# Influence of flame topology and Mach number on flame-shock interaction in a semi-closed channel

Emilie Yhuel, Guillaume Ribert, Pascale Domingo

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Emilie Yhuel, Guillaume Ribert, Pascale Domingo. Influence of flame topology and Mach number on flame-shock interaction in a semi-closed channel. Combustion and Flame, 2026, 283, pp.114526. 10.1016/j.combustflame.2025.114526. hal-05326676

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# Influence of flame topology and Mach number on flame-shock interaction in a semi-closed channel

Emilie Yhuel<sup>D</sup>, Guillaume Ribert<sup>D</sup>, Pascale Domingo<sup>D</sup>*

CORIA-CNRS, Univ. de Normandie, INSA de Rouen Normandie, Rouen, 76000, France

## ARTICLE INFO

**Keywords:**

Hydrogen combustion

Flame-shock interaction

DDT

Computational fluid mechanics

Numerical simulations

## ABSTRACT

The interaction between a laminar flame and a shock is examined in a numerical set-up based on a centimetric shock tube. A mixture of hydrogen and air at an equivalence ratio of 0.8 is first ignited at the end of the tube which leads to the development of a glove finger or a tulip flame towards which a shock is then sent. Realistic mechanisms for the chemistry are employed along with an accurate description of the diffusion including the Soret effect. Gravitational effects are also taken into account. The wall temperature is fixed at 300 K, which may seem a very stringent condition, but is better justified than the adiabatic condition because the very short interaction time limits the heat transfer with the wall. The impact of the description of the geometry (1D, 2D, and 3D) on the apparition of a transition to detonation is studied for two values of the incident shock Mach number. Depending on this Mach number, two different types of DDT are observed: one related to shock focusing occurring close to the axis of symmetry of the tube and a second type occurring in the vicinity of the reactive boundary layer. Finally, it has been shown that a small degree of asymmetry of the flame prior to its interaction with the shock can promote DDT, whereas excessive asymmetry tends to inhibit it.

**Novelty and significance statement**

The novelty of this research lies in the representativeness of the simulations of flame/shock interaction in terms of chemistry and transport modelling as well as numerical setup. These simulations offer new perspectives on the intricate links between compression waves and combustion enhancement, either by spontaneous self-ignition or by flame acceleration. An original relationship between the asymmetry of the flame front and the occurrence of a transition to detonation is demonstrated. The research contributes to efforts to reliably predict scenarios such as the transition from low-speed combustion to high-speed detonation.

## 1. Introduction

Green hydrogen has attracted growing interest in recent years as a flexible, decarbonised energy source with a wide range of potential applications such as propulsion in the aeronautical, automotive, or space industries as well as for green electricity storage. However, hydrogen is more prone to leaks and deflagration-detonation transition (DDT) than most fuels which have higher flammability limits. Understanding this risk is essential for the safe use of this carbon-free fuel but also in the detonation prevention in the industry, especially in the context of nuclear safety.

Detonation onset is a multiphysics and multiscale problem [1]. It means that gathering the scenario and the precise quantification of the conditions leading to DDT is highly challenging. Such work has been addressed through numerical and experimental studies on lab-scale configurations like shock tubes, obstacle-laden channels [2], or semi-confined channels, the latter being very important from the practical

point of view [3]. The role of shock reflection, shock focusing, instabilities, and mixing processes in the transition to detonation is highlighted in [4], with the conclusion that the mechanism for the detonation wave formation involves reactivity gradients that are generated in many ways (shock-induced, flames, etc.) [5]. For instance, Liu et al. [6] (see also [7]) studied hydrogen flame acceleration in a very narrow open 2D channel ($h < 2$ mm height) with numerical simulations featuring 1-step chemistry. They showed that an ultra-fast flame may develop in the boundary layer ahead of the initial flame front due to the viscous heating effect associated with wall friction, reaching Chapman–Jouguet velocity and detonation. In the work of Dzieminska and Hayashi [8] (see also Tang et al. [9]), auto-ignition has been observed by numerical simulation using detailed chemistry, in the boundary layer or between the flame and the shock wave in a 2D adiabatic smooth tube ($h = 2$ mm height), which will finally trigger a DDT. Flame and shock were moving in the same direction and the grid size did not exceed 3 µm.

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: pascale.domingo@insa-rouen.fr (P. Domingo).</footnote>
Yang and Radulescu [10,11] studied experimentally a hydrogen-air flame facing a shockwave (Mach number: $M_s = 1.9$) in a semi-closed channel (3400 mm long, 203 mm-tall, and 19 mm wide). They observed Richtmyer–Meshkov instabilities (RMI) on the flame front during the interaction, stretching the flame front, as also observed by Groom and Thornber [12] and Bambauer et al. [13,14] in their simulations (slip wall, 1-step chemistry). The second flame-shock interaction [10,11], coming after the shock reflection on the channel end wall, leads to further destabilisation of the flame front and a significant increase of the flame propagation velocity leading to DDT. Reported simulations featured a 2D cut of the experiment performed with 1-step chemistry. Oran and co-workers [5,15–17] studied numerically an ethylene-air flame-shock interaction ($M_s = 1.8$) in 2D and 3D rectangular channels with 1-step chemistry, highlighting the formation of reactive boundary layers and bifurcation shocks that interact with each other in 3D and generate hot spots that may be the germs of DDT. Similar behaviours were observed by Lipkowicz et al. [18,19] with a hydrogen/oxygen mixture using detailed chemistry and a mesh resolution of 50–100 µm. In their work, auto-ignition and DDT in a shock tube with cooled walls were observed linked to complex interactions between reflected shock waves and non-reactive turbulent boundary layers. Ignition and DDT can thus be controlled with shock focusing [20,21], where experimental and numerical studies were carried out to show the influence of Mach intensity and wedge reflectors angle on the ignition. Finally, in the context of using detonation for supersonic propulsion and for a non-homogeneous mixture of hydrogen and air, Wang et al. [22] recently evidenced the apparition of DDT starting either from a wrinkled flame front or behind a Mach stem in the fresh mixture.

This brief literature review demonstrates that the prediction of the detonation phenomenon in a closed environment requires capturing fine local phenomena but in interaction with near-wall phenomena and multiple wave reflections. It is therefore proposed in this study to simulate different combination in term of modelling to describe with accuracy a flame shock interaction (FSI) in centimetric shock-tube and the eventual occurrence of a transition to detonation (DDT) for a slightly lean hydrogen-air mixture. The 1D, 2D, and 3D approaches will be compared to know the limits of too much simplification. Two mechanisms controlling the apparition of DDT are evidenced and the role of gravity is clarified.

## 2. Numerical setup

A lean hydrogen-air flame (equivalence ratio $\phi = 0.8$, pressure 20 kPa, temperature 300 K) propagates from the right end of a semi-closed channel, facing a shock wave travelling at a Mach number of $M_s = 1.9$ or 2.4. This seemingly simple numerical setup enables the investigation of several significant phenomena, including shock-flame interaction involving shock waves travelling from fresh to burned gases and vice versa, the impact of boundary layers, and the effects of shock reflection on walls. Understanding the physics at play in this kind of configuration paves the way for future simulations of industrial scenarios involving much higher pressure levels and more complex geometries. The domain dimensions are given in Fig. 1. Non-slip isothermal walls at $T = 300$ K are enforced as in [18] to have more realistic flow conditions than with a slip adiabatic wall. Accordingly, as in [18], a subgrid model is activated, but only near the walls, at a thickness of 312.5 µm to compensate for the lack of resolution in the boundary layer. Currently the original Smagorinsky is used with $C_s = 0.15$. The model is not activated elsewhere to avoid its activation around shocks.

The hydrogen oxidation is described either by the *San Diego* (acronym SD) mechanism with 9 species and 23 reactions in its latest version [23] or by Mével et al. [24] with 11 species and 29 reactions.

The simulations are performed with the finite volume numerical code SiTCom-B, already fully detailed in [25,26] and validated on supersonic combustion flows [27–29].

<figure><img src="image_3.png" /><figcaption>Fig. 1. Sketch of the configuration under study. The H₂/air flame propagates from the closed end of the channel and the incoming shockwave moves at $M_s = 1.9$ or 2.4. $h = 45$ mm, $L = 270$ mm and $l = 21$ mm (for 3D simulations). In the fresh gases at rest, $P = 20$ kPa, $T = 300$ K.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Flame initialisation. Hemispherical flames, radius = 5 mm (a) centred on the symmetry axis. (b) offset by $h/32(= 1.4$ mm). (c) offset by $h/8(= 5.625$ mm). (d) planar flame located 14 mm from the end wall.</figcaption></figure>

In particular, SiTCom-B has a 4th order centred skew-symmetric-like scheme for the convective terms [30], a 4th order centred scheme for the diffusive terms, and a 2nd and 4th order artificial dissipation [31,32] to overcome spurious oscillations and damp high frequency modes. The resolution of the uniform mesh follows the recommendations of [26] with $\Delta x_i = 62.5$ µm, but a refinement up to 31.25 µm is achieved for 2D simulations. The grid resolution that has been retained is similar to that in Refs. [18,19]. Additionally, a resolution of 62.5 µm has recently been adopted for 3D simulations of flame-shock interactions at $M_s = 1.9$ [33], producing successful outcomes when compared with experimental results. Finally, the Hirschfelder and Curtiss model [34], which is used to model the diffusion velocity of species $k$, can be combined with thermal (Soret) diffusion, which is only activated for species with a molecular weight below $5 \text{ g mol}^{-1}$ (LS) [35]. The gravitational force (G) is activated in the direction $-\vec{y}$.

## 3. Flame propagation

The H₂/air mixture ($\phi = 0.8$) is ignited near the end wall [26]. Different ignition methods are considered as shown in Fig. 2. To mimic an ignition by electrodes, a hemispherical flame (5 mm radius) is constructed from a 1D flame solution. Three locations for the centre of these hemispherical flames were tested: on the symmetry axis, offset by $h/32(= 1.4$ mm) and by $h/8(= 5.625$ mm). This displacement from the centre of the wall makes it possible to reproduce any experimental uncertainty in the position of the electrodes and to characterise its effect on the FSI and DDT. A planar ignition as would be obtained with a grid of tungsten wires [10,11] is also studied as in [26], the 1D flame solution is then used to construct the initial planar premixed flame.

The flame is first allowed to spread freely to the left. The initial hemispherical flames transition to the so-called glove finger shape is shown in Figs. 3(a), (b), (c) and 4. The 3D flame is very stretched in the corners, with extinction zones due to the cold walls, especially in the x-y plane, as the flame first reaches the vertical walls before touching the horizontal ones. The 3D flame stretching in the corners is also visible in the numerical work of Sun et al. [36] in a square channel. For the two non-symmetric ignitions (cases (b) and (c) Fig. 2), the flames will reach
<table><caption>**Table 1**
Asymmetry factor $\zeta$ of the flames just before the first FSI. SD: San Diego. LS: Soret effect. G: Gravity. $\Delta x_i = 62.5 \ \mu\text{m}$.</caption><thead><tr><th></th><th>Case #</th><th>Initialisation</th><th>Shape</th><th>Kinetics</th><th>Hypothesis</th><th>ζ</th></tr></thead><tbody><tr><td>3D</td><td>9</td><td>Hemispherical centred</td><td>Glove finger</td><td>SD</td><td>LS G</td><td>0.00329</td></tr><tr><td>2D</td><td>5, 6, 10, 11</td><td>"</td><td>"</td><td>"</td><td>LS G</td><td>0.00208</td></tr><tr><td>"</td><td>17</td><td>"</td><td>"</td><td>"</td><td>–</td><td>0</td></tr><tr><td>"</td><td>15</td><td>"</td><td>"</td><td>"</td><td>LS</td><td>0</td></tr><tr><td>"</td><td>16</td><td>"</td><td>"</td><td>"</td><td>G</td><td>0.00204</td></tr><tr><td>"</td><td>19</td><td>Hemispherical 1/8 h</td><td>"</td><td>"</td><td>LS</td><td>0.10825</td></tr><tr><td>"</td><td>18</td><td>Hemispherical 1/32 h</td><td>"</td><td>"</td><td>LS</td><td>0.03503</td></tr><tr><td>"</td><td>7</td><td>Hemispherical centred</td><td>"</td><td>Mével</td><td>LS G</td><td>0.00241</td></tr><tr><td>"</td><td>8, 12</td><td>Planar</td><td>Tulip</td><td>SD</td><td>LS G</td><td>0.04448</td></tr><tr><td>"</td><td>20</td><td>"</td><td>"</td><td>"</td><td>LS</td><td>0</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Flame propagation (2D simulations) in the channel. San Diego mechanism. Ignition as: hemispherical flames, radius = 5 mm (a) centred on the symmetry axis. (b) offset by $h/32(= 1.4$ mm). (c) offset by $1/8 h(= 5.625$ mm). (d) planar flame.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Flame propagation in a H₂/air mixture ($\phi = 0.8$) from the end wall. Iso-surface Iso surface of the progress variable based on H₂O ($c = 0.5$) coloured by temperature at $t = 4.87$ ms. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

the upper wall first and the flame structure will remain non-symmetric during its propagation. The initial planar flame will take the so-called tulip shape (Fig. 3(d)).

The flame front position is plotted as a function of time in Fig. 5 (top) for different modelling strategies: 2D vs. 3D, SD vs. Mével, with and without gravity (G), with or without light Soret effect (LS) in transport and initial flame ignition. The 3D flame shows the fastest propagation due to the largest increase in flame surface area, i.e. its burning flame speed. Gravity leads to a very slight asymmetry in the flame, but without affecting the flame position (see [37]). The inclusion of the Soret effect (LS) slows down the flame propagation. Finally, the observed difference between the two chemical mechanisms comes from the slightly different values of the laminar flame velocity: $S_L = 1.5$ m s⁻¹ with Mével and $S_L = 1.54$ m s⁻¹ with SD when calculated for a 1D laminar flame at $\phi = 0.8$. As expected, the tulip flame propagates much slower than the finger flame.

<table><caption>**Table 2**
Overview of the simulations. $\Delta x_i$ is given in $\mu\text{m}$, and $\tau_{DDT}$ in $\mu\text{s}$.</caption><thead><tr><th></th><th>Case #</th><th>$M_s$</th><th>$\Delta x_i$</th><th>Kinetics</th><th>$\tau_{DDT}$</th></tr></thead><tbody><tr><td>1D</td><td>1</td><td>1.9</td><td>62.5</td><td>SD</td><td>–</td></tr><tr><td>"</td><td>2</td><td>2.4</td><td>62.5</td><td>Mével</td><td>270</td></tr><tr><td>"</td><td>3</td><td>2.4</td><td>31.25</td><td>Mével</td><td>292</td></tr><tr><td>"</td><td>4</td><td>2.4</td><td>62.5</td><td>SD</td><td>294</td></tr></tbody></table>

Following the work done in [26] to minimise the difference between all cases, the flame-shock interaction (FSI) is now examined for flames that have the same leading flame tip position, i.e. $14.55 \pm 0.05$ cm from the right, at the instant they encounter the shock. Such positions are identified by large dots on the profiles in Fig. 5 (top) and a representation of the resulting flame shapes is plotted in Fig. 5 (bottom). The incident shock wave will therefore encounter either a glove finger flame or a tulip flame, depending on the ignition method. The flame will be more or less symmetrical depending on the location of the spark and the presence of gravity.

The shock wave will therefore interact with the laminar flame (1st FSI), reach the end wall and generate a reflected wave that will interact with the flame again (2nd FSI). This scenario is studied with two Mach numbers ($M_s$), two kinetic schemes, different flame shapes and two mesh resolutions ($\Delta x_i$), through a 1D, 2D or 3D description, as summarised in Tables 2–6. As sketched in Fig. 1, a discontinuity in pressure and temperature is added to the left part of the domain, 14.9 cm from the right wall, corresponding to a normal shock propagating at Mach number $M_s$. The post-shock parameters of temperature ($T_s$) and pressure ($P_s$) imposed on the shocked fresh gases are determined by Rankine–Hugoniot relations: for $M_s = 1.9$, $T_s = 483$ K and $P_s = 0.8$ bar; for $M_s = 2.4$, $T_s = 612$ K and $P_s = 1.3$ bar.

The asymmetry factor of the flames, $\zeta$, just before the first FSI is determined as proposed by Ballossier et al. [38] and summarised in Table 1. The only perfectly symmetric cases ($\zeta = 0$) are those with no gravity and a centred ignition. All other flames have a rather weak asymmetry factor ($\zeta < 0.11$).

## 4. Flame - Shock Interaction (FSI), 1D approximation

From now on, the initial time ($t = 0$) corresponds to the time of the introduction of the discontinuity within the domain.

As mentioned in the introduction, it is legitimate to check whether 1D simulations can reproduce, even partially, the macroscopic flow characteristics, or at least the possibility of the occurrence of a transition to detonation, in order to avoid CPU-intensive 2D or 3D simulations wherever possible.

Fig. 6 shows the evolution of the maximum pressure in the domain as a function of time. For $M_s = 1.9$ (case 1), the pressure gradually increases to 3.3 times its initial value without any discontinuity. The FSIs are barely visible. This behaviour was confirmed with $\Delta x_i = 5 \ \mu\text{m}$.

As the Mach increases, the rise in maximum pressure after the 1st and 2nd FSIs is much more pronounced. When the pressure reaches 6 bar, a very sharp rise indicates auto-ignition followed by detonation
<figure><img src="image_7.png" /><figcaption>Fig. 5. Leading tip position versus time. Left: half-circular initial flame. Right: half-circular initial flame versus planar initial flame. Bottom: Iso surfaces of the progress variable based on H₂O (c = 0.5) for a similar flame leading tip position (14.55 cm from the right wall). SD: San Diego mechanism. Mevel: Mével mechanism. LS: light Soret approach activated. G: gravity included.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Evolution of the maximum pressure in the 1D computational domain.</figcaption></figure>

**Table 3**

Overview of the simulations including Soret effect and gravity. $\Delta x_i$ is given in µm, and $\tau_{DDT}$ in µs.

<table><thead><tr><th></th><th>Case #</th><th>Shape</th><th>M<sub>s</sub></th><th>Δx<sub>i</sub></th><th>Kinetics</th><th>τ<sub>DDT</sub></th></tr></thead><tbody><tr><td>2D</td><td>5</td><td>Glove finger</td><td>2.4</td><td>62.5</td><td>SD</td><td>195</td></tr><tr><td>"</td><td>6</td><td>"</td><td>"</td><td>31.25</td><td>"</td><td>198</td></tr><tr><td>"</td><td>7</td><td>"</td><td>"</td><td>"</td><td>Mével</td><td>210</td></tr><tr><td>"</td><td>8</td><td>Tulip</td><td>"</td><td>"</td><td>SD</td><td>220</td></tr><tr><td>3D</td><td>9</td><td>Glove finger</td><td>"</td><td>62.5</td><td>"</td><td>150</td></tr></tbody></table>

driven only by the piston effect. The exact time of detonation ($\tau_{DDT}$) is given in Table 2. Compared to the time between the 1st FSI and the onset of detonation, a variation of 10% is observed between the three 1D cases, with a small influence of the mesh resolution and the chemical mechanism. Finally, the pressure increase due to detonation is significant (>40 bar). A second pressure peak (black arrows in Fig. 6 on the right) with almost the same intensity as the first pressure peak is also observed and corresponds to the detonation travelling in the $+\bar{x}$ direction and reflecting on the end wall. Increasing the mesh resolution allows a better capture of the pressure maximum.

**5. Flame shock interactions at $M_s = 2.4$: detonation mechanism**

The detonating case ($M_s = 2.4$) is now studied through 2D and 3D simulations (Table 3). All these simulations include the Soret effect and gravity. The initial flames are then non-symmetric (see Table 1).

Cases 4, 5 and 9 are first discussed to compare the FSI in 1D, 2D and 3D with the same spatial resolution and kinetic mechanism.

The dynamics of the flow is illustrated in Fig. 7 for case 5. The first FSI ($t = 60$ µs) leads to the appearance of Richtmyer–Mehskov instabilities (RMI). Then the shock wave is reflected at the right end

<figure><img src="image_9.png" /><figcaption>Fig. 7. Temperature field and superimposed numerical schlieren for the case 5 (2D). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

wall, increasing pressure and temperature ($t = 110$ µs). The second FSI destabilises the flame front with the appearance of Kelvin–Helmholtz instabilities (KHI) near the walls and a reactive boundary layer (RBL) develops, hooked on the bifurcated shock, which grows with time (from $t = 150$ to $200$ µs). Shortly after the second FSI, the temperature in the fresh gases on the symmetry axis begins to rise as a result of the shock focusing. This temperature rise leads to auto-ignition ($t = 190$ µs), which rapidly evolves into detonation ($t = 195$ µs). This was to be expected as the 1D simulation predicted a detonation for this incident shock Mach number.

As in [17], the detonation occurs before any significant development of the reactive boundary layers, leading to quasi-identical ignition times ($\tau_{DDT}$) for both mesh resolutions in the 2D simulations (see Fig. 8 left, or $\tau_{DDT}$ in Table 3). However, the maximum pressure reached on the axis (Fig. 8 right) is about 10 bar lower in the less resolved 2D simulation. The influence of the chemical mechanism (cases 6 and 7) is more important but less than that obtained for the 1D simulations (see Table 2, cases 2 and 4).

The difference observed in the pressure maxima between 1D, 2D and 3D geometries can be explained by the presence of reflected waves. In 1D, these oblique waves are absent because there is no reflection from the side walls. In 2D, reflected waves on the top and bottom walls appear and in 3D they are enhanced by the presence of lateral walls. The presence of reflected waves, driven by reflexion off the side walls,
<figure><img src="image_10.png" /><figcaption>Fig. 8. Maximum pressure recorded in the domain (left) or on the symmetry axis (right).</figcaption></figure>

results in a faster pressure rise and consequently a shorter detonation delay.

The 3D simulation (see Fig. 1 in the Supplementary Material) follows the scenario depicted in Fig. 7, except that the mushroom-like shape observed at $t = 130$ µs is more elongated and narrower, and almost touches the end wall. The shock wave focusing in the fresh gas funnel appears earlier and is more constrained than in 2D due to the smaller width of the funnel. At 120 µs, its height is around one fifth of the channel height. While in 2D, the fresh gas funnel height at 130 µs is only slightly below the channel half-height. The reflected shock has then a stronger reflection angle in 3D than in 2D. This shock focusing is similar to that observed in [20], where a 60° reflector angle is more efficient in enhancing detonation than a 90° angle. In [20], Yang and Zhang have also demonstrated the crucial role of the occurrence of hot spots and oblique waves in sustaining the detonation. As a consequence, the ignition delay is shortened when moving from a 1D to a 3D description of the configuration (Fig. 8), and the transition to detonation occurs at 294 µs, 200 µs and 150 µs for the 1D, 2D and 3D simulations, respectively.

A zoom of the onset of detonation is shown in Fig. 9 for case 5 and Fig. 10 for case 6 and corresponds to a zoom of Fig. 7.

The local induction time, $\tau_i$, plotted in Fig. 9 is calculated with CAN-TERA, using the local thermodynamics conditions ($P, T, Y_k$) extracted from the simulation and the kinetic mechanism. When the temperature is above a given value $T + \Delta T$ (here $\Delta T = 400$ K), $\tau_i$ is reached and it becomes possible to construct auto-ignition maps for each instant of the simulation as in [18,19].

In Fig. 9 the temperature rise leads to a very small induction time ($10^{-7}$ s) between the oblique reflected shock and the flame front, and then auto-ignition occurs as in the Zeldovich gradient mechanism (or Shock Wave Amplification by Coherent Energy Release) [4].

In Fig. 10 the detonation occurs between $t = 196$ µs and $t = 198$ µs in a non-symmetric way compared to the less resolved case 5. At $t = 200$ µs, the detonation propagates in all directions, even in the burned gases (retonation), reaching the RBL (Reactive Boundary Layer) for part of it.

The initial stages of the ignition process differ between the two meshes. It is difficult to ascertain whether this is due to mesh refinement or to a random process, given that the shock structure regularly alternates between oblique shocks crossing with and without a Mach stem over time (as can be observed in Fig. 9 between instants 190 µs and 194 µs). In both cases, auto-ignition in the fresh gases begins behind a Mach stem (see Figs. 9 and 10 at $t = 190$ µs). In the less refined mesh, the remainder of the DDT development occurs behind the Mach stem, which is swiftly replaced by the detonation wave (Fig. 7, $t = 195$ µs). For the finest resolution, the Mach stem disappears from $t = 194$ µs (see

Fig. 10). Observing the appearance of DDT in the glove finger flame with high resolution and in the absence of gravity (case 20, Fig. 2, supplementary material) reveals that the entire AI process takes place behind the Mach stem. At $t = 190$ µs, the differences between cases with and without gravity (cases 6 and 20) are barely noticeable and will only become apparent later on. The differences between the fine (case 6) and less refined (case 5) meshes in the simulation with gravity may also be due to the coarser mesh hindering the development of instabilities due to flame asymmetry. If the analysis is carried out with fewer time-based details (see Fig. 4 in the supplementary material), the difference in behaviour between the two resolutions (cases 5 and 6) lasts for approximately 15 µs. In both cases of refinement, the apparition of AI in the fresh gases is clearly caused by the rise of pressure behind the reflecting oblique shocks in the vicinity of the axis.

The 3D simulation follows the scenario depicted in Fig. 7 except that the mushroom-like shape observed at $t = 130$ µs is flat, elongated, almost touching the end wall. A close-up of detonation's onset is shown in Fig. 12 and a more global view is given in Fig. 11.

In Fig. 12-left (section of the middle horizontal plane) a first increase of the $HO_2$ concentration in the fresh gases is observed just before the flame front and also behind the shock structure. This appearance of $HO_2$ in the fresh gases corresponds to the first pressure peak observed in Fig. 8. At another instant ($t = 142$ µs), the flame front accelerates and engulfs the newly formed radicals. However, there is no local pressure increase and the characteristics of a DDT do not appear. This is confirmed by the pressure drop in Fig. 8 down to 10 bar.

This decrease is quickly followed by a sharp increase in pressure up to 60 bar on the axis and 110 bar in the domain, an increase which is now a clear marker for a DDT appearance. The increase of $HO_2$ is first observed at $t = 148$ µs (Fig. 12-right) on both sides of the symmetry axis in a zone where both pressure and temperature are high. This second ignition in the fresh gases is now followed by an extremely rapid development of combustion and the reaction zone is preceded by a detonation wave ($t = 148$ µs).

After the detonation is initiated, it propagates and interacts with the RBL ($t = 152$ µs in Fig. 12). The detonation then overtakes the reflected shock (Fig. 7) and the bifurcated shocks until it passes the bifurcation foot. There is a transition from an overdriven detonation to a stable detonation as explained in [39]. It can be seen that the pressure of the stable detonation is the same in 1D and 2D. For the 3D simulation, the same tendency can be observed (Fig. 8), even if the simulation was stopped before the development of the stable detonation due to its high CPU cost. This pressure convergence is observed because the left shocked fresh gas conditions are the same regardless of the domain geometry hypothesis. The underlying mechanisms of flame-shock interaction and DDT are similar in 2D and 3D simulations,
<figure><img src="image_11.png" /><figcaption>Fig. 9. Top: temperature field and numerical schlieren. Bottom: induction time for the case 5 (2D).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Zoom on the detonation onset. Left: HO₂ mass fraction, middle: pressure, right: temperature, at the first instants of the DDT initiation. Numerical schlieren is superimposed on all the figures. Case 6.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. Zoom on detonation onset. Left: auto-ignition without DDT. Right: Successful DDT. Horizontal plane of symmetry. Numerical schlieren is superimposed on all the figures. Case 9.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. 3D view of the detonation onset. Iso surface of the progress variable based on H₂O (c = 0.5) coloured by temperature. Schlieren on vertical and horizontal planes of symmetry. Case 9. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

enabling relevant parametric studies in 2D. However, it should be noted that a 3D representation is necessary to retrieve experimental results or predict any incident. Case 8 corresponds to the FSI at Mach 2.4 when the flame has a tulip shape at the moment of interaction (Fig. 13). Unlike the previous cases, the flame topology leads to two RM instabilities at the flame front instead of one. After the second FSI, two funnels of fresh gas emerge, leading to two shock focusing zones (160 µs). The resulting hot spots (purple temperature field) generate multiple ignitions between 200 µs and 220 µs. These ignitions combine and develop into a detonation which propagates in the same way as the glove finger cases. DDT appears 20 µs after the first auto-ignition, compared to only 10 µs for the glove finger form. A significant asymmetry begins to appear at 220 µs. Although the genesis of the DDT is not exactly the same for the two initial flame shapes, there are no
<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Temperature field and superimposed numerical schlieren. Case 8 (2D). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<table><caption>**Table 4**

Overview of the 2D simulations with San Diego mechanism and $M_s = 1.9$. $\Delta x_i$ is given in µm, and $\tau_{DDT}$ in µs. Soret effect is included as well as gravity.</caption><thead><tr><th>Cas #</th><th>Flame shape</th><th>Ignition</th><th>Δx<sub>i</sub></th><th>τ<sub>DDT</sub></th></tr></thead><tbody><tr><td>10</td><td>Glove finger</td><td>Central</td><td>31.25</td><td>614</td></tr><tr><td>11</td><td>"</td><td>"</td><td>62.5</td><td>554</td></tr><tr><td>12</td><td>Tulip</td><td>"</td><td>31.25</td><td>-</td></tr></tbody></table>

significant differences once the DDT starts to propagate (see 246 µs Fig. 13 and 245 µs Fig. 7).

## 6. Flame shock interactions at $M_s = 1.9$: sensitivity to flame topology of the detonation mechanism

The incident shock Mach number is now lowered to 1.9 and the possible influence of the initial flame topology (glove finger or tulip) on the appearance of a DDT is investigated. All simulations presented in this section are 2D and include the Soret effect and gravity. The initial flames are then non-symmetric (see Table 1). The three configurations studied are summarised in Table 4.

### 6.1. Detailed study of DDT mechanism for the glove finger flame

The glove finger flame examined corresponds to case 10 ($M_s = 1.9$) detailed in Fig. 14 through the pressure and temperature fields.

The sequence of events is very similar to the one that has been observed for $M_s = 2.4$ (Fig. 7), except that a reactive boundary layer hooks onto the bifurcated shock, becomes turbulent and grows with time (from $t = 160$ to $520$ µs). This evolution is very close to the one observed in [26] for the same incident shock Mach number and a higher channel configuration. The fresh gas region between the RBLs gradually narrows and becomes a funnel in which reflected shocks and compression waves are observed in all directions, leading to the formation of focal points (purple temperature field from $t = 310$ to $520$ µs). This phenomenon was also observed by Oran and Gamezo [17], where ignition occurs in hot spots, occasionally followed by DDT.

However, in case 10 no auto-ignition occurs in the fresh gases, since the induction time $\tau_i$ in fresh gases (Fig. 15, zoom in on the area of interest corresponding to the black frame in Fig. 14) is between $10^{-2}$ s and $10^{-4}$ s, which is too long for ignition under the present conditions.

Instead, DDT in this case results from shock focusing on the flame front, as in [40], followed by flame acceleration (FA), as in [4,6,41].

A zoom of the detonation initiation is shown in Fig. 16, corresponding to the black frame in Fig. 14. At a very early stage ($t = 612$ µs) an increase in $HO_2$ concentration occurs at the flame front at the intersection of four shock waves. The flame surface then takes on a

protruding shape, as observed by H. Xiao and E. Oran in [40] and by Lai et al. [42], with a significant pressure variation across the interface between the fresh and burnt gases. At the following instant ($t = 616$ µs), the detonation propagates in all directions, consuming the fresh gases present in the funnel between the upper and lower RBLs.

At $M_s = 1.9$, the fact that auto-ignition occurs in 2D and not in 1D indicates that the reflected waves drive the auto-ignition and transition to DDT for this incident shock Mach number. To highlight the role of the shock focusing on the flame front, the pressure field after the second FSI along with the flame front are shown in Fig. 17. The $H_2O$ based progress variable is again used to locate the flame front.

Four main pressure peaks are observed over time. The local extremum #3 is higher than the other three from the start and continues to increase until the condition for DDT is reached.

The complete history of the pressure increase is plotted in Fig. 18 with the time evolution of the maximum pressure in the domain and on the symmetry axis. The pressure starts to increase quasi-linearly as soon as the shock is reflected at the end wall ($t \approx 100$ µs). Numerous small peaks of higher pressure are found, related to wave reflections and interactions, until $t \approx 580$ µs. When the maximum pressure in the domain reaches 10 bar, the pressure rise sharpens, corresponding to the transition to detonation. Differences in the maximum pressure reached in the domain and the time of DDT ($\tau_{DDT}$) are observed depending on the mesh resolution. The lower maximum pressure in the domain obtained for the less refined mesh (case 11) is certainly related to an insufficient mesh resolution. Regarding the pressure on the axis, the same maxima are obtained regardless of the spatial resolution. The ignition delay time depends mainly on the evolution of the RBL over time, which is influenced by the mesh resolution (see [26]), then an offset on $\tau_{DDT}$ can be expected between the two simulations (about 60 µs). Note that the simulations are stopped as soon as the RBLs reach the left boundary of the simulation domain, implying that the detonation has not reached its stable propagation mode.

### 6.2. Influence of the initial flame topology

No DDT is observed for case 12 where the flame has a tulip shape (Fig. 19, $t = 0$ µs). Different shock focussing can be observed, which does not result in a significant temperature rise close to the flame surface.

After the second FSI, a RBL develops as observed in case 10. Wave reflections in the fresh gas create preheating zones (violet temperature field in the fresh gas), visible from $370$ µs. However, these temperature rises are not sufficient to lead to auto-ignition in the fresh gases before the RBLs have reached the domain outlet on the left. As Ciccarelli and Dorofeev [4] and Karanam et al. [43] have shown the importance of strong density gradients during flame acceleration, to understand the differences between cases 10 and 12 and to highlight the critical conditions for FA and possible DDT, the density gradient is first extracted from the simulation.

The maximum of the density gradient in the flame front (determined as $0.01 < c < 0.85$), denoted $M_ρ$ ($= \max |\nabla ρ_{f1}|$), is plotted over time Fig. 20 for cases 10 and 12. $M_ρ$ starts to rise after the second FSI ($\approx 180$ µs) and reaches $20\,000$ kg m$^{-4}$ for both cases. However as no strong pressure rises are associated, no FAs are fostered. After $300$ µs, $M_ρ$ remains below $13\,000$ kg m$^{-4}$ for case 12, while for case 10, $M_ρ$ still shows strong discontinuous increases before reaching $20\,000$ kg m$^{-4}$ just before the DDT.

Since a local increase in the density gradient is not a sufficient indicator of the presence of auto-ignition or FA, the baroclinic term is now extracted from the simulation as it combines density and pressure gradients. The baroclinic term has been used to explain the formation of tulip flames, gravitational effects, RMI and vorticity generation by the misalignment between the pressure gradient $\nabla P$ associated with
<figure><img src="image_16.png" /><figcaption>Fig. 14. Pressure (top) and temperature (bottom) superimposed on schlieren field. Case 10.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Zoom on the induction time from Fig. 14. Case 10. GB; burnt gases. GF: fresh gases. White arrow: $10^{-7} \le \tau_i \le 10^{-6}$ s.</figcaption></figure>

a pressure wave or shock wave and the density gradient $\nabla \rho$ at the material interface [13,37,44–47]. It is defined as:

$$B = \frac{1}{\rho^2} \nabla \rho \times \nabla P = \frac{1}{\rho^2} \left( \frac{\partial \rho}{\partial x} \frac{\partial P}{\partial y} - \frac{\partial \rho}{\partial y} \frac{\partial P}{\partial x} \right) \quad (1)$$

As for the density gradient, the baroclinic term is extracted only in the flame front and its absolute value, $|B_{fl}|$, is plotted in Fig. 21 for three instants just before DDT for case 10. At 610 µs, the only region where $|B_{fl}|$ reaches high values (around $4.10^{13}$ s$^{-2}$) is in the centre of the domain where DDT will occur, consistent with observations made in previous sections. Other local increases in $|B_{fl}|$ of the order of $10^{13}$ s$^{-2}$, indicated by red arrows, are also visible but do not lead to FA.

<figure><img src="image_18.png" /><figcaption>Fig. 16. Zoom on detonation initiation from Fig. 14. Case 10. Shock: $M_s \approx 1.35$. Shock 2: $M_s \approx 1.17$. Shock 3: $M_s \approx 1.25$. Shock 4: $M_s \approx 1.13$. Superimposed numerical schlieren on all figures.</figcaption></figure>

To analyse the evolution of the flame front behaviour over time, $\max(|B_{fl}|)$ is plotted against time in Fig. 22. A critical threshold (indicated by the dotted lines) is observed at $4.10^{13}$ s$^{-2}$, which is reached at 610 µs in case 10, marking the onset of FA. At the following instants, 612 and 614 µs, the value of $|B_{fl}|$ decreases, but FA continues until the velocity reaches the Chapman–Jouguet (CJ) velocity, i.e. around $u_{CJ} \approx 1870$ m s$^{-1}$. In this context, the velocity of the combustion wave is $\omega_b \approx 1128$ m s$^{-1}$ in the wave reference frame and $u_b \approx 742$ m s$^{-1}$ in the laboratory reference frame, eventually leading to the transition to detonation. Case 12 never exceeds the $4.10^{13}$ s$^{-2}$ threshold. A peak at 400 µs close to this value is visible, but too transient to trigger FA.

In conclusion, the initial flame shape (glove finger or tulip) before the onset of the first flame-shock interaction (FSI) significantly influences the dynamics of the FSI and the transition to detonation (DDT), and the Mach number of the incident shock is not the only parameter to consider. A glove finger flame generates a single central Richtmyer–Meshkov (RM) instability, which favours intense reflections in a fresh
<figure><img src="image_19.png" /><figcaption>**Fig. 17.** Pressure field of case 10. Iso-line $c = 0.5$ (black). 1, 2, 3, 4: positions of local extrema of pressure.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 18.** Maximum pressure recorded in the domain (left) and on the symmetry axis (right). Cases 11 (62.5 µm) and 10 (31.25 µm).</figcaption></figure>

gas funnel. In contrast, a tulip flame generates two RM instabilities, splitting the reflected waves into two short, narrow funnels after the onset of the second FSI, before the development of reactive boundary layers (RBLs). These RBLs develop before merging, either gradually as in case 12, or abruptly through the DDT transition as observed in case 10. Exceeding a threshold for the baroclinic term ($|B_{fl}| > 4.10^{13} \text{ s}^{-2}$ in the case of $M_s = 1.9$) seems to be a sufficient condition to trigger FA.

## 7. Role of the gravity in the apparition of a DDT

The estimate of the Froude number based on the laminar premixed flame velocity ($Fr = S_l^2/(gh)$) is of the order of 5, suggesting that the inclusion of gravity in the modelling loop is not mandatory. However, when gravity is included, the laminar flame exhibits an asymmetry that is absent when gravity is not included (see Table 1). Some simulations were then performed to verify the influence of gravity on FSI and DDT.

Simulations of the FSI performed with the incident shock at Mach 2.4 do not change much when gravity is removed from the momentum

<figure><img src="image_21.png" /><figcaption>**Fig. 19.** Temperature field and superimposed numerical schlieren. Case 12.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 20.** $M_ρ = \max |\nabla ρ_{fl}|$. Black: case 10. Grey: case 12.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. 21.** Iso-line of baroclinic term conditioned by $c = 0.5$ (colours) superimposed on $|\nabla P|$ (black and white). Red arrows indicate local increases of baroclinic term. Case 10. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

equation (see Supplementary Material, Tab. 1, Fig. 2 and Fig. 3). However, for the incident shock at Mach 1.9, the appearance of a detonation is found to be associated with the presence of gravity. The Table 5 summarises the results obtained with an initial glove finger flame, with or without the addition of gravity and with or without the neglect of the Soret effect. The latter has almost no effect on the appearance of a detonation, except that the phenomena appear with an additional delay of a few µs (cases 10 and 16). The inclusion of the Soret effect in the simulation of hydrogen/air flames is important and gives more realistic results, but neglecting it does not significantly alter the results.
<figure><img src="image_24.png" /><figcaption>**Fig. 22.** Time evolution of max $|B_{f,l}|$. Black: Case 10. Grey: case 12. Red: threshold $4.10^{13} s^{-2}$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 5**</caption>

Role of gravity and Soret effect, summary of 2D simulations with San Diego mechanism and $M_s = 1.9$. $\Delta x_i = 31.25$ µm. $\tau_{DDT}$ is given in µs.

<table><caption>**Table 6**</caption><thead><tr><th>Case #</th><th>Flame shape</th><th>Hypothesis</th><th>Ignition</th><th>τ<sub>DDT</sub></th></tr></thead><tbody><tr><td>10</td><td>Glove finger</td><td>LS+G</td><td>Central</td><td>614</td></tr><tr><td>15</td><td>"</td><td>LS</td><td>"</td><td>–</td></tr><tr><td>16</td><td>"</td><td>G</td><td>"</td><td>620</td></tr><tr><td>17</td><td>"</td><td>–</td><td>"</td><td>–</td></tr></tbody></table>

Role of asymmetry, summary of 2D simulations with San Diego mechanism and $M_s = 1.9$. $\Delta x_i = 31.25$ µm. $\tau_{DDT}$ is given in µs. Flame shape: glove finger.

<table><thead><tr><th>Cas #</th><th>Hypothesis</th><th>Ignition</th><th>τ<sub>DDT</sub></th><th>ζ</th></tr></thead><tbody><tr><td>10</td><td>LS+G</td><td>Central</td><td>614</td><td>0.00208</td></tr><tr><td>18</td><td>LS</td><td>+1/32 h</td><td>590</td><td>0.03503</td></tr><tr><td>19</td><td>"</td><td>+1/8 h</td><td>–</td><td>0.10825</td></tr></tbody></table>

The strong influence of gravity on the occurrence of DDT is more surprising and needs to be explained.

As shown in the previous section, the baroclinic term on the flame front highlights areas of strong pressure and density gradients. The extremums of this term are plotted against time in Fig. 23 for the four cases studied. Cases without DDT do not exceed the $4.10^{13} s^{-2}$ threshold, which is bounded by the blue dashed lines.

It is then hypothesised that it is the slight asymmetry induced by the presence of gravity on the shape of the initial flame that leads to the appearance of DDT. Symmetric cases (cases 15 and 17) favour focusing in fresh gas along the axis of symmetry, thus reducing the probability of FA on the flame front. Naturally, if the shock tube were oriented vertically, the presence of gravity would not alter the intrinsic symmetry of the flame. To verify this hypothesis, two additional simulations without the addition of gravity but with an initially asymmetric flame are carried out and compared with case 10 (see Table 6).

For cases 10 and 18, detonation is observed (case 10: 614 µs, case 18: 590 µs) due to a slight asymmetry of the flame before FSI, either by taking gravity into account or by a slightly off-centre ignition (h/32). Case 10 was also run with gravity activated until the shock was introduced to the domain, after which it was shut down. No difference in the DDT appearance compared to the run with gravity activated throughout the simulation has been observed. Thus, gravity has an important role especially when the flame propagates before the FSI. Its impact is negligible during the FSI. On the other hand, case 19 does not lead to detonation despite the strong asymmetry of the initial flame. The maximum pressure in the domain is plotted in Fig. 24, showing the pressure rise mainly due to pressure wave reflections. Up to 400 µs the three cases reach similar maxima. Beyond 400 µs, the pressure of case 19 increases less rapidly than that of the other two cases, reaching 10 bar only 200 µs later (around 600 µs).

<figure><img src="image_25.png" /><figcaption>**Fig. 23.** Time evolution of max $|B_{f,l}|$. $dt = 2$ µs for cases 16 and 17, between 400 and 700 µs. Black: case 10, grey: case 17, red: case 15, orange: case 16, blue: threshold value $|B_{f,l}| = 4.10^{13} s^{-2}$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. 24.** Maximum domain pressure versus time for cases 10, 18 and 19. On the right, a zoom of the same graph.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>**Fig. 25.** Pressure field: (a) Case 18, (b) Case 19. Black: $c = 0.5$. 1 to 4: pressure focussing points.</figcaption></figure>

To illustrate the differences between the wave reflections of cases 18 and 19, the pressure field and flame front are shown from 400 µs in Fig. 25. As observed for case 15, the highly asymmetric evolution of the RBLs in case 19 hinders regular shock focusing in a centred fresh gas funnel, resulting in pressure dissipation and preventing it from exceeding 10 bar. On the contrary, the RBLs in case 18 remain sufficiently symmetric to allow pressure focusing, thus providing the right conditions for FA.

Finally, the maximum absolute value of the baroclinic term max $|B_{f,l}|$ for these three cases is plotted in Fig. 26. Again, the threshold $|B| = 4.10^{13} s^{-2}$ is not exceeded in case 19.

In conclusion, a small asymmetry favours the probability of having a shock focus at the flame front, but too large an asymmetry reduces this probability. In the case of centred ignition, the presence of gravity desymmetrizes the flame sufficiently before the first FSI to favour boundary layer development slightly off-centre with respect to the channel axis ($y = 0$). Wave reflections on these boundary layers are therefore off-centre with respect to the channel axis, favouring shock focusing on the flame front.

A simulation with the initial tulip shape was performed without gravity and still no DDT was observed (not shown for brevity).
<figure><img src="image_28.png" /><figcaption>**Fig. 26.** Time evolution of $\max(|B_{f1}|)$. Blue: threshold value $4.10^{13} \text{ s}^{-2}$. Case 18: $\Delta t = 2 \text{ µs}$ from 580 µs.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Fig. 27.** $\max(|B_{f1}|)$ function of the asymmetry factor $\zeta$. Star: DDT. Dashed line: threshold common to all the cases having an incident shock Mach number $M_s = 1.9$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

To summarise this study, the maximum $|B_{f1}|$ on the flame front for each case with $M_s = 1.9$ and $31.25 \ \mu\text{m}$ is plotted in Fig. 27, considering only the instants after the second FSI and before the DDT. Cases with DDT are marked with a star in the figure. Cases favourable to DDT have an asymmetry factor $\zeta$ between 0.002 and 0.035. The other cases are either symmetrical (indicated by the red ellipse) or too asymmetrical (indicated by the red rectangle). In addition, a threshold $|B| > 4.10^{13} \ \text{s}^{-2}$ is visible (blue dotted line), which conditions the activation of FA. This threshold on the baroclinic term is a criteria which could be used in conjunction with the critical expansion ratio as proposed by Dorofeev and Ciccarelli [48].

In the previous sections, the onset of a DDT was triggered by flame acceleration (FA), which is initiated by a local pressure rise at the flame front. This is a stronger double condition than the pressure rise in fresh gases of Mach $M_s = 2.4$. The presence of asymmetry (cases 10, 16 and 18) in the evolution of the reactive boundary layers facilitates the encounter of compression waves at a few points on the flame surface, whereas in the case of symmetric evolution (cases 15, 17 and 20) these pressure rises in fresh gases will occur predominantly on the axis and will not have the potential to auto-ignite the mixture for this Mach number. As the asymmetry of the initial condition increases (cases 12 and 19), followed by an increasingly asymmetric development of the reactive boundary layers, the occurrence of compressional wave crossings follows a more chaotic process, and so the probability of both a wave crossing leading to a significant increase in local pressure and a wave crossing close to a pre-existing flame front decreases.

These assumptions can be illustrated in terms of probabilities, as shown in Fig. 28. The number of events resulting in a pressure exceeding $P$ (ranging from 5 to 10 bar) within the flame front ($0.01 < c < 0.85$), is plotted as a function of time (between 500 and $620 \ \mu\text{s}$), for cases 10, 15, 18 and 19. Cases with DDT are shown as solid lines, and only pre-DDT times are plotted.

For (a) $P > 5$ bar, the probability of reaching 5 bar or more on the flame front is very high for all cases and increases with time (between

<figure><img src="image_30.png" /><figcaption>**Fig. 28.** Number of cells in the flame front ($0.01 < c < 0.85$) exceeding (a) $P > 5$ bar, (b) $P > 6$ bar, (c) $P > 7$ bar, (d) $P > 8$ bar, (e) $P > 9$ bar, (f) $P > 10$ bar. Solid/dash lines: w/wo DDT. $M_s = 1.9$.</figcaption></figure>

2000 and 12000 points). For (b) $P > 6$ bar, this probability is already lower (between 200 and 4000 points), but is still reached by all cases. Case 19 has a lower probability than the other cases for (b) $P > 6$ bar, due to its more asymmetric behaviour. Case 15 never reaches condition (e) $P > 9$ bar on the flame front. Finally, for (f) $P > 10$ bar, only cases 10 and 18 have between 25 and 300 points on the flame front reaching 10 bar or more, and therefore a high probability of FA.

## 8. Conclusion

Highly refined numerical simulations have been carried out to study hydrogen/air flame propagation ($\phi = 0.8$, $P = 0.2$ bar) and its interaction with a normal shock ($M_s = 1.9$ and $2.4$) in a semi-closed channel. Two detailed chemical mechanisms are compared. The diffusive flux closure includes the Soret effect and gravity is present. Depending on the method of flame ignition, two flame shapes are detected: glove finger flame and tulip flame. The main FSI stages [10,13,17,26] are retrieved: appearance of Richtmyer–Meshkov instabilities at the flame front, development of reactive boundary layers with lambda shocks hooked at their extremities. The occurrence of auto-ignition in the fresh gases followed by DDT was found to depend on the intensity of the incident shock, but also on the approximation of the problem dimensions, i.e. 1D, 2D or 3D. For the highest shock number, the appearance of DDT is associated with a shock focusing roughly on the symmetry axis of the channel in the fresh gases, far from the pre-existing flame fronts. The hypothesis on the domain dimensions and the initial flame shape only slightly modify the time at which DDT occurs. For a shock Mach number of 1.9, the problem is more complex. DDT, when it occurs, is the result of the crossing of sufficiently strong compression waves in the vicinity of the flame front. DDT is then preceded by flame acceleration. In this case, three conditions for obtaining DDT have been demonstrated: at least a two-dimensional geometric description of the domain, a glove finger flame and an asymmetry of the flame developed in the channel before its first encounter with the shock. This asymmetry can be the result of a non-symmetric ignition system or simply the result of gravity in a horizontal duct. It has also been shown that too much asymmetry of the initial flame prevents DDT, since in this case the scattering of the pressure waves reduces the probability of their meeting in the vicinity of the flame with sufficient compression energy. For this lowest shock Mach number, every detail of the setup counts when it comes to promoting or suppressing DDT. The initial distance between the flame tip and the tube end wall, or the initial flame surface topology, may also influence the process. A criterion based on the estimation of the maximum of the baroclinic term has been proposed to predict the occurrence of a DDT for the shock Mach number 1.9. The link between the evolution of the baroclinic term and the DDT triggered by flame acceleration is probably present in all configurations, regardless of the channel dimensions or mixture composition. However, the threshold value required is certainly configuration-dependent and would require further investigation.
## CRediT authorship contribution statement

**Emilie Yhuel:** Writing – review & editing, Writing – original draft, Visualization, Software, Investigation, Formal analysis. **Guillaume Ribert:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Investigation. **Pascale Domingo:** Writing – review & editing, Writing – original draft, Supervision, Project administration, Investigation, Funding acquisition, Formal analysis, 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.

## Acknowledgements

Part of this work was performed using computing resources of CRIANN (Normandy, France). Computing resources were also provided by IDRIS and CINES (grant numbers A0152B00152 and A0172B00152 (GENCI)). Emilie Yhuel has been founded by the Agence Nationale de la Recherche (ANR-20-CE05-0014 - PHYSSA).

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.combustflame.2025.114526.

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