Numerical evidence on deflagration fronts in a methane/n-dodecane dual-fuel shear layer under engine relevant conditions

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Kannan, Jeevananthan; Karimkashi, Shervin; Gadalla, Mahmoud; Kaario, Ossi; Vuorinen, Ville

Numerical evidence on deflagration fronts in a methane/n-dodecane dual-fuel shear layer under engine relevant conditions

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DOI:

10.1016/j.fuel.2023.128100 Published: 15/07/2023

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Kannan, J., Karimkashi, S., Gadalla, M., Kaario, O., & Vuorinen, V. (2023). Numerical evidence on deflagration fronts in a methane/n-dodecane dual-fuel shear layer under engine relevant conditions. Fuel, 344, [128100].

https://doi.org/10.1016/j.fuel.2023.128100

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Fuel 344 (2023) 128100

Available online 22 March 2023

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Full length article

Numerical evidence on deflagration fronts in a methane/n-dodecane dual-fuel shear layer under engine relevant conditions

Jeevananthan Kannan

^∗

, Shervin Karimkashi, Mahmoud Gadalla, Ossi Kaario, Ville Vuorinen

Aalto University, Department of Mechanical Engineering, School of Engineering, Otakaari 4, Espoo, 02150, Finland

A R T I C L E I N F O

Keywords:

Dual-fuel Flame initiation Deflagration Methane n-dodecane

A B S T R A C T

To date, high resolution spray-assisted dual-fuel (DF) studies have focused on capturing the ignition process while the subsequent post-ignition events have been largely neglected due to modeling requirements and high computational cost. Here, we use a simplified approach for studying ignition front evolution after ignition.

Three-dimensional scale-resolved simulations of igniting shear layers (0≤𝑅𝑒≤1500) are studied to better understand reaction fronts in engine-relevant conditions. We carry out quasi-DNS in a DF combustion setup consisting of premixed 𝑛-dodecane/methane/air/EGR at 700K as a fuel stream and premixed methane/air as the oxidizer at a pressure of 60 atmospheres and an ambient temperature of 900 K. The flow solution resolution is𝛿∕10, where𝛿=laminar flame thickness. The present study primarily focuses on the hypothesized flame formation and its characterization. Under these conditions, the simulations indicate two-stage ignition further leading to reaction front initiation and dual-fuel flame establishment. For𝑅𝑒 <1500, a reaction front resembling DF deflagration is demonstrated close to the auto-ignition timescales. At𝑅𝑒= 1500, mixing effects promote more rapid dilution and the DF deflagration front formation is slightly delayed although still observed.

For the first time, at rather short timescales of0.2 − 0.4 𝐼 𝐷𝑇 (ignition delay time) after the ignition, we provide numerical evidence on DF deflagration front emergence in shear-driven DF combustion processes via 3D numerical simulations for0< 𝑅𝑒≤1500.

1. Introduction

Internal combustion engines are the backbone of modern trans- portation and energy production, but their impact on the environment remains a pressing concern. The diesel-powered compression-ignition (CI) engines, in particular, present a major challenge due to their high levels of NOx, CO₂ and soot emissions [1–9]. As we move towards a more sustainable future, it is imperative that we explore innova- tive solutions to improve the efficiency and reduce the emissions of these engines. Low temperature combustion is one of the well-known solutions to control emissions in the diesel engines. For instance, homogeneous charge compression ignition (HCCI) [10,11] and reactivity controlled compression-ignition (RCCI) [12–15] are among the well- established low temperature combustion ideas. With relevance to the RCCI concept, dual-fuel (DF) combustion is considered in the present work.

In DF RCCI engines, interactive combustion of at least two fuels with different reactivity characteristics is utilized. In such a system, a low-reactivity fuel (LRF) is ignited using a small amount of a high- reactivity fuel (HRF). While the HRF (such as diesel) facilitates the ignition process of the LRF (such as methane), using the LRF leads to

∗ Corresponding author.

E-mail address: jeevananthan.kannan@aalto.fi(J. Kannan).

lower NOx and soot emissions compared to the conventional CI (diesel) engines. However, reactivity stratification and turbulence intensity may have significant effects on the mode of combustion after ignition in such DF systems. The combustion mode, either deflagration or auto- ignition, can affect the efficiency and emissions in engines. Thus, it is an important research topic to identify which mode of combustion is more probable in RCCI (DF) engines. The present study aims to investigate the effect of turbulence intensity on the post-ignition combustion mode under RCCI-relevant conditions using a simplified shear layer setup which has a close relevance to DF/RCCI and pilot spray assisted DF combustion.

A DF ignition process has been thoroughly studied using a DF spray setup together with large-eddy simulation (LES) by the present authors [16–20] and others [21–23]. Fig. 1(left panel) summarizes the main observations regarding the DF spray ignition based on our previous studies. In the DF spray setup, the cold liquid HRF (diesel) is sprayed to the high temperature ambient of premixed LRF/oxidizer, where the LRF is methane. In the first step (I), the HRF evaporates and starts to mix with the ambient mixture. In the second step (II), low-temperature chemistry (LTC) leads to the first-stage ignition. This

https://doi.org/10.1016/j.fuel.2023.128100

Received 20 November 2022; Received in revised form 8 February 2023; Accepted 7 March 2023

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Fig. 1. Left: Visualization of dual-fuel (DF) ignition process where the𝑛-dodecane spray (HRF) ignites the premixed methane/air -mixture (LRF). Right: Quasi-DNS of a DF shear layer with𝑛-dodecane/methane/air as the HRF and methane/air as the LRF. Four different regimes are noted from DF combustion: (I) fuel injection, mixing and evaporation, (II) low-temperature chemistry, and (III) high-temperature chemistry and flame initiation and (IV) flame front propagation (present topic).

step is characterized by the formation of many intermediate species, such as RO₂ (C₁₂H₂₅–OO), HO₂, and H₂O₂ which are formed from 𝑛-dodecane decomposition [24]. In the third step (III), onset of high- temperature chemistry (HTC) begins with fast temperature increase and the production of radicals such as OH which then lead to the second- stage ignition. Finally, in the fourth step (IV), there is an ignition front initiation from the hot ignition kernels, which might lead to flame formation.

While it is of high interest to study the characteristics of flame initiation and flame propagation processes (deflagration) under RCCI- relevant conditions [25], such a study requires either a very fine mesh resolution in the ambient, or an accurate turbulence chemistry interaction (TCI) model [21,26]. Considering the high computational cost of resolving the flame and also lack of a validated and accurate TCI model for such a DF spray setup, the majority of previous studies have not considered the post-ignition events (see e.g. [19,27]) or they have utilized the TCI models commonly used and validated for single-fuel conditions (see e.g. [28]). Here, we have used a shear layer setup as a simplified model of the DF spray, to study the combustion mode of the formed reactive fronts after ignition with a direct numerical simulation (DNS) -like mesh resolution. In the shear layer setup (the right panel in Fig. 1), a premixed mixture of the HRF and the LRF/oxidizer is considered within the fuel strip. In this mixture, we assume that some level of mixing is already taken place between the HRF and the ambient containing the LRF (to represent the first step in the DF spray setup which is evaporation and mixing). In this shear layer setup, the ambient is considered to be the premixed LRF/oxidizer mixture. The first- and second-stage ignition (II-III) and the onset of the ignition fronts (IV) are expected in this DF setup similar to the DF spray setup. Here, the main objective is to study the post-ignition event, i.e. the characteristics of the formed ignition fronts in step IV.

In the previous studies by the authors, the state-of-the art on DF spray-assisted ignition has been reviewed in detail [16–20,27,29,30].

Here, we briefly review the most recent papers on the DF spray- assisted ignition. Recently, Tekgul et al. [29] numerically investigated the effect of different DF split injection strategies on the ignition delay time (𝐼 𝐷𝑇) and heat release rate (HRR) characteristics in the RCCI context. It was concluded that the first injection provides the most important control over ignition and HRR. Furthermore, Karimkashi et al. [30] investigated the spray-assisted LES of DF diesel/methanol and diesel/methane under engine relevant conditions. They observed that the methanol/𝑛-dodecane mixture possesses a longer𝐼 𝐷𝑇 compared to the methane/𝑛-dodecane mixture. However, all of these works

were limited to the study of the combustion process until slightly after the ignition time without exploring the post-ignition reactive fronts.

Here, we use a DNS-like grid resolution and focus on the post-ignition reactive front characteristics together with a combustion mode analysis.

With respect to the DF combustion in a shear layer setup explained above, there is only one study in the literature, which is limited to a two-dimensional (2D) setup and a different set of fuels is used.

Yu et al. [31] investigated the ignition properties of a non-premixed n-heptane jet mixing in a surrounding iso-octane/air using 2D DNS under engine relevant conditions such that the ambient pressure level was increased during the combustion. They varied the jet velocity and found that the first-stage ignition is advanced with the increase of the relative velocity between the n-heptane jet and the premixed iso-octane/air charge due to the faster mixing process. Consequently, the second-stage ignition was also advanced in time. While the main focus of this study was on exploring the ignition characteristics, they also studied the formed flame fronts using chemically explosive mode analysis (CEMA) along with a budget analysis, which indicated the formation of a premixed flame.

Apart from the above-mentioned spray and shear layer simulations, there are a few studies in the literature on the characteristics of the reactive fronts after an auto-ignition event in DF systems using DNS in canonical configurations. Demosthenous et al. [32,33] demonstrated the initiation of a premixed methane flame by n-heptane autoignition in a simplified cubic domain. Recently, Karimkashi et al. [34,35]

investigated the combustion mode in locally stratified DF mixtures of methanol/n-dodecane and methane/hydrogen/n-dodecane under engine relevant conditions. They proposed a predictive theoretical tool to estimate the combustion mode for various fuel combinations under different turbulence levels. Furthermore, Zhou et al. [36] demonstrated the presence of both auto-ignition and flame formation in RCCI combustion modeling. Despite the above-mentioned studies, there is an evident research gap on identifying the combustion mode with high relevance to the spray ignited DF combustion. While RCCI combustion is commonly defined as a stratified low-temperature combustion mode with volumetric ignition, the presence of deflagrative fronts is still under debate [37–39]. The combustion modes in RCCI/DF engines may affect the combustion duration and hence emissions and efficiency. For instance, higher emissions and lower efficiency are reported from the auto-ignition mode compared to the deflagration mode [40]. Therefore, it would be of high relevance to better characterize the mode of combustion in DF combustion processes.

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The present work complements our previous works related to spray assisted ignition of DF𝑛-dodecane/methane [16,18,41–43]. Here, we utilize a simplified approach to study the flame initiation and propagation in the DF context. The main objective of this study is to characterize the post-ignition reactive fronts with relevance to the DF spray and RCCI approach. A simplified three dimensional (3D) shear layer setup is considered. In such mixtures, based on previous studies in simple 1D/2D setups [34], the mixing/stratification level, turbulence intensity, and fuel chemistry can switch the combustion mode between spontaneous and deflagration. The detailed objectives of the present paper are listed as follows:

1. Carry out a𝑅𝑒sweep on the 3D DF shear layer setup to identify the effect of turbulence intensity on the combustion mode.

2. Identify the combustion mode (auto-ignition versus deflagration) after ignition using the energy equation budget term analysis.

3. Investigate the characteristics of the formed post-ignition reactive fronts in the shear layer setup compared to ideal and normal deflagration fronts, i.e. one dimensional (1D) premixed methane/oxidizer flames.

2. Numerical methods

Since there are many similarities between the numerical methods used in this work and our previous DF spray simulations, here only a brief description is provided to give an overall insight. The interested reader is referred to any of our previous work on DF spray combustion for further details [16–20,27,29,30].

2.1. Governing equations

The governing equations for studying the reactive flows are the continuity, momentum, species transport and energy equations, which are solved on very fine space/time resolutions. Here, the equations are given in the standard non-filtered form since we directly resolve the low-Reynolds number flow including the flame thickness and since we do not apply any explicit subgrid scale modeling herein. Such simulations are termed as scale-resolved simulations, which are sometimes referred to as ‘quasi-DNS’ as well.

𝜕𝜌

𝜕𝑡 +𝜕𝜌𝑢_𝑖

𝜕𝑥_𝑖 = 0, (1)

𝜕(𝜌𝑢_𝑖)

𝜕𝑡 +𝜕(𝜌𝑢_𝑖𝑢_𝑗)

𝜕𝑥_𝑗 = 𝜕

𝜕𝑥_𝑗(−𝑝𝛿_𝑖𝑗+𝜏_𝑖𝑗), (2)

𝜕(𝜌𝑌_𝑘)

𝜕𝑡 +𝜕(𝜌𝑢_𝑖𝑌_𝑘)

𝜕𝑥_𝑖 = 𝜕

𝜕𝑥_𝑖 (

𝜌𝐷𝜕𝑌_𝑘

𝜕𝑥_𝑖 )

+𝜔̇_𝑘, (3)

𝜕(𝜌ℎ_𝑡)

𝜕𝑡 +

convection term

⏞⏞⏞⏞⏞

𝜕(𝜌𝑢_𝑗ℎ_𝑡)

𝜕𝑥_𝑗 =𝜕𝑝

𝜕𝑡+

diffusion term

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

𝜕

𝜕𝑥_𝑗 (𝜆

𝑐_𝑝

𝜕ℎ_𝑠

𝜕𝑥_𝑗 )

+

reaction term

⏞⏞⏞

̇ 𝜔_ℎ ,

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where𝜌, 𝑢_𝑖, 𝑝, 𝑌_𝑘, ℎ_𝑠 and 𝜏_𝑖𝑗, represent the density, velocity, pressure, mass fraction of species𝑘, sensible enthalpy and viscous stress tensor, respectively. In Eq. (4), the total enthalpy represents the sensible enthalpy and specific kinetic energy as given below.

ℎ_𝑡=ℎ_𝑠+𝑢_𝑖𝑢_𝑖

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We note that in this study for the shear layer problem, there is no liquid evaporation present and the corresponding source terms are zero since the fuel stream is considered to be in the gaseous state because it is a mixture of𝑛-dodecane (diesel surrogate) and methane/oxidizer at high temperature, with details provided in the following. Further, the reacting flow governing equations are closed by the ideal gas law.

Finally, the Lewis number of unity is considered for all the species considered based on chemical kinetics in the study. The unity Lewis approach has been justified in our previous spray simulations of 𝑛- dodecane/methane [16–20,27,29,30]. More importantly, since there

is not any light species like hydrogen in the reactants mixture, this assumption is considered reasonable here.

Here, the customized solver for reacting flows based on the recent work by Morev et al. [44] has been utilized for solving the gas phase governing equations. This solver was built in the OpenFOAM-8 frame- work based on the finite volume approach. The discretization methods are chosen so that they are consistent with our previous works (see e.g. [16–20,45]). The temporal discretization is carried out using a second order implicit time integration method while the convection term is discretized with a second order accurate non-linear scheme called the gamma scheme [46]. In order to discretize the diffusion terms, the standard second-order central scheme is used, namely linear interpolation. The current approach has been tested for many reactive and non-reactive flow systems in our previous studies [16–20]. As a remark, the PISO loop consisted of 5 inner iterations and 2 outer iterations, solved using the variable density solver with

reactingFoam

, a low Mach number compressible flow solver. Mach numbers were varied between 0.02 and 0.22.

2.2. Finite rate chemistry

In this work, operator splitting is used to separate the typically short chemical time scales from the fluid dynamical time scales. The thermo- chemical composition changes are represented as a set of ordinary differential equations (ODE). This has been carried out with the help of pyJac coupled with OpenFOAM [16]. The open-source platform pyJac provides the analytical reaction rate coefficients to the ODE solver [47].

Using DLBFoam, a fast reactive solver, a speedup for reactive flows has been achieved with the help of dynamical load balancing and reference cell mapping methods [48]. In a recent work by Morev et al. [44], DLBFoam was demonstrated for problems such as reactive sprays, a 2D DF shear layer, and Sandia flame D. A skeletal mechanism by Yao et al. [49] has been used in the present work (with 54 species and 96 reactions). The mechanism has been validated and utilized for DF 𝑛−dodecane/methane mixtures by Kahila et al. [17] and it is used in our other DF spray simulation works.

2.3. Simulation configuration

As a background motivation, we envision an𝑛-dodecane spray injection process in DF engine combustion using a shear layer DF setup as depicted inFig. 1. In the present model problem, it is assumed that the ambient premixed methane/oxidizer/EGR (exhaust gas recirculation) at the equivalence ratio𝜙=0.5 and T=900 K is uniformly distributed across the domain as shown inFig. 2a. This ambient oxidizer represents the ambient in a DF spray combustion where the LRF, oxidizer and EGR are premixed. The pressure is kept at 60 bar in all numerical simulations, to be relevant to engine conditions and the ECN spray A¹ conditions modified for DF spray combustion in our previous works [16–20]. Similar to the condition in our previous DF spray simulations, in all the simulations in this work, 15% O₂ by molar concentration is maintained in the oxidizer, while the molar compositions of other important species are listed inTable 1for the ambient oxidizer. Also, a stream of the𝑛-dodecane/methane/oxidizer blend moves in the middle of the domain with various Reynolds numbers (𝑅𝑒), defined based on the stream velocity and the initial strip height, generating the two- sided mixing layers for the fuel stream. The dimensions of the box and the middle strip considered for this shear layer setup are provided in Fig. 2a. As mentioned before, the mixture in the middle strip of the shear layer is in the gaseous state and it is considered such that it

1 Engine Combustion Network (ECN) Spray A is a benchmark test designed to evaluate diesel engines and fuel injection systems. Using a constant-volume combustion chamber, it simulates high pressure injection of diesel fuel at 150 MPa and a temperature of 363 K.

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Fig. 2. (a) Quasi-DNS shear-layer configuration details. (b) Computational domain with mesh refinement regions. Here, the mesh consists of uniform Cartesian cells which are refined down to 5μm for the flame initiation and propagation analysis. Initial velocity, species, and temperature profiles across the shear layer are modeled as smooth hyperbolic tangent functions.

Table 1

Specifications for the reacting shear layer simulations.

Conditions/Compositions Fuel strip Oxidizer

Temperature 700 K 900 K

Pressure 60 Bar 60 Bar

𝑛-C12H26 25% 0

N₂(molar) 53.53% 71.83%

O₂(molar) 11.25% 15.0%

CO₂(molar) 4.46% 5.95%

H2O(molar) 2.58% 3.46%

CH4(molar) 2.81% 3.75%

𝜙 0.3 0.5

represents some level of mixing between the HRF (𝑛-dodecane) and the LRF (methane)/oxidizer/EGR in the equivalent spray setup, see Fig. 1. The properties of this mixture are found considering inert mixing (non-reactive) between the envisioned𝑛-dodecane spray at 363 K and the ambient mixture at 900 K, following the ECN spray A conditions modified for DF spray combustion in our previous works [16–20]. As a result of this mixing, the fuel strip temperature is 700 K and the mixture composition is detailed inTable 1. The ambient has zero velocity while the strip velocity is varied from zero (𝑅𝑒 = 0) to higher velocities such that𝑅𝑒= 187, 375, 750, and 1500. In the present setup, initial velocity, temperature, and species composition profiles are modeled with a hyperbolic tangent function across the fuel strip and the ambient as shown in the schematics of the initial condition inFig. 2a. As remark, the equivalence ratio (𝜙= 0.5) is derived from premixed methane (LRF) air and the EGR. However, the mixture fraction is based on HRF.

Details of the computational grid with a corresponding cut-plane are depicted inFig. 2b. The considered mesh has five different refinement levels ranging from 160 μmon the outer region to5 μmin the inner region. The fine refinement region of 5 μm is the target region to capture the flame initiation and flame front propagation. In particular, the mixing, ignition and reactive fronts formation are all expected to take place in this fine region. The simulations are performed until the reactive front travels over the5 μmrefinement region. The target region is chosen based on 1D premixed flame analyses and the calculated laminar flame speed, which are further explained in Section3.2. In addition, the extended coarse mesh region in the 𝑦-direction is to reduce the

flame sensitivity to bottom and top boundaries. More importantly, pressure wave transmissive outlet boundary conditions are selected on the 𝑦-direction to reduce the pressure reflections effects. As a result of the combustion process, the pressure level would start to increase in a real combustion engine. However, in the present model problem we are not simulating an enclosed volume but rather an infinite volume. Hence, we aim at keeping the pressure level rather constant in the system via the pressure transmissive boundary conditions. Cyclic boundary conditions are considered in x- and𝑧-directions.

3. Results and discussion

In this study, the results and discussion of n-dodecane methane DF combustion are structured in a comprehensive manner. First, combustion is explored through zero-dimensional (0D) DF simulations at different n-dodecane mixture fractions and temperatures along the mixing line. This provides insights into the overall combustion behavior at various conditions. Next, the formation of deflegrative fronts with time evolution in the non-premixed setup is investigated through two types of simulations: (1) premixed DF flames at different mixture fractions using Cantera, and (2) 3D non-premixed DF shear layer setup at𝑅𝑒

=0 using OpenFOAM, where the flame remains intrinsically 1D. The grid resolution of the𝑅𝑒=0 OpenFOAM simulations are also analyzed to determine if the flame is fully resolved. Finally, 3D scale-resolved

‘quasi-DNS’ is used to study the DF combustion and reaction front propagation, with a focus on exploring the different combustion modes through an energy equation terms budget analysis and a scatter plot analysis of the intermediate species in the progress variable space.

3.1. Homogeneous reactor analysis

Here, we investigate DF ignition using 0D homogeneous reactor simulations along the mixing line [50] under the modified Spray A conditions with EGR (O₂ concentration 15%). In the so-called mixing line concept, the main idea is to emulate variations of the mixture fraction in a non-premixed setup using 0D homogeneous reactor. In this concept, the characteristics of the homogeneous mixture are defined by inert mixing of two envisioned streams of fuel and oxidizer (as in the non-premixed setup) at different ratios (representing the mixture fraction in the non-premixed setup). The mixture homogeneous

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Fig. 3. 0D Homogeneous reactor calculations: 𝐼 𝐷𝑇 vs. mixture fraction (𝑍) for the 𝑛-dodecane/methane/air -mixture. Here, 𝑍 describes the mixing extent of 𝑛- dodecane with the premixed methane-air stream while𝑍= 0implies zero𝑛-dodecane concentration. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

temperature is found based on the inert mixing of the streams according to their enthalpies. More details are available in [51]. Here, The fuel stream (𝑛-dodecane) is initially at 363 K while the ambient gas (oxidizer/methane/EGR) temperature is 900 K. The utilized Yao mechanism has been thoroughly validated and justified in our previous studies for such DF combustion studies [16,17,19]. We remind that in the present study, methane behaves as an LRF while𝑛-dodecane acts as an HRF and its purpose is to act as a high-reactivity ignition source.

From a practical point of view, there are two important aspects that need to be understood in order to achieve a smooth control over the operation of an efficient DF CI engine. First, in the studied temperature range, the premixed ambient methane should not auto-ignite. Secondly, it is necessary to investigate the effect of methane on DF ignition to ensure robust ignition. These questions can be explored via the mixing line concept in a homogeneous reactor-based 0D simulations for different mixture fraction values (𝑍).

Fig. 3 displays the𝐼 𝐷𝑇s, calculated based on the maximum temperature gradient in this work, for the DF mixtures in the mixing line as a function of 𝑛-dodecane mixture fraction (𝑍) for two different equivalence ratios𝜙= 0.5and 0.8. We note that two different𝜙are considered herein for the sake of comparison while 𝜙 = 0.5 is the relevant one in the 3D studied in this work. The following observations are made. First, 𝐼 𝐷𝑇 is slightly advanced in DF for 𝜙 = 0.5 in comparison to 𝜙 = 0.8 because the amount of the LRF is lower at leaner conditions; hence, IDT is less inhibited by the LRF, which is consistent with the findings in [16–20,27,52]. Second, envisioning a DF shear-layer simulation, the𝐼 𝐷𝑇−𝑍 plot can be characterized. In such a shear-layer DF setup, first,𝑛-dodecane mixes with the premixed ambient mixture, then, it ignites and finally, flame fronts form. With relevance to such an envisioned process, the𝐼 𝐷𝑇−𝑍 can be characterized by three different regions. In the rich part,𝑛-dodecane mixes (I) and starts to react with the premixed ambient methane/oxidizer/EGR oxidizer leading to the formation of complex intermediate species. In this region, IDT is still relatively long. In the flame initiation region (II), a reaction front is established starting from the ignition occurring at the most reactive mixture fraction corresponding to the minimum 𝐼 𝐷𝑇 of the system. Finally, although 0D simulations do not obviously capture any flame, in the flame propagation region (III), a flame front is hypothesized to burn the surrounding premixed methane/air at 𝑍 ≈ 0. We note that in the following non-premixed DF shear layer setup, the fuel stream is considered as a premixed mixture of𝑛- dodecane/methane/oxidizer/EGR as marked with the green arrow on

Fig. 4. 1D premixed flame simulations are carried-out using Cantera. Flame thickness, laminar burning velocity, and maximum temperature against𝑍 for modified Spray A conditions for𝑛-dodecane/methane/air -mixture. Here,𝑍describes the mixing extent of 𝑛-dodecane with the premixed methane-air stream while𝑍= 0implies pure methane- air mixture. The vertical dashed black line represents the stoichiometric mixture fraction.

the right side ofFig. 3, i.e. some level of mixing between the𝑛-dodecane and the ambient oxidizer is considered. In that setup, the oxidizer, marked with the purple arrow on the left side ofFig. 3, is the premixed mixture of methane/oxidizer/EGR at𝜙 = 0.5as previously shown in Fig. 2. We note that this symbolicFig. 3to better clarify the ideas we are discussing throughout the paper and the values for the boundaries are representative.

3.2. 1D𝑛-dodecane/methane/air flames

Here, the main motivation for simulating freely propagating 1D premixed laminar flames is to explore the characteristics of the hypothesized premixed flames (deflagrative fronts) initiated in the following DF shear layer case studies, including the flame speed and thickness.

Fig. 4shows the laminar flame speed, flame thickness and adiabatic flame temperature (𝑇_𝑚𝑎𝑥) distributions as a function of the mixture fraction calculated in Cantera using the Yao mechanism.

Fig. 4indicates that the flame thickness (𝛿) possesses a minimum value close to the stoichiometric mixture fraction where𝛿_𝑚𝑖𝑛= 50 μm.

In the present work, in order to resolve the flame in the following DF shear layer simulations, a grid resolution of𝛿_𝑚𝑖𝑛∕10is considered. It is also noted that the laminar burning velocities may range between 𝐿𝐵𝑉 = 0.3–2 m∕sdepending on the mixture fraction value. We note that in the 3D simulations in the subsequent sections, the initial DF shear layer flow velocity will be in the range of𝑈_𝑚𝑎𝑥 ∼ 10–75 m∕s depending on the Reynolds number (𝑅𝑒= 187–1500). However, after the initial mixing, the velocity scales decrease significantly to the order of the reported𝐿𝐵𝑉s.

Fig. 5provides a visual representation of the temperature evolution in a stagnant flow at 𝑅𝑒 = 0. In addition, the flame front displacement speed (𝑆_𝑑) is calculated using a temperature threshold of 1500 K and averaged to obtain the mean displacement speed. The mean displacement speed is then plotted over time and compared with the results from Fig. 4. The results demonstrate a increase in the mean displacement speed with time, providing a quantitative measure of the effect of fuel stratification on the flame speed.

Next, in order to have an initial comprehension on the characteristics of the formed flame fronts and their evolution with time in the following non-premixed DF shear layer cases, we simulate a shear layer case at𝑅𝑒 = 0. Then, we compare the characteristics of the formed flame fronts for𝑅𝑒= 0shear layer in OpenFOAM (3D) with those of the premixed flames from Cantera (1D), representing ideal deflagrative

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Fig. 5. 1D DF mixture at𝑅𝑒= 0shear layer: The blue line depicts the maximum temperature progression with respect to time in a 1D simulation using OpenFOAM, while the red inverted triangle displays the average flame displacement speed over time. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Time evolution of a DF mixture at𝑅𝑒= 0towards a premixed methane/air flame. Blue line: 1D premixed𝑛-dodecane/methane/air flame (Cantera). Green line:

1D premixed methane/air flame (Cantera). Symbols: 3D OpenFOAM result at different times. It is seen that the 3D OpenFOAM result is approaching the 1D premixed methane- air flame as time progresses for𝑅𝑒= 0. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

fronts (i.e. premixed flames). Here in Cantera (1D), the two extreme 1D premixed flame simulations are carried-out at two conditions i.e.𝑍= 𝑍_𝑠𝑡and𝑍= 0, which correspond to 1D DF and 1D SF (pure methane) flames, respectively. The comparison is conducted in the CH₄progress variable space, defined as follows:

𝐶= 𝑌_CH

4−𝑌_CH

4,𝑜

𝑌_CH

4,𝑝−𝑌_CH

4,𝑜

, (6)

where𝑌_CH

4,𝑜 corresponds to the ambient methane mass fraction and 𝑌_CH

4,𝑝is the methane mass fraction of the products (𝑌_CH

4,𝑝= 0for the present study since we have a lean premixed mixture).

Fig. 6displays the trends of flame establishment in a shear layer DF simulation at𝑅𝑒= 0as well as the two 1D premixed flames at𝑍=𝑍_𝑠𝑡 and 𝑍 = 0 from Cantera. The scatter data from OpenFOAM have been overlaid with the Cantera’s 1D premixed n-dodecane/methane/air flame. The𝑦-axis is normalized with the maximum value of CH₃and the plot is shown as a function of the CH₄progress variable. Initially, right after ignition at𝜏= 0.582 ms, it is noted that the results of the DF simulation carried out using 3D OpenFOAM exhibit similar characteristics

to those of a stoichiometric premixed𝑛-dodecane/methane/air flame.

With time progress, the results approach the 1D premixed methane/air flame computed by Cantera. We observe that the flame establishment time𝜏₃ ≈ 3 msin the𝑅𝑒= 0case. The considered example indicates that at𝑅𝑒=0, the present DF case develops a premixed methane/air flame outside the𝑛-dodecane containing region where𝑍= 0. Next, the 3D DF shear layers will be studied in detail for turbulent conditions, i.e.𝑅𝑒 ≫0.

3.3. Visual inspection of the 3D shear layers

In order to understand the characteristics of the 3D DF reacting shear layers,Figs. 7–9depict the spatial distribution of mixture fraction, temperature, and methyl radical (CH₃), respectively, for different 𝑅𝑒between 0 and 1500. The total duration of the 3D simulations in physical time varies between 0.75–1 ms or 1.4–1.8𝐼 𝐷𝑇. In all of the simulations, the expected process of mixing, ignition and flame front formation is observed, albeit with difference in timing and characteristics as discussed in the following. Here, the𝑅𝑒= 0case, which was discussed earlier inFig. 6, is shown for the sake of completeness.

The following observations are made inFigs. 7–9. For𝑅𝑒 ≥ 187, the flow is first triggered by a Kelvin–Helmholtz type instability with coherent vortex formation. As a general overview, at lower𝑅𝑒, ignition evolves smoothly into a reaction front, while at higher 𝑅𝑒, intense mixing promotes turbulence formation. In particular, for 𝑅𝑒 = 187, ignition appears at the vortex centres, which not only contain the most reactive mixture fraction but also sustain the heat for long enough time to promote the progress of reactions. For𝑅𝑒 > 375, mixing is more intense and the vortex lifetime is shorter. Hence, the ignition appears after the first main vortices have broken into smaller vortices.

Then, reaction fronts appear which finally propagate into the ambient mixture.

Qualitatively, it appears from Figs. 8 and 9that the higher the Reynolds number, the more the ignition shifts towards a ‘volumetric ignition’ mode. Two observations imply such a conclusion. First in Fig. 8, it is observed that at lower𝑅𝑒small ignition kernels are formed transitioning to thin flame fronts while at higher𝑅𝑒ignition tends to initiate spontaneously in a larger volume. This observation is consistent with the findings of [35,53] where higher turbulence is shown to shift the combustion mode in the stratified DF mixtures from deflagrative to volumetric. Similar observations were made in DF spray simulations in [29,54] with varying the injection strategies. At higher𝑅𝑒e.g. at 𝑅𝑒=1500, the scattered ignition kernels within a turbulent flow field resemble combustion in RCCI engines. Second inFig. 9, the spatial methyl radical patterns and their intensity change with increasing𝑅𝑒.

In particular, at lower𝑅𝑒the intensity of CH₃after ignition is stronger compared to𝑅𝑒=1500 wherein the intensity of CH₃remains weak or delayed at the time of consideration. These observations imply that the volumetric ignition mode in𝑅𝑒= 1500leads to burnt high temperature regions with low methyl radical traces compared to lower 𝑅𝑒 cases where high concentrations of the methyl radicals are still found at the latest time instances.

Fig. 10 depicts the spatial methyl radical patterns in a cut plane around the reaction fronts for different𝑅𝑒at 1.2𝐼 𝐷𝑇. Additionally, in order to detect the flame front, CH3 is used as the marker, since the radical exists only in thin regions of the HTC region [55–57].

Wrinkled patterns with flame-like structures are observed at higher 𝑅𝑒, whereas thicker and laminar-like patterns are observed for lower 𝑅𝑒. At 𝑅𝑒 = 0, pure diffusion/reaction controls the laminar flame progress while as 𝑅𝑒increases more mixing effects are observed. In particular, while diffusion plays an important role at different 𝑅𝑒, mixing effects increases with 𝑅𝑒. At 𝑅𝑒 = 187, the reaction fronts resemble qualitatively flame fronts as was already shown for𝑅𝑒= 0.

However, based on these visualizations, especially for𝑅𝑒 > 187, the reaction front structure remains unknown. In particular, due to stronger mixing at𝑅𝑒 >187, normal deflagration is still not fully established which might lead to lower CH₃concentrations specifically at𝑅𝑒= 1500.

This matter will be further explored in the following sections.

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Fig. 7. Temporal evolution of the mixture fraction field in 3D reacting shear layers at various𝑅𝑒.𝑌-axis coordinates are displayed in millimeters.

Fig. 8. Temperature evolution in the 3D reacting shear layers. The𝑛-dodecane/methane/air-mixture ignites and initiates the combustion. The coordinates labels on the𝑦-axis are in millimeters.

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Fig. 9. Temporal evolution of methyl radical (CH₃) field in the 3D reacting shear layer. CH₃is an indicator of CH₄dissociation and of the onset of high-temperature chemistry.

In the𝑦-axis, the coordinates labels are marked in millimeters.

Fig. 10. The evolution of the methyl radical field (CH₃) on a symmetry cut plane for varying Re (based on the strip height of the shear layer) at 1.2𝐼 𝐷𝑇.

3.4. Ignition in the 3D shear layers

Next, we investigate ignition in the 3D DF reacting shear layer cases by presenting numerical results for 𝑅𝑒 = 187, 375, 750 and 1500. In relevance to engine conditions, it is important to note that the mixing timescales may vary significantly, depending on the engine type and load. While high Reynolds numbers are expected in these scenarios, simulating such flows can pose significant challenges in terms of computational complexity and cost. In this study, we used a 3D shear layer model to understand the flow dynamics, with the results obtained using a quasi-DNS approach. Despite the limitations inherent in this approach, it provides valuable insight into the complex mixing processes that occur in high Reynolds number flows.

A characteristic feature of the DF mixtures is a two-stage ignition process which has been frequently reported also for the present Yao mechanism [16–19]. In particular, the first stage ignition corresponds to the dissociation of large hydrocarbons (such as𝑛-dodecane) at lower temperatures leading to the formation of intermediate species such as C₁₂H₂₅O₂, called RO₂ hereafter. The second stage ignition happens at higher temperatures leading to the initiation of high temperature reactions and consequently, maximum heat release rate. InTable 2, the

first stage (𝜏₁) and the second stage ignition delay time (𝜏₂) as well as the stoichiometric time (𝜏_𝑠𝑡) estimates from the present scale-resolving simulations are provided. Consistent with our previous studies, the first stage ignition (𝜏₁) is defined as the time to reach 20% of maximum RO₂ while the second stage ignition (𝜏₂) is defined as the time to reach the maximum gradient of𝑇_𝑚𝑎𝑥[16–19]. As an additional metric, we monitor the mixture fraction values near the reaction zones by considering the stoichiometric time (𝜏_𝑠𝑡) which is defined as the time for𝑍_𝑎𝑣𝑔 to reach the stoichiometric value. In the present study, the quantity𝑍_𝑎𝑣𝑔is defined as the conditional average of mixture fraction field interpolated at𝑇 = 1500 K. We note that𝑍_𝑎𝑣𝑔 is a simple metric to monitor the n-dodecane concentration at the reaction fronts. Nevertheless,𝑍_𝑎𝑣𝑔 is considered as a useful metric to characterize the flame formation time as noted later on in the paper. According toTable 2,1.67< 𝜏₂/𝜏₁<2.1 for the 3D cases. In addition,𝜏_𝑚𝑖𝑥(based on strip height/flow velocity) was calculated in order to quantify the relatively small mixing times being a factor of∼1/20–1/200 of the ignition delay time: the higher the𝑅𝑒the smaller the mixing time.

Fig. 11(a) depicts temporal evolution of the maximum temperature (T_max) for the 3D cases at different𝑅𝑒. It is noted that the first-stage and second stage ignition delay times from 3D simulations are longer than

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Table 2

Tabulated𝐼 𝐷𝑇 values from 3D quasi-DNS [ms]. In 0D the respective IDT are𝜏^0𝐷₁ = 0.252 msand𝜏₂^0𝐷= 0.459 ms.

𝑅𝑒 𝜏₁[ms] 𝜏₂[ms] 𝜏₂∕𝜏₁ 𝜏_𝑠𝑡[ms] 𝜏_𝑚𝑖𝑥[μs] 𝜏_𝑠𝑡/𝜏₂

0 0.267 0.582 2.17 0.615 ∞ 1.056

187 0.267 0.562 2.104 0.578 21.333 1.028

375 0.307 0.537 1.749 0.611 10.667 1.144

750 0.306 0.511 1.669 0.585 5.333 1.137

1500 0.271 0.561 2.070 0.787 2.667 1.402

Fig. 11. 3D reacting shear layer. (a) Temporal evolution of𝑇_𝑚𝑎𝑥. (b) Integrated heat release rate.

their respective 0D counterparts (𝜏₁^0𝐷= 0.252 msand𝜏^0𝐷

2 = 0.459 ms).

The primary reason for such a difference can be attributed to the 3D mixing process, which causes latency in the ignition process. The intermediate species such as RO₂are very sensitive in this high Re cases especially at 750 to 1500, and therefore, one can expect to observe slight differences and fluctuations in 𝜏₁ at those conditions. Further details and analyses on the IDT differences in 0D and 3D DF spray simulations are provided in [30]. In the present setup, the mixing in the systems also locally decreases the maximum temperature of the fuel/oxidizer mixture. This has been observed in the evolution of the maximum temperature for 𝑅𝑒= 1500 case at 0.5 ms. However, the

Fig. 12. Temporal evolution of𝑍_𝑎𝑣𝑔(conditioned with𝑇 =1500 K). Here, the black symbols mark the high temperature ignition time and the vertical dashed lines depict the time when the average mixture has reached stoichiometry.

effect of mixing on temperature does not appear to have a significant impact on the cases with𝑅𝑒 <1500studied in this work.

Heat release rate (HRR) is another important metric to understand ignition in any combustion process. The heat release rate for different 𝑅𝑒cases is shown in Fig. 11(b). It is seen that there are two peaks corresponding to the first and second stage ignition. HRR can be an indicator for calculating𝜏₁ and𝜏₂. However, for consistency with our previous works [19,20,27,44,52], we have used the definition mentioned in Section3.4. At earlier times, in particular before𝜏₂, both fuels are expected to contribute to the heat release characteristics. However, at later times when much of the𝑛-dodecane has reacted, the reaction fronts are hypothesized to resemble premixed methane/air flames. As a remark, despite the different Reynolds numbers, the fuel consumption rates appear similar for the different cases. This will be further explored in the following sections.

Fig. 12depicts the temporal evolution of𝑍_𝑎𝑣𝑔 (see above for its definition) for𝑅𝑒= 375, 750 and 1500. The stoichiometric time (𝜏_𝑠𝑡) is defined as when Z_𝑎𝑣𝑔crosses the stoichiometric mixture fraction value Z_𝑠𝑡 =0.023. Here,𝜏_𝑠𝑡 is considered to give a rough estimate on the time when reactions shift to the single fuel methane combustion. As can be seen in the figure,𝜏_𝑠𝑡ranges from values 0.611 ms (𝑅𝑒= 375) to 0.787 ms (𝑅𝑒= 1500). Furthermore, it is noted fromTable 2that 1< 𝜏_𝑠𝑡∕𝜏₂<1.4: at the highest Reynolds number𝜏_𝑠𝑡is the longest. This is consistent with the observations inFigs. 7–9, where𝐼 𝐷𝑇was delayed with increasing 𝑅𝑒 meaning that for 𝑅𝑒 = 1500, high 𝑛-dodecane concentrations persist for a longer time at the reaction zones than for 𝑅𝑒 = 375. This indicates that the potential flame initiation process may be delayed at higher𝑅𝑒, which will be discussed in the following sections.

3.5. Reaction analysis in the progress variable space

The evolution of reactions with time can be understood by tracking the evolution of different intermediate species scatter plots in the methane progress variable (𝐶) space. Here, considering the available information in the literature [32,33,35], CH₃ is chosen as an appro- priate marker for methane/air flames.Fig. 13shows scatter plots and their conditional mean of normalized CH₃mass fraction in the progress variable space for𝑅𝑒=187, 375, 750 and 1500. The presented data points are chosen in the range1⋅10⁻⁵≤𝑍≤𝑍_𝑠𝑡in order to focus on the reaction zones at the edge of the shear layer. The scatter plots are super- imposed with 1D premixed flame profiles (computed from Cantera) at 𝑍= 0(pure methane flame) and𝑍=𝑍_𝑠𝑡(𝑛-dodecane/methane flame), which can be considered to be bounding cases for the anticipated, potentially emerging, non-ideal flame profiles for the 3D DF shear

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Fig. 13. 1D premixed𝑛-dodecane/methane/air (blue line), 1D premixed methane/air (green line), 3D conditional mean (black line) and standard deviation (gray lines area) of CH3with respect to progress variable of CH4. Note:1𝑒⁻⁵< 𝑍 <0.03values where used to select/filter the scatter data points. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

layers, as earlier discussed inFig. 6. The normalization has been done with the maximum value of the CH₃ mixture fraction in each 3D simulation and in the considered 1D premixed laminar flames.

The following common observations can be made from Fig. 13 for 𝑅𝑒 = 187, 375, 750 and 1500. First, at early times (𝑡 ≈ 𝐼 𝐷𝑇), the conditional mean of normalized CH₃ mass fraction from the 3D quasi-DNS does not coincide with either one of the 1D premixed flame profiles. At this time, ignition is initiated with minimal production of CH₃ with low CH₄ consumption for all the cases. Hence, a flame structure cannot be observed yet. Second, as time goes on, the 3D profiles start to resemble the 1D premixed laminar flame profiles being clearly close to 𝑍 = 𝑍_𝑠𝑡 profiles. At 𝑡 ≈ 1.2𝐼 𝐷𝑇, the conditional averages are quite close to 1D DF premixed flames, whereas at 𝑡 ≥ 1.4𝐼 𝐷𝑇 the profiles become closer to the pure methane/air 1D profiles.

According to Fig. 13, the conditional means for 𝑅𝑒 ≤ 750 are initially closer to the DF 1D profiles but as time progresses, the profiles approach the 1D methane/air flame structures. However, this trend is not seen at 𝑅𝑒= 1500 because of different mixing/ignition patterns (see e.g.Fig. 8) as well as limitation of the simulation time to 1 ms.

This is consistent with our earlier discussions in Figs. 7–9, where higher turbulence was expected to delay the transition from volumetric ignition to deflagration. At𝑅𝑒 =1500, the strong mixing dilutes the mixture rapidly which slightly delays the flame initiation process (see Figs. 7, 8). By extending the simulation duration beyond 1 ms, the results will approach those of a methane/air flame at 𝑅𝑒 = 1500.

However, we are not able to provide a numerical evidence here since it needs our domain extension for𝑅𝑒=1500, which leads to higher computational costs. Therefore, the analysis provided in this subsection can be considered as the first evidence for establishment of reaction fronts resembling deflagrative fronts in all DF shear layer cases considered in

the present work. In addition, temperature distribution in the mixture fraction space for different Reynolds numbers are visually represented inFig. 14. This scatter data analysis provides a comprehensive overview of the temperature behavior in the mixture fraction space for varying Reynolds numbers. In addition to the scatter data, the conditional mean and standard deviation of temperature are also plotted inFig. 14.

At the 𝑡 = 𝐼 𝐷𝑇, the conditional mean of temperature deviates from the expected 1D premixed flame profile. This deviation occurs as the mixture experiences an initial release of heat energy, causing an increase in temperature. However, this is not a complete representation of the temperature behavior as the reaction progresses. As time ad- vances, the 3D conditional mean temperature profile begins to converge towards the expected 1D DF premixed flame by approximately t = 1.2𝐼 𝐷𝑇. This convergence indicates that the temperature behavior in the mixture fraction space is becoming more predictable and aligning with the expected 1D premixed flame profile. At 𝑡 = 1.4 IDT, a temperature of 2400 K is recorded, indicating intense heat release in the mixture. This high temperature is an indication of deflagration flames present within the range of 1.2–1.4 IDT for𝑅𝑒≤750. However, it may take longer to approach the 1D premixed flame at𝑅𝑒= 1500. Therefore, the deflagration flames are an important aspect of the reaction and contribute to the overall temperature behavior in the mixture fraction space.

When compared to the 1D single-fuel flame structures, we observe that at the latest time𝑡= 1.4𝐼 𝐷𝑇 the present conditionally averaged statistics are mostly in-between the two 1D limiting cases which can be understood since in 3D there will always be some finite concentration of𝑛-dodecane at the flame front as well as some combustion residues in the burned zone. To further elaborate on this point,Fig. 15depicts the mixture fraction cutplanes with temperature iso-lines𝑇 =950 K and

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Fig. 14. 1D premixed𝑛-dodecane/methane/air (blue line), 3D conditional mean (black line) and standard deviation (gray lines area) of T with respect to mixture fraction space.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Spatial distribution of mixture fraction (𝑍) at 1.3𝐼 𝐷𝑇. Temperature iso-lines𝑇 =950 K (blue line) and 1700 K (yellow line) are marked in the figure in order to point out the flame zone with finite𝑛-dodecane concentrations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

1700 K for𝑅𝑒=375, 750 and 1500. As can be seen, the reaction zones contain finite values of mixture fraction0 < 𝑍 ≤ 𝑍_𝑠𝑡(𝑍_𝑠𝑡 = 0.023).

This means that at 𝑡 = 1.3𝐼 𝐷𝑇, the reaction fronts still have some amount of 𝑛-dodecane i.e. we are considering a DF reaction front.

Therefore, the deflagration mode as ambient methane flame is not fully developed yet at this point. Well beyond t≥1.3𝐼 𝐷𝑇, the presence of 𝑛-dodecane would completely disappear, then forming pure methane based reaction fronts. Such fully developed methane reaction fronts were evident earlier in Fig. 6at𝑅𝑒 =0. As a remark, the presence of n-dodecane would diminish later on by mixing and combustion and the reaction fronts would be dominated by pure methane.

In summary, reaction fronts resembling deflagrative fronts are observed in all the studied cases. However, due to the short simulation time, the observed reaction front structures in the progress variable space still pose more DF than SF characteristics. To explore the flame

establishment in detail, an energy budget analysis is carried out in the next subsection.

3.6. Budget analysis

One of the well-known diagnostics tools for identifying the combustion mode of the reactive fronts is the energy equations terms budget analysis. Fig. 16 highlights the balance between the diffusion and reaction terms in a normal deflagration scenario, as observed within the reaction zone. According to Gordon et al. [58], this balance is crucial in maintaining energy equilibrium in the system. On the other hand the main characteristics of volumetric combustion is that the reaction term of the energy equation dominates the diffusion term being typically 1–2 order(s) of magnitude larger.

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Fig. 16. Energy budget analysis for 𝑅𝑒 = 0: A graphical representation of the normalized influence of diffusion (green line) and reaction (red line) on the energy equation at 𝑅𝑒 = 0, as analyzed along a sample line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Here, the diffusion and reaction rate terms of the energy equation, Eq.(4), are compared in the budget analysis. Given the characteristics of deflagrative versus volumetric combustion as explained above, in the non-premixed DF shear layer cases studied in this work, the following process is expected to occur. First, since auto-ignition is expected to pose the characteristics of volumetric ignition, it is expected that the reaction term dominates the diffusion term. As time marches and reactive fronts are formed, it is expected that the diffusion term starts to fall within the same order of magnitude with the reaction term. We note that in this problem, we typically do not expect to have an ideal normal deflagrative fronts, as previously discussed on the discussions of Figs. 13and15since pure methane flame is not established at higher𝑅𝑒 cases. From the earlier subsections, the scatter plots imply the existence of deflagrative fronts starting from1.1𝐼 𝐷𝑇for𝑅𝑒≤750cases while the flame establishment of 𝑅𝑒=1500 case occurs slightly delayed closer to𝑡= 1.2𝐼 𝐷𝑇. At𝑅𝑒=1500 the stronger mixing dilutes the mixture quickly which in particular delays the flame establishment process as noted also inFigs. 7,8, and9.

In order to conduct the budget analysis in this work, we consider several investigation lines normal to the reactive fronts in the 3D space as marked with𝑙− 0,𝑙− 1, and𝑙− 2on the depicted cutplanes inFig. 17 for𝑅𝑒=375, 750 and 1500. The cutplanes are at𝑡= 1.4𝐼 𝐷𝑇 to make sure that the fronts have evolved sufficiently in time, bearing in mind that the main objective of this study is to find out whether deflagrative fronts eventually form in such DF non-premixed cases. Otherwise, it is known from the literature that at 𝐼 𝐷𝑇, the reaction term always dominates the diffusion term, see [35,53]. Here, the investigation lines are selected randomly with the criteria of being normal to the flame fronts and stretched from the products to the reactants zone in the 3D domain, similar to the strategy taken in the literature for instance by Krisman et al. [59] and Krisman et al. [60]. Accordingly, the 1D profiles of temperature as well as the reaction and diffusion energy

budget terms along the investigation lines are provided in Fig. 17.

The budget terms are normalized with the peak value of the reaction term while the𝑥-axis represents the considered location along each line normalized with its own length.

The following observations are made fromFig. 17. First, along all the investigation lines, the temperature profiles present a monotonic behavior similar to 1D premixed flames with the highest temperature gradient within the reaction rate peak zone. Second, the reaction and diffusion terms along the sampling lines are in a same order of magnitude. Third, the structure of the reaction and diffusion terms are as expected in the premixed flames in almost all cases, i.e. the diffusion term is double picked. The smaller peak typically happens in the preheat layer of premixed flames and it is balanced with the convection term, whereas the larger peak is expected to happen in the reaction zone and balanced by the reaction term. The interested reader is referred to Figure 4 in [61] for the ideal structure of the reaction terms in a premixed flame (ideal normal deflagration) explained here.

We remind that here, we do not claim an ideal normal deflagration structure, but a structure very similar to that. Considering all the observations explained above, the budget analysis implies that for all the DF shear layer cases considered in this work, deflagrative fronts form long enough after𝐼 𝐷𝑇.

In summary, the quasi-DNS of the DF shear layer for different𝑅𝑒re- veals various stages. First, the ignition kernels form at the most reactive mixture around𝑡=𝐼 𝐷𝑇. These ignition kernels develop into a reaction front, which further develops into the DF deflagration mode around 𝑡 ∼ 1.2–1.4𝐼 𝐷𝑇, where reaction and diffusion terms of the energy equation are in order-of-magnitude balance. However, such an early state does not yet correspond to a fully developed premixed methane flame because the reaction fronts are formed in an environment with high levels of n-dodecane stratification. Over a short period of time, shown in 1D (𝑅𝑒= 0) to be around 𝐼 𝐷𝑇 ≈ 3 ms, the fronts develop into a genuine premixed methane/air flame i.e. ideal deflagration.

4. Conclusions

The present work extends the previous DF ignition studies by Kan- nan et al. [43] to better understand the combustion progression after the ignition. Due to the very high Reynolds number in fuel sprays, a simplified lower𝑅𝑒model problem was studied instead. We investigated methane/air ignition assisted by an𝑛-dodecane/methane/air/

EGR stream. The shear layer model problem was investigated in order to capture the reactive flow length/timescales using scale-resolving numerical simulations (quasi-DNS) where laminar flame thickness was resolved by approximately 10 grid points.

When Reynolds number increases, the results indicate that the ignition delay time is not changed significantly in the present model problem. At𝑅𝑒=187, we observe that the ignition is mainly triggered within vortex structures, which capture the most reactive mixture fraction leading eventually to flame initiation. For the larger Reynolds numbers, in particular𝑅𝑒=1500, the excessive mixing and dilution of the fuel mixture starts to delay the flame formation. The flame formation for the intermediate Reynolds numbers appear to be somewhere in between these extremes. As a remark, we have observed (not shown for brevity) that beyond𝑅𝑒 >1500, excessive mixing may prevent ignition and combustion in the present setup. However, this aspect could not be fully confirmed as the mixing patterns exceeded the refinement regions.

Based on the budget analysis and scatter plots of the intermediate species in the progress variable space, for 𝑅𝑒 ≥ 187, a deflagrative DF flame is established right after the auto-ignition similar to the case 𝑅𝑒= 0. However, the numerical findings indicate that within1.4𝐼 𝐷𝑇, finite𝑛-dodecane concentrations are still observed at the flame front and thus a pure methane flame has not been established yet for𝑅𝑒 >0.

At𝑅𝑒=0, the 1D flame profiles are perfectly established by𝑡= 3ms the latest; but for𝑅𝑒 >0, we were not able to simulate long enough to confirm pure methane air flame profiles yet. The present results may be

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Fig. 17. Energy budget analysis for375≤𝑅𝑒≤1500. Normalized contribution from the diffusion term (green line) and the reaction term (red line) in the energy equation for the sampling lines l-0, l-1, and l-2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

among the first to report 3D deflagrative DF flame structures occurring within 0.2–0.4IDT after the DF ignition delay time for the range of 𝑅𝑒 considered herein up to 1500. Finally, we note that it would be beneficial to revisit the present model problem results in the case of a real 3D spray assisted DF ignition.

CRediT authorship contribution statement

Jeevananthan Kannan:Conceptualization, Methodology, Software, Visualization, Writing – original draft.Shervin Karimkashi:Methodol- ogy, Software, Validation, Writing – original draft.Mahmoud Gadalla:

Validation, Writing – review & editing, Supervision.Ossi Kaario:Con- ceptualization, Writing – review & editing, Supervision.Ville Vuori- nen:Conceptualization, Writing – review & editing, Supervision, Fund- ing acquisition.

Declaration of competing interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgments

This study has been funded by the Academy of Finland (grant numbers 318024, 332784, and 297248). We acknowledge CSC, Finnish IT Center for Science for providing the computational resources.

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