Flame morphology boundaries and fundamental
combustion properties in unobstructed channels

Cristian C Mejía-Botero, Florent Virot, Luis Fernando Figueira da Silva,
Josué Melguizo-Gavilanes

► To cite this version:

Cristian C Mejía-Botero, Florent Virot, Luis Fernando Figueira da Silva, Josué Melguizo-Gavilanes. Flame morphology boundaries and fundamental combustion properties in unobstructed channels. Combustion and Flame, 2026, 286, pp.114806. 10.1016/j.combustflame.2026.114806. hal-05221478v2

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Flame morphology boundaries and fundamental
combustion properties in unobstructed channels

Cristian C. Mejía-Botero<sup>a,b</sup>, Florent Virot<sup>a</sup>, Luis Fernando Figueira da Silva<sup>a</sup>,
Josué Melguizo-Gavilanes<sup>a,c</sup>

Abstract

We investigated the effect of fundamental combustion properties (FCP) on the 3D morphology and dynamics of
flames and shocks during acceleration and transition to detonation in unobstructed channels. To achieve this, an
extensive experimental campaign was conducted using a simultaneous schlieren visualization set-up. The effect
of selected FCP was assessed by evaluating nine different mixtures of hydrogen, methane, and hydrogen/methane
blends, using oxygen with and without dilution by nitrogen, helium, or argon. The experimental results revealed
two characteristic flame evolution behaviors during flame acceleration (FA), depending on the mixtures: i) a sym-
metric flame inversion (tulip flame) during the early stages of FA, followed by a short, symmetric flame in the later
stages, with the formation of a precursor compression wave located relatively far from the flame, and ii) an asym-
metric, wrinkled flame during the early stages, which develops into a longer flame with the tip inclined toward a
corner of the channel, accompanied by the formation of multiple precursor compression waves ahead of the flame
in the later stages of FA. For a more robust statistical analysis, a morphology database was compiled from literature
sources reporting similar flame morphologies to those observed in our experiments. This database was analyzed
using the Feature Elimination Technique in conjunction with the Logistic Regression Model, which enabled the
identification of FCP boundaries between the observed flame morphologies. The analysis showed that the pairs
of properties most influencing flame morphology are the expansion ratio and the ratio of the laminar flame speed
to the sound speed in the combustion products, i.e., (σ, σ<sub>S</sub>/c<sub>b</sub>), as well as the latter ratio with the heat capacity
ratio, i.e., (σs<sub>L</sub>/c<sub>b</sub>, γ). Additionally, this methodology helped to identify experimental conditions where little or no
data is available in the literature, such as for mixtures with Lewis numbers smaller than unity, which are expected
to be affected by thermodiffusive instabilities. These boundaries can, therefore, serve as guidelines for selecting
experimental conditions that develop specific flame and shock morphologies and dynamics.

Keywords: Deflagration-to-detonation transition, flame morphologies, fundamental combustion properties, explosions risk assessment, flame-dynamics, morphology boundaries

Novelty and significance statement

This study establishes, for the first time, a di-
rect link between fundamental combustion properties
(FCP) and the observed flame and shock morpholo-
gies during flame acceleration in unobstructed chan-
nels from ignition to detonation onset. The results
offer predictive insights into flame morphology be-
havior (i.e., morphology boundaries as a function of
FCP), identify previously unexplored regimes, and
serve as a dataset for validation of numerical simula-

tions and provide guidance to target specific regimes
in experiments, thereby advancing the current under-
standing of flame acceleration and the deflagration-to-
detonation transition.

1. Introduction

Following an accidental ignition of a reactive
cloud, a deflagration is initiated, which then expands
and accelerates within the process enclosure. Under
certain conditions, the flame can accelerate and lead
to a detonation; this is known as the deflagration-
to-detonation transition (DDT). The detonation on-
set typically occurs in a small volume of unburned
mixture that has been thermodynamically conditioned
due to the flame acceleration (FA) process. However,
FA takes place on much larger spatial and temporal

<footnote><sup>a</sup>Institut Pprime, CNRS, ISAE-ENSMA and Université de Poitiers, Futuroscope Chasseneuil, France</footnote>

<footnote><sup>b</sup>Aix Marseille Université, CNRS UMR7342, Centrale Marseille, IRPHE, Marseille, France</footnote>

<footnote><sup>c</sup>Shell Global Solutions B.V., Major Hazards Management, Energy Transition Campus, 1031 HW Amsterdam, The Netherlands</footnote>

<footnote>*Flame morphologies in unobstructed channels
e-mail: cristian.mejia@irphe.univ-mrs.fr
(Cristian Mejía-Botero)</footnote>
scales. For this reason, the study of FA is just as important as the study of the detonation onset itself. Although DDT often occurs due to flame-shock-obstacle interactions, it can also arise in unobstructed and narrow spaces, such as gas compartments, hoses in fuel cells, battery packs, and electrolyzers, among others [1, 2].

One of the key factors influencing the FA process in unobstructed channels is the flame morphology, which directly affects the flame surface area. A larger flame surface area increases the reactants consumption rate and, consequently, the flame propagation speed during FA. This can also lead to the more readily formation of pressure waves, higher temperatures and pressures between the precursor shock and the flame, further promoting the transition to detonation. Interestingly, very little work has focused on characterizing the morphology of flames and shocks throughout the entire DDT process. Most studies have focused on the presence of the so-called tulip flame at the early stages of FA [3, 4, 5, 6], which was first studied by Clanet et al. [7], and some papers have evaluated flame morphology during the later stages of FA [8, 9, 10, 11, 12, 13]. While it is known that the DDT process is inherently three-dimensional, most experimental work focuses on 2D analysis, with only a few studies addressing 3D flame visualization [10, 11, 12, 13].

Fundamental combustion properties (FCP) play a crucial role in flame behavior, the most relevant properties being the expansion ratio ($\sigma$), laminar flame speed ($s_L$), laminar flame thickness ($\delta_f$), Zel'dovich number ($\beta$), Lewis number ($Le$), heat capacity ratio ($\gamma$), and sound speed relative to the combustion products ($c_b$). Indeed, several studies determined the effect of these properties on the behavior of laminar and turbulent flames. However, very few papers relate FCP to flame dynamics during DDT. One of the classical studies is that of Dorofeev et al. [14], which evaluated the effect of $\sigma$, $s_L$, $\delta_f$, and the hydraulic diameter ($D_h$) on flame acceleration behavior in obstructed channels for H₂ mixtures. Two different combustion regimes were observed: (i) slow and unstable flames characterized by $D_h/\delta_f < 500$ and $\sigma < 3.75$, and (ii) fast combustion for $\sigma > 3.75$ and $D_h/\delta_f > 500$. Flames in regime (i) exhibited global quenching, whereas flames in regime (ii) exhibited shock waves ahead of the flame and quasi-detonations. They concluded that the primary parameter affecting FA is $\sigma$, whose critical value is a function of $\beta$. However, no studies have been found in which these limits were evaluated in unobstructed channels, nor have any studies related FCP to flame morphology and dynamics during DDT. This gap is the main focus of the present paper.

The objective of this paper is to perform a complete characterization of flame morphology during DDT, and to relate it to the most relevant FCP using logistic regression techniques. Accordingly, this work is carried out using two different approaches: (i) an ex-

perimental campaign, in which the effect of the most relevant FCP on the 3D flame and shock morphology and dynamics is studied, and (ii) a data analysis, using a morphology database compiled from the literature, to enable a more in-depth statistical analysis. Logistic regression techniques are employed to propose flame morphology boundaries and determine the FCP that affect them during DDT. This paper is organized as follows: Section 2 details the simultaneous schlieren visualization set-up used to obtain the 3D morphologies and presents the experimental protocol, the mixtures' experimental design, and the methodology used for FCP calculation. Section 3 presents the results of the 3D flame morphologies observed during the experiments, where two morphologies were systematically observed for the early (ES) and late stages (LS) of the FA process. Section 4 introduces the morphology database, which includes studies reporting morphologies similar to those observed in our experiments. This database contains data for both early and late stages of FA. These data are analyzed separately, with Section 5 using the ES morphologies to identify flame morphology boundaries as a function of FCP, and Section 6 testing these boundaries against the LS data. Finally, Section 7 presents the main conclusions.

## 2. Methodology

### 2.1. *Experimental setup*

The experimental setup used to capture the 3D flame morphology during the entire DDT process corresponds to the simultaneous schlieren visualization set-up described in [12], which is schematically presented in Figure 1. The main component of the setup is the combustion chamber, which consists of a narrow polycarbonate channel with a square cross-section of $10 \times 10 \text{ mm}^2$ and a length of 1 m; the polycarbonate plates are 15 mm thick. The channel is closed at the ignition end and open at the other end. The ratio of the channel length to height is 100, so as to reduce the interaction between the flame and the reflected expansion wave stemming from the open end. The flame ignition is initiated by electrodes located 3 mm from the closed end to reduce the dead volume and its influence on flame propagation [15]. The arc produced by the electrodes deposits a small amount of energy in the gas (less than 1 mJ) to avoid direct detonation initiation.

The setup is equipped with two independent schlieren systems that allow simultaneous visualization of the flame and shock dynamics in the xy and xz planes of the channel (see Figure 1). Green (400 – 550 nm) and blue (450 – 700 nm) LED are used as light sources for the xy and xz planes, respectively. A methodology for LED selection applicable to DDT is described in [16]. The LED colors were selected based on the camera's spectral sensitivity for both views. With this technique, it is possible to obtain, for each viewing plane, an image of the flame morphology integrated along the line of sight. However,

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<figure><img src="image_2.png" /><figcaption>Fig. 1: Simultaneous schlieren visualization set-up. Indices c, 1, 2 and 3 refer to the condenser, collimator, focalization, and collection systems, respectively. See [12, 13] for more details.</figcaption></figure>

by combining the integrated images acquired from both directions, the exact position of the different features can be identified, thus providing an indirect view of the three-dimensional flame structure. For more information about the optical set-up, please consult [12, 13]. The field of view—the section of the channel that is measurable in the x-direction during each test—is 40 or 60 mm. The cameras used are: (i) for the xy plane view, the Shimadzu HPV-2, with a minimum frame rate of 125,000 FPS (1000 ns exposure time) and a maximum frame rate of 1,000,000 FPS (250 ns exposure time), and (ii) for the xz plane view, the Shimadzu HPVX-2 with a minimum frame rate of 200,000 FPS (200 ns exposure time) and a maximum frame rate of 10,000,000 FPS (50 ns exposure time).

Since the simultaneous schlieren visualization set-up only measures the flame morphology in a limited field of view, a third camera (not shown in Figure 1), the Photron FASTCAM SA-Z, is used to capture the full channel view in order to measure the flame position throughout the entire DDT trajectory. This camera has a maximum resolution of 1024 × 1024 pixels and a spectral sensitivity range of 400 – 1,000 nm, making it suitable for observing the visible light released during combustion. Its temporal resolution range is 20,000 – 120,000 FPS, with a value of 100,000 FPS used for each test. By tracking the position of the flame tip at different times, it is possible to reconstruct the *t*–*x* diagram, capturing all stages of FA and DDT. As this camera differs from those used for the simultaneous schlieren visualization set-up, it permits the simultaneous measurement of the complete *t*–*x* diagram for each test. This provides a substantial number of profiles for each experimental campaign, thereby enhancing the statistical significance of the dataset. For each mixture evaluated, approximately 20 profiles are measured to obtain a representative *t* – *x* distribution. For each *x*-position in the channel, the

average *t* value and its standard deviation are calculated using the 20 profiles. The flame position-speed, *u* – *x*, diagram is derived by numerically differentiating each individual *t* – *x* curve, where *u* = *dx*/*dt*.

## 2.2. Fundamental combustion properties (FCP) determination

The FCP determination is performed using Cantera [17]. The chemical mechanisms used are Mevel et al. [18] for the H₂ mixtures and, and San Diego [19] for the hydrocarbon mixtures. The *FreeFlame* object is used to calculate *s*<sub>L</sub>, which solves a boundary value problem specified by the 1D conservation equations for a given thermodynamic state and mixture composition.

To determine δ<sub>f</sub>, three classical definitions are considered. The first one is based on the maximum temperature gradient in the flame front, δ<sub>f</sub> = (T<sub>b</sub> – T<sub>u</sub>)/(dT/dx)<sub>max</sub>, where T<sub>b</sub> and T<sub>u</sub> are the burnt and unburnt temperature. The second definition considers the thermal diffusivity and s<sub>L</sub>, i.e., δ<sub>f,2</sub> = α/s<sub>L</sub>, where α = κ/ρc<sub>p</sub> is the thermal diffusivity of the fresh gases, being κ the thermal conductivity. Finally, the third definition considers s<sub>L</sub> also, but using the mixture kinematic viscosity (ν), i.e., δ<sub>f,3</sub> = ν/s<sub>L</sub>.

To calculate the Zel'dovich number, two definitions are used. The first one considers its original expression, i.e., β = E<sub>a</sub>(T<sub>b</sub> – T<sub>u</sub>)/RT<sub>b</sub><sup>2</sup>, where R is the universal gas constant, and E<sub>a</sub> is the activation energy, which is calculated using Arrhenius relation for the ignition delay time (τ) proposed in [20], and the second definition (β<sub>2</sub>) uses the formulation for the heat release rate (q̇) of laminar premixed flames recently proposed by [21].

Finally, for the calculation of the Lewis number, two definitions are also considered. The first one considers only the *Le* for the deficient reactant, i.e., the fuel for lean conditions; the oxidizer for rich conditions, *Le*<sub>D</sub> = α/*D*<sub>D,m</sub>, where *D*<sub>D,m</sub> is the mass diffusivity of the deficient reactant in the mixture. Although this definition is widely used, it is ambiguous near stoichiometric mixtures, i.e., φ = 1, for which there are no deficient reactant. For this reason, the effective Lewis number definition proposed in [22] is also used in this work, i.e., *Le*<sub>eff</sub> = 1 + ((*Le*<sub>E</sub> − 1) + (*Le*<sub>D</sub> − 1)A)/(1+A), where *Le*<sub>E</sub> is the *Le* for the excess reactant, and A = 1 + β(φ − 1), with φ = 1 for stoichiometric mixtures, φ = φ for rich mixtures (φ > 1), and φ = 1/φ for lean mixtures (φ < 1).

## 2.3. Mixtures selection

It may be speculated that, in the stages of FA where the flame is slower, i.e., the early stages, the FCP that could dominate the phenomenon are those associated to laminar flames, e.g., s<sub>L</sub>, δ<sub>f</sub>, Le, among others. During the formation of the precursor compression wave,
the morphologies could be controlled by other proper-
ties such as $\gamma$, which characterizes the gas compress-
ibility, or $c_b$, which is the sound speed in the com-
bustion products and is usually used as a criterion to
predict DDT. It is thus important to understand which
FCP affect each stage of DDT. To assess the effect of
a single property on DDT, one of the ways is to study
mixtures having the same value of the property of in-
terest. By doing this, it is possible to compare sim-
ilarities between their flame morphologies during all
the stages of DDT, determining a relationship between
the common morphologies and the specific FCP. In
this work, three experimental campaigns are designed
in which some FCP are kept constant: (i) mixtures
with the same $\sigma$ and different $\gamma$, (ii) mixtures with the
same $\sigma_{s_L}$ and different $Le_D$ ($Le_D > 1$, and $Le_D < 1$),
and (iii) mixtures with similar $\delta_f$ and different $\sigma$. For
more detailed information about the mixture selection
please consult the supplementary material.

For (i), 2H₂ + O₂ + ηᵢI⁻ mixtures are employed, with I = N₂, Ar, or He. These diluents are used since γ is smaller for diatomic (N₂, γ = 1.4) than for monoatomic gases (Ar and He, γ = 1.7). Additionally, the use of Ar and He leads to mixtures with the same γ and σ, but with a significantly different sound speed in the burnt gases, c_b. Furthermore, by selecting the adequate ηᵢI⁻ value, it is possible to obtain mixtures with the same σ. Initially, ηₙ₂ = 1 (σ = 7.9) is chosen, since this mixture was studied in Ballossier et al. [12] using the same experimental setup, and it is certain that DDT is obtained in the length of the channel. To maintain the same σ = 7.9 value for all the mixtures, ηᵣAr = ηᵣHe = 2.8 must be chosen. Thus, the mixtures selected in this experimental campaign are 2H₂ + O₂ + N₂ (γ = 1.4), 2H₂ + O₂ + 2.8Ar (γ = 1.6), and 2H₂ + O₂ + 2.8He (γ = 1.6). However, for the He-diluted mixture no DDT was obtained in this channel length. For this reason, the mixture 2H₂ + O₂ + 1.5Ar + 1.3He is analyzed also, which has σ = 7.9 and γ = 1.6. Furthermore, this mixture is selected to have the same c_b as the N₂-diluted mixture.

For the campaign (ii) mixtures that have the same $\sigma_{s_L}$ are considered while having $Le_D < 1$ and $Le_D > 1$. This can be effected by using H₂/O₂ with different equivalence ratios, $\phi$, since, for these mixtures, $\sigma_{s_L}$ has a non-monotonic behavior with $\phi$. In addition, $Le_D < 1$ for $\phi < 1$, and $Le_D > 1$ for $\phi > 1$ (see supplementary material). Accordingly, a lean ($\phi = 0.8$), and a rich ($\phi = 1.7$) mixture were selected.

Finally, for campaign (iii), mixtures with different $\sigma_{s_L}$ but with approximately the same $\delta_f$ are designed. One way to do this is by using H₂/CH₄ blends with stoichiometric O₂. To determine the required reactants concentration, the stoichiometric equation, i.e., $x_{H_2}H_2 + (1 - x_{H_2})CH_4 + \frac{4-3x_{H_2}}{2}O_2 \rightarrow (1 - x_{H_2})CO_2 + (2 - x_{H_2})H_2O$, is used, where the H₂, CH₄ and O₂ number of moles are a function of the H₂ blend mole fraction ($x_{H_2}$), where $x_{CH_4} + x_{H_2} = 1$. When computing $\delta_f$ as a function of H₂ content in the blend, i.e.,

$\delta_f = f(x_{H_2})$, it is observed (see supplementary material) that for $x_{H_2} \in [0, 0.8]$, $\delta_f$ does not present significant changes, whereas $\sigma_{s_L}$ increases significantly with $x_{H_2}$ due to the significant difference in the laminar flame speeds of H₂ and CH₄. Accordingly, mixtures with $x_{H_2} = 0, 0.5$ and $0.8$ are selected.

Table 1 shows a summary of the nine (9) mixtures selected in the three experimental campaigns, with their respective abbreviation names. The average run-up distances ($\bar{x}_{DDT}$) and times ($\bar{t}_{DDT}$) to detonation are also included, which are the distance and time elapsed from flame ignition and detonation onset. These are used to normalize the corresponding $t-x$ diagrams. Moreover, Figure 2 presents the FCP for the corresponding mixtures, where the FCP that were left constant for each experimental campaign are observed.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Fundamental combustion properties summary for the studied mixtures. The composition of each mixture is given in Table 1.</figcaption></figure>

### 3. Flame morphologies during DDT

During the study of the flame morphologies throughout the flame acceleration and DDT process of the three experimental campaigns designed, two characteristic transitions were systematically observed during FA. In these, the flame may take two different morphologies during the early stages (ES) and evolve equally to two different morphologies during the late stages (LS) of FA. These morphologies and their transitions are described in detail below for the corresponding mixtures.

#### 3.1. Early stage 1 to late stage 1 morphology transition (ES1 → LS1)

The first morphology transition, called ES1 → LS1, was observed for H₂/O₂ mixtures diluted with
<table><caption>Table 1: Summary of the mixtures studied in this work, their respective abbreviations, $\bar{x}_{DDT}$ and $\bar{t}_{DDT}$. All mixtures were tested at room temperature and pressure, i.e., ≈ 300 K and 1 bar.</caption><thead><tr><th>Mixture number</th><th>Mixture</th><th>Abbreviation</th><th>mixture name</th><th>x̄<sub>DDT</sub> [mm]</th><th>t̄<sub>DDT</sub> [ms]</th></tr></thead><tbody><tr><td>1</td><td>2H<sub>2</sub> + O<sub>2</sub> + N<sub>2</sub></td><td>N<sub>2</sub></td><td></td><td>633 ± 23</td><td>1.53 ± 0.03</td></tr><tr><td>2</td><td>2H<sub>2</sub> + O<sub>2</sub> + 2.8Ar</td><td>Ar</td><td></td><td>759 ± 9</td><td>3.27 ± 0.04</td></tr><tr><td>3</td><td>2H<sub>2</sub> + O<sub>2</sub> + 2.8He</td><td>He</td><td></td><td>NA</td><td>NA</td></tr><tr><td>4</td><td>2H<sub>2</sub> + O<sub>2</sub> + 1.5Ar + 1.3He</td><td>Ar+He</td><td></td><td>827 ± 13</td><td>3.59 ± 0.1</td></tr><tr><td>5</td><td>2H<sub>2</sub> + 1.25O<sub>2</sub></td><td>φ = 0.8</td><td></td><td>374 ± 7</td><td>0.63 ± 0.07</td></tr><tr><td>6</td><td>2H<sub>2</sub> + 0.58O<sub>2</sub></td><td>φ = 1.7</td><td></td><td>693 ± 18</td><td>1.04 ± 0.02</td></tr><tr><td>7</td><td>0.8H<sub>2</sub> + 0.2CH<sub>4</sub> + 0.8O<sub>2</sub></td><td>x<sub>H2</sub> = 0.8</td><td></td><td>366 ± 18</td><td>0.69 ± 0.08</td></tr><tr><td>8</td><td>0.5H<sub>2</sub> + 0.5CH<sub>4</sub> + 1.25O<sub>2</sub></td><td>x<sub>H2</sub> = 0.5</td><td></td><td>371 ± 11</td><td>0.77 ± 0.06</td></tr><tr><td>9</td><td>CH<sub>4</sub> + 2O<sub>2</sub></td><td>x<sub>H2</sub> = 0</td><td></td><td>467 ± 31</td><td>0.84 ± 0.13</td></tr></tbody></table>

Ar, He, and with Ar+He. Figure 3 shows representative characteristic morphologies obtained at different channel positions for these mixtures. Additionally, the $t/\bar{t}_{DDT} - x/\bar{x}_{DDT}$ and $u - x/\bar{x}_{DDT}$ diagrams are shown, in which the location of each morphology is indicated with numbers. In the case of the He-diluted mixture, the profiles are not presented since, as previously mentioned, this mixture did not reach DDT, and it is not possible to scale the FA trajectory with $\bar{x}_{DDT}$ and $\bar{t}_{DDT}$. The images were acquired using different camera frame rates according to the flame propagation speed: 200, 000 fps for the slower flames and up to 2, 000, 000 fps for the fastest ones.

Initially, the mixture is ignited in the closed end of the channel, yielding a hemispherical finger flame that moves towards the open end due to gas expansion. When the flame touches the channel walls, finger-shaped flame (image 1) remains, which rapidly accelerates due to the expansion at the closed end, as evidenced by the steep $u$ increase observed for both mixtures. Indeed, $u$ goes from 0 m/s to ≈ 180 m/s. Note that, in general, the behavior of the $t/\bar{t}_{DDT} - x/\bar{x}_{DDT}$ and $u - x/\bar{x}_{DDT}$ diagrams are very similar for both mixtures despite having different $\bar{x}_{DDT}$ and $\bar{t}_{DDT}$ (see Table 1), which shows that the run-up distance and time appropriately scale the flame acceleration during the DDT process for mixtures that present similar morphologies.

During this initial stage of propagation, the flame experiences a loss of area due to contact with the walls, which in turn reduces the reaction rate. This leads to flame flattening (images 2–4) and subsequent inversion (images 5–7), exhibiting what is known as the tulip-shaped flame, that is called in this work the ES1 morphology. Due to this inversion, the flame surface increases and an acceleration is observed. Simultaneously, Darrieus–Landau gas-expansion instabilities seem to occur, creating cellular structures (image 7), while maintaining the tulip shape. This morphology is composed of four stretched flame surfaces that propagate along the four corners of the channel, and are connected at the center leading to the characteristic tulip shape, both on the $xy$ and $xz$ planes. Each surface propagates at approximately the same speed in images 6–7, which is evidenced by the tip

<figure><img src="image_4.png" /><figcaption>Fig. 3: Characteristic evolution of the flame morphology systematically observed for $H_2/O_2$ mixtures diluted with He, Ar, and Ar+He, called the ES1 → LS1 morphology transition. Below, the $t/\bar{t}_{DDT} - x/\bar{x}_{DDT}$ and $u - x/\bar{x}_{DDT}$ diagrams are shown, where the corresponding morphologies are indicated with numbers.</figcaption></figure>

of each surface being nearly at the same $x$-position.

However, as the flame moves further along, the speed of each surface changes. This is evidenced in the $xz$ plane view of image 8, where the tips of the four flame surfaces are observed at different $x$ positions at each corner. This leads to an asymmetric flame, as shown in the $xy$ plane view. The flame surface that propagates along the left lower corner of image 8, when looking down the channel from ignition end, moves slightly faster than the other structures. From that moment on, this flame morphology prevails, leading to a structure inclined towards this corner, as shown in image 9. The subsequent flame morphology is significantly elongated, yielding to a
stretched flame that spreads along the top wall. It should be stressed that only with the xz plane view is impossible to differentiate whether this stretched flame propagates in one of the upper corners or along the wall. Only upon examination of the xy plane view, it is possible to observe that the flame propagates along both the corners and the walls, underscoring their importance in the FA process. The tip of the flame in the xz plane view shows a flame inversion at the center, as a consequence of the tulip shape of the preceding morphology. It is important to point out that the corner to which the flame is inclined can change from test to test, but the observed morphologies are always similar to those presented.

Significant changes in flame morphologies occur before the formation of the precursor compression wave. Indeed, after the long structures observed in image 9 of Figure 3, the flame shortens significantly (image 10) presenting a more symmetrical finger-shaped structure, but with a more corrugated surface. From 8 to 9, the flame exhibits a moderate increase in velocity, from 350 to 510 m/s. However, as the flame accelerates further, it induces a precursor shock wave, which is visualized as a flat shock propagating ahead of the flame in a position relatively far from the flame. The flame structure observed at this stage is hereby called the LS1 morphology, and is characterized by a relatively short flame length ($L_f \approx 1.5D_h$). Due to the formation of the preheating and compression zone downstream the precursor compression wave, an abrupt acceleration is observed until finally DDT occurs ($x/\bar{x}_{DDT} = 1$). The detonation onset mechanisms are not analyzed in this paper, and will be the object of a future publication.

The similarities observed in the morphologies for these mixtures throughout the entire FA process are noteworthy. One of the possible reasons for this is that, as seen in Figure 2, they have approximately the same $\sigma$ value (consistent with the experimental design), and similar $\sigma s_L$, which suggests that these FCP may be key to the morphology formation during the DDT process in smooth channels. However, other mixtures exhibit similar values of these FCP, but did not result in the ES1 → LS1 transition. As an example, the mixture diluted with $N_2$ has the same $\sigma$, and the mixture with $x_{H2} = 0$ has similar $\sigma s_L$ values, yet their morphologies were different, as will be discussed in the next sections. This indicates that $\sigma$ and $\sigma s_L$ alone are not sufficient to describe the morphological changes during the acceleration process. A more detailed analysis of this is thus carried out in Section 5 and Section 6.

## 3.2. Early stage 2 to late stage 2 morphology transition (ES2 → LS2)

The second morphology transition type observed during the experimental campaigns is hereby called the ES2 → LS2, observed for $x_{H2} = 0, 0.5, 0.8$ and $\phi = 0.8$ mixtures (see Table 1). Figure 4 shows

the characteristic morphologies together with their respective $t/\bar{t}_{DDT} - x/\bar{x}_{DDT}$ and $u - x/\bar{x}_{DDT}$ diagrams. As in Figure 3, these diagrams show a partial collapse, demonstrating again the importance of $\bar{x}_{DDT}$ and $\bar{t}_{DDT}$ for the scaling of the flame acceleration process during DDT. Figure 4 shows that, after the finger-shaped flame (image 1), a flame flattening occurs with the simultaneous formation of cellular structures (image 2). During this period the flame speed decreases, possibly due to a flame surface area decrease. These cellular structures apparently increase in size (images 3 and 4) favoring propagation along the lower left corner (image 5 and 6). This flame elongation is associated to an increase in surface area and, thus, to the flame propagation speed. These corrugated structures, that are possible due to flame instabilities, are herein labeled the ES2 morphology, which differs significantly from the ES1 structure that is characterized by the tulip shaped flame, as discussed in Section 3.1.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Characteristic evolution of the flame morphology systematically observed for H₂/CH₄ blends with $x_{H2} = 0, 0.5$ and 0.8, and for H₂/O₂ mixture with $\phi = 0.8$, called the ES2 → LS2 morphology transition. Below, the $t/\bar{t}_{DDT} - x/\bar{x}_{DDT}$ and $u - x/\bar{x}_{DDT}$ diagrams are shown, where the respective morphologies are indicated with numbers.</figcaption></figure>

In Figure 4, the elongated and asymmetric flame surface is seen further downstream (image 7), where a flame is seen whose tip is inclined towards the lower wall and spreads along the upper one. The xz plane view reveals that this flame structure has two parts: (i) one that spreads along the entire upper wall, and (ii) one that burns more centered in the channel, without spreading along the walls. Part (ii) is identified since in both views the surface is not inclined towards a corner. Subsequently, the flame further accelerates, leading to the formation of multiple precursor compression waves very close to the flame (image 8). At this stage, the tip of the flame is tilted towards one of the corners of the channel and exhibits a relatively

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long structure ($L_t \approx 5D_h$), which is referred to as the LS2 morphology. This morphology is quite different from LS1, which is characterized by a much shorter and symmetrical structure with the tip centered in the channel center-axis (see Section 3.1). When the precursor compression waves are formed, $u \approx 1100$ m/s and $x/\bar{x}_{DDT} = 0.8$. Subsequently, an abrupt acceleration occurs until DDT takes place at $x/\bar{x}_{DDT} = 1$.

The fact that these mixtures exhibit morphologies systematically similar to each other is noteworthy. Indeed, in Figure 2 the mixtures with $x_{H2} = 0, 0.5$ and $0.8$ have $\sigma > 8$ (higher values than the rest of the mixtures), indicating that the cellular structures observed during the early stages may be associated to gas expansion instabilities (i.e., Darrieus-Landau). In the case of the $\phi = 0.8$ mixture, although it exhibits a $\sigma \approx 7.9$, as the Ar, He and Ar+He-diluted mixtures (which presented the ES1 → LS1 evolution), $Le_D < 1$, suggesting that the cellular structures in this case may be due to thermo-diffusive instabilities. It is important to mention that a specific experimental study is needed in order to differentiate between the types of instabilities observed during the early stages of FA as well as the Le effect. Furthermore, the mixtures that exhibited the ES2 → LS2 morphology transition have similar values of $c_b \approx 1200$ m/s (see Figure 2), which may indicate that this property is also key in dictating the flame morphologies that develop during the DDT process, presumably in the late stages of FA. This last observation underscores the importance of carrying out a multi factorial analysis to characterize the morphologies.

### 3.3. Particular case: early stage 1 to late stage 2 morphology transition (ES1 → LS2)

While most of the mixtures exhibited the ES1 → LS1 or the ES2 → LS2 morphology transitions during DDT, a particular case was systematically observed for the rich H₂/O₂ mixture ($\phi = 1.7$), and those diluted with N₂, where an hybrid behavior was obtained, as shown in Figure 5. For these mixtures, the ES1 morphology is observed during the early stages of FA, a flame flattening (images 1–4), followed by a flame inversion or tulip flame, i.e., ES1 (images 5–7). Subsequently, an asymmetric structure forms (image 8), but unlike the ES1 → LS1 case, the flame evolves to a larger structure (images 9–10) with the tip inclined towards a corner of the channel. Eventually, this structure accelerates and produces multiple precursor compression waves close to the flame tip, which is characteristic of the LS2 morphology. This hybrid evolution suggests that these mixtures are in a morphology-transition stage. To improve the analysis, it is necessary to collect a larger amount of data, so that morphology types boundaries as a function of FCP can be proposed using statistical tools. Accordingly, data from the literature is collected and analyzed, as presented below.

<figure><img src="image_6.png" /><figcaption>Fig. 5: Characteristic evolution of the flame morphology systematically observed for the H₂/O₂ mixture diluted with N₂, and with $\phi = 1.7$, called the ES1 → LS2 morphology transition. Below, the $t/\bar{t}_{DDT} - x/\bar{x}_{DDT}$ and $u - x/\bar{x}_{DDT}$ diagrams are shown, where the respective morphologies are indicated with numbers.</figcaption></figure>

## 4. Morphology database

### 4.1. Criteria for morphology type classification

A systematic search was performed in Scopus and Google Scholar for scientific papers that include experimental images on flame morphology during FA in smooth channels. For the images to be included in the database, the corresponding experiments should meet all of the following criteria: (i) the channel must have at least one of the ends closed, (ii) the ignition should be effected at the closed-end, located at the center of the channel cross-section, (iii) the ignition source should be weak (low energy) to avoid ignition effects on the FA process, and (iv) only H₂ and/or hydrocarbons are considered as fuels. The images that presented at least one of the morphologies observed here during the early stages, i.e., ES1, or ES2, or during the late stages, i.e., LS1 or LS2, are compiled with the corresponding initial experimental conditions, so that their FCP can be computed. Since flame morphologies during FA are categorical variables, it is necessary to define selection criteria to permit traceability of the results.

Considering first the early stages (ES) morphologies, in the experimental results obtained in this work, it was observed that just after the flame ignition, a hemispherical flame was formed. After touching the channel walls, it took the morphology of a finger flame, and subsequently turned to the ES1 or ES2
morphologies, depending on the mixture evaluated.

Since flame morphology changes considerably as the

flame moves and speeds up in the channel, the main

selection criteria to classify a flame as ES is that the

images are taken after the finger flame, produced just

after ignition. To differentiate ES1 from ES2, the

overall flame curvature sign is considered. During the

formation of the tulip flame, i.e., ES1, the flame in-

version produces a change in the flame curvature sign,

going from positive (finger shape) to zero (flat flame),

and finally to negative when the tulip flame is formed.

Conversely, for the ES2 morphology there is flattening of the flame, where no change in the overall global

curvature is observed. Instead, the presence of flame

surface instabilities is evident.

The classification of a morphology as LS1 or LS2

is more complex, as it is difficult to define where the

late stage of FA begins. In our experiments, we ob-

served these morphologies before the formation of the

precursor compression waves. LS1 is characterized

by short flames, i.e., $L_f \approx 1.5D_h$, with the tip more

centered in the channel cross-section, whereas LS2 is

characterized by long flames, i.e., $L_f \approx 5D_h$, with the

tip inclined towards a corner. However, in the litera-

ture there are works such as those by Krivosheyev et

al. [9, 11], where the optical technique used to capture

the flame morphology, i.e., direct chemiluminescence,

does not permit to visualize pressure waves, thus the

classification for these cases should not be defined by

the precursor compression waves formation. For this

reason, here, the main criteria to classify LS1 and LS2

is the flame length, which is a quantifiable variable

that permits a more appropriate differentiation be-

tween the morphology types. In the search performed,

flames classified as LS1 had $L/D_h \in [1.6, 2.1]$, and

those classified as LS2 had $L/D_h \in [3.5, 5.9]$. A more

detailed analysis of flame length during the late stages

of FA will be presented in Section 6.

## 4.2. Mixtures and initial conditions

Table 2 shows the references and the initial con-

ditions of the collected flame morphology data. The

works are split between those that used closed/open

(C/O) and closed/closed (C/C) channel boundary con-

ditions for the different fuels. The fuels included are

$H_2$, $CH_4$, $C_2H_2$, $C_2H_4$, $C_3H_8$, or $CH_4/H_2$ blends. The

latter corresponds to the blends studied in the present

work and those evaluated by Yu et al. [5]. The ox-

idizer is $O_2$ for all the data points, and some mix-

tures are diluted with $N_2$, Ar, or He. All the works

are performed at room temperature, i.e., $T \approx 300$ K.

Channel cross-section (CS) shape is also considered,

where both circular and square were found, which en-

ables to address the question of whether the channel

shape plays an important role in determining the flame

morphology under the conditions studied. The results

suggest that the flame morphology does not depend

on the channel CS shape, since similar morphologies

were observed for channels with square and circular

cross sections. One example of this are the morpholo- gies reported in Krivosheyev et al. [9, 10, 11], dur- ing FA in $C_2H_2/O_2/Ar$ mixtures using circular tubes of 60 mm in diameter, where the ES2 → LS2 transi- tion is observed in spite of having a different shapes and scales when compared to our experiments. These similarities were first noted in [23]. This is in line with the results obtained in [24], where it was noted that $\bar{x}_{DDT}$ is affected by the aspect ratio of the chan- nel but not the cross section itself. As a result, further analysis will be restricted to the effect of channel size ($D_h$) and not of its shape. A total of 60 DP are ana- lyzed, which permits to investigate the effect of FCP on flame morphology under a wider range of experi- mental conditions.

## 4.3. Flame morphology distribution

Most of the references in the database focused the analysis on the early stages of FA only, and, thus, do not report the flame morphology at every stage. In to- tal there are 21 DP reporting the transition ES → LS; among which 9 DP are from the current work [13], 4 DP from [12], and 8 DP [9, 10, 11]. Most of the data yield the ES2 → LS2 transition (13 DP), 6 DP the ES1 → LS1, and only 2 DP the ES1 → LS2, corre- sponding to the mixtures $2H_2 + O_2 + N_2$, and $H_2/O_2$, $\phi = 1.7$, studied here. This suggests that there may be a relationship between the morphologies in the early and late stages of FA, i.e., ES1 with the LS1, and ES2 with the LS2. In addition, 32 DP reported flame mor- phologies during the early stages of FA only, one de- scribed the ES2 morphology; the remaining, the tulip- shaped flame (ES1) is described. Finally, 7 DP report the LS morphology only, all of which correspond to the LS2 morphology type.

Since most of the data is concerned with morpholo- gies during the early stages, i.e., the ES → LS transi- tion and ES only, statistical analysis can be carried out to determine the FCP that correlate with the mor- phologies described, and in doing so propose bound- aries that separate the ES1 and ES2 morphologies described in Section 5. Subsequently, the proposed boundaries will be tested against the LS data to verify whether the same boundaries that separate the ES data hold for the LS, and thereby correlate the morpholo- gies during the early and late stages of FA. This will be presented in Section 6.

# 5. Early stages analysis: morphology boundaries proposal

## 5.1. FCP feature elimination

To analyze the effect of the considered fundamen- tal combustion properties (FCP) on flame morphol- ogy during the early stages of flame acceleration, it is first necessary to determine and to eliminate auto- correlated properties. To this end, the three defini- tion of $\delta_f$, and the two definitions of $\beta$ described in Section 2.2 are considered in the analysis, as well as different expressions used for Le i.e., $Le_D$ and $Le_{eff}$.
### 5.3. Most important pairs of FCP for flame morphology estimation

The Recursive Feature Elimination analysis (Figure 7 (a)) shows that there is not a F1 score improvement when using more than two (2) FCP in the Logistic Regression Model, thus the analysis is focused on FCP pairs only. To determine which is the best FCP combination, the auto-F1 score matrix is used (Figure 7 (b)), which shows the F1 score for a given pair of FCP. The main diagonal values indicate the score using only a single FCP in the predictive model. Figure 7 (b) shows that the pairs involving $\sigma$ always have the highest F1 scores. Note that the F1 score with $\sigma$ alone (F1 = 0.94) is the same as that using $\sigma$ with either one of $\delta_f$, $D_h/\delta_f$, $\beta$, $\gamma$ or $Le_D$. It is thus clear that $\sigma$ is the parameter that influences the most the morphology changes during the early stages, under the conditions studied. The F1 score is maximum for the pairs $(\sigma, \sigma_{sL}/c_b)$ and $(\sigma_{sL}/c_b, Le_{eff})$, with F1 = 0.98, which also underscores the importance of these properties on the development of the flame morphology during the early stages of FA. The other pairs that show a large F1 score are $(\sigma_{sL}/c_b, Le_{eff})$, with F1 = 0.94, and $(\sigma_{sL}/c_b, Le_D)$ with F1 = 0.94. Considering only the effect of individual FCP, the best scores are obtained for $\sigma$ and $\sigma_{sL}/c_b$, but the $\gamma$ score is not negligible (F1 = 0.81). The latter suggests that $\gamma$ should affect the flame morphology, but it appears to be less important than either $\sigma$ or $\sigma_{sL}/c_b$.

<figure><img src="image_7.png" /><figcaption>Fig. 7: Recursive Feature Elimination results with the Logistic Regression model (a) Average F1 score, and (b) auto-F1 score matrix.</figcaption></figure>

### 5.4. FCP boundaries between the ES morphologies

In Section 5.3 the most important FCP pairs for the ES flame morphology estimation were determined. Now, boundaries between these pairs of properties are proposed based on the Logistic Regression Model.

#### 5.4.1. FCP against $\sigma$

As presented in Figure 7, the FCP that most affects flame morphology during the early stages of FA is $\sigma$ under the conditions studied. For this reason, the

<figure><img src="image_8.png" /><figcaption>Fig. 8: Optimal boundary between the ES1 and ES2 morphology types, determined with the Logistic Regression Model for the pairs of FCP that yield better F1-scores.</figcaption></figure>

results obtained with the Logistic Regression Model model are presented in Figure 8 for each of the FCP pairs against $\sigma$. This model determines the optimal boundary separating the ES1 and ES2 morphologies using a pair of FCP. The boundary is a straight line where, depending on its slope, the condition to obtain ES1 or ES2 differs. If the slope is negative, as in the case of the pair $(\sigma, \sigma_{sL}/c_b)$, the ES1 morphology region is below and the ES2 above the boundary line. On the contrary, if the slope is positive, as for $(\sigma, \beta)$, ES1 is found above, and ES2 below the boundary line. Table 3 shows the conditions to obtain morphologies ES1 or ES2 defined as the straight line for each pair of FCP. Note in Figure 8 that for all pairs there are black points (ES2) that fall on the red region (ES1), which shows that the boundaries proposed fail to describe all the experimental data. This is reflected in the F1 value obtained for each pair, which is always smaller than 1. The upper left image presents the $\sigma$ distribution for morphologies ES1 and ES2, showing a partial separation between these morphologies with $\sigma$ only. Note that, at $\sigma \approx 8$, there is an overlap, where mixtures with ES1 and ES2 are observed. In addition, there is a point at $\sigma \approx 6$, that shows ES2 morphology also. This confirms that $\sigma$ alone is not sufficient to describe the morphology transition. The two mixtures with the ES1 $\rightarrow$ LS2 transition are close to this intersection, which again suggests that lie at a transiting stage between ES1 and ES2.

As shown in the auto-F1 score matrix presented in Figure 7, the $(\sigma, \sigma_{sL}/c_b)$ pair improves the separation, and coherently Figure 8 shows that the border proposed for these variables can almost completely separate the ES1 and ES2 morphologies. Moreover, the two transition points ES1 $\rightarrow$ LS2 correspond to almost the same $\sigma_{sL}/c_b$, and are close to the mor-
<table><caption>Table 3: Coefficient of the boundary defined by the equation $y > a \sigma + b$ for which ES2 morphology is obtained.</caption><thead><tr><th>y</th><th>a</th><th>b</th></tr></thead><tbody><tr><td>σs<sub>L</sub>/c<sub>b</sub></td><td>-0.0095</td><td>0.13</td></tr><tr><td>-δ<sub>f</sub></td><td>-0.28</td><td>-2.4</td></tr><tr><td>D<sub>h</sub>/δ<sub>f</sub></td><td>-508</td><td>4962</td></tr><tr><td>-Le<sub>D</sub></td><td>-1.4</td><td>12.5</td></tr><tr><td>Le<sub>eff</sub></td><td>-1.72</td><td>17.9</td></tr><tr><td>-β</td><td>-1.14</td><td>7.67</td></tr><tr><td>-γ</td><td>-0.077</td><td>-0.65</td></tr></tbody></table>

pology transition boundary. When analyzing H₂ and HC mixtures separately for this pair, it can be observed that both fuel mixtures exhibit a similar trend but with a different slope. However, note that the point at $\sigma \approx 6$, that exhibits the ES2 morphology, is not well predicted by the model for any of the evaluated pairs of FCP. This point corresponds to a lean H₂/air mixture ($\phi = 0.42$) [4], the leanest H₂ mixture reported in the database, which are known to be affected by thermo-diffusive instabilities. Indeed, when analyzing the pair of $(\sigma, Le_D)$, it is observed that this mixture has $Le_D < 1$, so it is likely that the proposed boundaries are not able to classify flame morphologies that exhibit thermo-diffusive instabilities. The latter is possibly due to the lack of information on mixtures under these conditions. Note that, for the $(\sigma, Le_D)$ pair there is another black point with $Le_D < 1$ and $\sigma \approx 8$ that is wrongly predicted by the model, and corresponds to a moderately lean H₂/O₂ mixture ($\phi = 0.8$), studied in this work. Not surprisingly, the data suggests that the thermo-diffusive instabilities have a significant effect on flame morphology for lean H₂ mixtures, but more experimental data under ultra-rich and ultra-lean conditions are needed to determine more reliable FCP boundaries. For the rest of the FCP pairs, a clear morphology boundary is also observed, where only the two above mentioned data points that exhibit the ES2 morphology (lean H₂/O₂ ($\phi = 0.8$), and H₂/air ($\phi = 0.42$)), always fell on the ES1-morphology region.

### 5.4.2. FCP against $\sigma s_L/c_b$

Figure 9 shows the results using the Logistic Regression Model analysis with FCP against $\sigma s_L/c_b$; Table 4 presents the corresponding optimal boundaries. The upper left corner presents the $\sigma s_L/c_b$ distribution, where a partial separation of the morphologies is observed, and also that the two points that exhibit the ES1 → LS2 transition have similar $\sigma s_L/c_b$ values. Note that most of the data points collected in the database are characterized by $\sigma s_L/c_b < 0.02$, with each of them presenting the ES1 morphology, except the lean H₂/air mixture ($\phi = 0.42$), which has $\sigma s_L/c_b = 0.01$ but exhibited the ES2 morphology; again, the latter is not well predicted by any of the boundaries proposed, possibly due to the thermo-diffusive instabilities, as discussed above. However,

<figure><img src="image_9.png" /><figcaption>Fig. 9: Optimal boundary between the ES1 and ES2 morphology types, calculated with the Logistic Regression Model for the pairs of fundamental combustion properties that yield better F1-scores after ignoring the $\sigma$ effect.</figcaption></figure>

<table><caption>Table 4: Coefficient of the boundary defined by the equation $y > a \sigma s_L/c_b + b$ for which ES2 morphology is obtained.</caption><thead><tr><th>y</th><th>a</th><th>b</th></tr></thead><tbody><tr><td>σ</td><td>-104.2</td><td>13.56</td></tr><tr><td>-δ<sub>f</sub></td><td>-33.41</td><td>1.15</td></tr><tr><td>D<sub>h</sub>/δ<sub>f</sub></td><td>-12709</td><td>666.9</td></tr><tr><td>-Le<sub>D</sub></td><td>-171</td><td>6.4</td></tr><tr><td>-Le<sub>eff</sub></td><td>-169</td><td>5.51</td></tr><tr><td>-β</td><td>-4135</td><td>175</td></tr><tr><td>-γ</td><td>-2.2</td><td>-1.3</td></tr></tbody></table>

the other lean H₂/O₂ mixture ($\phi = 0.8$; $Le_D < 1$), is well predicted in this case (see the pair $(\sigma s_L/c_b, Le_D)$), since $\sigma s_L/c_b = 0.06$, which is far from the boundaries between ES1 and ES2.

The pair $(\sigma s_L/c_b, D_h/\delta_f)$ presents an interesting behavior, in which two different trends are observed for (i) $\sigma s_L/c_b < 0.02$, and (ii) $\sigma s_L/c_b > 0.02$. In (i) the data follows an almost vertical line, which shows that the ratio $D_h/\delta_f$ is independent of $\sigma s_L/c_b$, and in (ii) $D_h/\delta_f$ tends to increase with $\sigma s_L/c_b$. Note that all the points for $\sigma s_L/c_b < 0.02$ exhibit the ES1 morphology, which suggests that, under the conditions studied, these mixtures exhibit the tulip shape during the early stages of FA irrespective of the $D_h/\delta_f$ ratio. This indicates that mixtures that are sufficiently slow, i.e., small $\sigma s_L/c_b$, tend to yield the ES1 morphology regardless of channel size. Additionally, note that the optimal boundary lies between the two points that exhibited the ES1 → LS2 morphology transition. The remaining FCP pairs also are able to separate the morphologies, according to the F1-score reported in Figure 7. Note that the pair with the highest score is that of $(\sigma s_L/c_b, \gamma)$, where only the lean H₂/air mixture
<figure><img src="image_10.png" /><figcaption>Fig. 10: Optimal boundary testing with data that only reported morphology at the LS. Black and gray data points correspond to LS2 morphology, and are painted in different colors to better differentiate them for their similar values.</figcaption></figure>

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## 6. Late stages analysis

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### 6.1. Proposed boundaries performance against late stage data

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In the previous section, FCP boundaries were proposed using all the data that reported ES morphologies in the database. This was done since most of the reported data corresponded to ES morphologies, which permitted the proposed boundaries to cover a wide range of experimental conditions. In this section, these boundaries are used to test the seven (7) data points that reported the LS2 morphology only. Note that the points reporting ES → LS transitions are not considered in this test, as they were already used to determine the morphology boundaries.

The first properties pairs to be examined are those that led to the highest score, $F1$ = 0.98, with the ES morphologies in Figure 7, i.e., $(\sigma, \sigma_{SL}/c_b)$ and $(\sigma_{SL}/c_b, \gamma)$. The corresponding optimal boundaries obtained are shown in Figure 10. Black and gray data points correspond to the LS2 morphology, and are painted in different colors to better differentiate them for their similar values. Concerning $(\sigma, \sigma_{SL}/c_b)$, it is seen that all the HC have higher $\sigma$ values, compared to the H$_2$ mixtures, and are correctly predicted by the boundaries. However, the proposed boundary is not capable of predicting the $(\sigma \approx 7.5, \sigma_{SL}/c_b \approx 0.04)$, i.e., for the two H$_2$ mixtures considered, which corresponds to a lean mixture of H$_2$/O$_2$, at $p$ = 10 kPa and $\phi$ = 0.5. This point is also not estimated well by the pair $(\sigma_{SL}/c_b, \gamma)$, which, in addition, is not capable of predicting one of the HC data points. So, for this data set the pair that better predicts the morphology during the late stages of FA is $(\sigma, \sigma_{SL}/c_b)$, underscoring the importance of these properties in the FA process.

To better understand the characteristics of the mixtures that were wrongly predicted, Figure 11 shows the boundary between the pairs $(\sigma, D_h/\delta_f)$, $(\sigma, Le_D)$, $(\sigma, Le_{eff})$, and $(\sigma, \beta)$. These figures show that none of these pairs is able to predict the aforementioned H$_2$/O$_2$ point ($p$ = 10 kPa, $\phi$ = 0.5), which corresponds to a mixture with $Le_D$ = 0.42. Under this condition the flame is possibly influenced by thermodynamic effects, which is not well captured by the

<figure><img src="image_11.png" /><figcaption>Fig. 11: Optimal boundary testing with the data that only reported morphology at the LS. Black and gray data points correspond to LS2 morphology, and are painted in different colors to better differentiate them for their similar values.</figcaption></figure>

proposed boundaries due to the lack of data available for $Le_D$ < 1. An additional H$_2$ data-point is wrongly predicted by all the properties pairs, except for $(\sigma, Le_{eff})$ - a stoichiometric H$_2$/O$_2$ mixture, $p$ = 0.74 bar ($\sigma$ = 8.2). Note that mixtures with similar properties were also not well estimated by these FCP pairs with the ES morphology, as discussed in subsection 5.4.1. This shows that the proposed boundary limitations observed with the ES data still hold for the LS, suggesting a direct relationship between the morphologies in both stages of FA. It is thus concluded that the morphologies observed during ES correlate well with those observed during LS. Additionally, the evidence presented shows that the mixtures that exhibited the $ES1 \rightarrow LS2$ morphology, are in an intermediate transitional state between the $ES1$ and $ES2$ morphologies.

The proposed FCP boundaries can serve as a guide for future evaluations of characteristics observed in this study. As an example of potential research, the limited amount of data in the literature on the LS1 morphology highlights the significant potential for studying the detonation onset mechanisms in diluted mixtures with $\sigma$ < 8 and $\sigma_{SL}/c_b$ < 0.04. These mixtures exhibit simpler flame/shock structures when compared to those observed with the LS2 morphology, which permits for a more thorough study of the flame/shock interaction during the DDT process. On the other hand, if the interest is to study the early stages of FA, these boundaries suggests the selection of mixtures that exhibit flame inversion (tulip flame) or flame instabilities during the initial stages of acceleration. The nature of the different flame instabilities observed in this work remains an open question.

### 6.2. Flame length during LS

One of the main metrics that differentiate the morphologies seen during LS1 and LS2 is the flame

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<figure><img src="image_12.png" /><figcaption>Fig. 12: Normalized flame length during the late stages of flame acceleration as a function of the mixtures' expansion ratio, $\sigma$.</figcaption></figure>

length. Figure 12 shows $L_{f,LS}/D_h$ as a function of $\sigma$ for the mixtures that yield the LS1 and LS2 morphologies. The shorter flames (LS1) are, in general, characterized by smaller $\sigma$ value, and around $\sigma \approx 8$ a transition between shorter to longer flames is observed, which already has been noted when discussing the boundaries presented in Figure 8. It is interesting that the LS1 morphology has only been observed in our experiments and those performed by Ballossier et al. [12]. This is somewhat surprising since LS1 is usually correlated with ES1 (tulip flame), i.e. $ES1 \rightarrow LS1$, which is the most studied morphology during ES. This suggests that the behavior of flame and shock morphologies during DDT in slow and short flames, such as those documented for $H_2/O_2$ diluted with Ar or He in this work, has received little attention in the literature.

## 7. Conclusion

The effect of fundamental combustion properties (FCP) on the flame and shock morphology and dynamics during the deflagration-to-detonation transition in unobstructed channels was investigated. Three extensive experimental campaigns were carried out in which the effects of different FCP were evaluated: (i) the effect of the specific heat ratio, $\gamma$, for three mixtures with fixed expansion ratio, (ii) the effect of Lewis number, $Le$ ($Le < 1$ and $Le > 1$) for two mixtures with fixed product of $\sigma$ and the flame speed, $\sigma_{sL}$, and (iii) the effect of $\sigma_{sL}$ for three mixtures with fixed laminar flame thickness, $\delta_f$. A morphology database was compiled to enable meaningful statistical analysis. With this, FCP boundaries were proposed using Feature Elimination Techniques and Logistic Regression models. The main conclusions of this work are:

* Two different morphology types were observed during the early stage (ES) of flame acceleration: (i) inverted (tulip-shaped) flames (ES1), or (ii) flat, wrinkled flames (ES2). Thereafter, as the flame accelerated further, precursor compression waves formed in the late stage (LS) of FA, and two morphologies were also identified: (i) short, symmetrical flames with precursor compression wave formed relatively far from

the flame (LS1), or (ii) long flames with multiple precursor compression waves generated successively in front of the flame (LS2).

* Based on the Logistic Regression model and Recursive Feature Elimination applied to the compiled database, it was possible to propose boundaries between the different types of morphologies, showing that the pairs of FCP that optimally separate the observed morphologies are $(\sigma, \sigma_{sL}/c_b)$ and $(\sigma_{sL}/c_b, \gamma)$.

* A correlation was found between the morphologies during ES and LS, where for most cases the transitions $ES1 \rightarrow LS1$ and $ES2 \rightarrow LS2$ were observed. This was also confirmed with the data analysis, since the same optimal boundaries were able to separate both datasets independently.

* The mixtures $2H_2 + O_2 + N_2$, and rich $H_2/O_2$ ($\phi = 1.7$), which exhibited $ES1 \rightarrow LS2$ transitions, are in a morphological transition state. This was further supported by their proximity to the optimal boundaries.

* The boundaries proposed were not able to separate data from lean $H_2/O_2$ mixtures that are susceptible to thermodiffusive instabilities. This highlights the lack of information in the FA literature on mixtures with $Le < 1$, which opens up possibilities for future experiments.

* The proposed FCP boundaries can serve as a guide for researchers interested in investigating certain characteristics observed in this study during the early or late stages of FA, by selecting mixtures with appropriate FCP according to the limits proposed in this work.

As part of future work, more mixtures with $Le < 1$ will be considered with the aim of gathering more statistical information on the effect of thermo-diffusive instabilities on FA. This will lead to the proposal of boundaries that can predict a broader range of experimental conditions. Additionally, the different mechanisms of detonation onset observed during the study of the nine mixtures addressed in this work will be analyzed and correlated with the fundamental combustion properties.

## CrediT authorship contribution statement

C. Mejía-Botero: Conceptualization, Investigation, Formal analysis, Writing - Original Draft, Writing - Review & Editing. F. Virot: Investigation, Writing - Review & Editing, Supervision. L. F. Figueira da Silva: Investigation, Writing - Review & Editing, Supervision. J. Melguizo-Gavilanes: Conceptualization, Investigation, Formal analysis, Writing - Review & Editing, Supervision, Funding acquisition.
1 Declaration of competing interest

2 The authors declare that they have no known com-
peting financial interests or personal relationships that
could have appeared to influence the work reported in
this paper.
5

6 Acknowledgements

7 The authors gratefully acknowledge the financial
8 support from l'Agence Nationale de la Recherche
9 Program JCJC (FASTD ANR-20-CE05-0011-01) and
10 L'Institut Pprime. For the purpose of Open Access,
11 a CC-BY public copyright licence has been applied
12 by the authors to the present document and will be
13 applied to all subsequent versions up to the Author
14 Accepted Manuscript arising from this submission.
15

15 Supplementary material

16 The detailed methodology for the mixtures’ selec-
tion is presented as supplementary material.
17

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