# Lagrangian characterization of induction and reaction timescales in a cellular gaseous detonation

Hiroaki Watanabe, Akiko Matsuo, Ashwin Chinnayya, Noboru Itouyama, Ken Matsuoka, Jiro Kasahara

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Hiroaki Watanabe, Akiko Matsuo, Ashwin Chinnayya, Noboru Itouyama, Ken Matsuoka, et al. "Lagrangian characterization of induction and reaction timescales in a cellular gaseous detonation." Physics of Fluids, 2025, 37 (2), pp.026106. 10.1063/5.0249984. hal-04930061

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Induction and reaction times in a cellular detonation

AIP/123-QED

Lagrangian characterization of induction and reaction timescales
in a cellular gaseous detonation

Hiroaki Watanabe (渡部 広吾輝),¹ Akiko Matsuo (松尾 亜紀子),² Ashwin Chinnayya,¹
Noboru Itouyama (伊東山 登),³ Ken Matsuoka (松岡 健),³ and
Jiro Kasahara (笠原 次郎)³

¹)Institut Pprime - UPR 3346 - CNRS - ISAE-ENSMA - Université de Poitiers,
1 Avenue Clément Ader, BP 40109, 86961, Futuroscope-Chasseneuil Cedex,
France.

²)Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan.

³)Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi, 464-8603,
Japan.

(*Electronic mail: watanabe0204@keio.jp)

(Dated: 30 December 2024)
Induction and reaction times in a cellular detonation

A Lagrangian approach was proposed to analyze induction and reaction times in the cellular gaseous detonation. Two-dimensional simulations in an argon-diluted and non-diluted hydrogen-based mixtures were performed with detailed chemistry, along particle trajectories. The distribution of the induction and reaction times inside the cell was significantly different between the Eulerian and the Lagrangian perspectives, the latter showing non-monotonic behavior. Preferential thermodynamic paths laid along the Rankine-Hugoniot curve, and behind transverse waves (TW). All particles were ignited within half and one cell cycle for the diluted and non-diluted mixture, respectively. The ignition mechanisms were not only one-dimensional, but also multi-dimensional, with ignition behind the TW being the most important, and collision of TW and triple points being secondary. A new topology inside the cell could be drawn, from the intersection of the ignition front with TW. TW appeared as phase waves in the (x,t) diagram. Comparison of H₂O mass fraction between local and equilibrium values indicated that a local chemical disequilibrium remained (super-equilibrium), due to TW. Equating the mean sonic plane with thermochemical equilibrium in the non-diluted case is not completely accurate. Furthermore, the characteristic time scales for chemical and hydrodynamic phenomena were compared. The diffusive phenomenon did not make any contribution in the mixtures tested. In comparison to the Zel'dovich-von Neumann-Döring model, a shorter average induction time was found in the non-diluted mixture, which is not in line with the results from previous Favre approaches. The average reaction time was also shorter in both mixtures.

Keywords: Detonation, Cellular structure, Induction and reaction timescales, Lagrangian approach
Induction and reaction times in a cellular detonation

# I. INTRODUCTION

Gaseous detonations are supersonic combustion fronts¹. The Zel'dovich-von Neumann-Döring (ZND) steady one-dimensional (1D) model describes the coupling between the leading shock and a reaction zone. However, the exponential dependence of the reaction rates rapidly induces instabilities that ultimately draw cellular patterns, coming from triple point trajectories²⁻⁴. The latter connect a transverse wave, an incident shock, and a Mach stem. These observable cells are at the heart of system design, for applications related to the deployment of low-carbon fuels, which have to remain safe enough. Indeed, the dynamic parameters for the initiation, propagation, and quenching of detonations are empirically dependent on the cell structure, including its dimensions and irregularity⁵. More recently, Veiga-López et al.⁶ and Watanabe et al.⁷ showed that further understanding of this building block enabled to improve the efficiency of simplified chemical models. However, their studies were built upon the steady response of the detonation to losses. These studies also pointed out that unsteadiness should be added to the discussion and further design of reduced models. The first source of unsteadiness comes from the cellular structure. Indeed, this fascinating and hypnotic pattern consists of alternating incident shocks and Mach stems, bounded by triple points. The decaying shocks induce rapid expansion, which can quench locally the ignition process along trajectories⁸⁻¹³. Favre averaged results¹⁴⁻¹⁸ have been used to analyze the mean structure of detonations. However, they cannot completely capture the quenching process, which is local in nature. A Lagrangian approach is thus adopted here to investigate the induction and reaction timescales within the whole cellular structure, which will be compared to different hydrodynamic timescales.

The mixture instability influences the ignition mechanism downstream of the leading shock front. In weakly unstable mixtures with lower reduced activation energy $θ = E_a/(RT_vN)$ and $χ = θx_{ind}/x_{reac}$ parameter, the cellular structure is regular and the adiabatic compression of the leading shock front is the dominant mechanism for the ignition¹⁹⁻²¹. On the other hand, in the case of unstable mixtures with irregular cellular structure (higher $θ$ and $χ$), adiabatic shock compression is not sufficient to ignite all of the unburned mixture¹⁹,²²⁻²⁴, with about 50-60% of the shocked gas being considered to be burned by turbulent burning²²,²³. Moreover, several studies categorized the ignition occurrence inside the cell²²,²³,²⁵⁻²⁸. Mitrofanov²⁵ divided the cell into two: 50-60% in the first part of the cell was ignited by the shock, and the ignition in the second part was induced by other additional compression due to transverse waves. Hu et al.²⁷,²⁸ divided the cell into three
Induction and reaction times in a cellular detonation

regions for ignition in H₂/O₂/Ar mixture, whether the particle experienced the Mach stem, the incident shock, or the transverse wave.

To get insight on the process within the reaction zone of cellular detonation, the characteristic lengths/times such as induction length/time and the distribution of chemical species were measured ³,⁴,⁸,²⁹⁻³⁸. Jost et al.²⁹ were probably the first to measure the induction and reaction times from the emission of the spectral band with various initial pressures and mixtures. Edwards et al.³⁰ measured the induction length directly from the schlieren images and interferograms for a 2 H₂ + O₂ + 60% Ar mixture. They found a discrepancy in the induction delay time between experimental measurement and calculation at low temperatures below 1000 K, emphasizing the interaction between gas dynamics and chemistry. Lundstrom and Oppenheim⁸ showed experimentally and numerically that the induction length increased due to the unsteady expansion behind the leading shock. Dormal et al.³²⁻³⁴ obtained the distribution of the induction time inside the cell by the emission spectroscopy of hydroxyl radical and showed that the local induction time increased along the cell. Recently, Chavez et al.³⁵⁻³⁷ and Alicherif et al.³⁸ applied the planar laser induced fluorescence of nitric oxide (NO-PLIF) technique to visualize the induction zone, which revealed that its length varied during the cellular cycle. Short induction lengths were observed behind the Mach stem, while they become longer behind the incident shock.

Moreover, the coupling between the leading shock velocity and the reaction zone in terms of induction length was characterized by the comparison with steady model with leading shock velocity as an input³⁹,⁴⁰. In a weakly unstable mixture, the experimental and numerical induction length followed the trend of the steady model for all leading shock velocity. On the other hand, in an unstable mixture, the induction length followed the trend of the steady model only during high speed portion of front.

During the cell cycle, in the center line, the leading shock is curved and decays continuously from its first apex. The particles then experienced unsteady expansion, resulting in a longer induction time. In some cases, the quenching of the chemical reaction could occur⁸⁻¹³,²³,⁴¹ (see also recent review⁴² and supplementary material⁴¹). The pioneering work of Lundstrom and Oppenheim⁸ suggested that some secondary effects aided ignition. Later, Eckett et al.¹⁰ analyzed the competition among heat release, wave front curvature and unsteadiness in spherical direct initiation, and proposed a critical decay rate of the leading shock as a failure ignition criterion, based on the sole balance between unsteadiness and heat release. Furthermore, Radulescu and Maxwell¹¹ numerically and analytically analyzed the ignition with expansion given by a power
Induction and reaction times in a cellular detonation

law rate of decay behind the decaying shock. They obtained the critical condition in a closed form in terms of Damköhler number. Recently, Cheevers and Radulescu<sup>12,13</sup> proposed a criterion for ignition from the combination of the shock-change equation and asymptotic analysis. Ignition quenching was shown to be due to the shock unsteadiness for any mixture, the location of which depended on its regularity. This was in line with Frederik et al.<sup>41</sup>.

The majority of the studies have focused their analysis on two main curves in the cell structure, located on the center line and in the vicinity of the triple point. Then, these aforementioned efforts were compared to Eulerian results, i.e. measurement at a fixed point or single time snapshot. Taylor et al.<sup>43</sup> pointed out that the Eulerian approach was unable to track the history of gaseous particles to analyze the evolution of the chemical species and the thermodynamic properties. Some previous studies<sup>10,18,44–51</sup> used a particle tracking method to get more insight into the detonation front. However, none discussed the distribution of the induction and reaction times and ignition mechanism within a cellular structure, specifically when the center line and the triple point meet together at the last apex.

The remainder of this paper is structured as follows. Section II describes briefly the mathematical models and the numerical methodology employed. The problem statement is presented in the Section III. The results and discussions are presented in Section IV. Section IV A provides a characterization of the state of unburned mixture as well as global features of the flow fields. The dependency of initial condition on induction and reaction times in the condition relevant to detonation propagation is also evaluated through zero-dimensional (0D) constant pressure combustion modeling. Moreover, the distribution of induction and reaction times within the cell is obtained from both an Eulerian and a Lagrangian perspective and compared with each other in order to highlight their differences in a weakly unstable mixture, as presented in Section IV B. Furthermore, the time sequences of the induction and reaction zone from Lagrangian perspective are discussed to explain the distribution of induction and reaction times. Then, Section IV C presents the differences obtained with a mildly unstable mixture. In Section IV D, the ignition mechanisms are identified through a visual analysis of the induction and reaction zones. Additionally, Section IV E depicts a (x,t) diagram along the center line of cell and particle trajectories, thereby elucidating the wave dynamics and emphasizing the role of transverse waves in the relaxation to chemical equilibrium. Furthermore, the various characteristic time scales for chemical, hydrodynamic and detonation phenomenon along the center line of the cell are quantitatively compared in Section IV F to discuss their interplay. Moreover, the distribution of induction and
Induction and reaction times in a cellular detonation

reaction times from a Lagrangian perspective is compared with the prediction based on a steady
model in Section IV G. Finally, the main conclusions are drawn in Section V.

II. NUMERICAL SETUP

The details of the mathematical models and numerical methods can be found in our earlier
study by Watanabe et al.<sup>18</sup>. Therefore, only a brief description is presented here.

A. Governing equations

The governing equations for the gaseous phase were the two-dimensional (2D) reactive
compressible Navier-Stokes equations, with the ideal equation of state. The chemical reaction
mechanism proposed by Hong et al.<sup>52</sup>, which considers 9 species (H<sub>2</sub>, O<sub>2</sub>, H, O, OH, H<sub>2</sub>O, HO<sub>2</sub>,
H<sub>2</sub>O<sub>2</sub> and Ar) and 20 elementary reaction, was used.

The method proposed by Gordon et al.⁵³ was used for the estimation of the viscosity and
thermal conductivity for chemical species except for HO₂. The viscosity and thermal conductivity
for HO₂ were evaluated from the Chapman-Enskog method⁵⁴ and the Eucken method⁵⁵, respectively.
The multicomponent gas viscosity and thermal conductivity were estimated based on the pure
species value by Wilke method⁵⁶ and Wassiljewa method⁵⁷, respectively. The binary diffusion
coefficient was evaluated by the Chapman-Enskog method⁵⁴ and the mixing rule⁵⁸ was applied to
calculate the diffusion coefficient of a compound into the other compound.

To track massless gaseous particles, the particle tracking method was employed in this study.
The position of gaseous particle was updated using the velocity at the particle position.

There are some limitations to the model used. Indeed, the first one is the use of 2D modeling.
Indeed, Paknahad et al.⁵⁹ showed that three-dimensional (3D) simulations have a wider range of
distributions of leading shock velocity and thermodynamic states compared to 2D. There is also the
influence of geometry of confinement on regularity of marginal cellular structure⁶⁰,⁶¹. The second
limitation comes from the fact that none of the detailed chemistry available has been developed
in detonation conditions⁶². Thus, some uncertainties remain (see recent study by Sankar et al.⁶³
for discussions). The third limitation is that the mixtures tested can be considered as weakly and
mildly unstable. However, addressing these limitations is beyond the scope of this study.
Induction and reaction times in a cellular detonation

B. Numerical methods

A classical first-order operator-splitting method was employed to couple the hydrodynamics with the detailed chemistry. The spatial derivatives of the convective terms were discretized by a fifth-order advection upstream splitting method using pressure-based weight functions (known as AUSMPW+) improved by Kim et al.⁶⁴ based on a modified weighted essentially non-oscillatory scheme (known as MWENO-Z) by Hu et al.⁶⁵. A second order central differential scheme was applied to the discretization of the diffusive term. The time integration for the convective and diffusive terms was the third-order three-step total variation diminishing Runge-Kutta method⁶⁶, and the multi-time scale method⁶⁷ was used for the time integration of the chemical source term.

The first-order Euler method was used for the integration of the Lagrangian particles. The gas phase quantities around the Lagrangian particles were estimated by interpolating the surrounding three nearby Eulerian cell values by the barycentric interpolation⁶⁸.

C. Recording the variables for each Lagrangian particle

<figure><img src="image_2.png" /><figcaption>FIG. 1. Schematics of the time history from shock passage $\tau$ of thermicity $\sigma$ for one Lagrangian particle. $\tau_{ind}$ and $\tau_{reac}$ are the induction and reaction times, respectively. The time when the particle has experienced half of the maximum thermicity $\sigma_{max}$ from shock passage is $\tau_{reac,1}$.</figcaption></figure>

An approach to record induction and reaction times from a Lagrangian perspective during simulations was proposed. The variables of each Lagrangian particle were recorded during the course of their trajectories, being updated every time step. The time when the Lagrangian particles passed the leading shock front $t_{shock}$ was recorded by the first pressure jump experienced by the Lagrangian particle to estimate the time from the shock passage $\tau = t - t_{shock}$. In order to obtain the
Induction and reaction times in a cellular detonation

induction and reaction times for each Lagrangian particle, the same simulation was repeated two times. In the first simulation, the maximum thermicity value $\dot{\sigma}_{max}$ and its time for each Lagrangian particle were recorded to estimate the induction time $\tau_{ind}$ (see Fig. 1). After the first simulation, from which the maximum thermicity and the time when thermicity was maximum were recorded, the same simulation was conducted a second time with the same initial conditions and numerical methods, so that first and second half maximum thermicity time could be recorded to obtain the reaction time $\tau_{reac}$.

In this study, the induction time $\tau_{ind}$ was defined as the time from the leading shock passage to the time of maximum thermicity $\dot{\sigma}_{max}$, and the reaction time $\tau_{reac}$ was defined as the duration of the half thermicity pulse.

This proposed Lagrangian approach not only recorded induction and reaction times, but also visualised induction and reaction zones. In other numerical and experimental methods, such as the combination of OH and NO laser-induced fluorescence, the visualisation of reaction length was not possible$^{37}$. The proposed approach has the advantage of characterising the induction and reaction times behind a cellular detonation front, and was used for the visualisation in section IV B and IV C. In addition, recording during the simulations reduced the storage capacity for output files and allowed the distribution of induction and reaction times inside the cell to be obtained.

In addition, the transverse wave time $\tau_{tw}$ was defined as time from leading shock front passage to the time when the particle experienced a transverse wave for the first time. This time was estimated by post-processing the computed data and was determined when the pressure ratio from two successive output files was higher than 1.2. In the following, the characteristic time scale for cell cycle$^{69}$ as $\tau_{cell} = L_{cell}/D_{CJ}$, where $L_{cell}$ and $D_{CJ}$ are the average length of the detonation cell and Chapman-Jouguet (CJ) velocity, respectively, was also used.

D. Recording the induction time from Eulerian perspective

To make a comparison on the induction time between Eulerian and Lagrangian approaches, the induction time from Eulerian numerical probes $\tau_{ind,E}$, following the approach used by Dormal et al.$^{32}$, was evaluated. The induction time from Eulerian numerical probes was evaluated by the following equation

$$ \tau_{ind,E} = t_{mes} \frac{\rho_s}{\rho_0}. \qquad (1) $$
Induction and reaction times in a cellular detonation

<table><caption>TABLE I. Parameters of the reactive mixtures in the present conditions.</caption><thead><tr><th>Parameters</th><th>2H<sub>2</sub>-O<sub>2</sub>-7Ar</th><th>2H<sub>2</sub>-O<sub>2</sub></th></tr></thead><tbody><tr><td>D<sub>CJ</sub> [m/s]</td><td>1690.7</td><td>2834.3</td></tr><tr><td>M<sub>CJ</sub></td><td>4.8</td><td>5.3</td></tr><tr><td>x<sub>ind,ZND</sub> [µm]</td><td>76.6</td><td>48.6</td></tr><tr><td>x<sub>reac,ZND</sub> [µm]</td><td>409</td><td>72.5</td></tr><tr><td>τ<sub>ind,ZND</sub> [µs]</td><td>0.2</td><td>0.09</td></tr><tr><td>τ<sub>reac,ZND</sub> [µs]</td><td>0.1</td><td>0.02</td></tr><tr><td>E<sub>a</sub>/(RT<sub>v,N</sub>)</td><td>4.8</td><td>6.7</td></tr><tr><td>χ</td><td>0.9</td><td>4.5</td></tr><tr><td>γ<sub>N</sub></td><td>1.49</td><td>1.32</td></tr></tbody></table>

Here, $t_{mes}$, $\rho_s$, and $\rho_0$ are the time from the shock passage to the time when OH mole concentration became the half of its maximum value, the post shock density, and the initial density, respectively. To evaluate $t_{ind,E}$, the same simulation was repeated twice and these values were recorded every grid point. In the first simulation, the time when the leading shock passed, the maximum OH mole concentration value, and the time when the OH mole concentration was maximum, were recorded in each grid point. In the second simulation, the half of the maximum OH mole concentration time was recorded, which allowed the evaluation of $t_{mes}$. Then, the induction time from Eulerian numerical probes $\tau_{ind,E}$ was obtained from Eq. 1.

### III. PROBLEM STATEMENT

The present study simulated a cellular gaseous detonation propagation in a 2D straight channel. The massless gaseous particles were uniformly distributed in unburned mixture in the initial condition. The fully developed 2D cellular gaseous detonation propagated in unburned mixture with massless particles over a long distance for making statistical analysis, and the evolution of the particle trajectories was monitored. This allowed the induction and reaction times to be recorded from a Lagrangian perspective to analyze the induction and reaction zones structure of detonation. Two types of reactive mixtures have been studied to investigate the influence of mixture stability:
Induction and reaction times in a cellular detonation

70% diluted stoichiometric hydrogen oxygen mixture 2H₂–O₂–7 Ar and stoichiometric hydrogen oxygen mixture 2H₂–O₂ at atmospheric conditions (0.1 MPa and 300 K). The channel widths for 2H₂–O₂–7 Ar and 2H₂–O₂ mixtures were 2.6 mm and 2.0 mm, respectively, with adiabatic no-slip walls. The simulation conditions and results were the same as in our previous study¹⁸.

Table I lists the various parameters⁷⁰ for both mixtures characterizing detonation such as the CJ velocity *D*<sub>CJ</sub>, the CJ Mach number *M*<sub>CJ</sub>, the induction length *x*<sub>ind,ZND</sub>, the reaction length *x*<sub>reac,ZND</sub>, the induction time τ<sub>ind,ZND</sub>, the reaction time τ<sub>reac,ZND</sub>, the reduced activation energy *E*<sub>a</sub>/(RT<sub>vN</sub>), the χ=<i>E</i><sub>a</sub>/(RT<sub>vN</sub>) × *x*<sub>ind,ZND</sub>/*x*<sub>reac,ZND</sub> parameter, and the specific heat ratio at von Neumann (vN) state γ<sub>N</sub>. Following the definition by Radulescu¹⁹ and Ng et al. ⁷¹, the induction length *x*<sub>ind,ZND</sub> was defined as the distance from the leading shock front to the position where the thermicity was maximum, and the reaction length *x*<sub>reac,ZND</sub> was estimated by *u*<sub>CJ</sub>/σ̇<sub>max</sub> using the maximum thermicity σ̇<sub>max</sub> and the velocity at the CJ plane in the shock frame *u*<sub>CJ</sub>. The reduced activation energy was estimated using zero-dimensional constant volume (0DCV) simulations with varying shock speed by ±1% *D*<sub>CJ</sub>. The mixtures can be classified as weakly and mildly unstable mixture, according to the stability analysis⁴,¹⁰ based on the reduced activation energy and CJ Mach number. In the classification of the stability of cellular detonation in reduced activation energy and CJ Mach number plane by Weng et al.⁷², the diluted and non-diluted mixtures are located in weakly nonlinear unstable and nonlinear unstable regimes, respectively. Based on the χ parameter and CJ Mach number, the instability parameters lie slightly below and above the neutral stability curve, for the diluted and non diluted cases⁷¹. The study of highly unstable mixture is beyond the scope of this study.

The outflow boundary was used as the boundary condition for the downstream side of the detonation front. The grid was uniform and was equal at 2.0 µm and 1.6 µm from the region from the shock front up to 20.6 mm and 11.5 mm behind the front for the 2H₂-O₂-7Ar and 2H₂-O₂ mixture, respectively. Then, the grid was stretched. The grid resolution was about 38 and 30 points per the ZND induction length (*x*<sub>ind,ZND</sub>) for 2H₂-O₂-7Ar and 2H₂-O₂ mixture, respectively. This resolution has been shown to be largely sufficient to capture the mean structure<sup>73,74</sup>. In addition, this resolution is enough to reproduce the feature of the instantaneous flow field for weakly unstable mixture<sup>75</sup>. In fact, grid convergence was evaluated by comparing simulation results from lower resolution with a grid width two times larger than that used in the manuscript. The average propagation velocity, the average cell width, and 1D Favre average profiles were well captured in the present grid resolution (see our previous work<sup>18</sup>). In addition, the convergence for the trend
Induction and reaction times in a cellular detonation

of the distribution of induction and reaction times from a Lagrangian approach was evaluated in the supplementary material. The trend of the distribution of the induction and reaction times was not found to change with the grid resolution used. Therefore, the conclusions drawn in this study were not called into question by the grid resolution. The recycling block technique<sup>76</sup> was applied to enable the detonation to propagate a distance long enough to obtain statistical values. The length of the propagation for the statistical analysis was about 1000 $x_{ind,ZND}$ for 2H<sub>2</sub>–O<sub>2</sub>–7Ar mixture and 1200 $x_{ind,ZND}$ for 2H<sub>2</sub>–O<sub>2</sub> mixture. The instantaneous 2D flow fields were saved each time the detonation front propagates 0.5 $x_{ind,ZND}$. The average cell width in the simulations from the manual measurement was estimated as 1.3 mm and 0.7 mm for 2H<sub>2</sub>–O<sub>2</sub>–7Ar mixture and 2H<sub>2</sub>–O<sub>2</sub> mixture, respectively. The average propagation velocity agreed with the CJ velocity in the present conditions.

The massless gaseous particles were initially located in the fresh mixture in every grid point. The number of these particles inside the computational domain changed during the simulation due to the recycling block method and were around 34 million and 25 million for 2H<sub>2</sub>–O<sub>2</sub>–7Ar and 2H<sub>2</sub>–O<sub>2</sub> mixture, respectively. The total number of particles in the region where the detonation propagates is about 50 million and 60 million for 2H<sub>2</sub>–O<sub>2</sub>–7Ar and 2H<sub>2</sub>–O<sub>2</sub> mixture, respectively.

IV. RESULTS AND DISCUSSIONS

A. State for unburned mixture and 0D constant pressure combustion in p-T diagram

Figure 2(a,d) showed instantaneous H₂ mass fraction, superimposed with density Schlieren, coming from the classical cellular structure. Here, only a portion of the computational was shown. The strong transverse wave structure occurred in the second part of the cell<sup>77</sup> in 2H₂ – O₂ – 7Ar mixture, whereas the reactive and nonreactive transverse waves were observed in 2H₂ – O₂ mixture. Moreover, the detonation propagation velocity in the horizontal direction can be shown to decrease along the cell and larger variations in the leading shock front velocity was found in the non-diluted case (Fig. 2(b,e)). From the simulations, the state for the unburned mixture behind the leading fronts was mostly distributed along the inert Rankine-Hugoniot curve below the von Neumann state (Fig. 2(c,f)). In addition, in 2H₂ – O₂ – 7Ar mixture, some portion of the unburned gas also laid along the reflected shocked gas state at ~ 0.86 D<sub>CJ</sub> with the second shock Mach
<figure><img src="image_3.png" /><figcaption>FIG. 2. Top: 2H₂ – O₂ – 7Ar mixture, bottom: 2H₂ – O₂ mixture. Left: Instantaneous flow fields of H₂ mass fraction superimposed with Schlieren density. Middle: flow field of the horizontal normalized detonation velocity *D*<sub>x</sub>/*D*<sub>CJ</sub> with leading shocks in white lines. Right: joint pdf of the unburned mixture in the *p* – *T* diagram. The unburned mixture whose H₂ mass fraction was higher than 0.01277 and 0.111 for 2H₂ – O₂ – 7Ar and 2H₂ – O₂ mixture, respectively, and slightly lower than its initial value was taken into account. The white solid line in (c,f) represents states along the Rankine-Hugoniot curve. White broken lines are states behind the reflected shock waves, whose initial states are related to post-shock states at 0.8 and 1.2 *D*<sub>CJ</sub>, respectively. The white circle, triangle and square in (c,f) are the post-shock states at 1.0, 0.8, and 1.2 *D*<sub>CJ</sub>, respectively.</figcaption></figure>

number around 1.2, due to the presence of transverse waves in the cellular structure (Fig. 2(c)). These value of second shock Mach number were similar with the previous estimation³,⁷⁸. The similar feature was obtained for the non-diluted case (Fig. 2(f)), with the second shock Mach number up to 1.4. However, the states along the reflected shock were more distributed in 2H₂–O₂
<figure><img src="image_4.png" /></figure>

Induction and reaction times in a cellular detonation

mixture, due to the more irregular cellular structure (Fig. 2(c,f)).

<figure><img src="image_5.png" /><figcaption>FIG. 3. p-T diagram for 2H₂ – O₂ – 7Ar mixture (top) and 2H₂ – O₂ mixture (bottom). Induction (left) and reaction (right) times from 0D constant pressure combustion process. The white solid line represents the state along the Rankine-Hugoniot curve. White broken lines are states behind the reflected shock wave. The white circle, triangle and square are the pots-shock states at 1.0, 0.8, and 1.2 D<sub>CJ</sub>, respectively. The solid black thick lines correspond to integer powers of 10, whereas the thin lines are increments between two successive powers. The pink thick line is isoline of τ<sub>cell</sub>.</figcaption></figure>

Zero-dimensional constant pressure (0DCP) simulations⁷⁹ were performed to evaluate the
dependency of the chemical times on initial conditions and Fig. 3 depicts the induction and
Induction and reaction times in a cellular detonation

reaction times in *p*-T diagram. From Fig. 3(a,b) for the diluted mixture, the induction time showed a strong temperature dependency, changing by orders of magnitude with respect to the initial temperature (Fig. 3(a)). In addition, the effect of initial pressure on induction time was less significant. On the other hand, the dependency of the reaction time with respect to the initial pressure was higher to that of the initial temperature (Fig. 3(b)). The same observation on the global trend of induction and reaction time in *p* − *T* diagram could be made for the non-diluted mixture (Fig. 3(c,d)). However, the variation in the induction time was more pronounced in the non-diluted mixture than that in the diluted mixture (6 against 3) due to higher reduced activation energy (Fig. 3(a,c)).

B. Structure of the chemical time scales for the diluted mixture

As expected from the deceleration and curvature of the leading shock front in Figs. 2(b,e), the distribution of the induction and reaction times were non-uniform. Two methods were used to estimate the induction times inside the cell from Eulerian perspective. The first estimation came from the 0D constant pressure estimation (without the loss term due to the volumetric expansion), with initial conditions from the post-shock conditions, coming from the horizontal instantaneous detonation velocities. The second estimation was determined from Eulerian numerical probes, following the approach used by Dormal et al.<sup>32</sup>, Eq. 1. The estimation of the reaction time was made with the 0DCP using the post-shock state as an initial condition. The average cell length from the simulation results for 2H<sub>2</sub> − O<sub>2</sub> − 7Ar and 2H<sub>2</sub> − O<sub>2</sub> mixture was 2.1 mm and 1.1 mm, respectively. Then, the characteristic time for cell cycle τ<sub>cell</sub> for 2H<sub>2</sub> − O<sub>2</sub> − 7Ar and 2H<sub>2</sub> − O<sub>2</sub> mixture were evaluated as 1.23 µs and 0.40 µs, respectively.

The distribution of induction and reaction times from the first and second approach for 2H₂ – O₂–7Ar mixture is shown in Fig. 4 and 5, respectively. Both estimations for the diluted case shown here were consistent with the previous experimental³²,³³ and numerical⁸⁰ finding that the induction time in the center line of the cell increased along the cell. Similarly, the reaction time in the center line of the cell from 0DCP results in Fig. 4(b) exhibited an increase along the cell. The same observations were also made in 2H₂ – O₂ mixture (not shown here).

However, the estimation from the Eulerian perspective did not take into account induced motion
of gaseous particle after the passage of the leading shock front, and multi-dimensional effect such
as transverse wave and jets, and the volumetric expansion due to the deceleration of the leading
Induction and reaction times in a cellular detonation

<figure><img src="image_6.png" /><figcaption>FIG. 4. Flow fields of (a) induction and (b) reaction times for the diluted case, determined from 0D constant pressure estimations based on post-shock conditions.</figcaption></figure>

shock front and frontal curvature. It should be noted that the transverse wave time was estimated from the two successive output data and the serration pattern in Fig. 6(c) was generated from the fact that the transverse wave time was made discrete by this procedure. From the Lagrangian particles, the chemical time scales as well as transverse wave time were then compared to the previous Eulerian estimations, in Fig. 6. Differences were qualitatively striking with the two previous estimations.

The first observation was that the induction process finished within 1/2 of the cell cycle timescale (Fig. 6(a)), with a non-monotonic behavior. The maximum induction time could be found around the center of the cell in its second part. Monnier et al.⁸¹ also assumed in their model that the complete induction time was assumed to be τcell/2, which is in line with our observation for our regular mixture. Near the edges of the second part of the cell, the induction time was shorter.
Induction and reaction times in a cellular detonation

<figure><img src="image_7.png" /><figcaption>FIG. 5. Flow field of induction time for the diluted case, determined from Eulerian numerical probes.</figcaption></figure>

The second observation was that the reaction time was also non-monotonic. The reaction time increased then decreased from the first to the second part of the cell. The main heat release occurred within 1/5 of the cell cycle timescale (Fig. 6(b)).

The third observation was that all Lagrangian particles experienced the transverse wave within one cell cycle (Fig. 6(c)), which could not be inferred from the Eulerian perspective, see Figs. 4 and 5. The transverse wave time was longer at the center line of the cell and particularly near the start of the cell cycle. On the other hand, near the edges of the cell in its second part, they experienced the transverse wave before approximately the ZND induction time (not shown here).

The decrease of both induction and reaction times in the second part of the cell can be clearly related to the transverse wave passage. Even though the Mach number of these transverse waves was not very high (up to about 1.2), as compared to that of the leading shock, the topology of the chemical time scales in the *p*–*T* diagram (see Fig. 3(a,b)) made their influence non-negligible.

The ratio of the induction time to the reaction time $\tau_{ind}/\tau_{reac}$ was lower than one in the first part of the cell and around the edges of the cell in its second part. This ratio then increased up to 30 (see Subsect. IV F) locally due to much shorter reaction time in the regions where the collision of the transverse waves occurred. This is consistent with the results from Fig. 3, where the transverse waves were shown to decrease substantially the reaction time due to the increase of pressure, as compared to the less evident influence of pressure on induction times.

To explain the distribution of induction time from the Lagrangian perspective, the time sequence of 2D instantaneous particle flow fields of induction time is depicted in Fig. 7. In
Induction and reaction times in a cellular detonation

<figure><img src="image_8.png" /><figcaption>FIG. 6. Flow fields for (a) induction time $\tau_{ind}/\tau_{cell}$, (b) reaction time $\tau_{reac}/\tau_{cell}$, and (c) transverse wave time $\tau_{tw}/\tau_{cell}$. The Lagrangian data are displayed in their initial position. The mixture is 2H$_2$ – O$_2$ – 7Ar.</figcaption></figure>

the first part of the cell, the induction times was shorter due to higher initial temperature, behind
the leading Mach stem (Fig. 7(a,b) and 3(a)). In addition, the particles located initially just at
the start of the cell in the center line had the shortest induction time (Fig. 6(a)). In the second
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Induction and reaction times in a cellular detonation</figcaption></figure>

FIG. 7. Time sequence of instantaneous 2D flow fields of the induction time for 2H₂ – O₂ – 7Ar mixture. Only the Lagrangian particles whose time from shock front passage is less than the induction time are displayed. The lines are the density Schlieren and the gray contour in the background is the maximum pressure. The detonation propagated from the left to the right and the time passed from (a) to (f). The times of the snapshots are (a) t₀, (b) t₀ + 0.29 µs, (c) t₀ + 0.48 µs, (d) t₀ + 0.78 µs, (e) t₀ + 1.09 µs, and (f) t₀ + 1.21 µs.

part of the cell near its edges, the Lagrangian particles experienced transverse waves soon after the passage of the incident shock front, resulting in shorter induction time despite the lower initial temperature induced by the weaker leading shock front (Fig. 7(c,d,e,f)). As for the particles initially located around the center of the cell in the second part of the cell, the induction process was enhanced by the transverse wave passage, and also the transverse wave collision and the collision of triple points (Fig. 7(e,f)). The point of transverse wave collision behaved as a phase wave (see Subsect. IV E), which consumed the unburned gas behind the decaying incident shock.

Moreover, the present visualization of the induction zone from Lagrangian perspective by
Induction and reaction times in a cellular detonation

<figure><img src="image_11.png" /><figcaption>FIG. 8. Time sequence of instantaneous 2D flow fields of the reaction time for 2H₂ – O₂ – 7Ar mixture. Only the Lagrangian particles within half pulse width thermicity are displayed. The lines are the density Schlieren and the gray contour in the background is the maximum pressure. The detonation propagated from the left to the right and the time passed from (a) to (f). The times of the different snapshots are (a) t₀, (b) t₀ + 0.29 µs, (c) t₀ + 0.48 µs, (d) t₀ + 0.78 µs, (e) t₀ + 1.09 µs, and (f) t₀ + 1.21 µs.</figcaption></figure>

displaying particles before their induction time in Figs. 7 qualitatively agreed with the experimental visualization of the induction zone by NO-PLIF technique in the studies by Chavez et al.³⁶ and Alicherif et al.³⁸. Further experimental details can be found in Chavez et al.³⁶. The same features along the center line of the cell, such as shorter induction length behind the Mach stem, longer induction length behind the incident shock, and maximum induction length around the end of the cell cycle were observed. The keystone structure was also evident in the weakly unstable mixture as in previous experimental studies³,³⁶.

The flow field analysis further revealed that the ignition front behind the leading shock intersected the reactive transverse wave, resulting in the first pattern in Fig. 6(a). The second
Induction and reaction times in a cellular detonation

pattern came from the ignition by the transverse wave collision. The collision of the second triple points, separating the weak and strong transverse waves, generated the third pattern.

The time sequence of 2D instantaneous particle flow fields of reaction time is depicted in Fig. 8. The reaction time for particles in the first part of the cell gradually increased along the cell due to the decrease in the propagation velocity as seen in the trend in Fig. 3(b) (Fig. 8(a,b,c,d)). Once particles experienced the transverse wave before the end of induction process, the reaction time became shorter (Fig. 8(d,e,f)). The shortest reaction time was found for particles, which were ignited by the collision of the transverse waves (Fig. 8(f)).

Near the edge of the second part of the cell where the ignition was induced by the weak transverse wave, a small lagged region, according to the terminology of Hu et al.²⁷, was formed and the reaction time was longer than that for particles, which experienced the strong transverse wave (Fig. 8(d,e,f)).

The slight decrease in reaction time near the border line between the autoignition by Mach stem and reactive transverse wave after the passage of incident shock was probably due to enhanced reaction by collision of transverse waves (Fig. 8(f)).

C. Structure of the chemical time scales in the non-diluted mixture

The average cell length from the simulation results for 2H₂ – O₂ mixture was 1.1 mm. Then, the characteristic time for cell cycle τcell for 2H₂ – O₂ mixture was evaluated as 0.40 µs.

Figure 9 presents the distribution of induction time, reaction time and transverse wave time from Lagrangian perspective for this mildly unstable mixture. The same serration pattern as in Fig. 6(c) can be found in Fig. 9(c), due to the discrete numerical procedure. Not only similarities but also differences in the time scale distribution were found between the diluted and non-diluted mixtures (Figs. 6 and 9).

Firstly, larger variation in induction and reaction times was present in the non-diluted mixture than in the diluted mixture, due to higher reduced activation energy and irregularity of the cellular structure (Figs. 6(a,b) and 9(a,b)). Although the induction process completed within half of the cell cycle in the diluted mixture (Figs. 6(a)), the induction time in the non-diluted mixture in some cases was increased up to one cell cycle, due to the presence of the unburned gas pocket and to larger cells (Fig. 9(a)).

Indeed, the maximum induction time was found not only around the center of the cell as in the
<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>FIG. 9. (a) same caption as in Fig. 6(a), (b) same as Fig. 6(b), and (c) same as Fig. 6(c). The Lagrangian data are displayed in their initial position. The mixture is 2 H₂ – O₂.</figcaption></figure>

diluted mixture but also near the end of the second part of the cell for particles, which formed the
Induction and reaction times in a cellular detonation

unburned gas pockets. Besides, the reduction in the induction time by the forward and backward jets was also apparent, due to stronger jets by lower specific heat ratio$^{82-84}$. As for the cells with higher propagation velocity (Fig. 2(e)), the induction time was shorter throughout the cell.

Furthermore, reaction time in the second part of cell is shorter due to higher initial pressure by transverse wave. On the contrary, some particles in the second part of the cell had longer reaction times due to unburned gas pockets (Fig. 9(b)).

Moreover, the global trend of the distribution of the transverse wave time was similar between both mixtures (Fig. 6(c) and Fig. 9(c)). The maximum transverse wave time is on the order of the time required for one cellular cycle and could be found near the centerline at the beginning of the cell. The transverse wave time was shorter near the edges of the second part of the cell.

Figure 10 depicts the time sequence of 2D instantaneous particle flow field of induction time for 2H₂ – O₂ mixture. As in 2H₂ – O₂ – 7Ar mixture, the induction length was shorter behind the Mach stem and the longer induction length was found behind the incident shock as a global feature. On the other hand, more variation in the induction length behind the leading shock front was observed due to higher reduced activation energy. Some transverse waves were nonreactive and unburned gas pockets were formed, resulting in longer induction time than half of the cell cycle. In this mildly unstable mixture, the unburned gas pockets were consumed within the order of one cell cycle after the passage of few transverse waves (Fig. 10(d)). By contrast, the particles experiencing reactive transverse wave or transverse detonation completed the induction process soon after the passage of transverse wave/detonation (Fig. 10(d,e,f)).

The intersection of the transverse waves, which behave as a phase wave, interacts with the lagged regions, igniting them. Compression waves were then generated, which caught up with the Mach stem, enhancing its strength and decreasing the induction and mainly the reaction times (Fig. 10(a,b)).

In some cases, the induction process was shortened due to the interaction of the backward jet with the unburned gas pocket (Fig. 10(c,d)). The tip of the jet pushed the unburned pocket and increased its stagnation temperature$^{85}$.

The time sequence of 2D instantaneous particle flow fields of reaction time for 2H₂ – O₂ mixture is illustrated in Fig. 11. The reaction front shape was more irregular than that in the diluted case due to the increase in reduced activation energy$^{86}$ (Figs. 8 and 11). As in the diluted mixture, the reaction time was shorter behind the Mach stem and gradually increased along the cell in the first part of the cell (Fig. 11(a,b,c)). In the second part of the cell, particles experiencing
<figure><img src="image_14.png" /></figure>

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0249984

Induction and reaction times in a cellular detonation

<figure><img src="image_15.png" /><figcaption>FIG. 10. Time sequence of instantaneous 2D flow fields of the induction time for 2H₂ – O₂ mixture. Only the Lagrangian particles whose time from shock front passage is less than the induction time are displayed. The lines are the density Schlieren and the gray contour in the background is the maximum pressure. The detonation propagated from the left to the right and the time passed from (a) to (f). The times of the different snapshots are (a) t₀, (b) t₀ + 0.11 µs, (c) t₀ + 0.26 µs, (d) t₀ + 0.43 µs, (e) t₀ + 0.51 µs, and (f) t₀ + 0.53 µs.</figcaption></figure>

the transverse wave had shorter reaction time due to the higher initial pressure (Fig. 11(d,e,f)). Especially, the transverse detonation and the strong transverse wave induced the shortest reaction time (Fig. 11(d,e,f)), whereas both induction and reaction times became longer behind the weak transverse wave (Fig. 11(c,e,f)).

Moreover, from the reaction zone visualization, the unburned gas pocket were consumed from its edges, which meant that the autoignition inside the unburned gas pocket was not the main mechanism for the ignition (Fig. 11(a,d,e)). In this mildly unstable mixture, the unburned gas
<figure><img src="image_16.png" /></figure>

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0249984

Induction and reaction times in a cellular detonation

<figure><img src="image_17.png" /><figcaption>FIG. 11. Time sequence of instantaneous 2D flow fields of the reaction time for 2H₂ – O₂ mixture. Only the Lagrangian particles within half pulse width thermicity are displayed. The lines are the density Schlieren and the gray contour in the background is the maximum pressure. The detonation propagated from the left to the right and the time passed from (a) to (f). The times of the snapshots are (a) t₀, (b) t₀ + 0.11 µs, (c) t₀ + 0.26 µs, (d) t₀ + 0.43 µs, (e) t₀ + 0.51 µs, and (f) t₀ + 0.53 µs.</figcaption></figure>

pocket were burned within the order of one cell cycle after the passage of transverse waves. In addition, the reaction time for the unburned gas pocket was longer than that for other particles in the second part of the cell.

D. Ignition mechanisms

The analysis of induction and reaction zone visualization by Lagrangian particles highlighted
the ignition mechanisms in cellular gaseous detonation.
Induction and reaction times in a cellular detonation

The following four main factors contributed to the ignition in weakly unstable 2H₂ – O₂ – 7Ar mixture.

The first factor is the adiabatic shock compression by Mach stem. The adiabatic shock compression by Mach stem was strong enough to induce the ignition before the effect of the transverse wave.

The second factor was the strong transverse wave after the passage of the incident shock⁸⁷.

On the other hand, particles which passed the weak transverse wave had longer induction time and formed the lagged region²⁷. As a third factor, the ignition of the lagged region occurred, after the transverse wave collision.

The fourth factor for ignition was triple point collision, as explained by Libouton et al.³³.

As a result of all these processes, no unburned gas pockets were formed in the weakly unstable mixture.

In mildly unstable 2H₂ – O₂ mixture, four additional ignition mechanisms were present compared to ignition mechanisms for 2H₂ – O₂ – 7Ar mixture.

The first factor was the surface burning of the unburned gas pocket. Some of transverse waves became unreactive and the unburned gas pockets were formed behind the leading front⁸⁵. As seen in the reaction zone visualization in Fig. 11, the exothermic chemical reaction took place from the edges of the unburned gas pocket. Nevertheless, in this mildly unstable mixture, the consumption of unburned gas pockets was enhanced by passage of transverse waves and completed within order of 1 cell cycle.

The second additional factor came from the increased intensity of the forward and backward jets by the lower specific heat ratio at vN state. The increase in temperature inside unburned gas pocket due the deceleration of the gas by the backward and forward jets clearly favored the ignition⁸⁵.

Thirdly, one can also observe compression waves coming from the rapid consumption of the lagged region by the transverse wave collision and second triple point collision. These pressure waves enhance the induction and reaction processes, around the first quarter of the cell⁸⁸,⁸⁹.

Moreover, the fourth factor was the presence of transverse detonation with the reactant being consumed rapidly after its passage.

To sum up from the Lagrangian analysis, in the first part of the cell, Mach stem was sufficient to induce the ignition. In the second part of the cell, the incident shock was never found to be the sole factor for ignition within the cell time scale. There was always additional compression mechanism coming into play, either transverse waves (first role), transverse wave collision, or triple point collision (see also supplementary material for time history of variables for particles),
<figure><img src="image_18.png" /></figure>

Induction and reaction times in a cellular detonation

in line with the initial suggestion of Mitrofanov²⁵. As for the non-diluted mixtures, some of
the ignition mechanisms were the same, but with additional features like transverse detonations.
Minor mechanisms came from the surface burning mainly local shock-induced combustion after
the passage of transverse waves, compression waves coming from the consumption of the lagged
regions, and jets.

In the present weakly and mildly unstable mixtures, level of instability is not high enough for the
role of turbulent burning to be significant as in highly unstable mixture²²,²³.

E. (x,t) diagrams along the center line and particle trajectory

The trajectory of the particles and the reaction zone structure were analyzed using the (x,t)
diagram in the center line of the cell. Figure 12 shows (x,t) diagram of the magnitude of density
gradient in the mean shock fixed frame, together with the trajectory of selected particles until the
second half maximum thermicity time (τreac + τreac,1 in Fig. 1); the difference in H₂O mass fraction
between the instantaneous and the equilibrium values; and the leading shock velocity history for
the diluted and non-diluted mixtures. The initial position of the selected particles were shown in
the supplementary material.

In 2H₂-O₂-7Ar mixture, the pulsation of leading shock front ranged from about 0.85 ~
1.40 DCJ (Fig. 12(c)), corresponding to the classical 1D pulsating oscillations⁹⁰,⁹¹. However,
patterns that are indirect signature of multidimensional effects can also be observed. Indeed,
the transverse waves correspond to the slightly inclined, and more or less horizontal curves,
representing high velocity phase waves. Jets originate from the triple point collision. In the (x,t)
diagram, they can be assimilated as waves propagating at the material velocity (Fig. 12(a,b)). The
information that particles were experiencing jets remained over several cell cycles. The particle
trajectories highlighted that the ignition took placed due to adiabatic compression by the leading
shock (Fig. 12(a)), which is a Mach stem in the first part of the cell. This leading shock was greater
than the CJ velocity up to 0.45 cell length for this cell. On the other hand, the particles, which were
initially located after 0.6134 times cell length from the start of the cell, completed their induction
process with the help of the transverse wave. The end of induction process by transverse wave
around the end of cellular cycle agreed with the schematic by Libouton et al.³³. In addition, the
reaction time can be seen to be shorter for these particles due to higher initial pressure behind
transverse wave. Within the time cell cycle, more oscillations with smaller amplitude can be seen.
<figure><img src="image_19.png" /><figcaption>Induction and reaction times in a cellular detonation</figcaption></figure>

FIG. 12. (x,t) diagram along the center line of the detonation cell in mean shock frame for (a,b,c) 2H₂ – O₂ – 7Ar and (d,e,f) 2H₂ – O₂ mixtures. (a,c) Magnitude of density gradient with particle trajectories in red. (b,d) Difference in mass fraction of H₂O between the instantaneous and the equilibrium values with the contour line of magnitude of density gradient in white color. (c,f) Normalized propagation velocity. The particle trajectory until the first half maximum thermicity time, particle trajectory from the first half maximum thermicity time to the maximum thermicity time, and the particle trajectory from the maximum thermicity time to the second half maximum thermicity time are shown in red, blue and green lines, respectively. The 22 particles (every 0.05 times cell length from the start to the end of one particular cell and 1 particle with maximum induction time inside the cell are displayed in (a,d). The initial position of the particle with maximum induction time normalized by cell length of the particular cell for 2H₂ – O₂ – 7Ar and 2H₂ – O₂ mixture is 0.6134 and 0.5340 from the start of the cell, respectively. The cell length for the particular cell for 2H₂ – O₂ – 7Ar and 2H₂ – O₂ mixture is 2.08 mm and 1.60 mm, respectively.

This was due to the misalignment of the cell with the horizontal axis, and also due to the pressure
Induction and reaction times in a cellular detonation

waves induced by the combustion of the lagged region.

In 2H₂ – O₂ mixture, the leading shock front pulsated from about 0.80 ~ 1.45 D<sub>CJ</sub> (Fig. 12(f)). Compared to 2H₂ – O₂ – 7Ar mixture, more waves were present. For most of them, they can be assimilated to compression waves, which projection in the 1D plane can be assimilated to pressure waves propagating at speeds *u* ± *c* (Fig. 12(d,e)). As for particles which were initially located after 0.534 times cell length (Fig. 12(d)), transverse detonation finished the induction process near the end of the cell and the reaction took place rapidly in this particular cell.

The CJ theory in the ideal case assumed a chemical equilibrium at the end CJ state. However, the burned gases after the main heat release zone were found to be out of chemical equilibrium (Fig. 12(b,d)). In the diluted 2H₂ – O₂ – 7Ar mixture, just after the leading shocks, the fresh gases composition is frozen, the difference between the instantaneous and the equilibrium values was negative. Along the particle trajectories, this difference relaxed towards zero. Transverse waves markedly disrupted this equilibrium, especially when they interacted with the remnant of forward and backward jets. Indeed, with the increase of temperature, the equilibrium value of H₂O mass fraction decreases due to dissociation. Thus, water species, immediately downstream of the transverse wave, was found to be in excess, resulting in super-equilibrium. There was a time lag for the flow to relax towards its equilibrium value. The remnant of the forward and backward jets had closer state to the chemical equilibrium than the other part, and trait of them was seen along the entropy wave in (x,t) diagram. In addition, compression waves slightly affected the local condition from the chemical equilibrium. In the non-diluted case, an additional feature can be found. The contribution of the compression waves, propagating at ~*u* ± *c* was more pronounced for the disruption of the equilibrium states.

Figure 13 depicts the 2D instantaneous flow fields of difference in H₂O mass fraction between the local and equilibrium values, in a domain much larger than that of the (x,t) diagram. In the diluted case, the departure from the chemical equilibrium became lower as the distance from the leading shock front increased and the influence of transverse wave on the departure from chemical equilibrium weakened as they decayed (Fig. 13(a)). In the non-diluted case, although the departure from the chemical equilibrium became lower in far field from the leading shock front, a certain level of chemical disequilibrium still remained locally (Fig. 13(b)).

Note that from our previous study¹⁸, the hydrodynamic thickness, which is the distance between the leading shock and the sonic line and is determined from Eulerian mean values, was 8.8 mm and 6.3 mm for 2H₂ – O₂ – 7Ar and 2H₂ – O₂ mixtures, respectively. The region where the
<figure><img src="image_20.png" /></figure>

Induction and reaction times in a cellular detonation

<figure><img src="image_21.png" /><figcaption>FIG. 13. 2D instantaneous flow fields of difference in H₂O mass fraction between the local and equilibrium values for (a) 2H₂ – O₂ – 7Ar and (b) 2H₂ – O₂ mixture, respectively. The contour line of magnitude of density gradient was superimposed in the figure.</figcaption></figure>

departure from chemical equilibrium was reduced corresponded roughly to the region behind the mean sonic plane, this departure being more important for the more unstable case. Edwards et al.⁹² observed the attenuation of the transverse wave two to four cell length downstream of the front and suggested the link between the decay of the transverse waves and the establishment of the sonic plane from the comparison of the estimation of hydrodynamic thickness by Vasilev et al.⁹³. From this observation, Jarsalé et al.⁹⁴ also used the decay of the pressure signal to estimate the hydrodynamic thickness. The decay of the transverse wave was thought to be responsible for not only the attenuation of the mechanical fluctuations but also the reduction of the disturbances inducing chemical disequilibrium, leading to the establishment of the mean sonic plane. The numerical results presented here did not entirely align with the aforementioned assumptions, as a certain degree of chemical disequilibrium, albeit likely minimal, locally persisted.

F. Time scales analysis

Seven characteristic time scales along the center line of the cell were quantitatively compared with each other. Two were chemical time scales, i.e., induction time $τ_{ind}$ and reaction time $τ_{reac}$. Four came from the hydrodynamic phenomena and were the transverse wave time $τ_{tw}$, the expansion time $τ_{exp}$, the diffusive time inside the induction zone $τ_{diff,ind}$, and the diffusive time
Induction and reaction times in a cellular detonation

inside the reaction zone $\tau_{\text{diff,react}}$, respectively. The last one was the cell characteristic time $\tau_{\text{cell}}$, which was related with the cellular cycle.

The expansion time was defined as Eq. 2 using the material derivative of density

$$ \tau_{\text{exp}} = -\frac{1}{\rho} \frac{D\rho}{Dt}. \qquad (2) $$

The minimum expansion time, which was estimated from the shock passage to the first half maximum thermicity time $\tau_{\text{react},1}$ (see Fig. 1) was used later to discuss the possibility of quenching due to unsteady expansions.

Following the approach by Taileb et al.⁹⁵, the diffusive times were evaluated using the square of the length scale divided by thermal diffusivity.

$$ \tau_{\text{diff,ind}} = \frac{x_{\text{ind}}^2}{\alpha_{\text{max,ind}}}, \qquad (3) $$

$$ \tau_{\text{diff,react}} = \frac{x_{\text{react}}^2}{\alpha_{\text{max,react}}}, \text{ such as } x_{\text{react}} = \frac{u_{\text{CJ}}}{\dot{\sigma}_{\text{max}}}. \qquad (4) $$

Here, $x_{\text{ind}}$ is the induction length evaluated from the distance from the shock to the particle position at $\tau_{\text{ind}}$. $\alpha_{\text{max,ind}}$ was the maximum thermal diffusivity inside the induction zone.

The reaction length $x_{\text{react}}$ was estimated as¹⁹,⁷¹ by Eq. 4, and $\alpha_{\text{max,react}}$ was the maximum thermal diffusivity inside the reaction zone. $u_{\text{CJ}}$ is the velocity at the CJ plane in the shock fixed frame and $\dot{\sigma}_{\text{max}}$ is the maximum thermicity for each Lagrangian particle, respectively.

Furthermore, the Damköhler number $D_a$ defined by Eq. 5 was compared with the critical Damköhler number for exponential expansion $D_{a,cr,exp}$ and the critical Damköhler number for power law expansion $D_{a,cr,pow}$ to discuss the possibility of quenching due to volumetric expansion. The analytical formula for $D_{a,cr,exp}$ and $D_{a,cr,pow}$ were summarized by Radulescu and Maxwell¹¹ and given by Eqs. 6 and 7.

$$ D_a = \frac{\tau_{\text{exp}}}{\tau_{\text{ind}}}, \qquad (5) $$

$$ D_{a,cr,exp} = (\gamma_s - 1) \frac{E_a}{RT_s}, \qquad (6) $$

$$ D_{a,cr,pow} = (\gamma_s - 1) \frac{E_a}{RT_s} - \frac{1}{n}. \qquad (7) $$
Induction and reaction times in a cellular detonation

Here, n is a constant to model a self-similar expansion. In the case of spherical Taylor-Sedov blast waves, the analytical formulas for n for strong (near the center) and weak (far field) shock waves were derived by Radulescu and Maxwell¹¹ as in Eqs 8 and 9, from post-shock values.

$$n_{\text{strong}} = \frac{2(13 + 5\gamma_s)}{5(1 + \gamma_s)^2}, \qquad (8)$$

$$n_{\text{weak}} = \frac{6}{5\gamma_s}. \qquad (9)$$

There are some limitations to the above models (in Eqs. 6 and 7) for predicting quenching by rapid expansion. Eckett et al.¹⁰ considered that a constant post-shock expansion time. Paknahad et al.⁵⁹ also recently found that the shock decay in 2D and 3D detonation was lower than that predicted from blast-wave theory. The critical Damköhler number in Eqs. 6 and 7 was derived for one-step simplified chemistry. In the case of a detailed chemical model, the response of the ignition to volumetric expansion can be complex due to non-monotonous evolution of ignition time with pressure and temperature⁹⁶, which may lead to a different critical Damköhler number. Moreover, our simulations showed a non-constant Damköhler number (see supplementary material).

Figures 14 and 15 depict 1D plots of the characteristic time scales in the center line of the cell for 2H₂ – O₂ – 7Ar and 2H₂ – O₂ mixtures, respectively. The shock propagation velocity, time history of variables and Damköhler number can be found in supplementary material. The selected particles to evaluate these characteristic time scales were the same as those used in previous subsection IV E.

In 2H₂ – O₂ – 7Ar mixture, the induction and reaction process were completed within 1/2 cell cycle (Fig. 14(a)). The transverse wave time monotonically decreased along the cell and every particle experienced the transverse wave within one cell cycle. The expansion time was longer than the induction time and tended to increase along the cell.

The diffusive times inside the induction zone and reaction zone were much longer than the induction and reaction time, and $\tau_{\text{diff,ind}}/\tau_{\text{ind}}$ and $\tau_{\text{diff,react}}/\tau_{\text{react}}$ ranged from $10^2 \sim 10^4$, and $10^4 \sim 10^5$, respectively (Fig. 14(a,b)). This means that the diffusive process can be considered to be minor in the induction and reaction process in the diluted 2H₂ – O₂ – 7Ar mixture.

The ratio of transverse wave time to induction time became lower than 1 after 0.6134 cell length, which meant that the particles experienced transverse wave before the induction process in this region (Fig. 14(c)). After they experienced the transverse wave before the induction process, the
<figure><img src="image_22.png" /><figcaption>FIG. 14. Characteristic time scales in the center line of the cell for 2H₂ – O₂ – 7Ar mixture. (a) Characteristic time normalized by τcell, (b) characteristic time ratio, (c) τtw/τind and comparison between Lagrangian results and 0D constant pressure modeling results, and (d) ratio of Damköhler number to critical Damköhler number.</figcaption></figure>

induction and reaction times were clearly changed, with a substantial decrease (Fig. 14(c)). The fact that the ratio of transverse wave time to induction time was around 1 in the second part of the cell highlighted that the transverse wave greatly enhanced the chemical reaction, even in this weakly unstable mixture.

In addition, estimations of the chemical times scales can be made from 0D constant pressure modeling using post-shock states as initial conditions. They were then compared to the simulation results. There was a clear deviation for particles experiencing transverse wave before the induction process. On the other hand, the ratio of transverse wave time to induction time for particles in the first part of the cell was larger than 1, meaning that the transverse wave did not influence the ignition mechanism.

From the shock velocity, ignition timescales of 0D constant volume combustion could be estimated.
<figure><img src="image_23.png" /><figcaption>FIG. 15. (a) same caption as in Fig. 14(a), (b) same as Fig. 14(b), (c) same as Fig. 14(c), and (d) same as Fig. 14(d). The mixture is 2H₂ – O₂.</figcaption></figure>

In the first part of the cell (before 0.6134 $x/L_{cell}$), the Damköhler data coincide with the estimation from the simulations. No quenching due to expansion could be expected. In the second part of the cell, the estimations from the shock states indicate some degree of quenching, diverging from the simulation results, specially at the end of the cell. The analysis provided by the Damköhler data were thus only consistent for particles in the first part of the cell. Indeed, in the second part of the cell (after 0.6134 $x/L_{cell}$), the particles undergo the strong influence of the transverse waves, collisions of transverse waves and triple points, all of them being purely 2D effects.

Even though the ratio of induction time to reaction time which is the indicator of the stability of the cellular structure⁹⁷,⁹⁸ is 0.19 in ZND model, the ratio of induction time to reaction time increased up to about 30 in the second part of the cell even in this weakly unstable mixture (Fig. 14(b)).

Furthermore, in the mildly unstable 2H₂ – O₂ mixture, the induction and reaction process were completed within 1 cell cycle for the selected particles in this cell (Fig. 15(a)). The particles experienced transverse wave within 1 cell cycle, with the transverse wave time decreasing along
Induction and reaction times in a cellular detonation

the cell. The expansion time was about 1 to 100 times larger than the induction time (Fig. 15(a,b)). Although diffusive times inside the induction and reaction zones were closer to the characteristic time of the cell as compared to the diluted case, $\tau_{\text{diff,ind}}/\tau_{\text{ind}}$ and $\tau_{\text{diff,react}}/\tau_{\text{react}}$ ranged from $10^2$ to $10^3$, and $10^3$ to $10^4$, respectively. The diffusive process can be expected not to play a major role for induction and reaction process for these particles (Fig. 15(a,b)).

The ratio of induction time to reaction time became much larger than the inverse of the reduced activation energy (the $\chi$ parameter greater than 1) and up to 600 in the second part of the cell (Fig. 15(b)). As in 2H$_2$–O$_2$–7Ar mixture, the ratio of transverse wave time to induction time became closer to one after 0.534 cell length from the start of the cell cycle (Fig. 15(c)), which highlighted the significant role of transverse wave on the chemical reactions. Once the particles experienced transverse wave before the induction process, the trend of the distribution of induction and reaction times clearly changed and deviated even more from the results using 0D constant pressure combustion modeling. In addition, the thing to note is that in the second part of the cell, the induction time from 0D constant pressure modeling using post-shock condition as initial condition was larger than the time for one cell cycle (red cross in Fig. 15(c)), ratio of which reached two orders of magnitude around the end of the cell. This fact implied that the adiabatic shock compression was not sufficient enough to induce the ignition. Other mechanisms such as transverse wave for instance played a significant role in the ignition.

Furthermore, the ratio $D_a/D_{a,cr}$ based on induction time from 0D constant volume modeling became much lower than 1 after around 0.3 cell length. As compared to the diluted case, this trend is enhanced due to higher reduced activation energy and shorter expansion time. This meant that the quenching due to the volumetric expansion is more likely to occur. Indeed, Austin$^2$ also found the possibility of quenching around the end of the cell in unstable mixture by analyzing the numerical data of Gamezo et al.$^{99}$ in terms of critical decay rate model$^{10}$. This finding suggested that the adiabatic shock compression was not sufficient for ignition. In this non-diluted case, 2D effects were mandatory for ignition, with the following list of phenomena in order of importance: the first group being the strong influence of transverse waves, transverse detonations, their collisions as well as that of triple points, and the second group being the combustion of lagged regions, and jet effects. Here, we have to be cautious with the present conclusion as the expansion time was not constant during the particle trajectory.

The comparison of $\tau_{\text{diff,ind}}/\tau_{\text{ind}}$ and $\tau_{\text{diff,react}}/\tau_{\text{react}}$ suggested that the role of diffusive process in induction and reaction process was not major in the weakly and mildly unstable mixtures tested.
Induction and reaction times in a cellular detonation

In fact, Singh et al.¹⁰⁰ concluded from numerical study that the role of diffusion in high speed one-dimensional steady flame and detonation is negligible, which was in line with the present simulations. Nevertheless, this observation may not be valid for more unstable mixture with irregular structures (turbulent transport, larger unburned gas pockets, and longer hydrodynamic thickness).

G. Joint pdf of induction and reaction times

<figure><img src="image_24.png" /><figcaption>FIG. 16. Joint pdf between induction/reaction times from Lagrangian perspective and horizontal detonation velocity for (a,b) 2H₂ – O₂ – 7Ar mixture and (c,d) 2H₂ – O₂ mixture. (a,c) Induction time τᵢₘ₉/τᵣₐELL. (b,d) Reaction time τᵣₐELL/τᵣₐELL. The white line represents the result from 0D constant pressure combustion modeling using the post shock state as the initial condition. Solid black thick lines correspond to the integer powers of 10, whereas thin lines are increments between two successive powers.</figcaption></figure>
Induction and reaction times in a cellular detonation

<figure><img src="image_25.png" /><figcaption>FIG. 17. 2D flow fields for the ratio of Lagrangian results to results from 0D constant pressure combustion modeling using post-shock conditions in 2H₂ – O₂ – 7Ar mixture. (a) Induction time and (b) reaction time.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIG. 18. Same caption as in Fig. 17 for 2H₂ – O₂ mixture.</figcaption></figure>

The distribution of induction and reaction times was further compared with predictions from steady 0D constant pressure combustion with the leading shock front velocity, as initial conditions. Figure 16 depicts the joint pdf between the induction / reaction times and the horizontal detonation velocity. The 2D flow fields for the ratio of Lagrangian results to steady model predictions for the diluted and non-diluted mixtures are depicted in Figs. 17 and 18, respectively.

The simulation results deviated from those from the steady model, due to volumetric expansion and compression from transverse waves (Figs. 16, 17, and 18).

In 2H₂ – O₂ – 7Ar mixture, most particles had longer induction time (Figs. 16(a), 17(a)). However, some particles, which experienced a leading shock with about 0.9 ~ 1.1 D<sub>CJ</sub> showed
Induction and reaction times in a cellular detonation

similar trend than the steady model with a slight increase. The particles, with the leading shock
around 1.2 D<sub>CJ</sub> were located near the edges of the cell in the first part of the cell (Fig. 17(a)),
with longer induction time compared with steady model. On the other hand, some particles had
shorter induction time than that from the steady model. The induction process was enhanced from
transverse wave passage and their induction time became shorter than the steady prediction.

The distribution of the reaction time in 2H<sub>2</sub> – O<sub>2</sub> – 7Ar mixture showed two main regions different from the steady model: longer reaction time due to expansion, shorter reaction time due to transverse wave passage (Figs. 16(b), 17(b)). As seen in Figs. 6(b) and 8, particles which experienced transverse wave before induction process had shorter reaction time due to higher initial pressure as compared to post-shock conditions. This reduction in reaction time induced deviations from the steady model. Especially, collision of triple points induced significant reduction in reaction time (Fig. 17(b)). For particles, which do not experience the influence of transverse wave, the volumetric expansion caused an increase in reaction time.

Moreover, the deviation from the steady model prediction on induction time was also observed
in 2H₂ – O₂ mixture (Figs. 16(c), 18(a)). Compared with the diluted case, the distributions
were broader due to higher reduced activation energy and variations in cell size (Fig. 16). Most
particles had longer induction time due to expansion (Fig. 16(c)). On the other hand, not only
the presence of transverse wave and transverse detonation but also the forward and backward jets,
and compression waves from ignition of lagged region led to shorter induction time than that from
the steady model. In addition, particles, which formed unburned gas pockets showed a significant
reduction in induction time up to about two orders of magnitude as compared to steady model
prediction (Fig. 18(a)). This observation again indicated that adiabatic shock compression alone
was not enough to consume the unburned gas pockets.

Two peaks were observed in distribution of reaction time. One peak centered around 1.2 D<sub>CJ</sub> corresponded to longer reaction time due to volumetric expansion and presence of unburned gas pocket. The other peak around 0.9 D<sub>CJ</sub> came from the influence of transverse waves, transverse detonation, and compression waves.

The comparison of induction and reaction times from Lagrangian perspective with those based
on the steady model highlighted the role of leading shock and the multi-dimensional effects. In
the diluted cases, the latter manifest themselves through transverse waves, and jets (Fig. 17). In
the non-diluted case, additional effects were present: transverse detonations, compression waves
Induction and reaction times in a cellular detonation

<table><caption>TABLE II. Average and standard deviation for induction and reaction times from Lagrangian perspective. The values are normalized by $\tau_{cell}$.</caption><thead><tr><th></th><th>Value</th><th>2H<sub>2</sub>–O<sub>2</sub>–7Ar</th><th>2H<sub>2</sub>–O<sub>2</sub></th></tr></thead><tbody><tr><td rowspan="4">Induction time</td><td>ZND</td><td>0.15</td><td>0.23</td></tr><tr><td>Average</td><td>0.15</td><td>0.18</td></tr><tr><td>Standard deviation</td><td>0.09</td><td>0.18</td></tr><tr><td>Standard deviation / ZND</td><td>0.61</td><td>0.77</td></tr><tr><td rowspan="4">Reaction time</td><td>ZND</td><td>0.09</td><td>0.05</td></tr><tr><td>Average</td><td>0.07</td><td>0.03</td></tr><tr><td>Standard deviation</td><td>0.03</td><td>0.02</td></tr><tr><td>Standard deviation / ZND</td><td>0.32</td><td>0.41</td></tr></tbody></table>

from the consumption of lagged regions, and unburned gas pocket shock-induced consumption (Fig. 18). Thus, the prediction using steady model departed from Lagrangian results, for chemical times and specially induction times. For the better prediction of induction and reaction times inside the cell, influence of multi-dimensional effects such as transverse wave before induction process needs to be modeled in the second part of the cell.

Furthermore, arithmetic average induction and reaction times from Lagrangian perspective were compared with ZND model in Table II. The pdf for induction and reaction times for both mixtures to analyze their distribution can be found in supplementary material. Average induction time from Lagrangian result in the diluted mixture agreed well with the value from ZND model despite the distribution inside the cell (Fig. 6(a)). On the other hand, average induction time from Lagrangian results in the non-diluted mixture was 22% shorter than prediction by ZND model. As for reaction time, average reaction time from Lagrangian result was shorter than the ZND model result by 22% and 40% for the diluted and non-diluted mixtures, and deviation from ZND model results became larger in the non-diluted mixture.

From Figs. 6 and 9, distribution was wider in the non-diluted mixture than in the diluted mixture. In addition, standard deviation in induction time normalized by result from ZND model was larger than that in reaction time for both mixtures, due to greater sensitivity of the local
Induction and reaction times in a cellular detonation

conditions on induction time (Fig. 3).

The ZND induction length is usually used for the estimation of the cell such as in Ng model¹⁰¹. Is ZND induction length/time the proper length scale to characterize the cell? Indeed, the ratio of experimental cell width to induction length varied with different mixtures and initial conditions¹⁰¹. The disagreement observed between the average Lagrangian values of induction time and ZND induction time in mildly unstable 2H₂ – O₂ mixture may be one of the reasons for the variability of the ratio of experimental cell size to induction length. If the nonequilibrium effects and the uncertainty of chemical models are excluded as potential defects, there remain defects in the use of the ZND induction length, which may also affect this ratio: 3D cellular structure is not taken into account, and a single value for induction length is output, even though variability of cell width is present.

Also, we may question the meaning of average value of induction/reaction time from Lagrangian perspective. In Alicherif et al.³⁸, they thought that the induction length averaged in the first part of the cell was relevant to compare ZND induction length, because in the second part of cell, there was the influence of transverse wave on induction length. From our findings, this would infer discrepancies in the diluted case. Nevertheless, in the case of regular cells as modeled in Monnier et al.⁸¹, the average was assumed at the half cell length, consistent with our numerical results. Even if turbulent burning is not taken into account, which would be the case for the non-diluted case of the present study, the average was lower than the ZND result. This is in complete opposition of what could be expected from previous works, based on Favre averaging processes¹⁵,¹⁶,¹⁸, even if, when the isentropic coefficient becomes lower and jets become stronger, some exception can occur¹⁰².

V. CONCLUSIONS

The induction and reaction times of a cellular gaseous detonation were investigated along the Lagrangian trajectories, for a weakly and mildly unstable hydrogen-based mixtures at ambient conditions. The unburned gas states were mostly distributed along the inert Rankine-Hugoniot curve below the vN state. Some part were also located along the reflected shock state (at ~ 0.8 D<sub>CJ</sub>) due to the presence of transverse wave.

All the unburned gas mixtures were consumed within half and one cell cycle for the diluted and non-dilutes cases, respectively. The Lagrangian induction and reaction times showed a non-
Induction and reaction times in a cellular detonation

monotonic behavior. Therefore, a cautious approach is recommended when interpreting Eulerian
data. The Lagrangian induction times have drawn a new topology within the cellular structure,
bounded by triple points. The new patterns came from the intersection of the ignition fronts
behind the incident shock with the transverse waves, either reactive or from their collision, and the
collision of the second triple points.

In the mixtures tested, the Mach stem was sufficient for the ignition in the first part. In the
second part, the combination of the incident with the transverse wave in the second part of the
cell was the main factor for ignition. In addition, seven characteristic time scales for chemical,
hydrodynamic phenomena, and cell cycle along the center line of cell were quantitatively compared
for the first time. The ratio of the transverse wave time to the induction time in the second part
of the cell was very close to one, highlighting the dominant role of a transverse wave in ignition,
even for a weakly unstable mixture. The diffusive phenomenon did not contribute to the induction
and reaction processes.

Furthermore, the comparison of H₂O mass fraction between the local and the equilibrium values indicated that the chemical equilibrium was not locally established, mainly due to transverse waves, which induced a super-equilibrium state. The chemical disequilibrium was also present when the latter interacted with the remnants of the forward and backward jets. In the non-diluted case, the contribution of compression waves to chemical disequilibrium was more significant. Even if the transverse waves decayed far behind the leading shock front, it is not completely accurate to equate the mean sonic plane with the thermochemical equilibrium in the non-diluted case.

Moreover, the joint pdf between the chemical times and the leading shock showed that multi-
dimensional effects induced deviations from the 0D constant pressure combustion results. In
addition, the Lagrangian average value agreed with that from the ZND model for the diluted case
despite the distribution inside the cell. On the other hand, a shorter induction time was found in
the non-diluted mixture. Moreover, the average reaction time was shorter than the ZND reaction
time for both mixtures. These results are not completely in line with the results from previous
Favre approaches.

It would be interesting to investigate highly unstable mixtures with the present approach, if the
proper numerical resolution could be achieved. In addition, the Lagrangian approach questions the
experimental results, for the determination of reaction times, and can complement the experimental
results as the PLIF could be affected by vortical and unburned structures³⁸.
Induction and reaction times in a cellular detonation

## ACKNOWLEDGMENTS

This research was subsided by JSPS KAKENHI Grant number JP20K22391 (Grant-in-Aid for Research Activity Start-up), JP21K14094 (Grant-in-Aid for Early-Career Scientists), JP19J12758 (Grant-in-Aid for Specially Promoted Research), the Paloma Environmental Technology Development Foundation, and the CPER FEDER Project of Région Nouvelle Aquitaine and pertains to the French government program "Investissements d'Avenir" (EUR INTREE, reference ANR-18-EURE-0010). Part of this research was supported by the French National Research Agency (ANR) under the project TRACKDEMO ANR-23-CE05-0021. HW thanks the financial support from JSPS Overseas Research Fellowships from April 2022 to March 2024. For the purpose of Open Access, a CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

## AUTHOR DECLARATIONS CONFLICT OF INTEREST

The authors have no conflict to disclose.

## AUTHOR CONTRIBUTIONS

* HW: Conceptualization (lead), Methodology (lead), Software (lead), Resources (equal), Validation (lead), Investigation (lead), Visualization (lead), Data curation (lead), Formal analysis (lead), Funding acquisition (lead), Writing - original draft (lead), Writing - review & editing (equal)

* AM: Software (lead), Validation (lead), Writing - review & editing (equal)

* AC: Conceptualization (lead), Investigation (lead), Formal analysis (lead), Methodology (lead), Resources (equal), Supervision (lead), Project administration (lead), Funding acquisition (lead), Writing - review & editing (lead)

* NI: Resources (equal), Writing - review & editing (equal)

* KM: Resources (equal), Writing - review & editing (equal)

* JK: Resources (equal), Funding acquisition (lead), Writing - review & editing (equal)
Induction and reaction times in a cellular detonation

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

SUPPLEMENTARY MATERIAL

The initial position of the selected particles and the shock velocity at the position of the selected particles are given in the supplementary Material. In addition, the time history of not only the variables but also Damköhler number for selected particles, initially located in the center line of the cell are available. Furthermore, pdf for induction and reaction times are presented to understand the distribution. Moreover, the grid convergence for the distribution of induction and reaction times from a Lagrangian approach was assessed.

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