# Structure and speed of hydrogen-air triple flames

Kennie Chaplet, Laurent Selle, Thierry Poinsot

► To cite this version:

Kennie Chaplet, Laurent Selle, Thierry Poinsot. Structure and speed of hydrogen-air triple flames. Combustion and Flame, 2026, 283, pp.114579. 10.1016/j.combustflame.2025.114579. hal-05408340

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# Structure and speed of hydrogen-air triple flames

Kennie Chaplet<sup>ID*</sup>, Laurent Selle<sup>ID</sup>, Thierry Poinsot

Univ Toulouse, Toulouse INP, CNRS, IMFT, 2 All. du Prof. Camille Soula, Toulouse, 31400, France

## ARTICLE INFO

**Keywords:**

Hydrogen

Triple flames

Squished flames

## ABSTRACT

Triples flames play an important role in flame stabilization processes, especially for hydrogen flames. Being able to predict the speed $s_T$ of a triple flame is crucial in many hydrogen combustor designs as it controls flame stabilization and flashback. These speeds highly depend on the mixing field between fuel and air in front of the flame, usually measured by the thickness of the profile of mixture fraction $z$: $\Delta = 1/\max(\partial z/\partial X)$. In the limit of high $\Delta$ values, triple flames propagate at a speed $s_T$ given by theory. When $\Delta$ decreases, $s_T$ is modified and for conventional fuels, theory indicates that it decreases.

The present work focuses on the structure and speed of non-propagating H₂-air triple flames when $\Delta$ is varied. It uses a DNS of hydrogen-air combustion with detailed chemistry where the inlet velocity is set via a control loop ensuring that a steady triple flame is stabilized in a stream at a given value of $\Delta$ so that a ($s_T$ vs $\Delta$) curve can be built. Results confirm that, even for hydrogen, $s_T$ can be predicted for large $\Delta$, by the standard expression $s_T = (\rho_u/\rho_b)^{1/2}s_L$ if expansion ratio $\rho_u/\rho_b$ and laminar burning velocity $s_L$ are evaluated at their maximum value, respectively near stoichiometry and at equivalence ratio 1.6 for H₂. When $\Delta$ is decreased, the triple flame speed decreases significantly but extinction is not observed.

**Novelty and significance statement:** Few detailed studies on hydrogen triple flames exist, even though this information is of practical use for combustor design. This paper uses high-fidelity DNS to show that the theoretical expression for triple flame speeds $s_T = (\rho_u/\rho_b)^{1/2}s_L$ can work for hydrogen flames with simple and logical adjustments. It can then be used at any pressure by experimentalists and engineers to predict stabilization of hydrogen - air diffusion flames.

Moreover, this work studies triple flames stabilized in a steady mixing field, contrary to previous works focusing on the ignition and propagation of hydrogen triple flames. In these conditions, it assesses quantitatively, for the first time, how the gradient of mixture fraction decreases triple H₂/air flame speeds. However, quenching is not observed. A scaling is obtained and extended to engine-relevant pressures.

## 1. Introduction

The advent of hydrogen to replace conventional fossil fuels such as kerosene, natural gas or propane raises critical questions regarding flame stabilization. While conventional flames are often lifted and far from injectors in most systems, hydrogen flames are faster, thinner and can come much closer to the injector lips, burning in a diffusion mode. This attachment of the diffusion flame on injector lips [1,2] is a key phenomenon for design. Attached flames can increase thermal load on the burner lips [3], increase NOₓ emissions via local hot spots and lead to flashback when one of the streams is premixed.

Even though all these drawbacks might be mitigated, multiple hydrogen injection strategies favoring fast mixing and lifted flames have been proposed in the literature. In order to prevent flashback, hydrogen and air are often injected separately. For instance, the Micromix concept consists of a large number of small-scale fuel jets into a crossflow of

oxidizer [4,5]; in the HYLON injector, hydrogen is axially swirled and injected through a central tube into a radially swirling annular flow of air [1,6].

For all systems, predicting whether or not a hydrogen flame will attach to the lips of an injection system is a key issue. In the absence of swirl, this question has been studied extensively at atmospheric pressure for jets where flame lift-off distances and lengths have been measured experimentally for methane, hydrogen and other fuels [7–9]. An assessment of multiple flame stabilization theories has been proposed by [10], where liftoff is often driven by local quenching of strained flames [11–13]. More recently, the importance of ‘edge flames’ and more specifically triples flames in jet flame stabilization has been widely reported [14–17], with extensive reviews available [18,19]. These flames are more generally studied in mixing layers of fuel and oxidizer in laminar and turbulent cases [20–23]. For the design of

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: kennie.chaplet@imft.fr (K. Chaplet).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. The triple flame structure at the interface between a layer of methane (top) and air (bottom).</figcaption></figure>

Source: Adapted from [26].

practical burners, theories have been proposed to predict the attached or lifted regime assuming triple flame structures, such as the Triple Flame Upstream Propagation theory [24] based on earlier works [25].

### 1.1. Triple flames

Triple flames were historically observed by Phillips at the interface between a layer of methane and air [26], as seen in Fig. 1. They are composed of a rich-premixed branch on the fuel side, a lean-premixed branch on the oxidizer side, and a central diffusion branch resulting from excess air and fuel from the premixed branches diffusing into each other. The triple point is at the junction of the 3 branches. For conventional fuels, it coincides with the leading edge (fastest point of the flame) and is located near stoichiometry.

Triple flames propagate faster than the maximum laminar burning velocity of their fuel [27,28]. Though prior theoretical studies had assumed zero heat release rate [29,30], Ruetsch et al. [22] showed that this increased velocity was caused by heat release effects leading to flow divergence.

The structure and dynamics of triple flames are mainly controlled by two fields:

* First, the mixing field, via the topology of the zone in which the flame can propagate.

* Second, the flow field, which affects the flame through velocity gradients perpendicular to the flame propagation direction, as found in complex injectors.

This work only investigates the effects of the mixing field: triple flames propagating in a constant velocity field are considered and only a gradient of mixture fraction $z$ is imposed. For a 2D triple flame in the X-Y plane, propagating in the Y-direction, the mixing field is characterized by the mixing thickness $\Delta = 1/\max(\partial z/\partial X)$ [19,22].

To distinguish effects due to the mixing field and to the velocity fields, triple flames affected by the mixing field are here referred to as 'squished' flames and those affected by the flow field, following standard practice 'stretched' flames.<fnref n="1" /> As depicted in Fig. 2, low values of $\Delta$ lead to highly squished triple flames while low $\Delta$ values correspond to the theoretical limit of unsquished triple flames.

For fuels with unity Lewis numbers, decreasing $\Delta$ decreases the triple flame speed [22,31]: very squished triple flames propagate more slowly than low-squish flames. It should be pointed out that a theoretical derivation with the assumptions of unity Lewis number, equal

<figure><img src="image_4.png" /><figcaption>Fig. 2. The effect of mixing thickness on the resulting triple flame squishing.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. H₂/air triple flame structure and main mechanisms. The arrow sizes indicate the qualitative importance of the diffusion mechanisms of O and H species as well as heat.</figcaption></figure>

diffusivities for all species and single-step chemistry, reached the same conclusion regarding the effect of squishing [32].

Increasing $\Delta$ increases the propagation velocity by enlarging the flammable zone [22], decreasing curvature, which improves the conductive preheating of unburnt gases [29] and modifying fuel concentration on which premixed-branch laminar burning velocities depend [19].

In the limit of large mixing thicknesses $\Delta$, where $\Delta \gg \delta_L^0$, the triple flame speed $s_T$ can be predicted from theoretical conservation laws upstream and downstream of the flame [22]:

$$s_T = \left( \frac{\rho_u}{\rho_b} \right)^{\frac{1}{2}} s_L \quad (1)$$

where $\rho_u$ and $\rho_b$ are the unburnt and burnt gas densities, and $s_L$ is the laminar burning velocity (LBV). For unity Lewis fuels such as methane, expansion ratio and LBV are calculated at stoichiometry. For hydrogen/air flames, the calculation is different and will be discussed in Section 3.3.

### 1.2. Hydrogen triple flames

There are fewer works on hydrogen triple flames. Briones et al. [33] have studied numerically the effect of H₂ enrichment on CH₄/air triple flames while [23] focused on the ignition and propagation of triple flames in an H₂/air mixing layer. Premixed and triple hydrogen flames were compared in [34,35] and the effects of increasing pressure and air temperature were assessed. These works suggest that $s_T$ increases with the mixing thickness as for conventional fuels.

Eq. (1) for $s_T$ is in principle applicable to hydrogen. However, because of differential diffusion [36], there are significant differences between conventional fuels and hydrogen triple flames:

* an increased triple flame speed $s_T$ [33]

* a modified location of the flame leading edge in $z$ space: the fastest point is not near stoichiometry [23,33], since the maximum LBV of H₂ is reached for $\phi \approx 1.6$ [37]

* a specific structure [35] for hydrogen flames summarized in Fig. 3

These works were performed on propagating, non-stationary triple flames. The present paper aims to:

<footnote n="1">¹ Even without aerodynamical stretch due to the flow field, the mixing field effect applied to the triple flame through $\Delta$ might also be measured in terms of a 'flame stretch': indeed, small values of $\Delta$ lead to strongly curved flame tips (Fig. 2) and therefore to high stretch values. Throughout this paper, the results are interpreted in terms of $\Delta$ or 'squishing' only, without referring to stretch for clarity.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Triple flame domain, initial and boundary conditions.</figcaption></figure>

* simulate laminar H₂/air triple flames using a fully resolved, high-fidelity code at atmospheric conditions.

* stabilize these flames at constant mixing thicknesses in order to quantify the effect of Δ on triple-flame speed.

* clarify the use of Eq. (1) in the case of hydrogen

* discuss the possibility of flame extinction due to squishing [19] for hydrogen triple flames.

A direct application of the scaling laws provided in the next sections and verified at atmospheric pressure is that they allow predicting triple-flame speeds at high pressures as well (cf. Fig. 10).

## 2. Computational set-up

### 2.1. Domain and numerical methods

2D DNS of laminar H₂/air triple flames are performed at atmospheric pressure. The domain is a rectangular box of 75 × 45 mm in the X and Y directions, respectively. The initial and boundary conditions are described in Fig. 4. Initially and at the bottom inlet, the mixing thickness Δ₀ is prescribed with a hyperbolic-tangent profile of mixture fraction z between H₂ and air. z is calculated with Bilger's definition:

$$z = \frac{\beta - \beta_{ox}}{\beta_{fuel} - \beta_{ox}} \qquad (2)$$

$$\beta = \sum_{k} Y_{k} \frac{a_{H} N_{H}^{k} + a_{O} N_{O}^{k}}{W_{k}} \qquad (3)$$

where Y_k is the mass fraction of species k, N_H^k, N_O^k the number of H and O atoms in species k, W_k the molecular weight of species k, β_{ox} and β_{fuel} the values of β in the oxidizer and fuel streams respectively. The weighting coefficients are a_H = 1/2, a_O = -1. The equivalence ratio is $\Phi = \frac{\frac{\mathrm{zs}Y_{H_2}^f}{(1-z)Y_{O_2}^{ox}}}{\mathrm{Y}_{O_2}^{ox}}$ with s = 8 the stoichiometric air-to-fuel ratio, Y_{O_2}^{ox} the mass fraction of O_2 in the oxidizer stream and Y_{H_2}^f the mass fraction of H_2 in the fuel stream.

The flame ignites at the meeting point of H₂, air and a layer of hot gases placed in the downstream 10% of the domain. It then propagates upstream, in the negative Y-direction and is not confined.

A structured grid with constant mesh size ΔY = 25 µm is used. This value corresponds to 13 points in a φ = 1.6 one-dimensional premixed flame front, taking its thermal thickness δ_L = 317 µm as the reference. The mesh size ΔX evolves from 25 µm in the flame zone to 1 mm near the lateral boundaries. The flame zone is 15 or 30 mm wide depending on the value of Δ₀, resulting in 1.4 or 2.5 M cells, respectively.

The compressible code AVBP, developed by CERFACS, is used to solve the reactive Navier-Stokes equations (cerfacs.fr/avbp7x/). Chemistry is described by the San Diego mechanism with 9 species and 21 reactions [38]. For transport, a Wilke law approximation assuming a

<table><caption>**Table 1**

Schmidt number of each species.</caption><thead><tr><th>Species</th><th>H<sub>2</sub></th><th>H</th><th>O<sub>2</sub></th><th>OH</th><th>O</th><th>H<sub>2</sub>O</th><th>HO<sub>2</sub></th><th>H<sub>2</sub>O<sub>2</sub></th><th>N<sub>2</sub></th></tr></thead><tbody><tr><td>S<sub>c</sub></td><td>0.23</td><td>0.14</td><td>0.80</td><td>0.53</td><td>0.52</td><td>0.58</td><td>0.80</td><td>0.81</td><td>0.91</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Evolution of mixing thickness between Y₀ (inlet) and Y due to the advection-diffusion equilibrium in the two cases V_inlet ~ s_T and V_inlet >> s_T.</figcaption></figure>

binary H₂/air mixture is used, so that the dynamic viscosity of the mixture is given by:

$$\mu_m = \frac{X_{H_2} \mu_{H_2}}{X_{H_2} + (1 - X_{H_2}) \phi_{H_2,air}} + \frac{(1 - X_{H_2}) \mu_{air}}{X_{H_2} \phi_{air, H_2} + 1 - X_{H_2}} \qquad (4)$$

with

$$\phi_{H_2,air} = \frac{\left[ 1 + \left( \frac{\mu_{H_2}}{\mu_{air}} \right)^{1/2} \left( \frac{W_{air}}{W_{H_2}} \right)^{1/4} \right]^2}{2\sqrt{2} \left( 1 + \frac{W_{H_2}}{W_{air}} \right)^{1/2}} \qquad (5)$$

where X_i is the molar fraction of i, μ_i the dynamic viscosity of i, and W_i the molar mass of i. μ_{H_2} and μ_{air} are temperature-dependent and computed with a power law. The diffusion coefficient D_k of species k is then calculated with constant but species-dependent Schmidt numbers S_c_k: D_k = μ_m/(ρ_m S_c_k), with ρ_m the mixture density. The species Schmidt numbers are provided in Table 1. The mixture thermal conductivity λ_m is calculated with a constant Prandtl number from λ_m = C_{pm}μ_m/Pr, with C_{pm} the specific heat capacity of the mixture. A more detailed mixture-averaged transport model involving contributions of each species in the mixture and binary diffusion coefficients was tested and provided similar results.

### 2.2. Flame stabilization

Fig. 5 shows how the mixture fraction field imposed at the inlet diffuses downstream: if the flame is moving in the domain, for example from top to bottom, it is subjected to decreasing values of Δ. In order to study a steady-state configuration, the triple flame must be stabilized at a fixed position, i.e. fixed Δ. Because the mixing thickness at a given height Y depends on the inlet velocity, there is a feedback between flame speed, value of Δ at its location and inlet velocity: to control precisely the Δ value in front of the flame, a control loop is needed. Here a simple but robust dichotomy strategy is used where the inlet velocity is adjusted a few times to stabilize the flame position and
<table><caption>**Table 2**
List of cases: reduced mixing thicknesses at inlet ($\Delta_0^+$) and in front of the steady triple flame ($\Delta^+$).</caption><thead><tr><th>Case</th><th>C1</th><th>C2</th><th>C3</th><th>C4</th><th>C5</th><th>C6</th></tr></thead><tbody><tr><td>$\Delta_0^+$</td><td>32</td><td>10</td><td>5.3</td><td>3.2</td><td>0.8</td><td>0.8</td></tr><tr><td>$\Delta^+$</td><td>32</td><td>11</td><td>6.8</td><td>5.5</td><td>4.8</td><td>2.1</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Comparison of the heat release fields of cases C1 (left, low squishing: $\Delta^+ = \Delta/\delta_L = 32$) and C6 (right, high squishing: $\Delta^+ = \Delta/\delta_L = 2.1$). The gray lines are the streamlines. A $10\delta_L$ scale is provided for comparison, with $\delta_L = 317$ µm.</figcaption></figure>

therefore the value of $\Delta$ seen by the flame. As a result, by changing the target flame position $Y_{target}$ or the mixing thickness $\Delta_0$ at the inlet, the curve $s_T$ versus $\Delta$ can be constructed.

## 3. Results and discussion

A proper scaling for $\Delta$ is to compare it to the minimum thermal flame thickness of a premixed H$_2$/air 1D flame, which is $\delta_L = 317$ µm reached at an equivalence ratio of $\phi = 1.6$ [22]. The resulting reduced mixing thickness is $\Delta^+ = \Delta/\delta_L$ and $\Delta_0^+$ is the value of $\Delta^+$ imposed at the inlet. Table 2 summarizes the 6 cases studied in this work, and their corresponding reduced mixing thicknesses $\Delta_0^+$ and $\Delta^+$. C1 is the least squished flame and C6 the most squished flame. For each case, a corresponding non-reacting simulation is run with $V_{inlet} = s_T$ in order to compare the mixing and flow fields.

### 3.1. Effect of squishing on triple flames

First, the effect of squishing on the triple flame structure is discussed. Fig. 6 compares the heat release rate fields of the least and most squished cases studied: C1 and C6, respectively. The isocontours of $\phi = 1.6$ (white) and $\phi = 1$ (blue) are provided.

These flames propagate in the negative Y-direction and their leading edge is located around $\phi = 1.6$, with the diffusion branch set on the stoichiometry line, as depicted in Fig. 3. They exhibit a large difference in width, C6 being confined to a small flammable zone because $\Delta$ is small, hence the denomination ‘squished’. The streamlines are diverted over a longer distance for C6 which explains why it is faster than C1, following the reasoning of [22] on streamline divergence and its impact on propagation velocity.

C1 has three distinct branches, with a relatively weak diffusion branch. On the contrary, C6 exhibits what might be called a bi-brachial structure, with the lean and diffusion branches merging. This phenomenon can occur at high squishing as suggested by Chung [19], culminating in a potential extinction at extremely high squishing. Extinction was never observed in the present work, its possibility will be discussed in the next section.

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** For C4 only: Maximum $\partial z/\partial X$ with and without flame and maximum temperature with flame as a function of $Y^+ = Y/\delta_L$. The gray line indicates the flame position.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Maximum $\partial z/\partial X$ as a function of $Y^+ = Y/\delta_L$ for all cases C1-C6. The dots mark the $Y^+$ position of the flame for each case.</figcaption></figure>

### 3.2. Flame location and mixing thickness

In order to establish a link between triple flame speed and $\Delta = 1/\max(\partial z/\partial X)$, a definition for $\partial z/\partial X$ must be given: very close to the triple flame, $\partial z/\partial X$ is affected by the flame; because of species diffusion, very far upstream of the flame, $\partial z/\partial X$ is smaller than what the flame encounters. Fig. 7 displays the maximum value of $\partial z/\partial X$ versus Y and the maximum temperature along the Y axis for C4 ($\Delta^+ = 5.5$) to determine the flame position. The maximum $\partial z/\partial X$ along Y for the corresponding cold case, without flame, is also provided in blue. The Y dimension is scaled by the minimum flame thickness at $\phi = 1.6$, $\delta_L = 317$ µm: $Y^+ = Y/\delta_L$.

Without flame, $\partial z/\partial X$ decreases from the inlet ($Y^+ = 0$) to the outlet ($Y^+ = 142$), following the advection-diffusion equilibrium described in Section 2.2. For a reacting flow with a flame stabilized at $Y^+ \approx 59$, $\partial z/\partial X$ near the inlet boundary is the same as in the non-reacting flow. However, as $Y^+$ increases, $\partial z/\partial X$ undergoes a significant reduction in front of the flame. Upon crossing the flame, it increases again before settling as the outlet is reached. This trend is observed for all cases C1 to C6. The reduction of $\partial z/\partial X$ in front of the flame is caused by the deviation of the streamlines [22], which is the consequence of dilatation. This increases the resistance of the flame to squishing but also raises the following question: what is the appropriate location to define $\partial z/\partial X$ in front of the flame? One might define it a few $\delta_L$ in front of the flame, but this seems arbitrary, especially since, as will be seen later, cases C1 to C6 do not impact the z field to the same extent. Here, the value of $\partial z/\partial X$ in front of the flame was defined by its value in the cold flow, at the location of the flame (blue square in Fig. 7 for C4): this is how the values of $\Delta^+$ were determined a posteriori to build Table 2.

Fig. 8 presents the maximum gradient of z versus $Y^+$ for all reacting cases C1-C6. Note that the C6 flame in gray has to be stabilized close to the inlet to reach small values of $\Delta^+$.

All cases provide an evolution of $\partial z/\partial X$ similar to that analyzed in Fig. 7 for C4. However, when the flame is less squished (C1-C3), the reduction of $\partial z/\partial X$ in front of it spans over a longer reduced distance
<figure><img src="image_11.png" /><figcaption>Fig. 9. Ratio of triple flame speed $s_T$ over theoretical triple flame speed $s_{T,th}$ vs. reduced mixing thickness $\Delta^+$.</figcaption></figure>

$Y^+$ than it does for the more squished flames (C4-C6). Similarly, the more the flame is squished, the more $\partial z/\partial X$ is reduced in front of it.

To summarize, a very squished flame modifies the $z$ field considerably over a small distance, whereas less squished flames modify the field little over a large distance. This might be interpreted as a stabilizing effect [22]: the more the flame is squished, the more its resistance to squishing increases through the reduction of the gradient of $z$ in front of it. This effect provides a better resistance to quenching suggesting that sustaining a mixing thickness small enough to quench an $H_2$ triple flame might require an additional, aerodynamic stretch.

### 3.3. Effect of mixing thickness on the triple flame speed

The influence of reduced mixing thickness $\Delta^+ = \Delta/\delta_L$ on the speed of steady triple flames $s_T$, is presented in Fig. 9, which shows the ratio of $s_T$ to the theoretical triple flame speed $s_{T,th}$ (Eq. (1)) versus reduced mixing thickness $\Delta^+$ for the six cases C1-C6.

It must be pointed out that using Eq. (1) in the ‘classical’ way, that is with $\rho_u/\rho_b$ and $s_L$ computed at stoichiometry is not adapted to hydrogen flames. With this methodology, the triple-flame speed does not converge to $s_{T,th}$ for low squishing. This discrepancy is rooted in the fact that the leading edge of a hydrogen triple flame is not at $\phi = 1$ but rather around $\phi = 1.6$ (cf. Fig. 6), which corresponds to the maximum premixed flame speed. Nevertheless, because the divergence of the streamlines is driven by the dilatation caused by the diffusion flame, the ratio $\rho_u/\rho_b$ should still be computed at stoichiometry. Consequently, the following expression for $s_{T,th}$ should be used for hydrogen flames:

$$s_{T,th} = (\rho_u/\rho_b(\phi = 1))^{1/2}s_L(\phi = 1.6) \quad (6)$$

Fig. 9 shows that, when the reduced mixing thickness is sufficiently large, $s_T$ tends to $s_{T,th}$ as expected, with a small deviation of around 2%. Moreover, cases C5 and C6 at $\Delta^+ = 11$ and $\Delta^+ = 32$ respectively, have the same speed $s_T \approx s_{T,th}$. The criterion $\Delta^+ > 10$ therefore seems appropriate for the validity of the theoretical formula.

Fig. 9 also shows that $s_T$ increases with reduced mixing thickness $\Delta^+$, similarly to conventional flames and to theory [19,31]. The velocity of the most squished flame C6 at $\Delta^+ = 2.1$ is 30% smaller than $s_{T,th}$. This might be significant in the context of predicting attachment or lift-off in cases such as those described in Section 1.

In the context of hydrogen combustion at high-pressure, the scaling of Fig. 9 can be used to evaluate the triple flame speeds, which control flame stabilization. Fig. 10 provides dimensional values of triple flame speeds versus $\Delta$ at 10 and 20 bars. The procedure to obtain the high-pressure scaling is the following: (1) the fresh-gases temperature $T_1$ is calculated under the approximation of isentropic compression ($\gamma = 1.4$); (2) the value of $s_L^0$ at $T_1$ is computed with Cantera [39]; (3) the theoretical triple-flame speed $s_{T,th}$ is calculated with Eq. (1) and finally (4) by assuming that the ratio $s_T/s_{T,th}$ does not depend on the pressure, the value of $s_T$ at the target pressure is inferred.

Fig. 10 recalls that triple flames are much faster when pressure (and temperature) increase, reaching almost 20 m/s at 20 bars and

<figure><img src="image_12.png" /><figcaption>Fig. 10. Triple flame speed $s_T$ (m/s) vs. mixing thickness $\Delta$ (m) at 1, 10 and 20 bars assuming isentropic compression.</figcaption></figure>

suggesting that flames that are lifted at 1 bar may reattach when pressure is increased. Moreover, the $s_T$ curves also shift to the left, showing that, at high pressure, the theoretical expression for $s_T$ is valid over a larger range of mixing thicknesses, the condition $\Delta^+ > 10$ for its validity being more easily satisfied since high-pressure $H_2$ flames are very thin. It is suggested that Eq. (6) is a very simple and convenient method to evaluate triple flame speeds with confidence.

### 4. Conclusion

In this work a triple flame of hydrogen and air is stabilized in mixing layers characterized by their mixture fraction gradient field. Stiff cases are called squished flames and the level of squishing is measured through the thickness $\Delta$ of the mixture fraction field at the flame location in the non-reacting case. Numerical simulations are performed in a 2D DNS framework using detailed chemistry and a control loop on the inlet velocity to stabilize the triple flame.

For large values of $\Delta$ (low squishing), the DNS results converge to a slightly modified theoretical expression for the triple-flame speed: $s_{T,th} = (\rho_u/\rho_b(\phi = 1))^{1/2}s_L(\phi = 1.6)$ compared to usual fuels.

For small values of $\Delta$ (strongly squished flames), the triple-flame structure is altered significantly: the flame becomes bi-brachial. The theoretical expression for its speed does not hold and $s_T$ is not equal to $s_{T,th}$. No quenching due to squishing alone is observed mainly because of hydrogen's high diffusivity.

The differences in the shape and speed of $H_2$ triple flames compared to unity Lewis fuels are mainly due to $H_2$'s diffusivity ($Le < 1$): this causes the LBV to peak at $\Phi = 1.6$ which changes the leading-edge position (shape of the flame) and the theoretical formula for $H_2$ vs. conventional fuels.

The triple-flame speed increases with mixing thickness $\Delta$, and can be as low as 30% under the theoretical predictions for the most squished case studied here. The scaling between $s_T$ and mixing thickness $\Delta$ obtained at atmospheric pressure by DNS can be extrapolated to higher pressures, highlighting that triple flames will be much faster and thinner in high pressure, high inlet temperature burners, leading to possible flame attachment on injector lips.

### CRediT authorship contribution statement

**Kennie Chaplet:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **Laurent Selle:** Writing – review & editing, Supervision, Conceptualization. **Thierry Poinsot:** Writing – review & editing, Writing – original draft, Supervision, Conceptualization.

### 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 project has received funding from the H2TECH project led by SAFRAN, France. HPC resources from GENCI (allocation A0172B10627) are highly acknowledged. The authors also acknowledge the support of Félicia Garnier in setting up the studied case.

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