi$ on K become subtractive (additive).

3.2 Prediction of Weak/strong Combustion Modes

Many fuels with and without the NTC behavior were chosen to investigate the effect of temperature and equivalence ratio fluctuations on the combustion modes by varying the initial mean temperatures and different fluctuation levels [1, 20, 27, 28, 30, 75,76,77]. The initial conditions and physical parameters are chosen to be relevant to IC engine conditions [1, 33,34,35,36,37,38,39,40,41,42,43]. The DNS solution were processed to verify the predictive accuracy of the ignition criteria.

Figure 20 shows the homogeneous ignition delay time of DME/air mixtures and ethanol/air mixtures and the sensitivity of ignition delay time to temperature that is needed to compute Sa$_p$. DME exhibits the negative-temperature coefficient (NTC) regime in which $\tau _\mathrm{{ig}}$ increases with increasing temperature such that its $\tau _\mathrm{{ig}}$ varies non-monotonically with temperature. As such, within/near the NTC regime, the variation of d$\tau _\mathrm{ig}/\text {d}T$ for DME is small compared to that of single-stage fuels (e.g., ethanol), which in turn leads to a small $|\nabla \tau _\mathrm{{ig}}|$ for the same temperature fluctuation level. It suggests that within/near the NTC regime, the mixture with $T'$ only is more susceptible to spontaneous ignition as quantitatively predicted by Sa$_p$ in Fig. 21 [2].

Figure 22 shows a typical initial condition used in the simulations. These DNS cases were adopted to perform the statistical analysis of $\mathrm{{Sa}}$ such that the volume-average (mean) Sa, $\mathrm {\overline{{S}}a}$, the probability density function of Sa, $P(\mathrm {Sa})$, and the volume fraction of the regions with Sa <1, $F_{\mathrm{{Sa,}}S}$, extracted from the spatial Sa distribution are obtained. In addition, $\mathrm{{Sa}_p}$ for each simulation was also computed based on the information of the initial mean bulk mixture condition such as $T_0$, $p_0$, and $\phi _0$, and the RMS values, $T'$ and $\phi '$ as representatively shown in Fig. 21 as a function of temperature.

Fig. 20

a The 0-D homogeneous ignition delay time, $\tau _\mathrm{ig}$, and b the absolute value of the sensitivity of $\tau _\mathrm{ig}$ to temperature, $|\text {d}\tau _{\mathrm{ig}}/\text {d}T|$, as a function of temperature for ethanol/air mixtures (red) and DME/air mixtures (blue) at $\phi$ of 0.5 and 1.0, and pressure of 35 atm

Fig. 21

The predicted Sa number, Sa$_p$, as a function of temperature for ethanol/air mixtures (red) and DME/air mixtures (blue) at $\phi$ of 0.5 and 1.0, and pressure of 35 atm, and assuming the identical integral length scale of velocities and temperature field, $l_t = l_T = 1$ mm, $\tau _{t}/\tau _\mathrm{ig} = 1$, and $T' = 15$ K. The shaded region of Sa$_p < 1$ is for strong ignition, while the region of Sa$_p > 1$ is for weak ignition

Fig. 22

(taken from Luong et al. [2])

a–b initial fields of temperature and equivalence ratio with a negative T–$\phi$ distribution at the initial mean temperature and equivalence ratio of 900 K–0.5 with their fluctuation of $T'$ and $\phi '$ of 30 K and 0.15, respectively, and c–d the corresponding ignition delay time and $\mathrm{{Sa}}$ distribution. The black iso-lines of $\mathrm{{Sa}}$ = 1 delineates two distinct combustion modes of strong ($\mathrm{{Sa}}$ < 1) and weak ($\mathrm{{Sa}}$ > 1) ignition

The isocontours of HRR for some representative DME cases are shown in Fig. 23. Spontaneous ignition characterized by volumetric ignition is dominant even with $T'$ of 60 K at $T_0$ of 900 K due to a small variation of $\tau _{ig}\mathrm{}$with temperature within/near the NTC regime as shown in Fig. 20). On the contrary, deflagration propagation becomes dominant for the cases with $T_0$ of 680 and 1045 K outside the NTC regime. Similar results were found for n-heptane [20, 29,30,31, 75, 76, 78, 79].

Contrary to the $T'$-only cases, in the presence of both T and $\phi$, especially when they are negatively correlated, $T_0$ of 900 K results in a much lower HRR (due to the synergistic effect of $T'$ and $\phi '$) than those of $T_0$ outside the NTC regime (due to the subtractive effect of $T'$ and $\phi '$ for the cases with $T_0$ of 770 and 1045 K), as seen in the bottom row of Fig. 23.

Fig. 23

(adapted from Luong et al. [1])

2-D contours of the normalized HRR at the maximum spatial-averaged HRR occurrence for the DME/air mixtures at the initial pressure of 30 atm. The first (second) row corresponds to the cases with temperature fluctuations only, $T'$ of 60 K, (with both temperature and equivalence ratio fluctuations, $T'$ of 30 K and $\phi$ of 0.15 [2])

Quantitative assessment of combustion modes for the previous cases was also conducted. For each initial field of a DNS case, the exact distribution of Sa within the domain and its associated statistical quantities can be computed. For direct validations with the predictive quantities such as $F_S$ and $\mathrm {\overline{{S}}a}$, the fractional contributions of weak and strong ignition to the total heat release were extracted from the detailed DNS solution.

Figure 22a-b shows representative initial fields of temperature and equivalence ratio distribution. Based on these detailed DNS information, the spatial distribution of $\tau _{ig}\mathrm{}$ and Sa, and the corresponding probability density function (PDF) of Sa, $P(\mathrm {Sa})$ are computed in Figs. 22c–d and 24, respectively. The volume fraction of the regions attributed to strong ignition was also estimated as

$$\begin{aligned} F_{\mathrm{Sa},S} = \int \limits _0^{1} {P\mathrm {(Sa)}\text {d}\mathrm{Sa}} \,, \end{aligned}$$

(17)

which was proposed as a quantitative metric to predict the fractional cumulative heat release rate associated with strong ignition directly computed from DNS solution fields, $F_{\mathrm{Da},S}$ as [20, 27, 28, 30, 75, 76]

$$\begin{aligned} F_{\mathrm{Da},S} =\frac{\sum {\left\langle {\left. {\dot{q}} \right| \mathrm{{Da}} > \mathrm{{D}}{\mathrm{{a}}_0}} \right\rangle } }{\sum {\dot{q}} } \,, \end{aligned}$$

(18)

where the summation is operated over the total number of computational cells in the DNS domain, $\dot{q}$ is the HRR, and Da is Damköhler number defined as the ratio of the local reaction and diffusion terms for a selected scalar variable [16, 17, 20, 28, 30, 71, 75, 76, 80]. Strong ignition is dominant if $F_{\mathrm{{Sa,}}S}$ $\sim$ 1. Strong and weak ignition is delineated by a threshold value, Da$_\mathrm{{0}}$, of order unity that depends on the chosen initial condition. The significance of the Sa-based metric is that the prediction of combustion modes is based purely on the initial conditions.

Fig. 24

(taken from Luong et al. [2])

A representative probability density function of $\mathrm{{Sa}}$ distribution, PDF(Sa). SI and WI denote strong ignition and weak ignition, respectively with a threshold of $\mathrm{{Sa}}$ = 1

Consistent with the qualitative observations in the previous section, $\mathrm {\overline{{S}}a}$ and $F_{\mathrm{{Sa,}}S}$ for the DME cases of $T_0$ at 680 K, 900 K, and 1045 K are found to be, respectively, 4.2 & 0.20, 0.5 & 0.92, and 1.3 & 0.4. The cases with $T_0$ of 680 K and 1045 K have $\mathrm {\overline{{S}}a}$ greater than one, suggesting a weak-ignition dominance, while for the case with $T_0$ of 900 K, a strong ignition is expected. The predictive accuracy of $\mathrm {\overline{{S}}a}$ and $F_{\mathrm{{Sa,}}S}$ is further verified by the magnitude of $F_{\mathrm{{Da,}}S}$ as shown in Fig. 25.

Figure 25 shows a strong correlation between $\mathrm {\overline{{S}}a_n}$= $\mathrm {\overline{{S}}a}$/(1+ $\mathrm {\overline{{S}}a}$) and $F_{\mathrm{{Sa,}}S}$ regardless of $T_0$, $T'$, $\phi '$, and fuel types. A good correlation between $\mathrm{{Sa}_p}$ and $F_{\mathrm{{Sa,}}S}$ was also demonstrated, which is attributed to the agreement between the predicted $\mathrm{{Sa}_p}$ and the volume-average $\mathrm {\overline{{S}}a}$. Based on this universal correlation, the magnitude of either $\mathrm {\overline{{S}}a}$ or $\mathrm{{Sa}_p}$ can be used to quantitatively infer $F_{\mathrm{{Sa,}}S}$, which characterize the combustion intensity. In general, a higher $\mathrm {\overline{{S}}a}$ corresponds to a lower $F_{\mathrm{{Sa,}}S}$, which translates into a lower combustion intensity. In particular, $\mathrm {\overline{{S}}a_n}$ = 0.5 serves as a transition point from SI to WI dominance with 60% heat released by spontaneous ignition mode ($F_{\mathrm{{Sa,}}S}$ = 0.6). Spontaneous ignition dominance ($F_{\mathrm{{Sa,}}S}$ = 1.0) occurs at $\mathrm {\overline{{S}}a}$ < 0.3. On the contrary, weak ignition is expected to be dominant at $\mathrm {\overline{{S}}a}$ > 1.0. In summary, either $\mathrm {\overline{{S}}a}$ or $\mathrm{{Sa}_p}$ can be used to quantitatively predict the combustion modes over a wide range of initial conditions and different fuel types.

Fig. 25

(taken from Luong et al. [1])

The volumetric fraction of Sa < 1, $F_{\mathrm{Sa},S}$, as a function of the normalized $\mathrm {\overline{{S}}a}$, $\mathrm {\overline{{S}}a_n}$ $=$ $\mathrm {\overline{{S}}a}$/(1+$\mathrm {\overline{{S}}a}$) to fit within a range between 0 and 1, with 0.5 corresponding to Sa = 1

4 Summary

Recent developments in the theoretical study to predict detonation development in IC engine conditions were summarized. A brief review was given on the theoretical framework, starting from the original Zeldovich theory, and the regime diagrams per Bradley [46] and Im [11], and various proposed predictive criteria by Luong et al. [3, 4]. A large collection of DNS simulation data were used for a priori and a posteriori test for the validation of different predictive criteria based on the initial solution field, and its extension to large turbulent conditions with statistical average and variances has been suggested.

In real combustion engines, additional complexities arise from the front-to-wall and front-to-front interactions which may further enhance or suppress the occurrence and intensity of the detonation event. This is of strong practical significance and is currently being investigated by large-scale simulations [3, 4].