# Flame propagation limit and consumption speed of hydrogen-air flame in narrow channels

C C Mejía-Botero, L Pratabuy Tran-Quang, Z Chen, P Boivin, C Almarcha

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C C Mejía-Botero, L Pratabuy Tran-Quang, Z Chen, P Boivin, C Almarcha. Flame propagation limit and consumption speed of hydrogen-air flame in narrow channels. 41st International Symposium on Combustion, The Combustion Institute, Jul 2026, Kyoto (Japan), Japan. hal-05495064

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# Flame propagation limit and consumption speed of hydrogen-air flame in narrow channels

C.C. Mejía-Botero$^{a,*}$, L. Pratabuy Tran-Quang$^a$, Z. Chen$^b$, P. Boivin$^b$, C. Almarcha$^a$

## Abstract

This work presents a methodology for studying hydrogen-air propagation limits and flame consumption speeds in narrow Hele-Shaw channels using 2D Lattice Boltzmann simulations and experimental analysis. Experiments focused on flame propagation along the channel face, while simulations modeled propagation within the sub-millimetric gap $h$, reducing simulation time and capturing asymmetries. It was confirmed that as $h$ increases, the propagation limits widen and the consumption speed increases. In addition, a normal consumption speed, $S_{cn}$, based on the experimental flame length measurements, was introduced. The results show that 2D gap simulations predict both propagation limits and $S_{cn}$ with less than 5% deviation for most experimental conditions, enabling efficient prediction of flame behavior without simulating the full channel face.

Keywords: Flame propagation; Narrow channels; Quenching limits; Hydrogen safety; Micro-combustors

## 1. Introduction

The study of flame dynamics in narrow, confined channels is fundamental to the design of microcombustors, microscale propulsion systems, and portable power generators, where characteristic dimensions fall within the millimeter or sub-millimeter range [1–4]. Under such conditions, heat losses become comparable to heat release rates, and both flame propagation limits and instability mechanisms — such as hydrodynamic and thermo-diffusive effects — differ significantly from those observed in large-scale systems [5–8].

An interesting experimental configuration for investigating flame instabilities is the Hele-Shaw burner [9–14]. This device consists of two parallel plates of height $H$ and width $W$, separated by a gap distance $h$. By varying $h$, the influence of confinement on flame dynamics can be systematically examined by observing the flame front along the channel face $H \times W$, thus enabling visualization of characteristic instability patterns under quasi-2D conditions.

Numerical simulations of these systems [14–18] are increasingly used to validate models that capture multiple interacting instabilities. Typically, such studies perform two-dimensional simulations of the channel face, assuming that the flame remains steady and symmetric within the gap. However, this approach involves two primary limitations: (i) the dimensions $H \times W$ are much larger than the gap $h$, making these

simulations computationally demanding, and (ii) the assumption of symmetry and uniformity within the gap can lead to inaccuracies in estimating flame consumption speeds [19–22].

To address these challenges, the present study introduces a methodology for investigating hydrogen ($H_2$) air premixed flame propagation in Hele-Shaw channels based on numerical simulations of flame dynamics within the gap. The Lattice Boltzmann Method has previously demonstrated the ability to predict symmetry breaking in narrow channels in the absence of heat loss [23]. The objective of this work is to take into account heat loss and compare numerically predicted flame consumption speeds and propagation limits with the corresponding experimental measurements conducted in Hele-Shaw channels of fixed $H$ and $W$, and with varying sub-millimeter and millimeter gap sizes $h$.

## 2. Methodology

A schematic of the Hele-Shaw cell is shown in Figure 1(a). The burner has a height of $H = 30$ cm, a width of $W = 10$ cm, and gap distances of $h = 0.6$, 0.7, and 0.88 mm. For each $h$, the lean and rich propagation limits, $\phi_{lean}$ and $\phi_{rich}$, are determined both numerically and experimentally, and mixtures within this equivalence ratio range are investigated. Additional tests at $\phi = 1.2$ are conducted for $h \in [0.58, 1.1]$ mm, enabling a more detailed analysis of

<footnote>$^a$Aix Marseille Univ, CNRS, Centrale Med, IRPHE, Marseille, France</footnote>

<footnote>$^b$Aix Marseille Univ, CNRS, Centrale Med, M2P2, Marseille, France</footnote>

<footnote>*Corresponding author.</footnote>
the influence of $h$ on the flame consumption rate.

<figure><img src="image_2.png" /><figcaption>Fig. 1: (a) Schematic Hele-Shaw cell, (b) flame propagation in the gap $h$ (domain for numerical simulations), (c) example of flame propagation in the face for $\phi = 0.6$, $h = 0.88$ mm, captured using the schlieren technique, and (d) flame front detection and flame length $\eta$.</figcaption></figure>

Numerical simulations are performed to analyze flame propagation within the gap, while experimental measurements focus on propagation along the channel face. The corresponding procedures are detailed below.

## 2.1. Numerical

The numerical methodology follows that described in [23], previously employed to analyze symmetry breaking in narrow channels. The governing equations for mass, momentum, energy, and species transport are solved with the Lattice-Boltzmann solver ProLB [24–27]. Details of the formulation are given in [26]. The flow is assumed ideal, and combustion is modeled using a 12-step skeletal H₂-air mechanism [28, 29]. Transport properties use a constant Prandtl number and species-specific Schmidt numbers from the CERFACS database [30], validated for this mechanism. The effective Lewis and Zeldovich numbers follow [31] and [32], respectively.

The 2D computational domain, shown in Figure 1(b), represents the space between the glass walls

of the Hele-Shaw cell, separated by a gap $h$ in the $z$-direction ($z \in [0, h]$). The domain length $L_y$ is set at 10 times longer than the channel width $h$. Boundary conditions along $y$ direction are set as a constant pressure outlet for burnt gases and a non-reflecting inlet for fresh gases. The grid size is set at $10\mu m$ for an accurate resolution near the walls. The domain is initially filled with an H₂/air mixture at equivalence ratio $\phi$, and the flame propagates from the open to the closed end along $y \in [0, \infty]$. Contrary to previous 2D-simulations in the gap [23], the boundary conditions on the walls are now taken as isothermal, which corresponds to a physical asymptotic limit of infinite heat capacity or infinite conductivity and thickness of the walls. This drastic assumption gave however the same minimal quenching distance as when using the heat transfer parameter introduced in [19].

Propagation limits are determined by varying $\phi$ in increments of $\Delta\phi = 0.01$ until the lean and rich limits, $\phi_{\text{lean}}$ and $\phi_{\text{rich}}$, are identified. Outside this range, no self-sustained flame propagation occurs. Results are presented for $\phi \in [\phi_{\text{lean}}, \phi_{\text{rich}}]$ for each $h$.

Simulations are performed in a reference frame moving with the flame at the consumption speed $U_{\text{ref}} = S_{\text{num}}$, defined as $S_{\text{num}} = \frac{1}{\rho_u Y_{D,u}} \int \dot{\omega}_D dx$, where $D$ denotes the deficient fuel species, $\dot{\omega}_D$ its consumption rate, and $\rho_u$ and $Y_{D,u}$ the density and mass fraction of the unburned mixture.

The numerical results for $S_{\text{num}}$ and $[\phi_{\text{lean}}, \phi_{\text{rich}}]$ are compared with the corresponding experimental measurements, whose methodology is described below.

## 2.2. Experimental

For each test, the channel is filled with the desired H₂/air mixture via multiple uniformly distributed injection holes at the channel's bottom (see Figure 1(a)). H₂ and air flow rates are controlled by mass flow controllers, calibrated to achieve the target equivalence ratio, $\phi$, with an accuracy of $\pm 2\%$. Flow is stopped 35 ms before ignition using a solenoid valve to stabilize the mixture and prevent flame-flow interaction. Flame propagation along the channel face is recorded using the schlieren technique, with the optical setup described in [33].

Figure 1(c) shows an example of flame propagation on the Hele-Shaw cell face for a mixture with $\phi = 0.6$ and $h = 0.88$ mm. The consumption speed, $S_c$, is estimated from two consecutive images as $S_c = \frac{1}{W} \frac{dA_b}{dt}$, where $A_b$ is the area of burnt gases. This approach, commonly used in Hele-Shaw experiments [33], accounts for the increase in flame front area due to instabilities. However, numerical simulations within the gap do not capture the flame surface increase on the channel face. To address this, we define an average normal consumption speed, $S_{cn}$, as $S_{cn} = \frac{1}{\eta} \frac{dA_b}{dt}$, where $\eta$ is the total flame length on the channel face, obtained from contour recognition of schlieren images (see Figure 1(d)). Normalizing
by $\eta$ allows gap simulations to predict $S_{cn}$, effectively accounting for the flame area increase on the channel face. Finally, to determine the experimental propagation limits, $\phi$ was varied in steps of $\Delta\phi = 0.03$ near the propagation limits for each $h$, yielding the range [$\phi_{lean}, \phi_{rich}$].

# 3. Results and discussion

## 3.1. Propagation limits

Figure 2 shows the propagation limits, [$\phi_{lean}, \phi_{rich}$], for the three $h$-values, calculated numerically (×) and measured experimentally (solid line). The simulations capture the propagation limits well across all gap sizes. The largest deviation occurs for $\phi_{rich}$ at $h = 0.88$ mm, where the experimental value is 2.31 and the numerical prediction is 2.1, corresponding to an 8% difference. For the remaining limits, the discrepancy between numerical and experimental results is below 5%. The results also indicate that the propagation limits expand with increasing $h$, reflecting the higher heat and momentum losses at smaller gaps. These findings indicate that the isothermal wall assumption in the numerical model is consistent with the experimental conditions studied.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Comparison between numerical (×) and experimental (solid lines) $\phi$-propagation limits for different values of $h$.</figcaption></figure>

## 3.2. Propagation on the channel face

Figure 3 shows representative chronophotographs of experimental flame propagation: (a) for $\phi = 1.0$ at the three gap values $h$, and (b) for $h = 0.88$ mm at $\phi = 0.55, 1.20$, and 2.31, where $\phi = 0.55$ and 2.31 correspond to the lean and rich propagation limits for this gap, respectively. In (a), increasing $h$ allows faster flames and promotes Darrieus-Landau instability, which leads to longer flame lengths $\eta$, specifically $\eta = 117, 128$, and 141 mm for $h = 0.6, 0.7$, and 0.88 mm. In (b), the effect of $\phi$ at fixed $h$ is illustrated. Lean mixtures produce smaller cellular structures and longer flames due to thermo-diffusive instabilities, whereas richer mixtures result in shorter flame lengths, with $\eta = 155, 132$, and 109 mm for $\phi = 0.55, 1.20$, and 2.31, respectively.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Chronophotography of the propagation of H₂/air flames on the face of the Hele-Shaw for (a) $\phi = 1$ at different values of $h$, and (b) $h = 0.88$ mm and different values of $\phi$. Videos recorded at 250 FPS.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: (a) experimental consumption speed $S_c$ for the three $h$-values for $\phi$ within the respective propagation limits. (b) Experimental average normal consumption speed $S_{cn}$ (+) and numerical 2D propagation speed (solid line).</figcaption></figure>

## 3.3. Consumption speed

Figure 4 (a) shows the experimental consumption rates $S_c$ for three sub-millimetric gaps. For $h = 0.88$ mm, the propagation speed is already larger than the laminar flame speed, whatever the equivalence ratio $\phi$. This means that for such gap, the wrinkling of the flame front in the directions (x,y) and in the gap is more efficient than heat losses. As expected, the decrease in the gap width $h$ decreases the propagation speed due to an increase in the heat losses. However, we can note that the minimum flame speed, although not obtained with the same $\phi$ for different gaps, re-
mains around 1.5 – 1.75 m/s on the lean side. We can note a relative dispersion of the experimental results, especially for lean and stoichiometric mixtures. This may originate from the fluctuations in the flame surface due to intrinsic instabilities of these flames that are prone to develop Darrieus-Landau and Thermodiffusive instabilities. We can expect that these dispersions will be reduced when dividing the propagation speed by the flame surface (here the flame contour if the profile in the gap is stationary).

This is what is done on Figure 4 (b), where the consumption rate is normalized by the flame front length (which is associated with a flame surface) to form a normal flame speed averaged over the whole flame contour. The dispersion in the data is indeed reduced and we compare $S_{cn}$ to the 2D numerical simulation inside the gap. The two values differ by less than 8% whatever the equivalence ratio or gap width. We can conclude that at first order the main parameter driving the propagation speed is the profile taken by the flame in the gap. We observe a relative under-prediction of $S_{cn}$ by numerics on the lean side ($\phi < 1$) and an over-prediction on the rich side ($\phi > 1$) with a good agreement around stoichiometry. These differences may originate from thermodiffusive effects, provided that cusps have less influence in flame propagation than the flame fingers.

Finally, Figure 5 extends the analysis for the mixture $\phi = 1.2$ (corresponding to maximum flame velocity and smallest quenching distance), for different $h$-values. The simulations are performed in the range $h \in [0.56, 1.1]$ mm, and the experiments in $h \in [0.58, 1.1]$ mm with an uncertainty of 0.02mm. The smallest $h$-value slightly differs as it corresponds to the propagation limit for $\phi = 1.2$ obtained numerically and experimentally, with a difference of only 5% between both approaches. The results show, once again, that the simulations predict successfully the experimental $S_{cn}$. Furthermore, the difference between $S_{cn}$ and $S_c$ is found to increase significantly with $h$ over the studied conditions, since the flame velocity increases with $h$, thereby enhancing the Darrieus-Landau instability. It is important to note that, while $S_c$ is commonly used in the literature to measure flame consumption rates in Hele-Shaw cells, the $S_{cn}$ parameter introduced here provides a more accurate measure for comparing numerical results from gap simulations.

## 4. Conclusions

This work presents a methodology for predicting propagation limits and flame consumption speeds in Hele-Shaw channels using 2D numerical simulations. The experimental analysis focused on flame propagation along the channel face, while numerical simulations modeled propagation within the gap.

The numerical model accurately predicted both the lean and rich propagation limits, as well as the average normal consumption speed $S_{cn}$, with deviations below 5% for most of the tested conditions. The results demonstrate that increasing the gap width

<figure><img src="image_6.png" /><figcaption>Fig. 5: Consumption speeds for $\phi = 1.2$ at different $h$-values.</figcaption></figure>

$h$ expands the propagation limits and increases the consumption speeds, reflecting the influence of heat and momentum losses on flame dynamics in narrow gaps. Comparisons between experimental consumption speeds $S_c$ and $S_{cn}$ confirm that normalizing by the total flame length $\eta$ offers a better metric for comparing gap simulations with facial experimental measurements.

Overall, this study shows that 2D simulations within the Hele-Shaw gap can accurately reproduce key experimental measurements, including propagation limits and flame consumption speeds, without the need to simulate the entire channel face. This methodology provides an efficient and reliable framework for analyzing flame behavior in confined narrow channels, with potential applications in microcombustors, microscale propulsion systems, and portable power generators.

## 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.
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