Experimental–Numerical Comparison of H₂–Air Detonations: Influence of N₂ Chemistry and Diffusion Effects

Abstract

This study evaluates the performance of two-dimensional (2D) detonation simulations against recent experimental measurements for a stoichiometric hydrogen–air mixture at 25 kPa. The validation parameters rely on the average cell size ($\lambda$), the cell size variability ($2\sigma /\lambda$), and the dynamics of both the relative detonation speed (D/D_CJ) and the local induction zone length (${\Delta }_i$) along the cell cycle. We select Mével 2017’s and San Diego’s chemical models for 2D simulations, after evaluating 13 chemical models with Zeldovich–von Neumann–Döring (ZND) simulations. From this model selection, the effects of nitrogen chemistry and diffusion (Navier–Stokes or Euler equations) are evaluated on the validation parameters. The main findings are as follows: the simulations conducted with the Mével 2017 (with N₂ chemistry) model provide the best agreement with ${\lambda }_{\mathrm{mean}}^{\exp }$ (≈17%), while the experimental cell variability ($2\sigma /\lambda$) is reproduced within 20% by most simulation cases. This model (Mével 2017 with N₂ chemistry) also presents good agreement with both the ${\Delta }_i$ and D/D_CJ dynamics, whereas San Diego’s simulations under-predict them along the cell. Interestingly, the speed decay along the cell length exhibits self-similar behavior across all cases, suggesting independence from cell size variability, unlike the ${\Delta }_i$ dynamics. Finally, this study demonstrates the minimal impact of the diffusion on the simulation results.

1. Introduction

Hydrogen has emerged as a promising sustainable energy carrier for aviation due to its zero-carbon emissions, high energy density and rapid diffusivity. However, its wide ranging flammability limit (for example, in air, the flammable mixture composition range is 4–74%) and easy detonability introduce additional key challenges in its production, transportation, storage, distribution, and combustion systems [1]. In order to predict the consequences of a major safety hazard such as accidental high pressure H₂ release into the atmosphere, the storage unit contamination through air entrainment or flame acceleration in a sufficiently long storage vessel (larger length/diameter ratio) resulting in deflagration-to-detonation transition (DDT) requires further knowledge of the dynamic detonation parameters of such mixtures [2].

The detonation mode of combustion is characterized by a shock wave propagating at a speed of ∼2000 m/s for H₂–air mixtures, sustained by the energy released due to combustion. In this process, the combustion is initiated by the compression of the fresh mixture by the shock wave (typically 15–20 times the initial pressure for gaseous mixture [3]), making the detonation wave self-sustaining. This mechanism, wherein burning occurs immediately after compression by a shock, makes detonation an extremely efficient means of extracting energy from fuel [4]. The concept of leveraging this mode of combustion to enhance the thermodynamic efficiencies of conventional engines, which typically operate in deflagration mode, was first proposed by Zel’dovich [5]. Since then, the potential for harnessing the power of detonations has been well established [4,6], but it has only recently attracted significant attention [7], specifically in propulsion systems for aircraft/aerospace applications.

Recent milestones in the rotating engine (RDE) community have strengthened interest in this mode of combustion with several successful ground-level and in-space flight experiments. The longest ground-level experiment (251 s) was achieved in 2023 by NASA’s Marshall Space Flight Center with a LCH₄-LOx-fueled RDE [8], while the only in-space (≈150 km altitude) demonstrations of the RDE technology were conducted in 2022 [9] and 2024 [10] with GCH₄-GO₂-fueled and Lethanol-LN₂O RDEs, respectively.

The promise of harnessing detonation is particularly interesting for aerospace applications. Systems such as pulse detonation engines (PDEs) [11,12] and rotating detonation engines (RDEs) [13,14,15,16,17] aim to exploit the inherent high thermodynamic efficiency of detonation [3,6,18] to achieve higher thrust-to-weight ratios in comparison to traditional deflagration-based systems. However, this task comes with significant engineering challenges. The extremely rapid and high-pressure energy release associated with detonations necessitates advanced design strategies to ensure structural integrity [19], cooling requirements [20,21], and operational safety [22]. Consequently, research efforts are concurrently focused on optimizing engine designs that leverage detonation while mitigating the risks of uncontrolled or accidental detonations, especially in the context of hydrogen storage and handling [1,2,22,23,24].

In the classical theory of detonation, as described by the Zeldovich–von Neumann–Döring (ZND) model, a leading shock front compresses the reactive mixture, triggering exothermic chemical reactions in a subsequent finite reaction zone. Although extensive theoretical work has been devoted to explaining detonation phenomena, the ZND model itself is confined to a one-dimensional (1D), steady-state framework. In contrast, high-fidelity simulations that capture the full multidimensional and transient dynamics of detonations are computationally demanding because of the large disparity in time and length scales involved in resolving shock wave propagation and chemical kinetics. As a result, multidimensional (two- or three-dimensional) numerical simulations of detonations have historically been numerically demanding (to cite a few, see refs. [7,25,26,27,28,29,30,31,32,33,34,35]). Advances in numerical methods, such as direct numerical simulation (DNS) [32] and Large-Eddy Simulation (LES) [18,21], have enabled a deeper understanding of the intricate coupling between hydrodynamic shocks and chemical kinetics in detonation waves. Thanks to the advancements in computing power, detailed chemical kinetics models combined with adaptive mesh refinement [32,36] techniques have further refined our understanding of energy release mechanisms and shock front stability, reinforcing the potential of detonation-based combustion systems in aerospace propulsion.

Experimental investigations, such as those using confined geometries and shock tube setups, have demonstrated the critical role of geometry, turbulence, and initial conditions in determining the behavior of hydrogen–air detonations. In the literature, factors such as wall effects and channel configurations [31,37] can significantly affect detonation velocities and cell structures. These insights are crucial for designing propulsion systems where controlled combustion is desired. In addition, advanced diagnostics, such as high-speed imaging [38], schlieren imaging [39], laser-induced fluorescence [40], and combined diagnostics [41,42], have provided valuable insight into the detonation fronts, allowing qualitative and quantitative evaluations. These advancements further bridge the gap between theoretical predictions, numerical validation, and experiments.

Despite advances in theoretical modeling and high-fidelity simulations, our understanding of detonation dynamics remains limited, mainly due to the historical lack of quantitative experimental data for rigorous solver validation beyond the average cell widths (${\lambda }_{\mathrm{mean}}$). However, recent experiments on hydrogen detonations [40,41,42,43,44] now provide detailed data, enabling precise experimental–numerical comparisons. These detailed quantitative datasets enhance the predictive capability of computational tools by allowing rigorous solver validation, reducing uncertainties in detonation modeling, and refining the chemical kinetic mechanisms used in simulations. As a result, improved numerical models offer higher confidence for quantitative risk assessments, facilitating safer hydrogen fuel storage, distribution, and combustion system designs for aviation applications. This combined development in experimental validation and high-fidelity simulations strengthens the ability to predict and mitigate detonation risks in hydrogen-powered systems.

In this paper, we investigate stoichiometric H₂–air detonations by comparing the cell size metrics, along with the speed and induction zone length dynamics, predicted by our validated open-source OpenFOAM-based [45] solver against experimental data. Although previous studies have examined NO_x formation in RDEs [21,46,47] and PDEs [48,49], they focus primarily on emissions rather than the role of nitrogen chemistry in the detonation solution itself. Most hydrogen–air detonation studies treat nitrogen as inert (to name a few, refs. [27,32,33,50,51]), with limited exploration of its impact on the detonation structure. It is also important to emphasize that some dimensionality effect may arise both experimentally [37] and numerically [28]. Based on previous numerical studies [28] and the experimental conditions (i.e., CJ detonations with channel thickness that can accommodate more than three cells), dimensionality effects are expected to play a marginal effect in this study. Therefore, all the simulations that were performed in this study in a 2D rectangular domain have the primary objective of analyzing how nitrogen chemistry and diffusion effects influence key detonation characteristics such as cell size and induction zone length dynamics. Furthermore, this study assesses the influence of the chosen chemical kinetic model along with the role of diffusion (Eulerian vs. Navier–Stokes) on the simulation results.

2. Numerical Methodology

2.1. Governing Equations

In the present study, we perform two-dimensional (2D) numerical simulations of hydrogen detonations for a 2H₂+ O₂+3.76N₂ (stoichiometric hydrogen–air) mixture, at 25 kPa, 300 K. All simulations are performed using an open source OpenFOAM-based solver, reactingPimpleCentralFoam [52,53] in v2012. This solver is validated and verified for its capability in performing detonation simulations [45,54,55], meeting the current state-of-the-art requirements in the literature. We perform simulations considering diffusion (both viscous and heat conduction by using Navier–Stokes equations) and Euler equation solutions to quantify their influence for the mixture under study. In this semi-implicit solver for compressible flows, the convective fluxes are computed using the central-upwind schemes of Kurganov and Tadmor [56]. The governing equations integrated with detailed chemistry for this study are given below in tensorial notation:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(\rho \right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i\right)& =0\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(\rho u_j\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i{u}_j\right)& =-{\displaystyle \frac{\partial p}{\partial x_j}}+{\displaystyle \frac{\partial {\tau }_{\mathrm{ij}}}{\partial x_i}}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(\rho Y_k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho \left(u_i+V_{k,i}\right)Y_k\right)& ={\dot{\omega }}_k\mathrm{for}k=1\dots Ns\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(\rho h^{\mathrm{tot}}\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i{h}^{\mathrm{tot}}\right)-{\displaystyle \frac{\partial p}{\partial t}}& ={\dot{\omega }}_T-{\displaystyle \frac{\partial q_i}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_i}}\left({\tau }_{ij}{u}_j\right)\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill p-\rho R_u\sum _{k=1}^{N_s}{\displaystyle \frac{Y_k}{W_k}}T& =0\hfill \end{array}$$

(5)

Equations (1)–(5) are the differential forms of the governing conservation equations of mass, momentum, species, energy (total enthalpy, $h_{\mathrm{tot}}=\mathrm{sensible}\mathrm{enthalpy}+\mathrm{kinetic}\mathrm{energy}$ form) and ideal gas equation of state, respectively. Here, $N_s$, T, p, $\rho$, ${\tau }_{ij}$, $Y_k$, ${\dot{\omega }}_k$, q and ${\dot{\omega }}_T$, respectively, denote the number of species, temperature, pressure, density, viscous stress tensor, mass fraction of the $k^{\mathrm{th}}$ species, production rate, energy flux, and heat release rate. The energy flux $q_i$ is the sum of heat diffusion due to thermal conductivity ($\lambda$) given by Fourier’s law $\left(-\lambda {\displaystyle \frac{\partial T}{\partial x_i}}\right)$ and diffusion of species with different enthalpies for multi-component mixture $\left(\rho {\sum }_{k=1}^{N_s}{h}_k{Y}_k{V}_{k,i}\right)$. Note that this solver neglects the cross-diffusion effects and assumes a unity Schmidt number ($Sc=1$) to solver the species diffusion velocities, $V_k$ for all the species.

Table 1. Numerical schemes used in the simulations.

Mathematical Term	Scheme	Order
Partial time derivative, $\partial \varphi /\partial t$	CrankNicolson	$\mathcal{O}{\left(2\right)}_t$
Gradient, $\nabla \varphi$	Gauss linear	$\mathcal{O}{\left(2\right)}_x$
Divergence, $\nabla ·\Phi$	Gauss vanAlbada	$\mathcal{O}{\left(2\right)}_x$
Laplacian, ${\nabla }^2\Phi$	Gauss linear	$\mathcal{O}{\left(2\right)}_x$
Ordinary time derivative, $d(·)/dt$	rosenbrock34	$\mathcal{O}{\left(4\right)}_t$ (effectively)

summarizes the finite volume numerical schemes employed in our simulations in OpenFOAM notation. The keyword Gauss refers to the finite-volume discretization based on the divergence theorem. Then, the following keywords such as linear and vanAlbada refers to the interpolation and flux-limiting methods used for reconstructing the field values at the cell faces from the cell centers. Furthermore, all the ZND simulations conducted as a part of this study are performed using the MATLAB-based Shock and Detonation Toolbox (SDT) [57].

2.2. Chemical Kinetic Models

As previously indicated, we study a stoichiometric hydrogen–air mixture mixture at 25 kPa and 300 K. We assess several hydrogen–air oxidation models from the literature, some of which were previously studied under detonation conditions as indicated in

Table 2. Summary of important characteristic length scales for all the chemical models under consideration. Results are obtained with SDT [57]. The parameters are as follows: ${\Delta }_i$—induction zone length, ${\Delta }_R$—exothermic reaction length (defined as the thermicity half-width [75]), ${\tau }_{\mathrm{ind}}$—induction time, ${\lambda }_{\mathrm{Ng}}$—Ng’s correlation based SDT estimation of cell width, ${\theta }_{\mathrm{eff}}$—effective activation energy, $\chi$—stability parameter.

Models	${\mathit{\Delta }}_{\mathit{i}}$, $\mathsf{\mu }$m	${\mathit{\Delta }}_{\mathit{R}}$, $\mathsf{\mu }$m	${\mathit{\tau }}_{\mathbf{ind}}$, $\mathsf{\mu }$s	${\mathit{\lambda }}_{\mathbf{Ng}}$, mm	${\mathit{\theta }}_{\mathbf{eff}}$	$\mathit{\chi }$	Relevant Studies
Aramco 3.0	775	239	2.1	36	6.6	3.1	-
Burke 2012	792	241	2.2	37	6.6	3.1	Refs.  [33,54,76]
Zhu 2024	789	212	2.1	36	6.6	3.8	-
Hong 2011	783	233	2.1	36	6.9	3.3	Refs. [54,77,78]
KAUST-Mech-2023	807	205	2.2	36	7.0	4.0	-
Konnov 2019	751	194	2.0	34	6.7	3.7	-
Li 2015	708	218	1.9	33	6.5	3.3	-
Mével 2014	550	199	1.5	26	6.2	2.6	Refs. [54,55,66]
Mével 2017	553	207	1.5	26	6.2	2.6	Refs. [69]
FFCM-2	811	242	2.2	38	6.8	3.2	Refs. [54]
San Diego	753	208	2.0	34	6.4	3.8	Refs. [79,80]
GRI 3.0	781	217	2.1	36	6.8	3.3	-
Varga 2015	761	234	2.1	36	6.6	3.1	-

. Two quantitative parameters obtained from the ZND simulation, Ng’s cell size ($\lambda$ [58]) and induction zone length (${\Delta }_i$), are compared against the recently reported experimental data [41]. Figure 1a,b summarize this comparison for 13 different kinetic models: Aramco 3.0 [59], Burke 2012 [60], Zhu 2024 [61], Hong 2011 [62], KAUST-Mech 2023 [63], Konnov 2019 [64], Li 2015 [65], Mével 2014 [66], and Mével 2017 (i.e., which is a reduced model of Mével 2009 [67] for H₂–air combustion, available in [68,69]), FFCM-2 [70,71], Varga 2015 [72], San Diego 2016 [73], and GRI 3.0 [74]). From Figure 1a, it can be inferred that Mével’s models agree well with the experimental induction zone length at CJ speed, while all the other models predict induction zone length well beyond the experimental uncertainty. For the Ng cell size predictions (see Figure 1b), all the models are beyond twice the experimental mean cell width (that is, the accuracy of the Ng method is a factor of 2 [58]) , while Mével models have the lowest discrepancies and are near the Ng method’s uncertainties. Additional detonation parameters are summarized in Table 2 for each model, as well as the previous usage in detonation studies. The definition employed for estimating the effective activation energy (${\theta }_{\mathrm{eff}}$) and the stability parameter ($\chi$) is based on SDT [57]:

$${\theta }_{\mathrm{eff}}\equiv {\displaystyle \frac{1}{T_{\mathrm{vN}}}}{\displaystyle \frac{log{\tau }_{1.01D_{\mathrm{CJ}}}-log{\tau }_{0.99D_{\mathrm{CJ}}}}{1/T_{1.01D_{\mathrm{CJ}}}-1/T_{0.99D_{\mathrm{CJ}}}}}$$

(6)

$$\chi \equiv {\theta }_{\mathrm{eff}}{\displaystyle \frac{{\Delta }_i}{{\Delta }_R}}$$

(7)

In Equation (6), T and $\tau$ are the temperature and induction time, and these quantities are evaluated at the von Neumann ($\mathrm{vN}$) state, and at the shock speeds perturbed by $\pm 1\%$ (i.e., $1.01$D_CJ and $0.99$D_CJ). The ${\Delta }_R$ in the stability parameter, and $\chi$ in Equation (7) is the exothermic reaction length. In this study, the Mével 2017 and San Diego 2016 models are chosen for the 2D simulations because they effectively represent the entire set of models based on our ZND results (see Figure 1). Based on the effective activation energy in Table 2, the mixture under study is classified as a moderately unstable mixture. In addition to the direct experimental–numerical comparison, both viscous (Navier–Stokes) and inviscid (Euler) simulations are performed to highlight the role of diffusive effects on the solution. This sheds light on the suitable solver to be used for such mixtures. Additionally, the Mével 2017 model is employed with and without its NO_x sub-model to assess the impact of NO_x chemistry on detonation simulations.

Table 3. List of cases considered for 2D simulations. Here, E refers to Euler, NS stands for Navier–Stokes simulations, and ‘with NO_x’ and ‘w/o NO_x’ mean ‘with N₂ chemistry’ and ‘without N₂ chemistry’, respectively.

Case	Models	CJ Speed (m/s)	${\mathit{L}}_{\mathit{x}}/{\mathbf{\Delta }}_{\mathit{i}}\times {\mathit{L}}_{\mathit{y}}/{\mathbf{\Delta }}_{\mathit{i}}$	${\mathit{n}}_{\mathit{x}}\times {\mathit{n}}_{\mathit{y}}$	Grid Density in Uniform Resolution (Points/${\Delta }_{\mathbf{i}}$)
I	Mével 2017 (with NOx—E)	1938	$50\times 100$	$2474\times 7971$	80
II	Mével 2017 (with NOx—NS)	1938	$50\times 100$	$2474\times 7971$	80
III	Mével 2017 (w/o NOx—NS)	1938	$50\times 100$	$2474\times 7971$	80
IV	San Diego (w/o NOx—E)	1944	$50\times 50$	$1794\times 3028$	60
V	San Diego (w/o NOx—NS)	1944	$50\times 50$	$1794\times 3028$	60

lists the five cases considered for 2D simulations to address the points outlined above.

2.3. Computational Domain and Grid System

All simulations are performed in pseudo-shock-attached frame of reference (p-SFR) under Galilean transformation with a constant relative motion at CJ speed obtained using ZND simulation. Note that this frame of reference is employed in our previous studies [45,54], which show similar results as the classical lab frame of reference while significantly reducing the computational costs (see discussions in [45]). Table 3 indicates the CJ speed in each case for the converged solution. Figure 2 illustrates the schematics of the 2D computational domain initialized with steady-state ZND solution at $t=0$ s, the boundary conditions, and the non-uniform grid system used. The domain is initialized with the fresh gas mixture (2H₂ + O₂ + 3.76N₂ at 25 kPa and 300 K) at CJ speed, entering the domain from the left. We use the circular hot-spots (i.e., region of high pressure and temperature, ten times the initial temperature and pressure) to perturb the initial planar wave. We use slip wall boundary conditions on the top and bottom parts of the domain and zeroGradient boundary condition at the outlet. These boundary conditions are particularly relevant for simulating cellular structures far from the walls, which is in agreement with the experimental data employed in this study. We adopt a uniform grid size starting from the fresh gas and extending to a few induction zone lengths (∼10${\Delta }_i$) behind the shock wave. Then, the grid becomes non-uniform until the outlet with a grid-to-grid expansion ratio of ≃1%. Table 3 also summarizes the domain length and grid specifications, appropriately normalized by the ZND induction length (${\Delta }_i$). Similarly to our previous studies [45,54], these simulation parameters are selected after performing a systematic grid convergence study and a domain independence study to avoid mesh-dependent and mode-locked solutions [33,35,81]. Weng et al. [35] recommend to have 100${\Delta }_i$ of domain length based on their linear stability analysis of detonation mixtures using simplified chemistry. For the present study, we perform simulations with different combinations of domain sizes and find that the solution is independent of the domain length in x-direction for $L_x\ge 50{\Delta }_i$ with both chemical models. In the y-direction ($L_y$), the solution becomes domain independent for $L_y=50{\Delta }_i$ with San Diego’s model and for $L_y=100{\Delta }_i$ for Mével 2017 model. Similar to our previous studies [45,54], we define the convergence of the solution by monitoring that (i) the wave travels close to the CJ speed with allowed deviation of $\pm 2\%$ and (ii) the regularity of the detonation cell size does not change over time. Additionally, based on our previous studies, we confirm that variations in hotspot conditions (position, size, and number) do not significantly affect the cellular structure at sufficiently longer runs (>300 $\mathsf{\mu }$s), as all configurations converge to a similar steady-state solution in the considered weakly unstable mixture.

2.4. Experimental–Numerical Comparison of the Induction Zone

In the context of ZND simulations, the induction zone length (${\Delta }_i$) is defined as the distance between the shock front and the point of maximum heat release rate (point of maximum thermicity) [25]. Following this definition, Rojas-Chavez et al. [82] were able to measure ${\Delta }_i$ from 1D NO-LIF profile evolution in a NO-seeded (2000 ppm) H₂–air mixture, by exciting the NO at 225.12 nm, in a rectangular optical detonation duct (see [40,41,82] for details on the experimental setup and measurement procedure of ${\Delta }_i$). With this low concentration, NO tends to be non-reactive in the vicinity of ${\Delta }_i$ [44,45,82]. For the numerical counterpart, the procedure for measuring ${\Delta }_i$ in 2D simulations is identical to our previous work on argon diluted H₂-O₂ mixture [54,83]). Thus, we post-process the numerical 2D fields of temperature, pressure, and species mass fraction using our in-house spectroscopic code KAT-LIF with the same laser excitation wavelength and energy as in the experiments. Details on the spectroscopic code and fluorescence modeling are available in previous studies [54,82,84]. Note that this post-processing step may not be necessary for some chemical models for which the thermicity field can be used directly (see the discussions in the appendix of our previous study [54]).

3. Results and Discussion

This section compares the experimental and numerical results in the following three subsections: (1) comparison of cell size metrics, such as mean cell width (${\lambda }_{\mathrm{mean}}$) and its variability ($2\sigma /\lambda$); (2) comparison of the induction zone length (${\Delta }_i$) dynamics in the cell cycle; and (3) comparison of the relative shock speed decay in the cell cycle. In each case, the effects of the diffusion terms in the numerical solver and the influence of the N₂ chemistry on the simulation results are discussed.

3.1. Detonation Cell Size Metrics

Figure 3 illustrates the contours of the normalized detonation speeds (D/D_CJ, where D is the local shock speed estimated from the instantaneous shock position ($x_{\mathrm{shock}}$) as $d{x}_{\mathrm{shock}}/dt$), depicting the cellular structure characteristic of detonation waves for various simulation cases. Each contour represents the instantaneous local detonation speed normalized by the CJ speed, allowing for clear visualization of the wave dynamics across the computational domain. In these contour plots, the detonation front propagates from right to left, with the superimposed black solid lines indicating the triple point trajectories at selected time instances. Within a cell, the regions of higher relative speed (D/D_CJ > 1) indicate stronger local shocks typically associated with triple-point collisions and Mach stem, while areas of lower speed highlight weaker (incident) shock propagation and expansion regions. The boundaries of the cell reveal the triple point trajectories and their local speed. By comparing these cases, we can visually assess the influence of reactive nitrogen chemistry (N₂ chemistry) and diffusive effects (Euler versus Navier–Stokes simulations) on the cellular structures, shock front behavior, and its stability.

As can be inferred from Figure 3, the overall shock speed ranges from D/D_CJ = 0.7 to 1.6, in all the cases studied. This tends to demonstrate the marginal impact of the N₂ chemistry and the diffusion effects (i.e., Euler vs. Navier–Stokes) on the speed dynamics, at least for the chemical models and condition considered. The detonation speed decay is further analyzed in Section 3.3.

Table 4. Comparison of the detonation cell size metrics between numerical and experimental data. Here, E refers to Euler, NS stands for Navier–Stokes simulations, and ‘with NO_x’ and ‘w/o NO_x’ mean ‘with N₂ chemistry’ and ‘without N₂ chemistry’, respectively.

Case		${\mathit{\lambda }}_{\mathbf{mean}}$, mm	$2\mathit{\sigma }/{\mathit{\lambda }}_{\mathbf{mean}}$
Experiments [41]		13 ± 1	0.38 ± 0.2
	Mével 2017 (with NOx—E)	10.8	0.45
	Mével 2017 (with NOx—NS)	10.9	0.44
2D Simulations	Mével 2017 (w/o NOx—NS)	7.5	0.59
	San Diego (w/o NOx—E)	4.6	0.38
	San Diego (w/o NOx—NS)	4.5	0.32

summarizes the quantitative comparison of the mean cell widths (${\lambda }_{\mathrm{mean}}$) and the cell size variability ($2\sigma /{\lambda }_{\mathrm{mean}}$) between the numerical and experimental data. While the ${\lambda }_{\mathrm{mean}}$ predicted by all the models without N₂ chemistry deviates by a factor of 2–3, simulations conducted with Mével 2017’s model with N₂ chemistry have the lowest discrepancy (within 20%), for conditions with and without diffusion effects. When the N₂ chemistry is removed from Mével 2017, the predicted ${\lambda }_{\mathrm{mean}}$ is significantly reduced (≈−30%) as observed between Figure 3b,c, and deviates from the experimental one. For the cell size variability ($2\sigma /\lambda$), simulations conducted with Mével 2017’s models and N₂ chemistry or San Diego’s models (i.e., both without N₂ chemistry) reproduce the experimental cell variability within 20%. Taking into account both experimental and numerical uncertainties in $2\sigma /\lambda$, all these simulations are considered to be satisfactorily reproducing the variability of the experimental cell size. As observed previously for ${\lambda }_{\mathrm{mean}}$, removing the N₂ chemistry from Mével 2017 model increases the discrepancies with the experimental results.

This section reveals that the N₂ chemistry is critical for simulating H₂–air detonations, at least for the selected chemical models and pressure considered, to have close experimental–numerical agreement of the cell size metrics (that is, ${\lambda }_{\mathrm{mean}}$ and $2\sigma /\lambda$).

3.2. Experimental vs. Numerical Induction Zone Length Evolution Along the Cellular Cycle

Figure 4 compares the experimental [41] and numerical evolution of ${\Delta }_i$ over multiple cells as a function of the normalized detonation speed (D/D_CJ) for all the cases listed in Table 3. Numerically, ten cells are randomly selected from the ones represented in Figure 3 to extract the evolution of the induction zone over the cell cycle, after KAT-LIF post-processing (see Section 2.4). Experimentally, detection limits in the range of 10 pixels are typically encountered with the NO-PLIF technique, with larger experimental uncertainties in its vicinity. Thus, the observed plateau of induction zones, where ${\Delta }_i$ = 0.15–0.2 mm, is most probably related to this detection limit and/or the relatively large experimental uncertainties rather than an actual range of constant induction zone. For this reason, experimental induction zone lengths below 0.2 mm and D/D_CJ speed above 1.5 are not considered a reliable validation target in the following discussion. For all cases, the numerical data generally follow the expected trend of the induction zone length (${\Delta }_i$) dynamics by monotonically decreasing with D/D_CJ. Simulations conducted with Mével 2017’s model and N₂ chemistry (Figure 4a,b) show reasonable agreement with the experimental data, with a slight under-prediction of the induction zone length at higher D/D_CJ. All the other simulations, conducted without the N₂ chemistry (Figure 4c–e), show a more noticeable discrepancy in both the range of ${\Delta }_i$ and the value of ${\Delta }_i$ for fixed values of D/D_CJ. On average, the simulations conducted with San Diego’s model or with Mével 2017’s model without N₂ chemistry are underestimating the induction zone lengths. Furthermore, the inclusion of the N₂ chemistry makes ${\Delta }_i$ more sensitive to the D/D_CJ variations in the second part of the cycle (see, D/D_CJ < 1 in Figure 4a,b).

Based on the results of Figure 4, the evolution of the induction zone length is now represented as a function of the normalized cell length ($x/L$) in Figure 5a–e. Once again, the simulations conducted with Mével 2017 model and reactive nitrogen chemistry (N₂ chemistry) agree well with the experiment as opposed to the simulations conducted with the chemical models with inert nitrogen chemistry. In addition, a negligible effect of the diffusion is observed, when comparing Figure 5a,b or Figure 5d,e. It must be noted that the scatter of the data is highly correlated with the cell size variability (see Table 4; the higher the variability, the larger the scatter of the data). Such findings are in agreement with the experimental observations between two different mixtures in [41]. Moreover, the evolution of ${\Delta }_i$ for all cases appears to overlap for the first part of the normalized cell length, then starts to deviate for the second part of the cycle. This transition corresponds to the relative distance ($x/L$) at which the relative detonation speed transits from D/D_CJ > 1 to D/D_CJ < 1 (see Figure 6). This behavior is consistent among all models but more visible for Mével 2017’s model (see Figure 5a–c) that shows relatively higher cell size variability ($2\sigma /{\lambda }_{\mathrm{mean}}$). Overall, the inclusion of N₂ chemistry improves the experimental–numerical agreement by increasing the cell size variability and ${\Delta }_i$ dynamics, while no significant effects of the diffusion are found.

This section demonstrates that the N₂ chemistry is essential to reproduce the induction zone dynamics in H₂–air detonation, while the role of diffusion is less important. Note that both the Mével 2017 chemical models—with and without the NO_x chemistry—exhibit very similar thermodynamic states at the von Neumann and CJ conditions in both ZND and 2D simulations. However, significant differences in the cellular dynamics (Figure 3a,c) and local induction zone lengths (see Figure 4) are observed in the detailed 2D simulations. This indicates that the chemical kinetics associated with N₂ chemistry introduce new reaction pathways as discussed in [85] for subsonic combustion, playing a crucial role in defining the cellular structures and induction zone. Also note that the negligible role of diffusion is consistent with the recent results of Watanabe et al. [77] obtained on different mixtures with a different chemical model (Hong 2011 [62]).

3.3. Experimental vs. Numerical Shock Speed Evolution as a Function of Cell Length

Figure 6 illustrates the evolution of the relative detonation speed (D/D_CJ) as a function of the normalized cell length ($x/L$). All simulations agree reasonably well with the experimental speed decay, within or very close to the experimental uncertainties for most of the cell cycle. It should be noted that the shock speed at the beginning of the cell, near $x/L=0$, is relatively high for the experimental measurements compared to all the simulation results. The observed discrepancies can be associated with four phenomena: the rather high experimental uncertainties on the speed measurements near the triple point collisions; the indirect experimental measurements of $x/L$, relying on the conversion of distance between triple points into relative cell lengths for an average cell cycle (see the methodology section in [41]); the numerical methodology employed to analyze the cellular cycle, which tends to neglect points near triple point collisions (see methodology section in [54]); and the different experimental and numerical approaches employed to resolve the average cell cycle, which rely on 40 single-shot measurements (i.e., 40 different cell cycle) and 10 finely resolved cell cycles, respectively. Thus, drawing definitive conclusions on the experimental–numerical accuracy for very short $x/L$ distances (near 0) or high D/D_CJ values ($D/D_{\mathrm{CJ}}>1.5$) seems unrealistic. However, it can be concluded that both chemical models agree well with the experimental observations in terms of the speed decay, regardless of the usage of N₂ chemistry or diffusion effects. In fact, all speed profiles collapse onto a single curve with marginal cell-to-cell variabilities. Despite significant variations in the cell sizes, the normalized length scaling reveals a self-similar behavior, suggesting that the speed decay is independent of the cell size variability as opposed to the induction zone dynamics (see Figure 5).

4. Conclusions

In this study, we conducted 2D numerical simulations of a moderately unstable hydrogen–air mixture under stoichiometric conditions using two chemical models: Mével 2017 and San Diego 2016. The simulations were performed in a rectangular domain employing non-uniform grid strategies to optimize computational efficiency. The influence of diffusion effects (Euler vs. Navier–Stokes) and nitrogen chemistry on detonation cell size metrics and the evolution of both the relative detonation speed and the induction zone length over multiple cellular cycles were systematically analyzed and compared against experimental data. The key findings are summarized as follows:

• The comparison between numerical and experimental detonation cell size metrics highlights the crucial role of reactive nitrogen chemistry (i.e., N₂ chemistry). The chemical model incorporating reactive nitrogen (Mével 2017 with N₂ chemistry) yields mean detonation cell widths of 10.8 mm (Euler) and 10.9 mm (Navier–Stokes), closely matching the experimental mean of 13 mm. In contrast, models treating nitrogen as an inert species (Mével 2017 w/o N₂ chemistry and San Diego) significantly underestimate the cell size, predicting values of 7.5 mm and 4.5 mm, respectively. Although the best agreement is obtained for Mével 2017 with the N₂ chemistry, its impact on cell variability ($2\sigma /\lambda$) is less pronounced. These results suggest that reactive nitrogen chemistry is essential for accurately capturing the average cell structures, but it is less important to capture the cell variability.
• Analysis of the local induction zone length (${\Delta }_i$) dynamics as a function of the relative detonation speed (D/D_CJ) reveals that the simulations conducted with the Mével 2017 model with reactive nitrogen chemistry effectively reproduces both the range and the trend of ${\Delta }_i$ over a detonation cell cycle. In contrast, simulations conducted without N₂ chemistry tend to systematically under-predict ${\Delta }_i$. A similar trend is also observed when ${\Delta }_i$ is analyzed with respect to the normalized cell length ($x/L$), which further emphasizes the importance of nitrogen chemistry in accurately capturing H₂–air detonation wave dynamics.
• Comparisons between Euler and Navier–Stokes simulations indicate that viscous diffusion and thermal conduction have minimal impact on the detonation structure, irrespective of the chemical model used. This extends the recent observations reported by Watanabe et al. [77] obtained on different mixtures with a different chemical model.

Overall, this study emphasizes the significant impact of chemical kinetic model selection on accurately reproducing detonation characteristics. In particular, considering reactive N₂ is crucial for achieving better reproduction of the average cell structure (${\lambda }_{\mathrm{mean}}$), cell variability ($2\sigma /\lambda$), and cell dynamics (i.e., speed profile or induction zone evolution over the cellular cycle) in the present condition. As reported in our previous studies on argon-diluted mixtures, multiparameter measurements beyond cell-size metrics are essential for assessing the predictive performance of chemical models in numerical simulation of detonations. In future studies, detailed quantitative comparisons of H₂–air detonations should focus on comparing the performance of multiple detailed kinetic models that include N₂ chemistry in Eulerian-based simulations rather than employing simplified chemistry, without N₂ chemistry, with diffusion effects.

5. Computational Resource Utilization

The summary of the 2D computational grids and the corresponding number of CPU hours for all the cases are summarized in

Table 5. Summary of CPU hours used for performing 2D numerical simulations.

Chemical Models	Grid Density (Points/${\mathit{\Delta }}_{\mathit{i}}$)	Number of Grids † (${\mathit{n}}_{\mathit{x}}$ × ${\mathit{n}}_{\mathit{y}}$)	CPU Hours (in Millions)
Mével 2017 (with N2 chemistry)	80	2474 × 7971	0.25
	40–80	-	0.96 *
Mével 2017 (w/o N2 chemistry)	80	2474 × 7971	0.22
	40–80	-	0.56 *
San Diego (w/o N2 chemistry)	60	1794 × 3028	0.05
	40–60	-	0.21 *
Total (results presented in this study, see Table 3)			0.52
Total (all simulations *)			1.73

† Non-uniform grids (see [45]). * includes grid convergence and domain dependency tests.

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