# Cell size distributions of hydrogen-air flames during propagation in Hele-Shaw burners

C C Mejía-Botero, L Pratabuy Tran-Quang, C Almarcha

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C C Mejía-Botero, L Pratabuy Tran-Quang, C Almarcha. Cell size distributions of hydrogen-air flames during propagation in Hele-Shaw burners. 41st International Symposium on Combustion, The Combustion Institute, Jul 2026, Kyoto, Japan. hal-05495094

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Cell size distributions of hydrogen-air flames during
propagation in Hele-Shaw burners

C.C. Mejía-Botero$^{a,*}$, L. Pratabuy Tran-Quang$^a$, C. Almarcha$^a$

ªAix Marseille Univ, CNRS UMR7342, Centrale Marseille, IRPHE, Marseille, France

**Abstract**

Flame instabilities in narrow channels produce cellular structures that strongly affect flame propagation. One of the key distinguishing factors between thermodiffusive and hydrodynamic instabilities is the characteristic cell size formed on the flame surface. This study quantifies flame cell size distributions in a Hele–Shaw burner for hydrogen–air mixtures across a wide range of equivalence ratios (0.4 ≤ $\phi$ ≤ 2) and channel gaps (0.58 ≤ $h$ ≤ 7 mm), using schlieren imaging combined with a cusp-detection algorithm. For very lean mixtures, thermodiffusive instabilities dominate, producing small, highly branched cells that are largely independent of $h$. In contrast, for rich mixtures, the cell size increases markedly with $h$ under the conditions examined. Furthermore, the ratio of consumption speed to laminar flame speed exhibits a clear linear decrease with the ratio of mean cell size to laminar flame thickness. This methodology enables direct comparisons between experiments and simulations, offering a framework to connect flame morphology with propagation dynamics in confined channels.

Keywords: Hele-Shaw cells; flame propagation; narrow channels; hydrogen safety; flame instabilities, flame morphology

*Corresponding author.

**1. Introduction**

Hydrodynamic and thermodiffusive instabilities strongly affect flame propagation in narrow channels. These instabilities generate cellular structures that can amplify during propagation, increasing the flame speed by up to four times compared with a laminar, unstretched flame [1–4]. One of the experimental setup used for investigating such instabilities is the Hele-Shaw burner [5–9]. This burner consists of a confined region between two flat, parallel plates separated by a millimeter-scale gap $h$. Such a configuration enables the study of instabilities under quasi-2D conditions and facilitates the visualization of characteristic flame patterns.

Hydrodynamic instabilities arise from the density jump between unburned and burned gases. In contrast, thermodiffusive instabilities result from the mismatch between thermal diffusivity ($\alpha$) and mass diffusivity of the limiting reactant ($D_m$), characterized by the Lewis number $Le = \alpha/D_m$. When $Le < 1$, flames are prone to destabilization through this mechanism. This condition holds for lean hydrogen ($H_2$)–air mixtures, where both instabilities act simultaneously, generating complex morphologies. However, this concurrence complicates the experimental isolation of the individual contributions of each instability. To address this, numerical studies [10–20] have

examined their separate effects. These works show that purely hydrodynamic instabilities promote large-scale cusps with fractal-like flame fronts, whereas thermodiffusively unstable flames undergo chaotic formation and destruction of small cellular structures.

These studies indicate that the size and distribution of the cellular structures emerging during propagation represent a clear differentiating feature between these instabilities. Despite its importance, only a few studies have quantified cell size distribution in Hele-Shaw cells [21, 22]. However, these studies do not relate cell sizes to consumption rates, and were developed for propane-air mixtures. No studies were found that examine cell size distributions for $H_2$/air mixtures, which are more prone to thermodiffusive instability. Addressing this gap is the main objective of the present study. This analysis complements the classical dispersion relation analysis (DRA), which correlates the growth rate of surface perturbations, $\sigma$, with their wavenumber, $\kappa$, along the flame front. DRA is one of the most widely used approaches for studying flame instabilities [3, 5, 13–20, 23–28]. Nevertheless, the determination of the growth rate is limited to the linear stages whereas most flames are in a saturated nonlinear stage with a morphology not always directly linked to $\sigma(\kappa)$. For these reasons, analyzing the cell sizes distribution can provide complementary insights into the behavior of hydrodynamic and ther-
modiffusive instabilities, and the methodology proposed can be applied to a wide range of experiments and numerical simulations.

## 2. Methodology

Experiments were performed in a Hele–Shaw burner of 30 cm height and 10 cm width, with gaps $h \in [0.58, 7]$ mm. The channel is open at the top, where ignition is triggered by tungsten electrodes. Premixed H₂/air, with equivalence ratios $\phi \in [0.4, 2.3]$, were introduced from the bottom through niformly distributed small holes. Mixtures were prepared using Bronkhorst EL-Flow Select mass flow controllers, ensuring an accuracy of $\pm 2\%$ in $\phi$. To avoid flame-flow interaction, a PARKER LEGRIS solenoid valve shut off the reactant supply 35 ms prior to ignition. Flame propagation was visualized using a Z-type schlieren setup with two 30 cm–diameter, 2 m–focal-length concave mirrors and a green LED light source. Images were recorded with a Photron Fast Cam MiniAX. A resolution of 0.2 mm/px was used, which defines the lower limit of flame cell sizes that can be reliably measured. Further details of the optical arrangement and the fundamental combustion properties of the selected mixtures are provided in [29].

The methodology for measuring cell size is illustrated in Figure 1. In a) it is shown the flame propagation for $\phi = 0.8$ and $h = 0.88$ mm at two different times. The cell size, $\lambda_{cell}$, is defined as the distance between adjacent cusps formed by instabilities; thus, cusp detection is required and performed frame by frame. For each image, the flame front contour is extracted, yielding the green curves shown in Figure 1 b).

<figure><img src="image_2.png" /><figcaption>Fig. 1: Cusp detection methodology.</figcaption></figure>

Since cusps correspond to points of maximum curvature, the angle $\theta$ between consecutive contour points is computed, and cusps are identified as local maxima of $d\theta/dx$, as illustrated in Figure 1 c). Once cusp coordinates are extracted, the cell size is defined

as $\lambda_{cell} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$, where subscripts 1 and 2 denote adjacent cusp points. This procedure is applied over the entire flame front, yielding a distribution of $\lambda_{cell}$ values for each frame. For each $(\phi, h)$ pair, a global distribution is obtained by analyzing the entire $\lambda_{cell}$-distributions from all frames acquired during the propagation. Each condition was analyzed over roughly 1000 frames ($\approx$ 200 frames per test across five replicates), ensuring robust statistics for the cell-size distributions. The same methodology was applied to the numerical simulations of Berger et al. [20], who modeled a 10 cm–wide Hele–Shaw burner (identical to that used here) with H₂/air mixtures at $\phi = 0.4$. Using the supplementary video provided in their work, the present algorithm was applied to all frames, enabling direct comparison with our experimental results.

## 3. Results and discussion

### 3.1. Flame cell sizes distribution

Figure 2 presents chronophotographs of flame propagation for mixtures with $\phi = 0.4$ and 1.2 in channel gaps of $h = 0.58$, 1.5, and 3 mm. These cases enable comparison of flame patterns for mixtures with $Le < 1$ ($\phi = 0.4$) and $Le > 1$ ($\phi = 1.2$), thereby highlighting the influence of thermodiffusive instabilities under varying confinement. For $\phi = 0.4$, no flame propagation is obtained at $h = 0.58$ mm, since this gap is below the quenching limit of this mixture. At $h = 1.5$ and 3 mm, similar flame structures are visualized: the front initially appeared corrugated, and during propagation some perturbations grew into finger-shaped flames which contain multiple cellular structures. These structures repeated, producing branching with fractal-like patterns, with stronger ramifications at $h = 1.5$ mm than at $h = 3$ mm. However, no significant variation in the cell sizes are observed. In contrast, flames with $\phi = 1.2$ displayed notable variations in cell size with the gap. At $h = 0.58$ mm (the propagation limit for this mixture), elongated structures with small corrugations are observed, without finger-shaped flames. As $h$ increased, finger-shaped structures emerged, accompanied by a clear growth in cell size.

To analyze the $\lambda_{cell}$ distribution, Figure 3 a) shows the cumulative distribution function (CDF) of cell sizes normalized by the flame thickness, i.e., $\lambda_{cell}/\delta_f$, for mixtures with $\phi = 0.4, 0.7, 1.2$, and 2, at gaps of $h = 1.5$ and 3 mm. For comparison, numerical results from Berger et al. [20] are also included. The agreement between experiments and simulations is particularly close for $\phi = 0.4$, highlighting both the predictive capability of the simulations and the suitability of the present methodology for experimental validation of numerical studies.

The influence of $\phi$ and $h$ on flame cell size is evident in Figure 3 a). For a given $h$, $\lambda_{cell}$ increases markedly with $\phi$, in agreement with previous numerical studies [13, 15–20], which attribute smaller structures to thermodiffusive instabilities. Furthermore, as
<figure><img src="image_3.png" /><figcaption>Fig. 2: Examples of flames propagating in Hele-shaw cells. Videos recorded at 250 FPS.</figcaption></figure>

$\phi$ increases, the difference in $\lambda_{cell}$ between $h = 1.5$ and $h = 3$ mm becomes more pronounced, consistent with the trends observed in Figure 2. Figure 3 b) presents the CDF of $\lambda_{cell}$ normalized by its mean value, $\bar{\lambda}_{cell}$, showing a partial collapse of the data. This indicates that only the mean cell size varies, while the relative distribution remains unchanged. The latter shows that $\bar{\lambda}_{cell}$ can be used to analyze the effect of cell size on the flame consumption rate, as will be analyzed later.

To further assess the effect of $h$ under lean and rich conditions, Figure 4 presents the probability density function (PDF) of $\lambda_{cell}/\delta_f$ for a) $\phi = 0.4$ and b) $\phi = 1.2$ at different gaps. For $\phi = 0.4$, gaps up to $h = 7$ mm were investigated, showing negligible variation in the $\lambda_{cell}$ distribution. This indicates that, for ultra-lean mixtures, thermodiffusive instabilities strongly affect cell sizes, largely independent of the $h$ value and the associated hydrodynamic effects. In contrast, for $\phi = 1.2$, tests with $h \in [0.58, 3]$ mm reveal a clear increase of $\lambda_{cell}$ with $h$, which shows that the flame cell size is more dependent on the gap for rich mixtures than for lean mixtures.

Figure 5 shows the variation of $\lambda_{cell}/\delta_f$ with $\phi$ for all experiments. For all gaps, $\lambda_{cell}/\delta_f$ increases with $\phi$, consistent with previous observations, but a more pronounced increase in $\lambda_{cell}/\delta_f$ with $\phi$ is observed for lean mixtures.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Cumulative distribution function of $\lambda_{cell}$ for selected cases normalized by a) the laminar flame thickness $\delta_f$, and b) the mean cellular flame size $\bar{\lambda}_{cell}$</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: Probability density function of $\lambda_{cell}/\delta_f$ for a) $\phi = 0.4$, and b) $\phi = 1.2$ for different $h$ values.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5: $\bar{\lambda}_{cell}/\delta_f$ as a function of $\phi$.</figcaption></figure>

## 3.2. Relation between $\bar{\lambda}_{cell}$ and consumption speed

One of the most notable effects of instabilities on flame propagation is the enhancement of the consumption rate ($s_{cons}$). In this section, the relation between $\bar{\lambda}_{cell}$ and $s_{cons}$ is studied. $s_{cons}$ is calculated by linear interpolation of the burned-gas area over time, divided by the channel width. Figure 6 presents $s_{cons}$.
as a function of $\bar{\lambda}_{cell}$ for different $h$. The variations in $\bar{\lambda}_{cell}$ for each $h$ are due to the use of mixtures with different $\phi$. A clear $s_{cons}$ maximum is observed for each $h$, whose corresponding $\bar{\lambda}_{cell}$ value increases with $h$.

<figure><img src="image_7.png" /><figcaption>Fig. 6: Flame consumption speed $s_{cons}$ as a function of $\bar{\lambda}_{cell}/\delta_{f}$.</figcaption></figure>

However, Figure 7 shows that the normalization $s_{cons}/s_{L}$ against $\bar{\lambda}_{cell}/\delta_{f}$ reveals a linear trend in which $s_{cons}/s_{L}$ decreases with increasing $\bar{\lambda}_{cell}/\delta_{f}$, provided that $s_{cons}/s_{L} > 1$, i.e. when heat losses effects are not the ultimate dominant effect. This shows that flames that produce smaller cell structures generate higher consumption rates. In addition, the consumption rate increases significantly with $h$, which is due to the decrease of the momentum and heat losses when $h$ increases.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Normalized flame consumption speed $s_{cons}/s_{L}$ as a function of $\bar{\lambda}_{cell}/\delta_{f}$.</figcaption></figure>

This study investigated the cell sizes distribution of H₂/air flames in narrow Hele-Shaw channels to study flame instabilities. Using high-resolution schlieren imaging and a frame-by-frame cusp-detection methodology, the size and distribution of cellular structures were quantified across a wide range of equivalence ratios ($\phi$) and channel gaps ($h$).

For lean mixtures ($Le < 1$), thermodiffusive instabilities dominated flame morphology, producing small, highly branched cellular structures whose size was largely independent of the channel gap. In contrast, for richer mixtures ($Le > 1$), cell size increased significantly with both $\phi$ and $h$.

Analysis of the cumulative and probability density functions allowed a direct link between cell sizes and propagation dynamics. The relative distribution of cell sizes was found to scale with the mean size value, which can then be used to analyze flame dynamics.

The $s_{cons}/s_{L}$ against $\bar{\lambda}_{cell}/\delta_{f}$ relationship presented a linear trend for all $h$ values, showing that flames that produce smaller cell structures generate higher consumption rates.

This work provides a quantitative framework for relating flame cell morphology to propagation dynamics in confined channels. The methodology is broadly applicable to both experimental and numerical studies and offers a tool to better understand the role of instabilities in flame behavior, and to validate numerical results.

### Declaration of competing interest

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

### Acknowledgments

This research was supported by the ECOSAFE ANR project ANR-21-CE05-0028.

## 4. Conclusions
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