A Comparative Study of the Hydrogen Auto-Ignition Process in Oxygen–Nitrogen and Oxygen–Water Vapor Oxidizer: Numerical Investigations in Mixture Fraction Space and 3D Forced Homogeneous Isotropic Turbulent Flow Field

Abstract

In this paper, we analyze the auto-ignition process of hydrogen in a hot oxidizer stream composed of oxygen–nitrogen and oxygen–water vapor with nitrogen/water vapor mass fractions in a range of 0.1–0.9. The temperature of the oxidizer varies from 1100 K to 1500 K and the temperature of hydrogen is assumed to be 300 K. The research is performed in 1D mixture fraction space and in a forced homogeneous isotropic turbulent (HIT) flow field. In the latter case, the Large Eddy Simulation (LES) method combined with the Eulerian Stochastic Field (ESF) combustion model is applied. The results obtained in mixture fraction space aim to determine the most reactive mixture fraction, maximum flame temperature, and dependence on the scalar dissipation rate. Among others, we found that the ignition in ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures occurs later than in ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{N}}_2$ mixtures, especially at low oxidizer temperatures. On the other hand, for a high oxidizer temperature, the ignitability of ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures is extended, i.e., the ignition occurs for a larger content of ${\mathrm{H}}_2\mathrm{O}$ and takes place faster. The 3D LES-ESF results show that the ignition time is virtually independent of initial conditions, e.g., randomness of an initial flow field and turbulence intensity. The latter parameter, however, strongly affects the flame evolution. It is shown that the presence of water vapor decreases ignitability and makes flames more prone to extinction.

1. Introduction

Water plays a key role in combustion systems, influencing ignition and extinction processes, increasing efficiency, controlling emissions, and keeping critical components cool. As a ubiquitous substance, water has unique properties such as high specific heat capacity, thermal conductivity, and the ability to undergo phase changes, thanks to which it can significantly affect the behavior of the flame. The addition of water can influence combustion phenomena both physically and chemically, with the magnitude of these effects depending on how water is introduced into the combustion environment. The review paper of Dryer [1] perfectly summarizes the initial applications of water addition in practical combustion systems, such as gas turbines, internal combustion engines, and external combustion systems, describing its physical and chemical effects. The present research focuses on the auto-ignition problem in a hydrogen–oxygen–water vapor mixture. Hydrogen has recently become increasingly important due to its potential to play a crucial role in the decarbonization of the energy production sector. However, integrating hydrogen, hydrogen-enriched, or diluted hydrogen fuels into existing facilities presents significant challenges and carries considerable consequences. Hydrogen’s burning velocity, combustion temperature, and diffusivity are much higher than those of carbon-based fuels like methane or natural gas, and its wider flammability limits substantially alter the combustion regime. This can lead to combustion instability in gas chambers or engines, such as flame flashback, thermoacoustic instabilities, blow-off, or knocking. Levinsky [2], in a paper intriguingly titled “Why can’t we just burn hydrogen?”, discusses the challenges and prospects of hydrogen combustion.

Early experimental studies have shown that water vapor accelerates the oxidation of hydrogen and hydrocarbon mixtures in slow, non-explosive reactions [3,4,5]. Several decades later, Le Cong and Dagau [6,7] studied in detail the effect of water vapor on the kinetics of the oxidation of hydrogen and methane mixtures for a wide range of temperatures, equivalence ratios, and pressures, including jet-stirred reactor, shock tube, and gas turbine conditions. In contrast to previous studies, they showed that the presence of water vapor tends to inhibit the oxidation of hydrogen and methane. In fact, the effect of water or steam on the oxidation of hydrogen and hydrocarbons may vary and depends on detailed reaction conditions, including temperature, pressure, and the presence of catalysts. Water can dilute the concentration of fuel or oxidizer, or it can absorb heat, which can lower the temperature and slow the reaction. On the other hand, at high temperatures, water can dissociate into hydroxyl radicals (OH), which are highly reactive and can significantly accelerate the oxidation by providing alternative, faster reaction pathways. Wang et al. [8] showed that the ignition delay time of ${\mathrm{H}}_2$–air–steam mixtures in shock tube experiments is strongly dependent on the temperature and steam concentration but weakly dependent on pressure changes. Seiser and Seshadri [9] performed experimental and numerical studies of the chemical influence of water on critical conditions of extinction and auto-ignition of hydrogen and methane flames in a counterflow burner. They showed that increasing the amount of water leads to easier extinguishment of both premixed and non-premixed hydrogen flames, as well as difficulty in igniting the latter. Many experimental and numerical studies have also been devoted to the burning velocities of hydrogen and methane mixtures at wet conditions, showing a linear decrease in the burning velocity with increasing water vapor content [10,11,12,13]. Additionally, water or steam injection has been shown to reduce NO_x formation, which is particularly desirable in gas turbine combustors [14,15,16,17,18].

Numerical investigations in the framework of computational fluid dynamics (CFD) devoted to wet-combustion systems mainly focus on humidified gas turbine applications. For instance, Hiestermann et al. applied the Reynolds-Averaged Navier–Stokes (RANS) approach [19] and large eddy simulation (LES) method [20] to investigate the effect of high steam loadings on methane/kerosene combustion in aero-gas turbines. It has been shown that the main effect of added steam is to reduce the maximum temperature and the formation of NO_x. Krüger et al. [21], in the LES study of a hydrogen-fueled gas turbine, showed that adding steam spreads the heat release peak, thickens the flame front, and extends the flame further downstream. Rosiak and Tyliszczak [22], and Wawrzak and Tyliszczak [23], applied LES for modeling turbulent hydrogen jet flame in oxy-combustion regimes in which the oxidizer consisted of oxygen and water vapor. They showed that by changing their proportions, one can control a flame lift-off height and the flame temperature. Recently, Palulli et al. [24] studied the characteristics of the wet hydrogen/air swirl-stabilized flame using LES. The study showed promising results, with steam-diluted hydrogen flames that did not exhibit any hot spots or flashbacks and characterized by flameless combustion and lean NO_x-reducing mixture composition. Palulli et al. [25] also applied reduced-order LES modeling to investigate the so-called humid blowout in a swirl-stabilized gas turbine, i.e., the extinction of the flame when the steam dilution exceeds the permissible limit.

Ignition and extinction are the two most important phenomena in combustion processes. Accurate experimental analysis of these phenomena requires sophisticated equipment and advanced measurement techniques. Investigating ignition and extinction also poses significant challenges for CFD, and achieving accurate numerical results necessitates the use of direct numerical simulations (DNS) or LES combined with advanced combustion models, e.g., the Conditional Moment Closure (CMC) [26,27,28,29], Eulerian Stochastic Fields (ESF) [30,31,32], and Partially or Perfectly Stirred Reactor (PaSR, PSR) [33,34,35]. These models have shown considerable success in predicting ignition, flame propagation/stabilization, and extinction [29,32,36,37,38,39,40,41,42]. An important discovery regarding the ignition phenomenon was made by Mastorakos et al. [43]. Using DNS with a single-step chemical mechanism, they investigated methane auto-ignition in a 2D turbulent decaying homogeneous isotropic turbulent (HIT) flow field. They found that the ignition process is a stochastic and local phenomenon, and occurs at the most reactive mixture fraction, where the scalar dissipation rate is low. Moreover, they demonstrated that the time of ignition spot appearance is almost independent of the turbulent time scale, shows minimal variation across different realizations of the same flow, and decreases with partial premixing and reduction in the fuel stream width. The research and findings of Mastorakos et al. [43] initiated several subsequent numerical investigations devoted to the ignition problem. Im et al. [44] conducted similar 2D DNS studies on the ignition of a hydrogen–air mixing layer in decaying HIT using a detailed chemical kinetic mechanism. They analyzed a wide range of turbulence intensities and showed that the ignition delay time is almost insensitive to the turbulence intensity. Hilbert and Thévenin [45] confirmed these results by applying a detailed reaction scheme and taking multicomponent diffusion velocities and a non-unity Lewis number. Moreover, they showed that the turbulent Reynolds number ($R{e}_{\lambda }=u_{rms}\lambda /\nu$, $u_{rms}$—root mean square of the velocity field, $\lambda$—Taylor length scale, and $\nu$—kinematic viscosity) has a minimal and statistically insignificant impact on the ignition time. The observation of Mastorakos et al. [43]—that auto-ignition occurs around the same mixture fraction and away from stoichiometry conditions—has been confirmed by numerous additional DNS results using both simplified and complex chemistry in 2D and 3D simulations, see a review paper by Mastorakos [46] and the works cited therein. A common aspect of the works cited above is that they focused on the ignition problem in homogeneous and isotropic turbulent flow fields, which do not necessarily reflect the flow conditions in real applications. Recently, Wawrzak and Tyliszczak [47] performed a 3D LES-ESF of the hydrogen auto-ignition process in a mixing layer with the initial homogeneous isotropic (HIT) and anisotropic (HAT) turbulent fields. The ignition time, similarly to previous studies, was shown to be independent of $R{e}_{\lambda }$ in the HIT regime; whereas, in the HAT conditions, it was found to be pronouncedly dependent on $R{e}_{\lambda }$. Furthermore, the evolution of the flame temperature, the flame size, and its growth with time were also found to depend on $R{e}_{\lambda }$ for both initial conditions.

In contrast to the aforementioned studies assuming the initial velocity field is a decaying HIT/HAT field, much less research has been conducted on auto-ignition under conditions where the turbulent kinetic energy (TKE) of the flow does not decay with time. This may seem somewhat surprising considering that in real combustion devices, fuel and oxidizer streams are continuously supplied to the combustion zone, thereby maintaining the TKE level even after ignition occurs. An example of research aiming at mimicking these regimes is the experimental work of Pan et al. [48], who investigated the influence of a forced HIT on auto-ignition and combustion modes under engine-like conditions. It has been shown that in the forced HIT, the influence of a thermochemical mechanism on the combustion process is weaker compared to the unforced flow and manifests by the extended auto-ignition time, reduced burning rate, and weakened cool flame behavior, especially in scenarios with a longer chemical timescale. Recently, Caban et al. [49] and Boguslawski et al. [50] performed LES studies using an approximate deconvolution method [51,52,53,54] and sub-grid scale estimation approach [55,56] to model the ignition and extinction of nitrogen–diluted-hydrogen flame in a 3D forced HIT. The aim of these investigations was not to analyze ignition phenomena but to develop and verify new combustion models.

In the present research, we focus on the auto-ignition problem in a turbulent flow field in which hydrogen mixes with an oxygen–water vapor mixture in a forced HIT configuration at different turbulence intensities. To the best of the author’s knowledge, no studies have been conducted to analyze the auto-ignition in hydrogen–oxygen–water vapor mixtures, and this work is a step in this direction. It is worth noticing that the assumed fuel/oxidizer composition constitutes a perfectly clean combustion system in which the water vapor is the only combustion product when the combustion process is complete. For reference, we also consider an equivalent configuration with water vapor replaced by nitrogen that corresponds to conventional combustion systems. The investigations are conducted using the LES method combined with the ESF model. We formulate a computational framework (3D forced HIT and the fuel/oxidizer initialization process) that can be easily adopted by other researchers to continue and extend the present research. The proposed flow configuration ensures a constant ‘injection’ of energy to the flow and mimics real conditions in which flame kernels grow and evolve in space and time in highly turbulent regimes. LES of the auto-ignition process in 3D HIT is preceded by its comprehensive analysis in mixture fraction space by solving unsteady flamelet equations [57,58]. It aims to determine the auto-ignition delay times, the most reactive mixture fractions, and the ignitability of hydrogen–oxygen–nitrogen (${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{N}}_2$) and hydrogen–oxygen–water vapor (${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$) mixtures at different contents of nitrogen and water vapor in the oxidizer stream. It is shown that at low oxidizer temperatures, the ignition delay time in both configurations differs significantly and, in the ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures, it is significantly longer. Moreover, in these cases, the ignition occurs only in a limited range of low water vapor content. At higher oxidizer temperatures, the situation changes fundamentally. The ignition delay times are similar and the range of ignitability of the ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixture is much wider. The LES-ESF analysis shows that for both oxidizer compositions, the ignition occurs at the most reactive mixture fraction in the regions of a low scalar dissipation rate. It is found that the maximum temperature in ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures is lower and that these mixtures exhibit lower ignitability and are more prone to extinction. When this happens, the temperature of the combustion products decreases faster than in ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{N}}_2$ mixtures.

The paper is organized as follows. The mathematical model is discussed in the next section, including a description of the flamelet approach in mixture fraction space and the LES method combined with the ESF combustion model. Details of the numerical algorithms used to solve the flamelet/LES-ESF equations are presented in Section 3. The results of the simulations are discussed in Section 4, which is divided into two parts. First, we focus on the analysis of auto-ignition in mixture fraction space, followed by a discussion of the 3D LES results. This section also provides a detailed description of the numerical settings, such as temporal and spatial discretization schemes, and computational configurations. Concluding remarks and an outlook for future research are presented in Section 5. The paper concludes with three appendices, where the applied chemical kinetics schemes, detailed values of ignition delay times, and most reactive mixture fractions are presented in tables.

2. Mathematical Modeling

2.1. Unsteady Flamelet Model

The analysis of unsteady flame evolution in mixture fraction space is based on the adiabatic flamelet approach, which relies on solving a set of equations for species mass fractions $Y_{\alpha }$ ($\alpha =1,\cdots ,N_s$, $N_s$—number of species) given as [57,58]

$$\rho {\displaystyle \frac{\partial Y_{\alpha }}{\partial t}}={\dot{\omega }}_{\alpha }+{\displaystyle \frac{1}{2}}\rho \chi {\displaystyle \frac{{\partial }^2{Y}_{\alpha }}{\partial {\xi }^2}}$$

(1)

where $\xi \in [0,1]$ represents a mixture fraction space in which an oxidizer stream composition is defined at $\xi =0$ and a fuel stream composition at $\xi =1$. The variable $\rho$ denotes the density, ${\dot{\omega }}_{\alpha }(Y_1,\cdots ,Y_{N_s},T)$ stands for chemical source terms representing the production/consumption of species, and $\chi$ represents the scalar dissipation rate. As the main chemical kinetics scheme, we adopt the one defined by Mueller et al. [59] with 9 species (${\mathrm{H}}_2$, ${\mathrm{O}}_2$, ${\mathrm{N}}_2$, ${\mathrm{H}}_2\mathrm{O}$, $\mathrm{H}{\mathrm{O}}_2$, ${\mathrm{H}}_2{\mathrm{O}}_2$, $\mathrm{OH}$, $\mathrm{O}$, $\mathrm{H}$) and 19 reactions, defined in Appendix A in

Table A1. ${\mathrm{H}}_2/{\mathrm{O}}_2$ reaction mechanism of Mueller et al. [59] with the rate constants for the Arrhenius equation $k=A{T}^{\beta }exp(-E_a/RT)$, $R=1.9872$ [cal/mol K]—gas constant.

		A [cgs units]	$\mathit{\beta }$	${\mathit{E}}_{\mathit{a}}$ [cal/mol]
1.	$\mathrm{H}+{\mathrm{O}}_2\rightleftharpoons \mathrm{O}+\mathrm{OH}$	1.915$\times {10}^{14}$	0.0	16,439
2.	$\mathrm{O}+{\mathrm{H}}_2\rightleftharpoons \mathrm{H}+\mathrm{OH}$	0.508$\times {10}^5$	2.67	6290
3.	${\mathrm{H}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+\mathrm{H}$	0.216$\times {10}^9$	1.51	3430
4.	$\mathrm{O}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons \mathrm{OH}+\mathrm{OH}$	2.970$\times {10}^6$	2.02	13,400
5.	${\mathrm{H}}_2+\mathrm{M}\rightleftharpoons \mathrm{H}+\mathrm{H}+\mathrm{M}$	4.577$\times {10}^{19}$	$-1.40$	104,380
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
6.	$\mathrm{O}+\mathrm{O}+\mathrm{M}\rightleftharpoons {\mathrm{O}}_2+\mathrm{M}$	6.165$\times {10}^{15}$	$-0.50$	0
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
7.	$\mathrm{O}+\mathrm{H}+\mathrm{M}\rightleftharpoons \mathrm{OH}+\mathrm{M}$	4.714$\times {10}^{18}$	$-1.00$	0
8.	$\mathrm{H}+\mathrm{OH}+\mathrm{M}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+\mathrm{M}$	2.212$\times {10}^{22}$	$-2.00$	0
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/6.3/
9.	$\mathrm{H}+{\mathrm{O}}_2+\mathrm{M}\rightleftharpoons {\mathrm{HO}}_2+\mathrm{M}$	1.475$\times {10}^{12}$	0.60	0
	LOW/3.482$\times {10}^{16}$ $-0.411$ −1.115$\times {10}^3$/
	TROE/0.5 1$\times {10}^{-30}$ 1$\times {10}^{30}$/
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
10.	${\mathrm{HO}}_2+\mathrm{H}\rightleftharpoons {\mathrm{H}}_2+{\mathrm{O}}_2$	1.66$\times {10}^{13}$	0.0	823
11.	${\mathrm{HO}}_2+\mathrm{H}\rightleftharpoons \mathrm{OH}+\mathrm{OH}$	7.079$\times {10}^{13}$	0.0	295
12.	${\mathrm{HO}}_2+\mathrm{O}\rightleftharpoons \mathrm{OH}+{\mathrm{O}}_2$	0.325$\times {10}^{14}$	0.0	0
13.	${\mathrm{HO}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+{\mathrm{O}}_2$	2.890$\times {10}^{13}$	0.0	$-497$
14.	${\mathrm{HO}}_2+{\mathrm{HO}}_2\rightleftharpoons {\mathrm{H}}_2{\mathrm{O}}_2+{\mathrm{O}}_2$	4.200$\times {10}^{14}$	0.0	11,982
	DUPLICATE
	${\mathrm{HO}}_2+{\mathrm{HO}}_2\rightleftharpoons {\mathrm{H}}_2{\mathrm{O}}_2+{\mathrm{O}}_2$	1.300$\times {10}^{11}$	0.0	$-1629$
15.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{M}\rightleftharpoons \mathrm{OH}+\mathrm{OH}+\mathrm{M}$	2.951$\times {10}^{14}$	0.0	48,430
	LOW/1.202$\times {10}^{17}$ 0.0 4.55$\times {10}^4$/
	TROE/0.5 1$\times {10}^{-30}$ 1$\times {10}^{30}$/
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
16.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{H}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+\mathrm{OH}$	0.241$\times {10}^{14}$	0.0	3970
17.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{H}\rightleftharpoons {\mathrm{H}}_2+{\mathrm{HO}}_2$	0.482$\times {10}^{14}$	0.0	7950
18.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{O}\rightleftharpoons \mathrm{OH}+{\mathrm{HO}}_2$	9.550$\times {10}^6$	2.0	3970
19.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+{\mathrm{HO}}_2$	1.000$\times {10}^{12}$	0.0	0
	DUPLICATE
	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+{\mathrm{HO}}_2$	5.800$\times {10}^{14}$	0.0	9557

. It has been used in a number of studies involving hydrogen combustion and enabled accurate prediction of complex phenomena, e.g., lifted flame [60,61], auto and spark ignition [41]. For comparison purposes, test computations were also performed using the scheme of Li et al. [62], which additionally includes $\mathrm{CO}$, ${\mathrm{CO}}_2$, $\mathrm{Ar}$, and $\mathrm{He}$ and is defined by 23 reactions (see

Table A2. ${\mathrm{H}}_2/{\mathrm{O}}_2$ reaction mechanism of Li et al. [62] with the rate constants for the Arrhenius equation $k=A{T}^{\beta }exp(-E_a/RT)$, $R=1.9872$ [cal/mol K]—gas constant.

		A [cgs units]	$\mathit{\beta }$	${\mathit{E}}_{\mathit{a}}$ [cal/mol]
1.	$\mathrm{H}+{\mathrm{O}}_2\rightleftharpoons \mathrm{O}+\mathrm{OH}$	3.547$\times {10}^{15}$	$-0.406$	16,599
2.	$\mathrm{O}+{\mathrm{H}}_2\rightleftharpoons \mathrm{H}+\mathrm{OH}$	0.508$\times {10}^5$	2.67	6290
3.	${\mathrm{H}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+\mathrm{H}$	0.216$\times {10}^9$	1.51	3430
4.	$\mathrm{O}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons \mathrm{OH}+\mathrm{OH}$	2.970$\times {10}^6$	2.02	13,400
5.	${\mathrm{H}}_2+\mathrm{M}\rightleftharpoons \mathrm{H}+\mathrm{H}+\mathrm{M}$	4.577$\times {10}^{19}$	$-1.40$	104,380
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
	$\mathrm{CO}$/1.9/${\mathrm{CO}}_2$/3.8/
	$\mathrm{Ar}$/0.0/$\mathrm{He}$/0.0/
6.	$\mathrm{O}+\mathrm{O}+\mathrm{M}\rightleftharpoons {\mathrm{O}}_2+\mathrm{M}$	6.165$\times {10}^{15}$	$-0.50$	0
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
	$\mathrm{Ar}$/0.0/$\mathrm{He}$/0.0/
	$\mathrm{CO}$/1.9/${\mathrm{CO}}_2$/3.8/
7.	$\mathrm{O}+\mathrm{H}+\mathrm{M}\rightleftharpoons \mathrm{OH}+\mathrm{M}$	4.714$\times {10}^{18}$	$-1.00$	0
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
	$\mathrm{Ar}$/0.75/$\mathrm{He}$/0.75/
	$\mathrm{CO}$/1.9/${\mathrm{CO}}_2$/3.8/
8.	$\mathrm{H}+\mathrm{OH}+\mathrm{M}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+\mathrm{M}$	3.800$\times {10}^{22}$	$-2.00$	0
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
	$\mathrm{Ar}$/0.38/$\mathrm{He}$/0.38/
	$\mathrm{CO}$/1.9/${\mathrm{CO}}_2$/3.8/
9.	$\mathrm{H}+{\mathrm{O}}_2+\mathrm{M}\rightleftharpoons {\mathrm{HO}}_2+\mathrm{M}$	1.475$\times {10}^{12}$	0.60	0
	LOW/6.366$\times {10}^{20}$ $-1.72$ 5.248$\times {10}^2$/
	TROE/0.8 1$\times {10}^{-30}$ 1$\times {10}^{30}$/
	${\mathrm{H}}_2$/2.0/${\mathrm{H}}_2\mathrm{O}$/11/
	${\mathrm{O}}_2$/0.78/$\mathrm{CO}$/1.9/${\mathrm{CO}}_2$/3.8/
10.	${\mathrm{HO}}_2+\mathrm{H}\rightleftharpoons {\mathrm{H}}_2+{\mathrm{O}}_2$	1.66$\times {10}^{13}$	0.0	823
11.	${\mathrm{HO}}_2+\mathrm{H}\rightleftharpoons \mathrm{OH}+\mathrm{OH}$	7.079$\times {10}^{13}$	0.0	295
12.	${\mathrm{HO}}_2+\mathrm{O}\rightleftharpoons \mathrm{OH}+{\mathrm{O}}_2$	0.325$\times {10}^{14}$	0.0	0
13.	${\mathrm{HO}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+{\mathrm{O}}_2$	2.890$\times {10}^{13}$	0.0	$-497$
14.	${\mathrm{HO}}_2+{\mathrm{HO}}_2\rightleftharpoons {\mathrm{H}}_2{\mathrm{O}}_2+{\mathrm{O}}_2$	4.200$\times {10}^{14}$	0.0	11,982
	DUPLICATE
	${\mathrm{HO}}_2+{\mathrm{HO}}_2\rightleftharpoons {\mathrm{H}}_2{\mathrm{O}}_2+{\mathrm{O}}_2$	1.300$\times {10}^{11}$	0.0	$-1629$
15.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{M}\rightleftharpoons \mathrm{OH}+\mathrm{OH}+\mathrm{M}$	2.951$\times {10}^{14}$	0.0	48,430
	LOW/1.202$\times {10}^{17}$ 0.0 4.55$\times {10}^4$/
	TROE/0.5 1$\times {10}^{-30}$ 1$\times {10}^{30}$/
	${\mathrm{H}}_2$/2.5/${\mathrm{H}}_2\mathrm{O}$/12/
	$\mathrm{CO}$/1.9/${\mathrm{CO}}_2$/3.8/
	$\mathrm{Ar}$/0.64/$\mathrm{He}$/0.64/
16.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{H}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+\mathrm{OH}$	0.241$\times {10}^{14}$	0.0	3970
17.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{H}\rightleftharpoons {\mathrm{H}}_2+{\mathrm{HO}}_2$	0.482$\times {10}^{14}$	0.0	7950
18.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{O}\rightleftharpoons \mathrm{OH}+{\mathrm{HO}}_2$	9.550$\times {10}^6$	2.0	3970
19.	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+{\mathrm{HO}}_2$	1.000$\times {10}^{12}$	0.0	0
	DUPLICATE
	${\mathrm{H}}_2{\mathrm{O}}_2+\mathrm{OH}\rightleftharpoons {\mathrm{H}}_2\mathrm{O}+{\mathrm{HO}}_2$	5.800$\times {10}^{14}$	0.0	9557

). The scalar dissipation rate is a quantity that relates the flow field to the Equation (1) in flamelet-type models. It is defined as $\chi =2D\left({\displaystyle \frac{\partial \xi }{\partial x_i}}{\displaystyle \frac{\partial \xi }{\partial x_i}}\right)$, where D is the mass diffusion coefficient. Here, for analysis of the flame structure in $\xi$-space, we assume $\chi$ according to the Amplitude Mapping Closure model [63], representing the scalar dissipation rate in a counter-current diffusion flame. It is defined as

$$\chi ={\chi }_{0}exp\left(-2{\left[{\mathrm{erf}}^{-1}(1-2\xi )\right]}^2\right)$$

(2)

where ${\chi }_0=a/\pi$ is the maximum scalar dissipation rate and a is the strain rate. The set of Equation (1) is complemented by the equation for the enthalpy h defined as

$$h=\sum _{\alpha =1}^{N_s}{Y}_{\alpha }{h}_{\alpha };h_{\alpha }={\int }_{T_0}^T{C}_{p,\alpha }\mathrm{d}T+\Delta H_{f,\alpha }^0$$

(3)

where $h_{\alpha }$ are the enthalpies of individual species, $T_0$ is a reference temperature, $\Delta H_{f,\alpha }^0$ are formation enthalpies of individual species, and $C_{p,\alpha }$ are the temperature-dependent specific heats at constant pressure. For an adiabatic system, enthalpy does not change with time. Hence, knowing $h\left(\xi \right)$ and $Y_{\alpha }\left(\xi \right)$ allows us to compute the temperature from (3). Note that, alternatively, one can calculate the temperature from an equation analogical to (1). Then, however, the conservation of energy (enthalpy) cannot be guaranteed, i.e., $h\left(\xi \right)\approx \mathrm{const}+\mathcal{O}(\Delta t^m,\Delta {\xi }^n)$, where $m,n$ denote the orders of discretization of (1) in time and mixture fraction space.

2.2. Large Eddy Simulation

LES is widely used to model unsteady combustion phenomena, including ignition and extinction processes [22,29,36,37,39,60,64]. Compared to direct numerical simulation (DNS), in LES, only large flow scales are solved directly on a numerical mesh, while small scales have to be modeled. This scale separation is achieved through spatial filtering defined as [65,66] $\overline{f}(\mathbf{x},t)={\int }_{\Omega }\mathcal{G}(\mathbf{x}-{\mathbf{x}}^{\prime },\Delta )f(\mathbf{x},t)\mathrm{d}{\mathbf{x}}^{\prime }$, where f is any flow variable, $\Omega$ is the flow domain, $G(\mathbf{x},\Delta )$ is a filter function such that ${\int }_{\Omega }\mathcal{G}(\mathbf{x},\Delta )\mathrm{d}\mathbf{x}=1$, and $\Delta =Vo{l}^{1/3}$ is a filter width based on a local mesh volume. Chemically reacting flows experience density changes and, in such situations, density-weighted filtering, called Favre filtering, is commonly used. It is defined as $\tilde{f}(\mathbf{x},t)=\overline{\rho f}/\overline{\rho }$ [58].

We consider 3D low Mach number reactive flows for which the governing equations in the framework of LES are as follows:

$${\displaystyle \frac{\partial \overline{\rho }}{\partial t}}+{\displaystyle \frac{\partial \overline{\rho }{\tilde{u}}_j}{\partial x_j}}=0$$

(4)

$${\displaystyle \frac{\partial \overline{\rho }{\tilde{u}}_i}{\partial t}}+{\displaystyle \frac{\partial \overline{\rho }{\tilde{u}}_i{\tilde{u}}_j}{\partial x_j}}=-{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+\overline{\rho }{f}_i+{\displaystyle \frac{\partial {\tau }_{ij}}{\partial x_j}}+{\displaystyle \frac{\partial {\tau }_{ij}^{sgs}}{\partial x_j}}$$

(5)

$${\displaystyle \frac{\partial \overline{\rho }{\tilde{Y}}_{\alpha }}{\partial t}}+{\displaystyle \frac{\partial \overline{\rho }{\tilde{u}}_j{\tilde{Y}}_{\alpha }}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left(\overline{\rho }{D}_{\alpha }{\displaystyle \frac{\partial {\tilde{Y}}_{\alpha }}{\partial x_j}}\right)+{\displaystyle \frac{\partial J_{\alpha }^{sgs}}{\partial x_j}}+\overline{\rho {\dot{\omega }}_{\alpha }}$$

(6)

$${\displaystyle \frac{\partial \overline{\rho }\tilde{h}}{\partial t}}+{\displaystyle \frac{\partial \overline{\rho }{\tilde{u}}_j\tilde{h}}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left(\overline{\rho }{D}_h{\displaystyle \frac{\partial \tilde{h}}{\partial x_j}}\right)+{\displaystyle \frac{\partial J_h^{sgs}}{\partial x_j}}$$

(7)

The set of Equations (4)–(7) completes the equation of state $p_0=\overline{\rho }R\tilde{T}$, where $p_0$ is the thermodynamic pressure (constant in time and space for open flows), R is the gas constant, and T is the temperature. The symbol $u_i$ in Equations (4)–(7) represents the i-th component of the velocity; p is the hydrodynamic pressure; ${\tau }_{ij}=\mu \left[\left({\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial {\tilde{u}}_j}{\partial x_i}}\right)-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial x_k}}\right]$ is the viscous stress tensor, where $\mu$ is the molecular viscosity determined by Sutherland’s law; and ${\delta }_{ij}$ is the Kronecker delta. The symbols $D_{\alpha }=\mu /\left(\overline{\rho }\mathrm{Sc}\right)$ and $D_h=\mu /\left(\overline{\rho }\mathrm{Pr}\right)$ in species and enthalpy transport equations denote the mass and heat diffusivities, where $\mathrm{Sc}$ and $\mathrm{Pr}$ are the Schmidt and Prandtl number. In this study, we assume unity Lewis number ($\mathrm{Le}=\mathrm{Sc}/\mathrm{Pr}$) and take $\mathrm{Sc}=\mathrm{Pr}=0.7$ for all species [28,64].

As a result of filtering the nonlinear convective terms, unresolved sub-grid terms—denoted by the superscript ${·}^{sgs}$—appear in Equations (5)–(7). They are defined as ${\tau }_{ij}^{sgs}=\overline{\rho }\left({\tilde{u}}_i{\tilde{u}}_j-\widetilde{u_i{u}_j}\right)$, $J_{\alpha }^{sgs}=\overline{\rho }(\widetilde{u_j}{\tilde{Y}}_{\alpha }-\widetilde{u_j{Y}_{\alpha }})$, and $J_h^{sgs}=\overline{\rho }(\widetilde{u_j}\tilde{h}-\widetilde{u_{j}h})$. In the present study, the sub-grid stress tensor is modeled as ${\tau }_{ij}^{sgs}=2{\mu }_t{S}_{ij}+{\tau }_{kk}^{sgs}{\delta }_{ij}/3$, where $S_{ij}={\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial {\tilde{u}}_j}{\partial x_i}}$ is the strain-rate tensor of the resolved velocity field and ${\mu }_t$ is the sub-grid viscosity. The diagonal terms ${\tau }_{kk}^{sgs}$ of ${\tau }_{ij}^{sgs}$ are added to the pressure $\overline{P}=\overline{p}-{\tau }_{kk}^{sgs}{\delta }_{ij}/3$ and computed with the help of the pressure correction method [65], while ${\mu }_t$ is computed using the Vreman model [67]. The terms $J_{\alpha }^{sgs}$ and $J_h^{sgs}$ are modeled as $J_{\alpha }^{sgs}=\overline{\rho }{D}_{\alpha ,t}{\displaystyle \frac{\partial {\tilde{Y}}_{\alpha }}{\partial x_j}}$ and $J_h^{sgs}=\overline{\rho }{D}_{h,t}{\displaystyle \frac{\partial \tilde{h}}{\partial x_j}}$ with the sub-grid diffusivities $D_{\alpha ,t}=D_{h,t}={\mu }_t/\left(\overline{\rho }{\sigma }_t\right)$, where ${\sigma }_t$ stands for the turbulent Schmidt/Prandtl number equal to 0.7 [28,32,68].

The chemical source terms $\overline{\rho {\dot{\omega }}_{\alpha }}=\overline{\rho {\dot{\omega }}_{\alpha }(\mathbf{Y},T)}$ ($\mathbf{Y}={[Y_1,\cdots ,Y_{N_s}]}^T$) are highly non-linear. Their calculation as $\overline{\rho {\dot{\omega }}_{\alpha }(\mathbf{Y},h)}=\overline{\rho }\widetilde{{\dot{\omega }}_{\alpha }(\mathbf{Y},h)}\ne \overline{\rho }{\dot{\omega }}_{\alpha }(\tilde{\mathbf{Y}},\tilde{h})$, called a laminar chemistry model or no-model approach, may lead to substantial errors when the computational mesh is coarse and sub-grid fluctuations of species mass fractions and temperature are large. In this work, we apply the Eulerian Stochastic Field (ESF) approach [30,69] to model the combustion process. We note that, unlike other combustion models such as Conditional Moment Closure [26,27], and models based on the flamelet approach [57] or Partially and Perfectly Stirred Reactors [33,34,35], the ESF approach enables the modeling of multi-regime combustion processes, encompassing both premixed and non-premixed mixture regions.

In the first step of deriving the ESF model, Equations (6) and (7) are substituted with a corresponding evolution equation for the joint density-weighted filtered probability density function (pdf) [70,71,72]:

$$\tilde{\mathcal{P}}(\phi ,\mathbf{x},t)={\int }_{\Omega }{\displaystyle \frac{\rho ({\mathit{x}}^{\prime },t)\mathcal{F}(\phi ,{\mathit{x}}^{\prime },t)\mathcal{G}(\mathbf{x}-{\mathbf{x}}^{\prime })}{\overline{\rho }(\mathit{x},t)}}d{\mathbf{x}}^{\prime }$$

(8)

where $\mathcal{F}(\phi ,\mathit{x},t)={\prod }_{n=1}^{N_s+1}\delta [{\phi }_n-{\tilde{\varphi }}_n(\mathbf{x},t)]$ is the fine-grained pdf of the scalar quantities ${\tilde{\varphi }}_n=({\tilde{Y}}_1,{\tilde{Y}}_2,\cdots ,{\tilde{Y}}_{N_s},\tilde{h})$ and $\phi$ is their composition in space. Therefore, $\tilde{\mathcal{P}}(\phi ,\mathbf{x},t)$ is the probability that ${\varphi }_n$ falls within the range $[\phi ,\phi +d\phi ]$ at a specific spatiotemporal location within the LES filter size. The evolution equation for $\tilde{\mathcal{P}}(\phi ,\mathbf{x},t)$ is given as [73,74,75]:

$$\begin{array}{cc}\hfill \overline{\rho }{\displaystyle \frac{\partial \tilde{\mathcal{P}}\left(\phi \right)}{\partial t}}& +\overline{\rho }{\tilde{u}}_j{\displaystyle \frac{\partial \tilde{\mathcal{P}}\left(\phi \right)}{\partial x_j}}-\sum _{n=1}^{N_s+1}{\displaystyle \frac{\partial }{\partial {\varphi }_n}}\left(\overline{\rho }{\dot{\omega }}_n\tilde{\mathcal{P}}\left(\phi \right)\right)\hfill \\ & ={\displaystyle \frac{\partial }{\partial x_i}}\left[\left({\displaystyle \frac{\mu }{\sigma }}+{\displaystyle \frac{{\mu }_t}{{\sigma }_t}}\right){\displaystyle \frac{\partial \tilde{\mathcal{P}}\left(\phi \right)}{\partial x_i}}\right]-{\displaystyle \frac{\overline{\rho }}{\tau }}\sum _{n=1}^{N_s+1}{\displaystyle \frac{\partial }{\partial {\varphi }_n}}\left(({\phi }_n-{\tilde{\varphi }}_n)\tilde{\mathcal{P}}\left(\phi \right)\right)\hfill \end{array}$$

(9)

where $\tau =C_D\left(\mu +{\mu }_t\right)/\overline{\rho }{\Delta }^2$ is the so-called micromixing time scale, for which we set $C_D=2$ [32]. Note that the chemical source terms ${\dot{\omega }}_n\left(\phi \right)$ are not filtered in this approach and do not require modeling. However, due to the extremely high dimensionality of the above equation, it is impractical to solve it using conventional discretization methods. Valiño [30] proposed solving (9) by applying the Monte Carlo approach and Ito’s formulation of the stochastic integral in which $\tilde{\mathcal{P}}\left(\phi \right)$ is represented by an ensemble of $1\le n\le N$ stochastic fields for each of $N_s$ species (${\xi }_{\alpha }^n(\mathbf{x},t)$) and enthalpies (${\xi }_h^n(\mathbf{x},t)$). In effect, Equation (9) is replaced by an equivalent system of equations given as

$$\begin{array}{cc}\hfill \overline{\rho }\mathrm{d}{\xi }_{\alpha ,h}^n& +\overline{\rho }{\tilde{u}}_i{\displaystyle \frac{\partial {\xi }_{\alpha ,h}^n}{\partial x_i}}\mathrm{dt}-{\displaystyle \frac{\partial }{\partial x_i}}\left[\left({\displaystyle \frac{\mu }{\sigma }}+{\displaystyle \frac{{\mu }_t}{{\sigma }_t}}\right){\displaystyle \frac{\partial {\xi }_{\alpha ,h}^n}{\partial x_i}}\right]\mathrm{dt}\hfill \\ & =\overline{\rho }\sqrt{{\displaystyle \frac{2}{\overline{\rho }}}\left({\displaystyle \frac{\mu }{\sigma }}+{\displaystyle \frac{{\mu }_t}{{\sigma }_t}}\right)}{\displaystyle \frac{\partial {\xi }_{\alpha ,h}^n}{\partial x_i}}\mathrm{d}{W}_i^n-{\displaystyle \frac{\overline{\rho }}{2\tau }}\left({\xi }_{\alpha ,h}^n-{\tilde{\varphi }}_{\alpha ,h}\right)\mathrm{dt}+\overline{\rho }{\dot{\omega }}_{\alpha ,h}\left({\xi }^n\right)\mathrm{dt}\hfill \end{array}$$

(10)

where ${\xi }^n={[{\xi }_1^n,{\xi }_2^n,\cdots ,{\xi }_{N_s}^n,{\xi }_h^n]}^T$, ${\tilde{\varphi }}_{\alpha ,h}$ stands for ${\tilde{Y}}_{\alpha }$ or $\tilde{h}$, and $\mathrm{d}{W}_i^n={\eta }_i\sqrt{dt}$ represents the Wienner process with a dichotomous random vector ${\eta }_i=\{-1,1\}$ different for each spatial direction (i) and each stochastic field (n). As $N\to \infty$, Equation (10) becomes equivalent to Equation (9) in terms of one-point statistics [76]. This implies that the filtered scalars can be calculated by averaging over the stochastic fields as follows:

$${\tilde{Y}}_{\alpha }={\displaystyle \frac{1}{N}}\sum _{n=1}^N{\xi }_{\alpha }^n,\tilde{h}={\displaystyle \frac{1}{N}}\sum _{n=1}^N{\xi }_h^n.$$

(11)

Recently, Valiño et al. [69] reformulated Equation (10) by eliminating the molecular viscosity from the stochastic terms and the micromixing time scale ($\tau$). This correction removes an undesired spurious stochastic effect in laminar flow regions where ${\mu }_t\to 0$ and gradients of ${\xi }_{\alpha ,h}$ occur. In this work, we use this revised approach.

3. Numerical Methods

3.1. Solution Method of the Flamelet Equations

The ignition analysis in mixture fraction space relies on solving the flamelet Equation (1). For this purpose, the discretization in mixture fraction space is based on the 2nd-order central discretization method to approximate the second derivative. The chemical source terms are computed using the CHEMKIN interpreter [77], and the time integration is carried out using the Variable-coefficient Ordinary Differential equation solver with the Preconditioned Krylov method (VODPK) [78]. We note that the applied 2nd-order discretization method and the use of the CHEMKIN interpreter and VODPK solver is a standard approach in the context of the solution of the flamelet model in 1D, see for instance [64,79].

3.2. 3D LES/DNS Solver

We perform 3D LES using an in-house SAILOR code [80,81], in which the filtered Equations (4)–(7) are solved on a Cartesian half-staggered grid. The reference DNS are also computed using this code. All terms except the convective terms in the species and enthalpy transport equations are discretized in space by the 6th-order compact difference method [82,83]. The abovementioned convective terms are discretized using the 5th-order Weighted Essentially Non-Oscilatory (WENO) scheme [84]. The hydrodynamic pressure is computed from the projection method [85]. The time integration is performed using an operator splitting strategy, in which the convective/diffusive terms are solved separately from the chemical terms. The former are integrated in time by applying a predictor–corrector approach based on the 2nd-order Adams–Bashforth and Adams–Moulton methods. After obtaining intermediate solutions for the species and enthalpy, the chemical source terms are integrated in the same way as during the solution of the flamelet equations, i.e., using the CHEMKIN interpreter [77] and VODPK procedure [78]. The SAILOR code has been thoroughly verified in previous studies devoted to combustion problems [22,39,41,42,47,86,87] and always characterized an excellent accuracy.

4. Results

4.1. Auto-Ignition in Mixture Fraction Space

The analysis of the ignition process in mixture fraction space aims to determine the most reactive mixture fraction (${\xi }_{\mathrm{MR}}$) [43,46] and ignition delay time ($t_{\mathrm{ign}}$), including its dependence on the scalar dissipation rate (${\chi }_0$). The fuel stream defined for $\xi =1$ is assumed to be pure hydrogen ($Y_{{\mathrm{H}}_2}=1$) at the temperature $T_{\mathrm{F}}=300$ K. The oxidizer stream is a mixture of oxygen and water vapor with $Y_{{\mathrm{O}}_2}=0.1-0.9$, $Y_{{\mathrm{H}}_2\mathrm{O}}=1-Y_{{\mathrm{O}}_2}$ at $T_{\mathrm{O}}=$ 900–1500 K. Additionally, for comparison purposes, ${\xi }_{\mathrm{MR}}$ and $t_{\mathrm{ign}}$ are determined for the cases with an oxidizer composed of oxygen and nitrogen with $Y_{{\mathrm{O}}_2}=0.1-0.9$, $Y_{{\mathrm{N}}_2}=1-Y_{{\mathrm{O}}_2}$.

Table 1. The stoichiometric mixture fraction (${\xi }_{\mathrm{ST}}\times {10}^2$) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{N}}_2$ and ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures. The symbol $Y_{\alpha }$ denotes the mass fraction of ${\mathrm{N}}_2$ and ${\mathrm{H}}_2\mathrm{O}$ in the mixture.

${\mathbf{Y}}_{\mathit{\alpha }}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
	10.2	9.16	8.11	7.03	5.93	4.80	3.64	2.85	2.46	1.24

presents the values of the stoichiometric mixture fraction (${\xi }_{\mathrm{ST}}$) for all analyzed cases. Moreover, this table defines the contents of $Y_{{\mathrm{N}}_2}$ and $Y_{{\mathrm{H}}_2\mathrm{O}}$ considered in this work in ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{N}}_2$ and ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures, which we will later refer to as Case${\mathrm{s}}_{{\mathrm{N}}_2}$ and Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$.

The most reactive mixture fraction (${\xi }_{\mathrm{MR}}$) is calculated assuming ${\chi }_0=0$. The procedure of determining ${\chi }_{\mathrm{MR}}$ is adapted from [64,79]. It relies on finding $\xi$ for which either the temperature rises 1% above the initial value or the mass fraction of $\mathrm{OH}$ exceeds $2\times {10}^{-4}$. In [64,79,88], it was emphasized that these criteria are almost equivalent.

Three computational meshes were used. A basic one (M1) consisted of $2\times {10}^3$ uniform cells with $\Delta \xi =5\times {10}^{-4}$. Test computations performed on this mesh showed that ${\xi }_{\mathrm{MR}}$ for all analyzed cases are localized on a very lean side in $\xi <0.012$ and depend slightly on the temperature. Therefore, a hybrid mesh (M2) was created to increase the accuracy of the ${\xi }_{\mathrm{MR}}$ prediction. It consisted of 1200 uniform cells in $\xi \le 0.012$ and 800 non-uniform cells in between $0.012<\xi \le 1$, of which the sizes changed exponentially such that $\Delta \xi =7\times {10}^{-3}$ at $\xi =0.1$. Additionally, to evaluate solutions’ dependence on the mesh density, test computations were performed on a uniform mesh M3 with $5\times {10}^3$ cells with $\Delta \xi =2\times {10}^{-4}$. The time steps on the mesh M1 and refined meshes M2 and M3 were set to $\Delta t=5\times {10}^{-7}$ s and $\Delta t=1\times {10}^{-7}$ s. In all cases, the simulations were performed up to $t=2\times {10}^{-2}$ s. Initial conditions were taken as inert mixing lines with linear distributions of the species mass fractions and enthalpy. The initial profiles of $Y_{{\mathrm{H}}_2}$, $Y_{{\mathrm{O}}_2}$, $Y_{{\mathrm{N}}_2}$, $Y_{{\mathrm{H}}_2\mathrm{O}}$, enthalpy, and temperature for the cases with $Y_{{\mathrm{N}}_2}{|}_{\xi =0}=0.767$, $Y_{{\mathrm{H}}_2\mathrm{O}}{|}_{\xi =0}=0.767$ at $T_{\mathrm{O}}=1200$ K and $Y_{{\mathrm{N}}_2}{|}_{\xi =0}=0.5$, $Y_{{\mathrm{H}}_2\mathrm{O}}{|}_{\xi =0}=0.5$ at $T_{\mathrm{O}}=1200,1500$ K are shown in Figure 1. Comparing the enthalpy profiles, one can see that it changes oppositely along $\xi$ depending on whether nitrogen or water vapor are present in the oxidizer stream. Moreover, the absolute differences $|h_{\mathrm{O}}-h_{\mathrm{F}}|$ are significantly larger in the latter case. The temperature changes non-linearly due to a non-linear dependence of $C_{p,\alpha }\left(T\right)$ and is larger when the water vapor is present in the oxidizer.

Figure 2 and Figure 3 show $t_{\mathrm{ign}}\left(\xi \right)$ in Case${\mathrm{s}}_{{\mathrm{N}}_2}$ and Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$, calculated by applying the chemical kinetic mechanism of Mueller et al. [59]. The minima on the presented profiles correspond to $t_{\mathrm{ign}}$ at ${\xi }_{\mathrm{MR}}$ and its detailed values are given in Appendix B in

Table A4. Ignition delay time ($t_{\mathrm{ign}}$ [ms]) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{N}}_2$ mixtures at temperature $T_{\mathrm{O}}=$ 900–1500 K. The results labeled (D) were obtained on the dense mesh. The values separated by ‘/’ represent $t_{\mathrm{ign}}$ determined based on the temperature and OH mass fraction. A minus sign indicates the lack of ignition.

	${\mathit{Y}}_{{\mathbf{N}}_2}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
900		-	-	-	-	-	-	-	-	-	-
1000		0.330/0.339	0.345/0.343	0.358/0.356	0.374/0.371	0.396/0.392	0.427/0.422	0.475/0.471	0.527/0.523	0.564/0.561	0.808/0.812
1100		0.079/0.070	0.084/0.074	0.089/0.079	0.096/0.086	0.105/0.094	0.117/0.106	0.136/0.124	0.158/0.144	0.172/0.157	0.266/0.248
1200		0.037/0.031	0.039/0.033	0.043/0.035	0.048/0.047	0.052/0.043	0.058/0.048	0.069/0.058	0.078/0.069	0.088/0.074	0.138/0.190
1300		0.023/0.017	0.025/0.018	0.026/0.020	0.029/0.022	0.032/0.024	0.036/0.027	0.042/0.032	0.049/0.038	0.054/0.042	0.085/0.068
1400		0.015/0.010	0.017/0.012	0.018/0.012	0.020/0.014	0.022/0.015	0.025/0.017	0.029/0.020	0.034/0.024	0.037/0.026	0.059/0.042
1500		0.011/0.007	0.012/0.006	0.013/0.008	0.015/0.009	0.016/0.010	0.018/0.012	0.022/0.014	0.025/0.016	0.028/0.018	0.045/0.028

and

Table A7. Ignition delay time ($t_{\mathrm{ign}}$ [ms]) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures. The meaning of symbols and numbers are as in Table A4.

	${\mathit{Y}}_{{\mathbf{H}}_2\mathbf{O}}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
900		-	-	-	-	-	-	-	-	-	-
1000		-	-	-	-	-	-	-	-	-	-
1100		0.179/0.174	0.611/0.656	2.179/2.592	4.974/5.731	7.261/8.034	9.041/9.836	10.59/11.42	11.63/12.42	12.16/12.93	14.12/14.68
1200		0.049/0.043	0.068/0.059	0.092/0.083	0.123/0.130	0.163/0.152	0.216/0.205	0.288/0.276	0.353/0.341	0.394/0.380	0.606/0.585
1300		0.026/0.020	0.030/0.024	0.035/0.028	0.041/0.034	0.048/0.040	0.057/0.048	0.071/0.059	0.084/0.071	0.094/0.079	0.153/0.131
1400		0.016/0.012	0.018/0.013	0.020/0.015	0.023/0.017	0.026/0.019	0.030/0.023	0.037/0.028	0.044/0.034	0.049/0.038	0.082/0.063
1500		0.012/0.008	0.012/0.008	0.014/0.009	0.015/0.010	0.017/0.012	0.020/0.014	0.024/0.017	0.029/0.020	0.032/0.022	0.055/0.038

. Additionally, the red circles denote the locations of ${\xi }_{\mathrm{MR}}$, calculated based on $\mathrm{OH}$ mass fraction criterion (see also

Table A3. The most reactive mixture fraction (${\xi }_{\mathrm{MR}}\times {10}^3$) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{N}}_2$ mixtures at $T_{\mathrm{O}}=$ 900–1500 K. The results denoted with a label (D) are obtained on the dense mesh. The values separated by ‘/’ denote ${\xi }_{\mathrm{MR}}$ determined based on the temperature and OH mass fraction. The minus sign denotes the lack of ignition.

	${\mathit{Y}}_{{\mathbf{N}}_2}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
900		-	-	-	-	-	-	-	-	-	-
1000		3.3/3.4	3.3/3.4	3.3/3.4	3.3/3.3	3.3/3.3	3.2/3.3	3.2/3.2	3.1/3.1	3.0/3.0	2.7/2.7
1100		5.4/5.3	5.6/5.1	5.4/5.0	5.3/5.0	5.3/4.9	5.0/4.8	5.0/4.6	4.6/4.4	4.6/4.2	4.0/3.7
1100(D)		5.6/5.4	5.6/5.2	5.4/5.0	5.4/5.2	5.2/5.0	5.2/5.0	5.0/4.8	4.8/4.4	4.6/4.4	4.0/3.8
1200		6.8/6.2	7.2/6.5	6.7/6.0	6.7/5.9	6.5/5.8	6.4/5.6	5.9/5.5	5.7/5.2	5.6/5.1	4.8/4.2
1300		8.0/7.3	7.7/6.7	7.8/6.6	7.4/7.0	7.8/6.5	7.1/6.5	6.9/6.1	6.7/5.7	6.2/5.4	5.4/4.7
1300(D)		8.0/7.2	8.6/6.8	7.8/6.6	8.2/7.0	7.8/6.6	7.2/6.4	7.0/6.0	6.6/5.8	6.6/5.8	5.6/4.6
1400		8.7/7.6	8.8/7.7	8.9/7.3	8.6/7.2	8.4/6.9	8.1/6.6	7.7/6.6	7.1/6.3	7.2/5.9	6.0/4.8
1500		9.9/8.1	9.6/7.7	9.6/7.4	9.5/7.7	9.1/7.3	8.6/7.0	8.1/6.4	8.0/6.3	8.0/6.6	6.6/5.1

and

Table A5. The most reactive mixture fraction (${\xi }_{\mathrm{MR}}\times {10}^3$) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures. The meaning of symbols and numbers are as in Table A3.

	${\mathit{Y}}_{{\mathbf{H}}_2\mathbf{O}}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
900		-	-	-	-	-	-	-	-	-	-
1000		-	-	-	-	-	-	-	-	-	-
1100		4.2/4.2	2.9/3.3	1.8/2.3	1.7/2.7	2.2/3.3	2.6/3.8	3.0/4.3	3.2/4.6	3.4/4.8	3.9/5.2
1100(D)		4.2/4.2	3.0/3.2	1.8/2.4	1.8/2.8	2.2/3.4	2.6/3.8	3.0/4.4	3.2/4.6	3.4/4.8	3.8/5.2
1200		6.4/6.2	6.2/5.7	7.8/5.5	5.3/5.1	4.9/4.8	4.2/4.3	4.5/4.5	4.0/4.1	3.9/4.0	3.6/3.7
1300		7.7/7.0	7.9/6.9	7.9/7.1	7.7/7.1	7.7/7.0	7.4/6.8	6.5/7.0	7.3/6.5	7.1/6.4	6.5/5.8
1300(D)		7.8/6.9	7.9/6.9	7.9/7.2	7.8/7.2	7.8/6.9	7.9/6.9	7.8/6.8	7.6/6.8	7.4/6.6	6.8/5.9
1400		8.6/7.7	9.1/7.7	9.7/7.9	9.1/8.1	9.3/8.6	9.3/8.0	8.0/7.0	9.4/7.6	9.4/7.6	8.7/6.9
1500		10/8.1	9.9/7.9	10/8.9	11/8.6	11/8.7	11/8.4	10/6.5	11/8.2	11/8.3	10/7.8

in Appendix B). First, it should be noted that both criteria of determining ${\xi }_{\mathrm{MR}}$ give very similar values. The impact of the chemical kinetic mechanism is analyzed for Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ with $T_{\mathrm{O}}=1100$ K and $T_{\mathrm{O}}=1300$ K. Figure 3a,c show $t_{\mathrm{ign}}\left(\xi \right)$ for the cases with $Y_{{\mathrm{H}}_2\mathrm{O}}=0.1$ and $Y_{{\mathrm{H}}_2\mathrm{O}}=0.9$ (blue lines), calculated by applying the chemical kinetic mechanism of Li et al. [62]. It can be observed that compared to the mechanism of Mueller et al. [59], $t_{\mathrm{ign}}\left(\xi \right)$ is slightly shorter, particularly for $T_{\mathrm{O}}=1100$ K and $\xi >0.005$. A shorter ignition delay time obtained using the mechanism of Li et al. [62] was also reported in [79] in case of auto-ignition of nitrogen-diluted hydrogen in hot air. For $T_{\mathrm{O}}=1300$ K, differences are smaller and the predicted ${\xi }_{\mathrm{MR}}$ are very similar. Their detailed values are given in

Table A6. The most reactive mixture fraction (${\xi }_{\mathrm{MR}}\times {10}^3$) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures calculated by applying the chemical mechanism of Li et al. [62]. The meaning of symbols and numbers are as in Table A3.

	${\mathit{Y}}_{{\mathbf{H}}_2\mathbf{O}}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
1100		4.6/4.5	3.3/3.6	2.2/2.7	1.6/2.5	1.9/3.0	2.3/3.5	2.6/4.0	2.8/4.3	3.0/4.4	3.4/4.9
1100(D)		4.6/4.6	3.4/3.6	2.2/2.6	1.6/2.4	2.0/3.0	2.2/3.6	2.6/4.0	2.8/4.2	3.0/4.4	3.4/4.8
1300		7.9/7.6	7.9/7.2	8.1/7.0	8.1/7.1	8.4/7.5	8.1/7.2	8.0/7.2	7.7/6.7	7.5/6.8	7.1/5.9
1300(D)		8.6/7.6	8.6/7.2	8.2/7.6	8.2/7.2	8.2/7.4	7.9/7.2	7.8/6.9	7.6/6.9	7.8/6.6	6.9/6.2

. There, it can be seen that ${\xi }_{\mathrm{MR}}$ are minimally larger than those calculated using the mechanism of Mueller et al. [59] (cf. Table A5). This trend is consistent with the solutions presented in [79]. Hence, taking into account a relatively small impact of the chemical kinetic scheme on $t_{\mathrm{ign}}\left(\xi \right)$ and ${\xi }_{\mathrm{MR}}$, the following analysis is limited to the discussion of the results obtained using the mechanism of Mueller et al. [59]. This also applies to the 3D LES presented in Section 4.2.

Regarding the impact of the oxidizer temperature, it can be easily seen that its increase causes faster ignition and extends the range of $\xi$ for which it occurs in the assumed simulation time. Comparing Case${\mathrm{s}}_{{\mathrm{N}}_2}$ with Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$, one can notice that in the latter case, the impact of $Y_{{\mathrm{H}}_2\mathrm{O}}$ is larger than $Y_{{\mathrm{N}}_2}$. At low oxidizer temperature ($T_{\mathrm{O}}=1100$ K), ignition in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ takes place only in a narrow range ($\xi <0.017$) and occurs much later than in Case${\mathrm{s}}_{{\mathrm{N}}_2}$. The situation changes when the oxidizer temperature is high. For example, at $T_{\mathrm{O}}=1400$ K in Case${\mathrm{s}}_{{\mathrm{N}}_2}$, ignition is limited to $\xi \approx 0.057$, while in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$, it occurs up to $\xi \approx 0.087$ and proceeds faster. The reasons for this are the reactions in which ${\mathrm{H}}_2\mathrm{O}$ reacts with oxygen and hydrogen radicals and the dissociation of water into highly reactive hydroxyl radicals ($\mathrm{OH}$) [6,7] (see Table A1). In both configurations, in lean mixtures (small $\xi$), ignition occurs faster at higher oxygen content (smaller $Y_{{\mathrm{N}}_2}$, $Y_{{\mathrm{H}}_2\mathrm{O}}$). This tendency changes for larger $\xi$ and, e.g., for $T_{\mathrm{O}}=1400$ K (Figure 2d and Figure 3d) around $\xi =0.05$, ignition occurs later when the oxygen content is higher. The differences are particularly visible in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$. It should be noted that the location of ${\xi }_{\mathrm{MR}}$ for high oxidizer temperature ($T_{\mathrm{O}}\ge 1300$ K) and for $Y_{{\mathrm{N}}_2}\ge 0.5$, $Y_{{\mathrm{H}}_2\mathrm{O}}\ge 0.5$ is almost constant. For $T_{\mathrm{O}}<1300$ K, the general trend is that it tends towards lower $\xi$ values as the oxygen content decreases. The exception to this tendency are Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ for $T_{\mathrm{O}}=1100$ K. Here, the minimum value of ${\xi }_{\mathrm{MR}}$ occurs at an intermediate content of $Y_{{\mathrm{O}}_2}/Y_{{\mathrm{H}}_2\mathrm{O}}$.

The ignition for the cases with ${\chi }_0>0$ starts at ${\xi }_{\mathrm{MR}}$ and spreads along mixture fraction space due to the diffusion process. This scenario is similar in Case${\mathrm{s}}_{{\mathrm{N}}_2}$ and Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$. Figure 4 shows the temporal evolution of the ${\mathrm{HO}}_2$ and $\mathrm{OH}$ species mass fractions and temperature in mixture fraction space in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ for $Y_{{\mathrm{H}}_2\mathrm{O}}=0.5$ at $T_{\mathrm{O}}=1200$ K and with ${\chi }_0=100$ ${\mathrm{s}}^{-1}$. Initially, a peak of $Y_{{\mathrm{HO}}_2}$ appears on the lean side. Then, it moves towards the rich side, reaches a maximum value in the intermediate $\xi$ range, and then decreases smoothly. As time passes, the $Y_{{\mathrm{HO}}_2}$ level becomes very low and its maximum is again located near $\xi =0$. The evolution of $Y_{\mathrm{OH}}$ is different. Initially, $\mathrm{OH}$ also appears on the lean side; however, later, its peak moves only in a narrow range of $\xi$ and finally reaches a maximum near ${\xi }_{\mathrm{ST}}$. Along with the evolution of $Y_{{\mathrm{HO}}_2}$ and $Y_{\mathrm{OH}}$, the temperature changes and increases successively along mixture fraction space, starting from ${\xi }_{\mathrm{MR}}$. It is worth noting that the maximum temperature ($T_{\mathrm{MAX}}$), occurring at $\xi$ slightly larger than ${\xi }_{\mathrm{ST}}$, stabilizes relatively quickly compared to the temperature stabilization on the rich side. The variability of $T_{\mathrm{MAX}}$ in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ for all considered $Y_{{\mathrm{H}}_2\mathrm{O}}$ is presented in Figure 5 for $T_{\mathrm{O}}=1200$ K with ${\chi }_0=10$ ${\mathrm{s}}^{-1}$ and ${\chi }_0=1000$ ${\mathrm{s}}^{-1}$. It can be seen that the ignition occurs later for ${\chi }_0=1000$ ${\mathrm{s}}^{-1}$, particularly when $Y_{{\mathrm{H}}_2\mathrm{O}}$ increases. Figure 6a shows the ignition delay time for the cases with ${\chi }_0=1,10,100,1000$ ${\mathrm{s}}^{-1}$ and $T_{\mathrm{O}}=1100-1500$ K, measured as the time moment when the temperature at arbitrary value of $\xi$ reaches a level 1% higher than $T_{\mathrm{O}}$. In the range of ${\chi }_0\le 100$ ${\mathrm{s}}^{-1}$, the differences in ignition time are small, whereas when ${\chi }_0=1000$ ${\mathrm{s}}^{-1}$, the ignition time increases significantly. In the $log-\mathrm{lin}$ plot, it can be relatively well approximated as $t_{\mathrm{ign}}\approx a·b^{Y_{{\mathrm{H}}_2\mathrm{O}}}$, where $a,b$ are empirically adjusted constants. Sample approximation lines for the cases with ${\chi }_0\le 100$ ${\mathrm{s}}^{-1}$ ($a=0.0000385$, $b=22$) and ${\chi }_0\le 1000$ ${\mathrm{s}}^{-1}$ ($a=0.0000415$, $b=46$) are shown in red color. Dependence of the ignition time on $T_{\mathrm{O}}$ is shown in Figure 6b. It should be noted that for large values of ${\chi }_0$ and $Y_{{\mathrm{H}}_2\mathrm{O}}$, the ignition occurs only for a high oxidizer temperature. For ${\chi }_0\le 100$ ${\mathrm{s}}^{-1}$ and $T_{\mathrm{O}}=1100$ K, it takes place when $Y_{{\mathrm{H}}_2\mathrm{O}}\le 0.6$. For ${\chi }_0=1000$ ${\mathrm{s}}^{-1}$, the ignitability for $T_{\mathrm{O}}=1100$ K is limited to $Y_{{\mathrm{H}}_2\mathrm{O}}\le 0.2$, see Table A6. In Figure 6b, it can be seen that $t_{\mathrm{ign}}$ for $T_{\mathrm{O}}=1100$ K rises along with the increasing $Y_{{\mathrm{H}}_2\mathrm{O}}$, significantly faster than for $T_{\mathrm{O}}\ge 1200$ K. In this range of oxidizer temperatures, for $Y_{{\mathrm{H}}_2\mathrm{O}}\le 0.8$, $t_{\mathrm{ign}}$ can be well approximated by the formula analogical to the one proposed above.

After the ignition, the temperature increases rapidly and attains a constant level, as shown in Figure 5. The temperature rise is steeper for a larger ${\chi }_0$. This is attributed to a stronger diffusion, which causes a faster shift of the temperature peak from ${\xi }_{\mathrm{MR}}$ towards ${\xi }_{\mathrm{ST}}$. On the other hand, a larger ${\chi }_0$ causes $T_{\mathrm{MAX}}$ to be lower; yet, its dependence on ${\chi }_0$ is small compared to its variability in the function of $Y_{{\mathrm{H}}_2\mathrm{O}}$. Figure 7 shows $T_{\mathrm{MAX}}$ for ${\chi }_0=10$ ${\mathrm{s}}^{-1}$ and ${\chi }_0=1000$ ${\mathrm{s}}^{-1}$ at $T_{\mathrm{O}}=$ 1100–1500 K. Additionally, the results obtained for Case${\mathrm{s}}_{{\mathrm{N}}_2}$ with $T_{\mathrm{O}}=1100$ K and $T_{\mathrm{O}}=1500$ K are included in this figure. It can be seen that, in these cases, the temperature is significantly higher than when the water is present in the oxidizer stream, particularly for ${\chi }_0=10$ ${\mathrm{s}}^{-1}$. The observed non-linear decrease in $T_{\mathrm{MAX}}$ was reported by Wawrzak and Tyliszczak [23] in case of oxy-combustion of a nitrogen-diluted hydrogen. Here, it can be seen that changing $Y_{{\mathrm{H}}_2\mathrm{O}}$ from $0.1$ to $0.9$ causes the reduction in $T_{\mathrm{MAX}}$ from the range 2960–3080 K to 1440–1920 K. It should be also noted that for large $Y_{{\mathrm{H}}_2\mathrm{O}}$, the values of $T_{\mathrm{MAX}}$ are much more dependent on $T_{\mathrm{O}}$ than when $Y_{{\mathrm{H}}_2\mathrm{O}}$ is low. This is because at large $Y_{{\mathrm{H}}_2\mathrm{O}}$, the stoichiometric mixture fraction is closer to the boundary conditions at $\xi =0$ (see Table 1) and is, thus, more dependent on $T_{\mathrm{O}}$.

4.2. Auto-Ignition in a 3D Forced HIT

The results presented in the previous section demonstrated that the auto-ignition is strongly influenced by the oxidizer temperature, the content of nitrogen or water vapor in the oxidizer stream, and the scalar dissipation rate. The ignition time and the range of ignitability varied significantly depending on the presence of nitrogen or water vapor in the fuel/oxidizer mixture. In this section, we examine how these findings apply to auto-ignition in a turbulent flow. For this purpose, we define a forced HIT flow configuration, which represents an idealized turbulent flow where turbulence is statistically uniform and directionally independent, and the total kinetic energy does not vanish in time. This state is maintained by continuously injecting energy to counterbalance the dissipation of energy by viscous forces at small scales. Unlike classical decaying HIT, forced HIT more accurately represents a real scenario where fuel and oxidizer streams are continuously supplied to the combustion zone, thereby providing energy to the flow. A common approach to force HIT involves adding a forcing term into the Navier–Stokes equations, which ‘injects’ the energy within a specific wavenumber shell $k_L\le \left|\mathbf{k}\right|\le k_U$ or sphere $\left|\mathbf{k}\right|\le k_f$ [89], where k denotes the wavenumber and the subscripts ${·}_L,{·}_U$ stand for the lower and upper limit of the forcing range. These shells are most often confined to the lowest wavenumbers, ensuring that the force impacts the large-scale structures of the flow. Consequently, the injected energy cascades down to smaller and smaller scales, where it is ultimately dissipated by viscous effects. When the Navier–Stokes equations are solved in Fourier space, the forcing is expressed as $\widehat{\mathbf{f}}(\mathbf{k},t)=A\widehat{\mathbf{u}}(\mathbf{k},t)$, where the parameter A is adjusted to maintain a constant energy injection rate [90,91,92,93]. In contrast to the low wavenumber forcing method, Lundgren [94] proposed a linear forcing method in which forcing proportional to velocity is applied over the entire wavenumber space. Rosales and Meneveau [89] extended this approach to physical space, making it convenient to use in numerical codes formulated based on finite difference or finite volume discretization schemes. The forcing term in the Navier–Stokes Equation (5) in the linear forcing method is defined as [89]

$$f_i=A{\tilde{u}}_i\mathrm{for}i=1,2,3$$

(12)

where the A parameter has the dimension of ${\mathrm{s}}^{-1}$ and can be taken constant, $A=\theta$ ($\theta$—the strength of the forcing), dependent on a time-varying velocity fluctuation level; $A=\theta /3u_{rms}^2$ ($u_{rms}^2=\langle u_j{u}_j\rangle /3$), or related to an initial energy dissipation level, $A=\theta \epsilon /3u_{rms}^2$, where $\epsilon =-\nu \langle u_i{\displaystyle \frac{{\partial }^2{u}_i}{\partial x_j^2}}\rangle$ is the mean energy dissipation rate per unit mass. In this study, we follow the first option and assume a constant value of A throughout the simulation, which is equivalent to imposing a given eddy turnover time [89].

4.2.1. HIT Initialization

The computational domain is a cubic box (0.01³ m³) with periodic boundary conditions. Two computational meshes are used: a dense one consists of 256³ uniformly spaced nodes and a coarse mesh of 64³ nodes. The simulation procedure is split into two parts. In the first one, the non-reactive flow is modeled starting from random disturbances of the velocity field on the dense mesh. The results of these simulations are then used to initialize the fuel/oxidizer distribution for the reactive flow simulations, as will be explained momentarily. The applied forcing causes the flow to evolve in such a way that, after some time, turbulent spatiotemporally correlated vortical structures develop. Figure 8 shows the time evolution of the total kinetic energy (E) as a function of forcing parameter A equal to $A=100,300,500$ ${\mathrm{s}}^{-1}$. Initially, in all cases, the kinetic energy of the flow rapidly decreases to a very low level, then begins to increase at a rate dependent on A. After this transient phase, it stabilizes at nearly constant levels determined by the forcing strength A. The larger the value of A, the higher and more fluctuating the kinetic energy becomes. The dashed lines indicate time moments $t_1=0.15$ s, $t_2=0.225$ s, and $t_3=0.3$ s, for which the initial solutions for reactive flow simulations are generated.

Table 2. Characteristic parameters of the initial flow fields.

	$\mathit{A}=100$ ${\mathbf{s}}^{-1}$				$\mathit{A}=300$ ${\mathbf{s}}^{-1}$				$\mathit{A}=500$ ${\mathbf{s}}^{-1}$
Time	${\mathit{t}}_{\mathbf{1}}$	${\mathit{t}}_{\mathbf{2}}$	${\mathit{t}}_{\mathbf{3}}$	$\langle [{\mathit{t}}_{\mathbf{1}},{\mathit{t}}_{\mathbf{3}}]\rangle$	${\mathit{t}}_{\mathbf{1}}$	${\mathit{t}}_{\mathbf{2}}$	${\mathit{t}}_{\mathbf{3}}$	$\langle [{\mathit{t}}_{\mathbf{1}},{\mathit{t}}_{\mathbf{3}}]\rangle$	${\mathit{t}}_{\mathbf{1}}$	${\mathit{t}}_{\mathbf{2}}$	${\mathit{t}}_{\mathbf{3}}$	$\langle [{\mathit{t}}_{\mathbf{1}},{\mathit{t}}_{\mathbf{3}}]\rangle$
$R{e}_{\lambda }$	57	50	62	52	90	110	94	91	111	121	115	120
$u_{rms}$	0.95	0.87	0.99	0.91	2.65	3.06	2.92	2.76	4.50	5.31	4.38	4.75
$h/\lambda$	0.042	0.044	0.040	0.044	0.074	0.073	0.077	0.076	0.101	0.092	0.097	0.098
$h/\eta$	0.62	0.61	0.63	0.062	1.37	1.52	1.47	1.42	2.09	2.01	2.03	2.12
${\tau }_t\times {10}^{-4}$	9.87	10.16	9.69	9.79	1.99	1.74	1.73	1.86	0.86	0.79	0.92	0.83

presents the characteristic parameters of these flow fields, including $R{e}_{\lambda }$, $u_{rms}$, $h/\lambda$, $h/\eta$ (h—computational mesh size, $\eta$—the Kolmogorov length scale), and the eddy turnover timescale ${\tau }_t=\lambda /u_{rms}$ for all analyzed cases. Columns marked $\langle [t_1,t_3]\rangle$ correspond to time-averaged values in the interval $[t_1,t_3]$.

Figure 9a displays the evolution of the kinetic energy spectrum for the case with $A$ = 100 ${\mathrm{s}}^{-1}$. The blue line corresponds to the initial conditions with most of the energy concentrated at small scales (high wavenumbers). The fact that they are random and uncorrelated causes their fast dissipation. This behavior is typical for randomly generated disturbances assumed as initial velocity field or at the inlet boundary of spatially developed flows, e.g., jet flows [95]. The decay of energy at the smallest scales is accompanied by an increase in energy at large scales (small wavenumbers), as indicated by the gray lines and arrows marked (1). After $t\approx 0.03$ s, the energy at higher wavenumbers also increases, as shown by dashed lines and arrows (2). This is due to the cascade mechanism of energy transfer from large to small scales. Finally, after $t\approx 0.08$ s, the energy spectrum oscillates around the red line over time, as indicated by arrows (3).

This red line represents the time-averaged spectrum computed based on the spectra in subsequent time moments in the $[t_1,t_3]$ range. The evolution of the energy spectra for the cases with $A=300$ ${\mathrm{s}}^{-1}$ and $A=500$ ${\mathrm{s}}^{-1}$ is analogous to the one presented for $A=100$ ${\mathrm{s}}^{-1}$, but individual stages (1), (2), and (3) are reached earlier as A increases. The time-averaged energy spectra for $A=100$ ${\mathrm{s}}^{-1}$, $A=300$ ${\mathrm{s}}^{-1}$, and $A=500$ ${\mathrm{s}}^{-1}$ are shown in Figure 9b. It can be seen that the higher the A value, the wider the range of wavenumbers in which the energy spectrum stabilizes at the non-negligible level. In these cases, the flow field characterizes the presence of small turbulent scales. Figure 10 shows the contours of the instantaneous u-velocity component (in the x-direction) and vorticity modulus ($|\Omega |$) in the main ‘x-y’ cross-section of the computational domain. It can be seen that as A increases, the absolute values of u and $|\Omega |$ become larger and smaller vortices appear.

4.2.2. Species and Temperature Initialization

As shown in Table 2, the solutions at time instances $t_1$, $t_2$, and $t_3$ differ slightly for each value of A. In terms of $R{e}_{\lambda }$, these flow fields can be classified as weakly ($A=100$ ${\mathrm{s}}^{-1}$), moderately ($A=300$ ${\mathrm{s}}^{-1}$), and highly ($A=500$ ${\mathrm{s}}^{-1}$) turbulent. Performing computations starting from the initial conditions at $t_1$, $t_2$, and $t_3$ for each A adds credibility to the conclusions regarding the auto-ignition process at different $R{e}_{\lambda }$. Drawing conclusions based on a single test case for a particular A ($R{e}_{\lambda }$) would be questionable.

Regarding the mesh resolution, the simulations for the weakly turbulent flow ($A=100$ ${\mathrm{s}}^{-1}$) on the dense mesh can be treated as DNS because, in this case, $h/\eta <1$ (see Table 2), meaning that all flow scales are accurately resolved. Therefore, the solutions obtained for $A=100$ ${\mathrm{s}}^{-1}$ will be used to verify the LES-ESF model. The mixture composition and enthalpy distribution are initialized based on the vorticity modulus shown in Figure 10a. Its normalized value ${\Omega }_{\mathrm{norm}}=\left(\right|\Omega |/{|\Omega |}_{\mathrm{MAX}})\in [0,1]$ is used to define the mixture fraction distribution according to the formula

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}tanh\left[\delta \left({\displaystyle \frac{{\Omega }_{\mathrm{norm}}}{0.5}}-{\displaystyle \frac{0.5}{{\Omega }_{\mathrm{norm}}}}\right)\right]\end{array}$$

(13)

where $\delta$ determines the width of a mixing zone between fuel and oxidizer. Figure 11a shows the profile of $\xi \left({\Omega }_{\mathrm{norm}}\right)$. It can be seen that the oxidizer stream at $\xi =0$ is associated with low values of ${\Omega }_{\mathrm{norm}}$. This choice is motivated by the fact that in combustion devices, a high oxidizer temperature causes the rise in molecular viscosity, which makes the flow field ‘smoother’ and less turbulent; thus, it characterizes a lower vorticity level. Conversely, a region in which a cold fuel is present characterizes the occurrence of small flow scale and large velocity gradients (→ large vorticity). The contours of the initial mixture fraction distribution in the main ‘x-y’ cross-section (a semi-transparent blue cross-section in Figure 12a) in the case with $A=100$ ${\mathrm{s}}^{-1}$ at the time moment $t_1$ are presented in Figure 11b. It can be seen that in the region where $\xi \approx 1.0$ occurs only locally and intuitively, one can say that the mixture contained in the domain is lean, in the global sense. The values of equivalence ratios calculated based on a spatially averaged mixture fraction $\langle \xi \rangle$ ($\varphi ={\displaystyle \frac{\langle \xi \rangle }{1-\langle \xi \rangle }}{\displaystyle \frac{1-{\xi }_{\mathrm{ST}}}{{\xi }_{\mathrm{ST}}}}$) for $A=100,300,500$ ${\mathrm{s}}^{-1}$ are given in

Table 3. Spatially averaged equivalence ratio ($\varphi \times {10}^2$) for Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$. The symbol $Y_{\alpha }$ denotes the mass fraction of ${\mathrm{N}}_2$/${\mathrm{H}}_2\mathrm{O}$ in the oxidizer stream.

	${\mathit{Y}}_{\mathbf{\alpha }}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
A
100		6.61	7.44	8.51	9.93	11.9	14.8	19.8	25.5	29.7	59.8
300		0.88	0.99	1.14	1.33	1.59	1.99	2.66	3.43	3.99	8.02
500		0.34	0.39	0.44	0.52	0.62	0.78	1.05	1.35	1.57	3.16

. It can be observed that as the content of ${\mathrm{N}}_2$/${\mathrm{H}}_2\mathrm{O}$ in the oxidizer stream increases (and the ${\mathrm{O}}_2$ content decreases), the mixture becomes richer. The opposite trend is observed with increasing A. This is attributed to the fact that the vortices are smaller for larger A, and regions with high vorticity, where $\xi \approx 1$, occupy less space. Nevertheless, it is worth noting that the ratios of maximum to minimum $\varphi$ for each A are very similar: 9.04 ($A=100$ ${\mathrm{s}}^{-1}$), 9.11 ($A=300$ ${\mathrm{s}}^{-1}$), and 9.29 ($A=500$ ${\mathrm{s}}^{-1}$). It can be concluded that the proposed method of generating initial conditions relates the mixture composition with the flow field such that its ignitability decreases with increasing A due to a leaning of the mixture.

The initial enthalpy and species distributions are computed based on a linear dependence on $\xi$, similarly as in the analysis in the mixture fraction space. They are defined as

$$\begin{array}{cc}\hfill Y_{\alpha }=(1-\xi )& ·Y_{\alpha ,\mathrm{O}}+\xi ·Y_{\alpha ,\mathrm{F}}\hfill \\ \hfill h=(1-\xi )& ·h_{\mathrm{O}}+Z·h_{\mathrm{F}}\hfill \end{array}$$

(14)

where the subscripts ${·}_{\mathrm{O}}$ and ${·}_{\mathrm{F}}$ denote the oxidizer composition ($Y_{{\mathrm{O}}_2}$ = 0.1–0.9, $Y_{{\mathrm{N}}_2}=1-Y_{{\mathrm{O}}_2}$, or $Y_{{\mathrm{H}}_2\mathrm{O}}=1-Y_{{\mathrm{O}}_2}$, $h_{\mathrm{O}}\left(T_{\mathrm{O}}\right)$) and fuel ($Y_{{\mathrm{H}}_2}=1$, $h_{\mathrm{F}}\left(T_{\mathrm{F}}\right)$). The temperature at $\xi \in (0,1)$ is computed based on the local mixture composition and enthalpy, as described in Section 2.1.

4.2.3. Auto-Ignition Process in 3D Forced HIT

First, we concentrate on the case with $A=100$ ${\mathrm{s}}^{-1}$, for which the Kolmogorov length scale is correctly captured on the dense mesh, see Table 2. For this flow regime, we perform DNS for Case${\mathrm{s}}_{{\mathrm{N}}_2}$ for which the auto-ignition occurs relatively fast compared to Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ (Cf. Table A4 and Table A7). The obtained DNS results are used to validate the correctness of the LES-ESF approach, which is then used both for Case${\mathrm{s}}_{{\mathrm{N}}_2}$ and Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ with different $Y_{{\mathrm{N}}_2}$, $Y_{{\mathrm{H}}_2\mathrm{O}}$ contents and for $A=100$ ${\mathrm{s}}^{-1}$, $A=300$ ${\mathrm{s}}^{-1}$, and $A=500$ ${\mathrm{s}}^{-1}$. We note that the LES-ESF simulations start from initial solutions identical to those generated for DNS. The latter are projected from the dense mesh onto the coarse mesh. Figure 12 shows isosurfaces of the Q-parameter ($Q=15$ ${\mathrm{s}}^{-2}$) colored by temperature for the cases with $Y_{\mathrm{N}2}=0.5$ and $T_{\mathrm{O}}=1100$ K. The Q-parameter is commonly used to indicate organized vortical structures. It is defined as $Q=1/2({\Omega }_{ij}{\Omega }_{ij}-S_{ij}{S}_{ij})$, where $S_{ij}$ and ${\Omega }_{ij}$ are symmetrical and anti-symmetrical parts of the velocity gradient tensor. The computations started from the initial conditions at $t_1=0.15$ s (see Figure 8). The presented solutions aim to visualize the ignition process with reference to Figure 13, which shows the evolution of the maximum temperature values and the mass fractions of $\mathrm{OH}$ and ${\mathrm{HO}}_2$ species found in the computational domain. Please note that the time axis has been reset to zero, i.e., $t=0$ s in Figure 13 refers to $t_1=0.15$ s in Figure 8. Points marked on the temperature profile correspond to the following time instances: shortly before ignition (a); at the ignition moment, when the maximum temperature reaches 1% above $T_{\mathrm{O}}$, i.e., $T_{\mathrm{MAX}}=1111$ K (b); during the temperature rise (c–e); and when the flame extinguishes, causing the temperature to decrease (f). It should be noted that according to the analysis of the ignition in mixture fraction space (see Figure 4), the rise in $Y_{{\mathrm{HO}}_2}$ precedes the growth of $Y_{\mathrm{OH}}$ and temperature. An analysis of Figure 12a,b shows that the differences between the results are only quantitative before ignition. The flow evolves, and the vortical structures change in shape and position. A qualitative change in the solution occurs after ignition, as the rise in molecular viscosity at high temperatures reduces the dynamics of large-scale motion, allowing smaller structures to prevail.

Figure 13. Temperature and species mass fraction ($Y_{\mathrm{OH}}$, $Y_{{\mathrm{HO}}_2}$) profiles for the case $Y_{{\mathrm{N}}_2}=0.5$ and $T_{\mathrm{O}}=1100$ K with $A=100$ ${\mathrm{s}}^{-1}$.

Figure 13. Temperature and species mass fraction ($Y_{\mathrm{OH}}$, $Y_{{\mathrm{HO}}_2}$) profiles for the case $Y_{{\mathrm{N}}_2}=0.5$ and $T_{\mathrm{O}}=1100$ K with $A=100$ ${\mathrm{s}}^{-1}$.

Figure 14 presents the isosurface of the most reactive mixture fraction ${\xi }_{\mathrm{MR}}=5.3\times {10}^{-3}$ colored by the scalar dissipation rate and temperature at the time moment shortly after the ignition occurred (point (b) in Figure 13). To clearly show a distribution of small $\chi$ values where the ignition is expected to occur [43], the color range of $\chi$ is limited to $\chi \le 0.5$ ${\mathrm{s}}^{-1}$ and larger values are displayed in gray. The maximum value found in the domain equals ${\chi }_{\mathrm{MAX}}=154.5$ ${\mathrm{s}}^{-1}$. An analysis of the presented solution clearly shows a correlation between the regions of small $\chi$ and large temperature. The inset figures show enlarged views where $\chi \le 0.05$ ${\mathrm{s}}^{-1}$ and $T_{\mathrm{MAX}}\ge 1080$ K.

A detailed analysis of the ignition process and an early flame evolution are shown in Figure 15 by the contours of the ${\mathrm{HO}}_2$, $\mathrm{OH}$ species mass fractions and temperature at time moments (a–f) marked in Figure 13. It can be seen that the appearance of ignition kernels manifested by temperature growth is preceded by the increasing ${\mathrm{HO}}_2$ mass fraction at time instant (a). Although not clearly visible here, it is accompanied by an increase in the $\mathrm{OH}$ mass fraction just before ignition and, consequently, an increase in temperature. After reaching the maximum around point (e), the temperature begins to decrease and the flame extinguishes. After point (f), the heat release drops to zero and only hot combustion products mix with the oxidizer and residue of fuel.

Figure 16 shows the evolution of the maximum temperature in the DNS and LES-ESF model on four stochastic fields ($N=4$ in Equation (11)) for Case${\mathrm{s}}_{{\mathrm{N}}_2}$ with $Y_{{\mathrm{N}}_2}=0.5,0.767,0.9$ and $A=100$ ${\mathrm{s}}^{-1}$ for two oxidizer temperatures, $T_{\mathrm{O}}=1100$ K and $T_{\mathrm{O}}=1300$ K. It is worth noting that the ignition time is predicted very well by the LES-ESF approach.

The discrepancies between the DNS and ESF results occur only after the maximum temperature peak is reached and appear earlier for cases with larger and steeper temperature growth ($Y_{{\mathrm{N}}_2}=0.5,0.767$). They are due to an insufficient resolution of the flame front, which becomes thinner for these conditions. The flame thickness can be roughly estimated as ${\delta }_l=D/S_l=2\times {10}^{-5}$ m [58], where $D=2.512\times {10}^{-5}{\mathrm{m}}^2/\mathrm{s}$ is the heat diffusion coefficient for the stoichiometric mixture and the burning velocity of diluted hydrogen equals $S_l=1.3\mathrm{m}/\mathrm{s}$ [96]. Comparing ${\delta }_l$ with the mesh size $h=3.9\times {10}^{-5}$ m, it can be readily inferred that the flame front cannot be properly resolved on the assumed mesh and, at this stage, DNS fails. On the other hand, we assume that after the ignition occurs, the ESF combustion model shows the correct solution, as it has been designed for modeling ignition and extinction processes. As one can see, the ignition occurs later as $Y_{{\mathrm{N}}_2}$ in the oxidizer stream increases. Conversely, in agreement with the analysis in mixture fraction space, the larger temperature (Figure 16b) significantly speeds up the ignition. Note that in the case where $Y_{{\mathrm{N}}_2}=0.5$, the simulations performed using DNS are unstable due to insufficient resolution of the flame thickness. Hence, knowing that DNS can only be considered reliable up to the moment of ignition, the following investigations and analysis are limited to the solutions obtained using the LES-ESF method.

Figure 17 presents the LES-ESF results for Case${\mathrm{s}}_{{\mathrm{N}}_2}$ with $Y_{{\mathrm{N}}_2}=0.5,0.767,0.9$ obtained based on various initial solutions determined based on the cases initialized by the solutions at the time moments $t_1$, $t_2$, and $t_3$. Additionally, the results obtained using various forcing parameters A are included for the cases initialized at $t_1$ and $t_2$. It can be observed that the variability in the ignition time and the rate of an initial temperature growth in the cases differing by the initial field and A is small. This observation agrees well with the findings of Hilbert and Thèvenin [45], who showed that $R{e}_{\lambda }$ and randomness of the initial flow field has only minor impacts on the occurrence of ignition. The differences start to be seen after reaching the maximum temperature. First, its values for larger A ($A=300$ ${\mathrm{s}}^{-1}$, $A=500$ ${\mathrm{s}}^{-1}$) are lower, which is connected with higher values of $\chi$. This is due to the fact that for larger A the flow characterizes larger turbulence intensity (larger $R{e}_{\lambda }$, see Table 2) and, thus, larger vorticity gradients. This translates to larger gradients of the mixture fraction and an increase in $\chi$, which causes a decrease in the maximum temperature, as shown in the simulations in mixture fraction space (see Figure 5 and Figure 7). Additionally, for a leaner initial mixture, e.g., for $Y_{{\mathrm{N}}_2}=0.5$ (see Table 3) and $A=300$ ${\mathrm{s}}^{-1}$, $A=500$ ${\mathrm{s}}^{-1}$, the flame extinguishes after the maximum temperature is reached. This is a combined effect of an intense mixing at larger $R{e}_{\lambda }$ and a significant amount of oxygen causing fast burning of hydrogen in the domain. Regarding the impact of the initial conditions (solid, dashed, and dash-dotted lines) on the temperature evolution after ignition, it can be observed that the solutions differ quantitatively but remain consistent from a qualitative perspective. The existent quantitative differences can be attributed to the nature of turbulent flow.

A qualitative similarity of the results is found also when comparing Case${\mathrm{s}}_{{\mathrm{N}}_2}$ and Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$. The presence of whether in the oxidizer stream does not change the ignition mechanism. The analysis of the obtained LES-ESF results shows that it occurs at the most reactive mixture fraction in the regions of small values of the scalar dissipation rate. The ignition time depends very little on A and the initial conditions. This is presented in Figure 18 showing the evolution of the maximum temperature in the LES-ESF results obtained for Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ for $Y_{{\mathrm{H}}_2\mathrm{O}}=0.5,0.767,0.9$ with $T_{\mathrm{O}}=1300$ K and $A=100$ ${\mathrm{s}}^{-1}$ and $A=500$ ${\mathrm{s}}^{-1}$. These results were obtained starting from the initial solution at time $t_1$. Additionally, in this figure, the LES-ESF results obtained for Case${\mathrm{s}}_{{\mathrm{N}}_2}$ with the equivalent amount of ${\mathrm{N}}_2$ are included (black lines with solid symbols). Consistently with the analysis in mixture fraction space, the ignition time is longer when the water vapor is present in the oxidizer stream and the maximum achievable temperature is lower. For instance, for $Y_{{\mathrm{H}}_2\mathrm{O}}=Y_{{\mathrm{N}}_2}=0.5$ with $A=100$ ${\mathrm{s}}^{-1}$ (black lines with open/solid square symbols in Figure 18a) at the time moment when the maximum temperature stabilizes ($t=0.001$ s), $T_{\mathrm{MAX}}=2753$ K (the case with $Y_{{\mathrm{H}}_2\mathrm{O}}=0.5$) and $T_{\mathrm{MAX}}=2974$ K ($Y_{{\mathrm{N}}_2}=0.5$). A similar trend can be observed in the solutions in mixture fraction space presented in Figure 7. For comparison, the evolution of the temperature in mixture fraction space in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ is also included in Figure 18a) for the cases with $\chi =10$ ${\mathrm{s}}^{-1}$ and $\chi =100$ ${\mathrm{s}}^{-1}$ (red and blue lines with open symbols). Regarding the ignition time, one can observe that the turbulence delays it minimally compared to the solutions in mixture fractions space, regardless of the assumed value of $\chi$. Comparing the rate of the temperature growth in the initial ignition phase, it can be seen that the LES-ESF results are closer to those obtained in mixture fraction space for $\chi =100$ ${\mathrm{s}}^{-1}$. This would be expected given the results in 3D in the analyzed case ${\chi }_{\mathrm{MAX}}=154.5$ ${\mathrm{s}}^{-1}$, as reported earlier. The impact of A, though very small on the ignition time, becomes evident at later times. For $A=300$ ${\mathrm{s}}^{-1}$ and $A=500$ ${\mathrm{s}}^{-1}$, the maximum values of the scalar dissipation rates are ${\chi }_{\mathrm{MAX}}=233.4$ ${\mathrm{s}}^{-1}$ and ${\chi }_{\mathrm{MAX}}=317.3$ ${\mathrm{s}}^{-1}$. Larger values of $\chi$ and the fact that for large A, the mixtures are leaner, cause extinction. Figure 18b presenting the results for the case with $A=500$ ${\mathrm{s}}^{-1}$ shows that for Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$, this happens much faster than for Case${\mathrm{s}}_{{\mathrm{N}}_2}$.

An analysis of the solutions in mixture fraction space (

Table A8. Ignition delay time ($t_{\mathrm{ign}}$ [ms]) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures at $\chi =1$ ${\mathrm{s}}^{-1}$.

	${\mathit{Y}}_{{\mathbf{H}}_2\mathbf{O}}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
900		-	-	-	-	-	-	-	-	-	-
1000		-	-	-	-	-	-	-	-	-	-
1100		0.181/0.174	0.621/0.664	2.244/2.568	5.060/5.615	7.462/8.091	9.361/10.04	11.03/11.72	12.12/12.81	12.70/13.35	14.76/15.23
1200		0.049/0.043	0.068/0.059	0.092/0.083	0.123/0.130	0.163/0.153	0.217/0.206	0.289/0.277	0.355/0.0342	0.396/0.382	0.609/0.588
1300		0.026/0.020	0.030/0.024	0.035/0.028	0.041/0.034	0.048/0.039	0.057/0.048	0.071/0.059	0.084/0.071	0.094/0.080	0.153/0.134
1400		0.016/0.012	0.018/0.013	0.020/0.015	0.023/0.017	0.026/0.019	0.030/0.023	0.037/0.028	0.044/0.034	0.049/0.038	0.082/0.063
1500		0.012/0.008	0.012/0.008	0.014/0.009	0.015/0.010	0.017/0.012	0.020/0.014	0.024/0.017	0.029/0.020	0.032/0.022	0.055/0.038

,

Table A9. Ignition delay time ($t_{\mathrm{ign}}$ [ms]) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures at $\chi =10$ ${\mathrm{s}}^{-1}$.

	${\mathit{Y}}_{{\mathbf{H}}_2\mathbf{O}}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
900		-	-	-	-	-	-	-	-	-	-
1000		-	-	-	-	-	-	-	-	-	-
1100		0.185/0.178	0.672/0.705	2.450/2.615	5.442/5.733	8.237/8.617	10.53/10.96	12.55/12.99	13.87/14.30	14.55/14.95	16.93/17.18
1200		0.049/0.043	0.069/0.059	0.093/0.083	0.125/0.139	0.166/0.155	0.222/0.210	0.296/0.284	0.365/0.0351	0.407/0.392	0.629/0.608
1300		0.026/0.020	0.030/0.024	0.035/0.029	0.041/0.034	0.048/0.039	0.057/0.048	0.071/0.059	0.085/0.072	0.094/0.080	0.155/0.132
1400		0.016/0.012	0.018/0.013	0.020/0.015	0.023/0.017	0.026/0.019	0.030/0.023	0.037/0.028	0.044/0.034	0.049/0.038	0.082/0.063
1500		0.012/0.008	0.012/0.008	0.014/0.009	0.015/0.010	0.017/0.012	0.020/0.014	0.024/0.017	0.029/0.020	0.032/0.022	0.055/0.038

,

Table A10. Ignition delay time ($t_{\mathrm{ign}}$ [ms]) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures at $\chi =100$ ${\mathrm{s}}^{-1}$.

	${\mathit{Y}}_{{\mathbf{H}}_2\mathbf{O}}$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
TO
900		-	-	-	-	-	-	-	-	-	-
1000		-	-	-	-	-	-	-	-	-	-
1100		0.207/0.199	0.939/0.954	3.899/4.001	8.591/9.297	12.93/16.35	17.23/18.97	-	-	-	-
1200		0.052/0.044	0.072/0.063	0.099/0.088	0.134/0.123	0.181/0.169	0.245/0.232	0.332/0.318	0.411/0.396	0.462/0.445	0.730/0.708
1300		0.026/0.020	0.031/0.025	0.035/0.029	0.042/0.035	0.049/0.041	0.059/0.049	0.074/0.062	0.088/0.075	0.098/0.083	0.164/0.139
1400		0.016/0.012	0.018/0.013	0.020/0.015	0.023/0.017	0.026/0.019	0.031/0.023	0.038/0.029	0.044/0.034	0.049/0.038	0.086/0.065
1500		0.012/0.008	0.012/0.008	0.014/0.009	0.015/0.010	0.017/0.012	0.020/0.014	0.024/0.017	0.029/0.020	0.032/0.022	0.057/0.039

and

Table A11. Ignition delay time ($t_{\mathrm{ign}}$ [ms]) for ${\mathrm{H}}_2$-${\mathrm{O}}_2$-${\mathrm{H}}_2\mathrm{O}$ mixtures at $\chi =1000$ ${\mathrm{s}}^{-1}$.

T	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.767	0.8	0.9
900	-	-	-	-	-	-	-	-	-	-
1000	-	-	-	-	-	-	-	-	-	-
1100	0.381/0.371	2.053/2.879	-	-	-	-	-	-	-	-
1200	0.061/0.052	0.089/0.079	0.129/0.118	0.191/0.178	0.282/0.268	0.404/0.389	0.546/0.529	0.663/0.644	0.732/0.712	1.099/1.067
1300	0.028/0.022	0.034/0.027	0.039/0.033	0.048/0.039	0.057/0.047	0.069/0.058	0.087/0.074	0.106/0.091	0.119/0.103	0.216/0.188
1400	0.017/0.012	0.019/0.014	0.020/0.016	0.025/0.018	0.028/0.021	0.033/0.025	0.041/0.031	0.049/0.038	0.056/0.043	0.099/0.078
1500	0.012/0.008	0.013/0.009	0.014/0.010	0.016/0.011	0.018/0.012	0.021/0.015	0.026/0.018	0.032/0.022	0.036/0.024	0.064/0.043

) showed that for low temperature of the oxidizer, the ignition occurs only when the scalar dissipation is low. For instance, for $T_{\mathrm{O}}=1100$ K, it takes place for all analyzed $Y_{{\mathrm{H}}_2\mathrm{O}}$ values when $\chi \le 10$ ${\mathrm{s}}^{-1}$. In 3D turbulent flow, $\chi$ varies and the situation in which it has small values on ${\xi }_{\mathrm{MR}}$ is very very likely. It could be seen in Figure 18 that when $T_{\mathrm{O}}=1300$ K and $A=100$ ${\mathrm{s}}^{-1}$ for which ${\chi }_{\mathrm{MAX}}=154.5$ ${\mathrm{s}}^{-1}$, the ignition occurs and flame evolves in the domain. Figure 19 shows the solutions obtained for the analogical cases but for $T_{\mathrm{O}}=1100$ K. It can be seen that ${\chi }_{\mathrm{MAX}}$ of the order of 100 ${\mathrm{s}}^{-1}$ occurring somewhere in the computational domain does not preclude the ignition. It occurs for all three initial conditions; however, it manifests by only a small rise in the temperature (approximately 250 K), which remains at this level. At this stage, the elevated temperature is not the effect of combustion. It corresponds to the temperature of combustion products, which spread in the domain. This can be readily inferred based on the analysis of the heat release rate ($\dot{Q}$) presented in Figure 18b. It can be observed that $\dot{Q}$ suddenly rises and reaches very large values at the moment of ignition, but then almost immediately drops to zero, indicating that the chemical reactions cease. Hence, unlike in mixture fraction space, the ignition at low oxidizer temperatures in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ is possible but the development of the flame is prevented by turbulent flow. Based on these results and the solutions presented in Figure 18, it can be concluded that mixtures containing water vapor exhibit lower ignitability and are more prone to extinction.

5. Conclusions

The presented research aimed at investigations of hydrogen auto-ignition in cases in which the oxidizer stream is composed of oxygen mixed with nitrogen or water vapor. The presented results were split into two parts. First, a comprehensive analysis of the ignition process in 1D mixture fraction space was performed, which was then followed by 3D LES-ESF simulations. For this purpose, we developed the 3D flow configuration (forced HIT), allowing for easy control of turbulence intensity and mixture composition in a wide range of global equivalence ratios. The analysis in mixture fraction space was focused on determining the most reactive mixture fractions (${\xi }_{\mathrm{MR}}$), ignition delay times ($t_{\mathrm{ign}}$), maximum temperatures ($T_{\mathrm{MAX}}$), and their dependence of the scalar dissipation rate ($\chi$). This research relied on solving the unsteady flamelet-type equations. To increase the reliability of the solutions, different density meshes and times steps were applied. The obtained results show the following:

• ${\xi }_{\mathrm{MR}}$ is weakly dependent on $Y_{{\mathrm{N}}_2}$/$Y_{{\mathrm{H}}_2\mathrm{O}}$ when the contents of ${\mathrm{N}}_2$/${\mathrm{H}}_2\mathrm{O}$ in the oxidizer stream are high.
• For Case${\mathrm{s}}_{{\mathrm{N}}_2}$, ${\xi }_{\mathrm{MR}}$ decreases with $Y_{{\mathrm{N}}_2}$ and the temperature.
• For Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$, ${\xi }_{\mathrm{MR}}$ decreases with $Y_{{\mathrm{H}}_2\mathrm{O}}$ at a high temperature and shows non-linear behavior in a function of $Y_{{\mathrm{H}}_2\mathrm{O}}$ at a low temperature.
• At a low temperature, the ignition delay time for Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ is significantly longer; in a lin-log plot, it shows a linear dependence versus $Y_{{\mathrm{H}}_2\mathrm{O}}$.
• At a high temperature, the ignition delay times are similar but the ignitability range in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ is substantially wider.
• The maximum temperature decreases with increasing $Y_{{\mathrm{N}}_2}$/$Y_{{\mathrm{H}}_2\mathrm{O}}$, and the observed dependence is non-linear.
• The maximum temperature in Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ is lower than in Case${\mathrm{s}}_{{\mathrm{N}}_2}$.
• High values of the scalar dissipation rate delay the ignition and lower the maximum temperature but cause its faster growth in the initial ignition phase.

The 3D LES-ESF analysis of the ignition process was preceded by validation against DNS results obtained for cases with oxidizer streams composed of oxygen mixed with nitrogen, where ignition occurs earlier, making DNS feasible given the limited computational resources. It has been shown that LES-ESF predicts the ignition time very well. The lack of agreement between LES-ESF and DNS results was found at the latter times and was attributed to an insufficient resolution of the flame front on the DNS mesh. It allowed accurate resolving of the Kolmogorov length scale before the ignition but was too coarse to resolve a hydrogen flame thickness. The obtained results showed the following:

• Regardless of the oxidizer composition, the ignition occurs in regions of the most reactive mixture fraction where the scalar dissipation rate is low.
• The ignition time is almost independent of the randomness of the initial conditions and turbulence intensity but largely affected by the oxidizer composition and its temperature.
• Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ ignite later and the maximum flame temperature in these cases is lower.
• Case${\mathrm{s}}_{{\mathrm{H}}_2\mathrm{O}}$ exhibit lower ignitability and are more prone to extinction.

In conclusion, the presented research confirms the current understanding of the auto-ignition process and demonstrates that, in a qualitative sense, it also applies to hydrogen–oxygen–water vapor mixtures. The novelty of this research lies in identifying several quantitative differences between ignition in a conventional environment (e.g., air) and in a water vapor-diluted oxidizer stream. The formulated forced 3D HIT configuration and mixture initialization method enable advanced and repeatable research on ignition in flow regimes with almost identical statistical properties but varying instantaneous conditions. The detailed description of the proposed computational framework allows other researchers to replicate and extend the investigation.