# Effects of the channel cross-section shape on detonation re-initiation downstream of a single-hole obstacle

Vianney Monnier, Léa Vilasi, Vincent Rodriguez, Pierre Vidal, Ratiba Zitoun

► To cite this version:

Vianney Monnier, Léa Vilasi, Vincent Rodriguez, Pierre Vidal, Ratiba Zitoun. Effects of the channel cross-section shape on detonation re-initiation downstream of a single-hole obstacle. Shock Waves, 2025, 10.1007/s00193-025-01246-0. hal-05238488

HAL Id: hal-05238488

https://isae-ensma.hal.science/hal-05238488v1

Submitted on 3 Sep 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Effects of the channel cross-section shape on detonation re-initiation downstream of a single-hole obstacle

V. Monnier<fnref n="1" /> · L. Vilasi<fnref n="1" /> · V. Rodriguez<fnref n="1" /> · P. Vidal<fnref n="1" /> · R. Zitoun<fnref n="1" />

Received: 22 March 2025 / Revised: 22 July 2025 / Accepted: 1 August 2025

© The Author(s) 2025

## Abstract

We identify six scenarios of detonation re-initiation downstream of an obstacle with a single-centered hole in square and round tubes. The tubes have the same cross-sectional area of 16 cm², and the hole has the same shape as the tube cross section, but different open area ratios. The test mixtures investigated were 2 H₂ + O₂ + 2 Ar and 2 H₂ + O₂. We used soot foil recordings, high-speed schlieren, and chemiluminescence imaging to obtain longitudinal and frontal views of the diffraction phenomena. Depending on the initial pressure, one supercritical, four critical, and one subcritical scenarios were observed. The supercritical and critical scenarios were more likely to occur in the square tube than in the round tube, all other parameters being equal. Most of these transient, three-dimensional effects of the cross-sectional shape can be observed even at initial pressures for which there is no effect on steady propagation, e.g., without an obstacle or far from it. This raises the question of what dimensionality refers to in experiments in terms of global and local detonation dynamics.

**Keywords** Detonation · Re-initiation · Cross-section shape · Obstacle

## 1 Introduction

Detonation diffraction is an important topic because of its applications in industrial safety and the design of initiation devices in aerospace equipment, e.g., the chamber of rotating detonation engines. A large number of studies [1] have addressed detonation diffraction for various configurations, in particular the shape of the obstacle and the number or shape of holes in the obstacle [2]. However, to our knowledge, studies have not addressed the effect of the cross-sectional shape of the tube. In this article, we detail our experimental investigation [3, 4] on the diffraction of a steady detonation from a single-hole obstacle depending on the cross-sectional shape of the tube, namely, square or round. Square channels are indeed convenient for visualization in the laboratory, but round tubes are the rule in industry, both for practical and economic reasons.

Successful detonation propagation in premixed reactive media requires the coupling of the combustion wave with its

preceding shock wave [5, 6]. Their decoupling unbalances the energy production and loss and can lead to the quenching of the self-sustained detonation [7–10]. Such decoupling occurs when the detonation diffracts during its transmission to a confinement of greater dimensions [11–15].

Zeldovich et al. [16] introduced the notions of sub- and supercritical behaviors in detonation transient dynamics. Subcritical behavior refers to detonation quenching with no possibility of re-initiation. The accepted experimental indication is the disappearance of the cellular structure typical of detonation reaction zones. Critical behavior refers to the cases of detonation quenching followed by re-initiation, i.e., a discontinuous cellular structure, which first disappears and then reappears.

Studies on detonation diffraction have described two critical scenarios, e.g., [15, 17], and are limited to two dimensions [18, 19]. For the critical behavior, detonation re-initiation in a tube generally results from a hot spot from which a self-accelerating flame emerges. These heterogeneities can have several causes [20], possibly combined, such as wall defects, turbulent boundary layers, or gradients of initial state (pressure, temperature). Compression waves that form ahead of the flame front coalesce to form a shock wave. This shock wave can become strong enough to initiate an explosion in the mixture between the shock and the flame. Detonation

Communicated by G. Ciccarelli.

<footnote>V. Monnier
vianney.monnier@utoulouse.fr</footnote>

<footnote n="1">¹ Institut Pprime, UPR 3346 CNRS, ISAE-ENSMA, 86961 Futuroscope-Chasseneuil, France</footnote>
initiation can then occur if the shock-combustion coupling condition is maintained. This canonical mechanism is called SWACER (Shock Wave Amplification by Coherent Energy Release) [21, 22]. The supercritical behavior involves the continuous propagation of the detonation, whether its velocity, cell size, or regularity vary or not. Local extinction may occur, typically in the vicinity of the area change, i.e., just after the obstacle.

The cell mean width $\bar{\lambda}$ obtained from measurements on a longitudinal (wall) soot recording remains a convenient characteristic length for detonation dynamics analysis, in particular, for predicting the detonation behavior during transient phenomena [23].

Studies following the one by Zeldovich et al. lead to a criterion for the successful transmission of a detonation (that is, critical to supercritical behaviors) based on cell-to-tube widths ratio, the critical diameter of the donor section being $d_c \simeq 10$ to $30\bar{\lambda}$ [24], depending on the cellular structure regularity [25–27], mixture initial temperature [28], or if the channel is 2D or 3D [29]. That criterion is used for pre-sizing detonation arresters, but other parameters are also taken into account, e.g., obstacle shapes, spacings, open area ratio, number of holes, sizes, etc. [30]. If the obstacle is several cell lengths thick, it behaves as a confinement reduction, and if its open area transverse dimension is smaller than one mean cell, the detonation quenches within the obstacle. If the shock wave does not decouple from the combustion wave within the obstacle, the sudden change in area causes the detonation to diffract.

The cross-sectional shape and area (A) of the channel may influence the propagation of a self-sustained detonation [31]. This influence manifests itself by a difference in $\bar{\lambda}$ from one confinement geometry to another, with all initial thermodynamic (initial pressure $p_0$ and temperature $T_0$) and chemical (composition, equivalence ratio) parameters of the reactive mixture being equal. However, if the parameters (A, $p_0$) are sufficiently large [31, 32], the geometry no longer affects $\bar{\lambda}$, and the head-on collision soot foil recordings of the detonation show an irregular pattern. This results in a large number ($O(100)$) of cells per unit area on the detonation front. In this situation, the cells can be termed as self-confined [3], being independent on the transverse dimension and the cross-sectional shape of the tube.

Section 2 presents the experimental setup and methodology. Section 3 details the results from the diffraction and the re-initiation of the detonation passing through a single-hole obstacle for two tube shapes. Section 4 discusses the results and provides conclusions.

## 2 Experimental setup and methodology

We performed experiments in two tubes with a round or square cross section of the same area of $16 \pm 4\% \text{ cm}^2$, see

<figure><img src="image_2.png" /><figcaption>**Fig. 1** Square tube (or round) with $16 \text{ cm}^2$ cross-section area. A: ignition, S: Shchelkin spiral, P: piezoelectric pressure gauges, O: obstacle, L: soot-coated surface or quartz windows, R: steel wool. Dimensions are in mm</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Fig. 2** Schlieren setup of the square tube for this study. LS: light source, M: mirror, C: converging lens, D: iris shutter, C': collection lens, HSC: Shimazu HPVX-2 high speed camera, P: piezoelectric transducer (the signal amplifier is not represented for clarity), dotted arrow: triggering information</figcaption></figure>

Fig. 1. A spark plug or an exploding wire (Fig. 1, A) was used to ignite the reactive mixture, and a Shchelkin spiral (Fig. 1, S) to shorten the transition to detonation within 1 m from ignition. The obstacle (Fig. 1, O) was a 5-mm-thick plate, placed 2750 mm away from ignition. It had a single hole of the same shape as the channel cross section, i.e., square in the square channel and round in the round tube. In this study, the opening area ratio (OAR) is defined as the ratio of the cross-sectional area of the orifice to the cross-sectional area of the tube, and it was varied from OAR = 10% to 75%. A steel wool ball (Fig. 1, R) was positioned at the end wall of the tube, opposite to ignition, to dampen the shock in the cases of non-successful transmission, before the shock reflects.

We recorded pressure signals upstream of obstacles with Kistler 603B piezoelectric pressure transducers (Fig. 1, P) coupled to Kistler 5018A electrostatic charge amplifiers whose triggering times indicated both the detonation steadiness and its average velocity. Longitudinal soot foils (Fig. 1, L) recorded the cellular structure of the detonation upstream and downstream of the obstacle. In the square channel, quartz windows replaced the soot-coated surfaces to enable high-speed schlieren (Fig. 2) or chemiluminescence recordings.

For chemiluminescence imaging, a Shimadzu HPVX-2 camera recorded 256 images of emitted light from the burnt gases with an opening time of 200 ns and a frequency ranging from 0.5 to 2 MHz. The camera started recording the
<figure><img src="image_4.png" /><figcaption>**Fig. 3** Average detonation velocities $\bar{D}$ in square (□) and round (○) 16-cm² tubes as a function of initial pressure $p_0$ [3]. Dotted lines are Chapman–Jouguet velocities. Experimental uncertainties are 115 m/s [33]</figcaption></figure>

images after a delay from the triggering time of the closest upstream pressure transducer (Fig. 2, P). The delay time was set according to detonation velocities measured in previous experiments [31] and compiled in Fig. 3. For schlieren imaging, a 500-W mercury vapor lamp (Fig. 2, LS) emitted a conical light beam toward a concave mirror (Fig. 2, M) of 1.5-m focal length and 150-mm diameter. The converging lens and collecting lens (Fig. 2, C and C') had 750- and 65-mm focal lengths and 150- and 50-mm diameters, respectively. The iris shutter had a 1-mm-diameter opening.

The schlieren and chemiluminescence techniques show integrated information across the camera depth of field, hence require end-view recordings to fully appreciate the three-dimensional aspects of the wave dynamics, that is, differentiate information in the width of the tube. In both the round and square channels, removing the damping steel wool (R) and replacing the end flange with a quartz window enabled end-view chemiluminescence recordings downstream of the obstacle.

The reactive mixtures were stoichiometric undiluted and argon-diluted mixtures of hydrogen-oxygen, i.e., 2H₂ + O₂ and 2H₂ + O₂ + 2Ar. They were prepared in a separate tank using the partial-pressure method and then injected at the initial pressure $p_0$ after vacuuming the tubes at $T_0 = 294 \pm 3$ K.

## 3 Experimental results

The experiments showed very high repeatability, thus allowing superposition of results from different experiments, based on soot recordings, schlieren, and chemiluminescence imaging.

<figure><img src="image_5.png" /><figcaption>**Fig. 4** Detonation wave diffraction and decoupling of the combustion wave after passing through the obstacle with OAR = 10% (square cross-section tube). Superposition of colorized chemiluminescence and schlieren images, camera frequency 250 kHz. The mixture is 2H₂ + O₂ at $p_0$ = 60 kPa</figcaption></figure>

Six diffraction scenarios were identified: one subcritical, four critical, and one supercritical. Each is detailed in Figs. 5 to 14, based on cross-comparisons of schlieren imaging, soot foil recordings, and end-view schematics. For each mixture, the parameters (OAR, $p_0$) determine the occurrence of a scenario. The common feature of each scenario is that the shock generated by diffraction through the obstacle interacts symmetrically with the tube wall(s). Analysis of the recordings shows that this interaction generates four Mach shocks in the square tube and one toroidal Mach shock in the round tube. Each wave propagates along the wall(s) and converges toward the tube axis. The scenarios differ in whether and how the reflected wave participates in the re-initiation of the detonation. These scenarios are:

- No re-initiation of the detonation downstream of the obstacle (Sect. 3.1). This subcritical scenario occurs at low OAR or $p_0$.

- Re-initiation of detonation by the convergence of shocks in the centerline of the tube, only observed in the square tube (Sect. 3.2).
<figure><img src="image_6.png" /><figcaption>**Fig. 5** Schematic and longitudinal wall recordings of the diffraction process downstream of the obstacle in the round tube. Blue: diffracted shock, green: reflected shock, yellow: upstream detonation quenching</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 6** Subcritical case. Diffraction with OAR = 25%. The mixture is 2H₂ + O₂ at *p*₀ = 15 kPa. Schlieren images (1 to 7, left): longitudinal recordings, longitudinal soot recording (bottom), end-view schemes (right); Δ*f*images = 6.5 µs. The soot foil recording and the schlieren images have the same length scale</figcaption></figure>

- Re-initiation of detonation by convergence of shocks in section corners, only observed in the square tube (Sect. 3.2).

- Re-initiation of detonation by deflagration-to-detonation transition (Sect. 3.2.1). This critical scenario occurs at low OARs (OAR ≤ 25% in round tube and OAR = 10% in square tube).

- Re-initiation of detonation by shock reflection on the wall (Sect. 3.2.2). This critical scenario occurs at high initial pressures or intermediate OARs.

- Supercritical transmission of detonation without extinction (Sect. 3.3). This scenario occurs for OAR ≥ 50% for the round tube and OAR = 75% for the square tube.

Schlieren and chemiluminescence imaging (Fig. 4) and soot recordings provide an overview of the diffraction dynamics over the entire cross section in the square tube, and the soot recordings provide information about the cell structure on the walls of both the round and square tubes.

**3.1 Detonation diffraction, quenching, and subcritical scenario**

Figure 5 shows a schematic of the axisymmetric diffraction process over a soot recording in the round tube, and Fig. 6 shows schlieren images with the corresponding soot recording from the square tube. When the shock wave diffracts,
<figure><img src="image_8.png" /><figcaption>**Fig. 7** Critical case: re-initiation by convergence of shocks in the centerline. Diffraction with OAR = 25%. The mixture is 2H₂ + O₂ + 2Ar at p₀ = 12.5 kPa. Schlieren images (1 to 7, left): longitudinal recordings, longitudinal soot recordings (bottom), end-view schemes (right); Δt images = 8.5 µs. The soot foil recording and the schlieren images have the same length scale</figcaption></figure>

the detonation loses strength as the expansion wave moves
toward the center of the tube. This reduction in strength
causes the temperature downstream of the shock to drop,
increasing the induction time [34]. The reaction zone even-
tually decouples from the shock [9, 35–37] and the detonation
quenches (Fig. 5, a and Fig. 6, 1). The diffracted shock
wave then reflects on the tube wall(s). In the round tube,
the diffracted shock reflects on the circle defined as the inter-
section between the obstacle and the tube wall, reflecting a
toroidal shock wave sweeping along the tube wall and toward
the tube axis of symmetry (Fig. 5, a to c). In the square tube,
the diffracted shock reaches the four tube walls closest to the
orifice, i.e., the points located in the center of the walls (Fig.
6, 2), resulting in four reflected shocks propagating toward
the centerline of the tube and along the walls of the tube
(Fig. 6, 3). Those shock waves then combine by pairs at the
corners of the tube (Fig. 6, 4–5). In both round and square
channels, the reflected shock(s) focus at the center of the tube
and form a Mach shock (Fig. 5, d and Fig. 6, 6) that prop-
agates outwards to the tube wall(s) to imprint on the sooted
surface (Fig. 5, d-e and Fig. 6, 7). The cycle repeats peri-

<figure><img src="image_9.png" /><figcaption>**Fig. 8** Critical case: re-initiation by convergence of shocks at the corners. Diffraction with OAR = 25%. The mixture is 2H₂ + O₂ at p₀ = 30 kPa. Superposition of colorized chemiluminescence and schlieren images (1 to 5, left): longitudinal recordings, chemiluminescence end-view recordings (middle), soot recording (bottom), and end-views schemes (right); Δt images = 3 µs. The soot foil recording and the schlieren images have the same length scale</figcaption></figure>

odically until the shocks fade or reflect on the end wall. In
the phenomena described above, there is no re-initiation of
the detonation downstream of the obstacle, i.e., this scenario
is subcritical. Removing the damping wool (Fig. 1, R) may
result in re-initiation when the inert shock reflects on the end
wall, as in [38–40], for high enough initial pressures. Those
cases are not within the scope of the present study.

**3.2 Shock-shock interactions critical scenarios**

Figure 7 shows the case of a detonation in 2H₂ + O₂ + 2Ar at p₀ = 12.5 kPa diffracted by the obstacle with OAR = 25% in the square tube. The waves reflect from the walls and converge at the tube axis. This convergence produces a wave strong enough to ignite the mixture and initiate a diverging detonation (Fig. 7, 5–6) that propagates partly in the pre-shocked medium, where fresh and burnt gases are present, and partly in the fresh mixture. This divergent detonation reaches the walls after expansion and becomes flat again. Soot recordings (Fig. 7) show a cellular structure that is more irregular downstream of the obstacle than upstream. This irregularity is due to the presence of burnt gases along with
<figure><img src="image_10.png" /><figcaption>**Fig. 9** Critical case: hybrid re-initiation by convergence of shocks in the corners and in the centerline; OAR = 50%, mixture: 2H₂ + O₂ + 2Ar, p₀ = 15 kPa. Images 1 to 7: schlieren longitudinal recordings (left, Δtimages = 6 µs) and end-view schemes (right). Bottom: longitudinal soot recording. The schlieren images and the soot foil recording have the same longitudinal length scale</figcaption></figure>

fresh mixture. As the re-initiated detonation propagates, it stabilizes in fresh mixture.

Figure 8 shows the case of a detonation of 2H₂ + O₂ at p₀ = 30 kPa crossing the obstacle with OAR = 25%. Shocks propagate along the walls after reflection of the diffracted shock and interact in pairs at the four corners of the tube (Fig. 8, 2). This interaction produces four "hot spots" where overdriven detonations ignite the gases pre-shocked by waves propagating along the adjacent walls (Fig. 8, 3–4). The resulting detonation catches up with the waves reflected downstream of the obstacle, propagates into the fresh mixture, and re-stabilizes such as before the obstacle (Fig. 8, 5). Soot recordings show the symmetry of the phenomenon on one of the tube walls. Smaller cells than upstream of the obstacle indicate the overdriven detonation.

These two mechanisms of re-initiation by shock-shock interaction (at the corner and the center of the square tube)

can both occur during the same experiment, as in Fig. 9, where the detonation initiated at the corner quenches before re-initiation by the convergence of the shocks at the centerline of the tube.

Re-initiation by shock-shock interaction were never observed in the round tube at our conditions.

### 3.2.1 Deflagration-to-detonation transition scenario

Figures 10 and 11 show deflagration-to-detonation transition (DDT) scenarios in the round and square tubes. The wall(s) are sites of hot spot ignition in both the round and square channels. The resulting flame is slow compared to the preceding shock(s), so they are not coupled. The deflagration propagates toward the end wall of the tube and partly around the walls, producing conical soot traces, that is, dark triangular regions typical of deflagration regimes. In the round tube, the flame self-accelerates and eventually undergoes DDT. When the transition occurs, a bright spot is captured by the high-speed imaging, and an overdriven detonation propagates on the entire cross section. Repeated experiments in the round tube with the same mixture and initial parameters systematically showed DDT on the walls. It is difficult to anticipate the number of hot spots, the azimuth angle, the distance from the obstacle and the DDT distance from one experiment to another conducted under the same parameters because of the stochastic action of the Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instabilities. Broadly speaking, these are typical of any DDT phenomenon, but our experiments were not equipped to characterize their respective contributions. The initiation of the detonation appears instantaneous compared to that of the deflagration. In the square tube, e.g., in Fig. 11 the deflagration ignition occurs at the center of a wall before being caught up by a re-ignited detonation in the corners. This deflagration is only faintly visible on schlieren images, as the pressure gradients are smaller than those of the shock waves and burnt gases. Therefore, soot recordings are needed to confirm its occurrence.

### 3.2.2 Shock-reflection scenario

Figures 12 and 13 show cases of shock-reflection re-initiation. The reflected shock waves are strong enough to initiate an overdriven detonation. These detonations recombine to form a self-sustaining detonation similar to the one upstream of the obstacle. In the round tube, the diffracted shock wave is reflected off the walls, creating a toroidal detonation initiation. In practice, a small defect on the concentricity of the tube walls with the obstacle, e.g., the soot foil placed around the wall made the cross-section shape slightly drop-shaped instead of round, as in Fig. 12, induces the shock to reflect first on a point of the foil instead of a line, hence the non-axisymmetric recording.
<caption>**Fig. 10**</caption>

Deflagration-to-detonation transition re-initiation in the round tube. Top: Chemiluminescence imaging downstream of the obstacle in the round tube (OAR = 25%, mixture 2H₂ + O₂, p₀ = 30 kPa, camera frequency 15.5 kHz). The arrows show a hot spot. The camera is slightly off-axis of the tube. Middle: Longitudinal soot recordings downstream of the obstacle (OAR = 25%, mixture 2H₂ + O₂ + 2Ar, p₀ = 50 kPa). Vertical arrows: hot-spot deflagration ignition, horizontal arrow: transition location. Bottom: Qualitative schemes of the process

<figure><img src="image_11.png" /><figcaption>**Fig. 11** Soot recording showing deflagration at the center of a wall and overdriven detonations at the corners. Full lines: successive positions of fronts, dashed lines: separations of propagation domains. Yellow: highly overdriven detonations emanating from the tube corners, identified by very small cells. Red: overdriven detonation in the fresh mixture. Blue: deflagration</figcaption></figure>

<caption>**Fig. 12** Shock-reflection re-initiation scenario in the round tube. Top: Chemiluminescence imaging downstream of the obstacle in the round tube with OAR = 25%. The mixture is 2H₂ + O₂, p₀ = 80 kPa, and camera frequency is 15.5 kHz. Gray arrow: hot spot. The camera is slightly off-axis of the tube. Middle: Longitudinal soot recordings from the downstream of the obstacle with OAR = 25% in the round tube. The mixture is 2H₂ + O₂ + 2Ar at p₀ = 60 kPa. Bottom: Qualitative schemes of the process</caption>
<figure><img src="image_12.png" /><figcaption>**Fig. 13** Critical case: re-initiation by shock reflection. Diffraction with OAR = 25%. The mixture is 2 H₂ + O₂ + 2 Ar at p₀ = 70 kPa. Superposition of colorized chemiluminescence and schlieren images (1 to 5, left): longitudinal recordings, chemiluminescence end-view recordings (middle), longitudinal soot recording (bottom), and end-view schemes (right); Δtimages = 4 µs. The soot foil recording and the schlieren images have the same length scale</figcaption></figure>

### 3.3 Supercritical scenario

Figure 14 shows the case of a detonation in 2H₂ + O₂ at p₀ = 50 kPa crossing the opening obstacle with OAR = 75%. The decoupling process of the combustion and the diffracted shock is too slow to quench the detonation before the shock reflects at the walls. The detonation cells may be larger downstream of the obstacle than upstream, before re-stabilizing to their initial width.

From p₀ = 7.5 to 50 kPa, the detonation is transmitted in the round and square tubes for the obstacle with OAR = 75%.

## 4 Discussion and conclusions

This article details a description of the three-dimensional experimental transients that follow the diffraction of a steady cellular detonation through a single-hole obstacle in straight tubes. The focus is on the influence of the cross-sectional shape of the tube, namely round or square, which extends our previous study on the cellular structure of a steadily-propagating detonation [4, 31]. We have identified six diffraction scenarios, comprising 1 subcritical, 4 critical, and 1 supercritical behaviors. Figure 15 summarizes these scenarios with open area ratios (OAR) below or equal to 50%, as only supercritical behaviors are observed at OAR = 75%.

As for the critical cases, no re-ignition by shock refocusing occurred in the round tube. Only extinction without re-ignition, DDT, re-ignition by wall-shock interaction, and transmission scenarios occurred, as already observed by

<figure><img src="image_13.png" /><figcaption>**Fig. 14** Supercritical case. Superposition of colorized chemiluminescence and schlieren images and soot recording of a detonation after diffraction passing through the obstacle with OAR = 75%. The mixture is 2 H₂ + O₂ at p₀ = 50 kPa; Δtimages = 3 µs. The soot foil recording and the schlieren images have the same length scale</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 15** Diffraction scenarios based on the open area ratio and the cross-sectional shape of the tube. Dark gray bands: round cross section. Light gray bands: square cross section. Crosses: no re-ignition. Empty squares: re-ignition by tube-center shock interaction. Green squares:</figcaption></figure>

re-ignition by corner-shock interaction. Red pentagons: re-ignition by wall-shock interaction. Yellow triangles: re-ignition by DDT. Black circles: transmission
Ciccarelli et al. [34]. This is due to the concentricity of the tube and the diffracted shock. The latter reflects along a line, i.e., the perimeter of the section, and not at one or several points. This produces a more powerful toroidal reflected shock wave than in the square tube because of its negative curvature and, therefore, inward-converging propagation direction. Additionally, the absence of corners in the round tube limits the promoting effects of turbulence and hot spots caused by friction at the walls.

A different diffraction scenario can occur in the square and round tubes depending on the open area ratio of the obstacle (OAR = 10, 25, and 50%), all other parameters being equal. A first explanation would be to invoke the difference in cell widths $\bar{\lambda}$ measured from one tube to the other, when the initial pressure is low enough ($p_0 \le 30$ kPa for the tubes of this study), that can reach 100% [31]. However, for OAR = 10%, the scenario differs from one cross-sectional shape to the other, despite the negligible influence of this shape for steady propagation of detonation [31]. Therefore, the tube cross-section shape influences transient phenomena on a wider range of initial pressures than quasi-steady propagation. The criteria on the number $O(100)$ of self-confined cells on the front surface—i.e., for transverse dimensions of channels large enough they no longer influence the detonation dynamics—does not apply to transient phenomena. Thus, for $2H_2 + O_2 + 2Ar$ at $p_0 = 80$ kPa, re-ignition occurs by DDT in the round tube and by shock-shock interaction in the square tube. In addition, Fig. 15 shows the observed diffraction scenarios for each mixture, based on the open area ratio (OAR) and the cross-sectional shape of the tube, as functions of the initial pressure $p_0$, the cell mean width $\bar{\lambda}$ upstream of the obstacle, and the ratio of the hydraulic diameter $d_h/\bar{\lambda}$ of the aperture to the cell mean width $\bar{\lambda}$ (the hydraulic diameter $d_h$ is the diameter for a circle, and the side length for a square, that is, in this work, $d$ and $a$ for the round and square apertures, respectively, cf. Fig. 1). In particular, the hydraulic diameter is insufficient for predicting the diffraction scenario for our conditions, as its numerical value alone does not provide information about the cross-sectional shape. Generally, depending on their transverse dimensions (e.g., $d$, $a$), channels with different cross-sectional shapes may have the same hydraulic diameter.

The extinction without re-ignition parameters are the same with OAR = 25% and $2H_2 + O_2$, or with OAR = 10% and $2H_2 + O_2 + 2Ar$, for the same cell width. Overall, the range of cell widths for re-ignition is higher in square than round tubes. The subcritical scenario has a wider observation range, i.e., occurs up to higher $p_0$ when detonating a more unstable mixture, as reported by Kuznetsov et al. [26]. The purpose of the study is not to discuss in depth the effect of detonation cell regularity depending on the cross-sectional shape, but rather to demonstrate that this parameter and others have effects which may differ from one shape to the other. As for the cell

regularity (i.e., the Ar dilution in this work), the plots in Fig. 15 show quantitative differences, but the same diffraction scenario. Finally, from a phenomenological point of view, the cell width and the initial pressure vary reciprocally with respect to each other. The plots in the first and second rows of Fig. 15 clearly demonstrate the effect on the diffraction scenario.

For our conditions, we cannot identify any criterion using the hydraulic diameter-to-cell width ratio as for diffraction in unconfined volumes [41], nor the distance between the closest wall and the aperture as in the 2D phenomena studied by Sun et al. [37], that guarantees re-ignition. Our choice was to compare tubes of equal cross-section areas, but results show that the re-initiations by shocks downstream of the obstacle are a function of their strength, therefore with an implication of distances from the obstacle rather than surfaces. DDT occurs at walls within boundary layers, hence comparison between tubes should also involve equal longitudinal surfaces of the confinement rather than equal cross-sectional surfaces, with special attention on interacting boundary layers, e.g., in channels with small aspect ratios or at corners. The cell size alone cannot be used for predicting transients in three-dimensional configurations.

Our observations lead to a discussion of dimensionality for transient detonation dynamics. In the round tube, the phenomena are 2D axisymmetric, except for the DDT scenario where the wall surface is involved. In the square tube, diffraction through a square hole can induce two pairs of opposing fronts originating independently of each other from the opposite walls (Fig. 13), or four fronts originating from the corners, depending on the interaction of adjacent shocks (Fig. 8). The first case is as a pair of 2D re-initiations, but the latter is 3D, and would not have occurred if the hole in the obstacle was a slit. Therefore, some care should be taken when classifying experimental channels as two- or three-dimensional depending on their aspect ratio.

In fact, any setup initially two-dimensional remains two-dimensional along its entire length if its width does not vary, regardless of possible variations in its height. Thus, a backward-facing step [42], or a perforated plate with a distribution and density of holes that make the transverse directions indistinguishable from each other [30] can be seen as an extruded two-dimensional setup, regardless of their width. This study suggests that the simple configurations used in laboratory experiments to facilitate the identification of fundamental phenomena are not always applicable to the prediction of transients in industrial configurations.

**Supplementary Information** The online version contains supplementary material available at https://doi.org/10.1007/s00193-025-01246-0.
**Acknowledgements** This work was supported by the Ministry of Higher Education, Research, and Innovation (France) and the CPER-FEDER Project of *Région Nouvelle Aquitaine*.

**Data Availability** The data that support the conclusions of this study are available from the corresponding author upon reasonable request. The schlieren and chemiluminescence video recordings are provided as the supplementary material.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## References

1. Wen, J.X., Hecht, E.S., Mevel, R.: Recent advances in combustion science related to hydrogen safety. Prog. Energy Combust. Sci. **107**, 101202 (2025). https://doi.org/10.1016/j.pecs.2024.101202

2. Liu, Y., Lee, J., Knystautas, R.: Effect of geometry on the transmission of detonation through an orifice. Combust. Flame **56**(2), 215-225 (1984). https://doi.org/10.1016/0010-2180(84)90038-5

3. Monnier, V.: Aspects tridimensionnels de la détonation cellulaire: des observations expérimentales et un modèle. Ph.D. thesis, École Nationale Supérieure de Mécanique et d'Aérotechnique (2023). (in French) https://doi.org/10.13140/RG.2.2.17111.85921

4. Monnier, V., Rodriguez, V., Vidal, P., Zitoun, R.: Three-dimensional dynamics of detonation diffraction: effects of the tube cross-section shape. Proc. 29th ICDERS, Paper 59 (2023)

5. Vieille, P.: Étude sur le role des discontinuités dans les phénomènes de propagation. J. Phys. Theor. Appl. **9**, 621-644 (1900). (in French) https://hal.science/jpa-00240479v1/file/ajp-jphystap_1900_9_621_0.pdf

6. Zeldovich, Y., Barenblatt, G.: Theory of flame propagation. Combust. Flame **3**, 61-74 (1959). https://doi.org/10.1016/0010-2180(59)90007-0

7. Vidal, P.: A necessary condition for adiabatic explosion behind a shock of arbitrary dynamics. Comptes Rendus de l'Académie des Sciences, Paris, Série IIB **327**(1), 95-102 (1999)

8. Vidal, P., Khasainov, B.: Critical slow dynamics of detonation in a gas with non-uniform initial temperature and composition: a large-activation-energy analysis. Intern. J. Spray and Combust. Dynamics **1**(4), 435-471 (2009). https://doi.org/10.1260/175682709789685822

9. Pintgen, F., Shepherd, J.: Detonation diffraction in gases. Combust. Flame **156**(3), 665-677 (2009). https://doi.org/10.1016/j.combustflame.2008.09.008

10. Arienti, M., Shepherd, J.: A numerical study of detonation diffraction. J. Fluid Mech. **529**, 117-146 (2005). https://doi.org/10.1017/S0022112005003319

11. Soloukhin, R., Ragland, K.: Ignition processes in expanding detonations. Combust. Flame **13**(3), 295-302 (1969). https://doi.org/10.1016/0010-2180(69)90007-8

12. Edwards, D.H., Thomas, G.O., Nettleton, M.A.: The diffraction of a planar detonation wave at an abrupt area change.

J. Fluid Mech. **95**(1), 79-96 (1979). https://doi.org/10.1017/S002211207900135X

13. Edwards, D., Thomas, G., Nettleton, M.: Diffraction of a planar detonation in various fuel-oxygen mixtures at an area change. In: Oppenheim, A.K., Manson, N., Soloukhin, R.I., Bowen, J.R. (eds.) Gasdynamics of Detonations and Explosions. Progress in Astro. and Aero., AIAA, vol. 75, pp. 341-357 (1981). https://doi.org/10.2514/5.9781600865497.0341.0357

14. Knystautas, R., Lee, J., Guirao, C.: The critical tube diameter for detonation failure in hydrocarbon-air mixtures. Combust. Flame **48**, 63-83 (1982). https://doi.org/10.1016/0010-2180(82)90116-X

15. Klein, J., Samimi-Abianeh, O.: Simultaneous schlieren and direct photography of detonation diffraction regimes in hydrogen mixtures. Combust. Flame **272**, 113845 (2025). https://doi.org/10.1016/j.combustflame.2024.113845

16. Zeldovich, Y.B., Kogarko, S., Simonov, N.: An experimental investigation of spherical detonation in gases. Soviet Phys. tech. Phys. **1**, 1689 (1956)

17. Ohyagi, S., Obara, T., Hoshi, S., Cai, P., Yoshihashi, T.: Diffraction and re-initiation of detonations behind a backward-facing step. Shock Waves **12**, 221-226 (2002). https://doi.org/10.1007/s00193-002-0156-z

18. Bhattacharjee, R., Lau-Chapdelaine, S., Maines, G., Maley, L., Radulescu, M.: Detonation re-initiation mechanism following the Mach reflection of a quenched detonation. Proc. Combust. Inst. **34**(2), 1893-1901 (2013). https://doi.org/10.1016/j.proci.2012.07.063

19. Shi, X., Pan, J., Jiang, C., Li, J., Zhu, Y., Quaye, E.K.: Effect of obstacles on the detonation diffraction and subsequent re-initiation. Int. J. Hydrogen Energy **47**(10), 6936-6954 (2022). https://doi.org/10.1016/j.ijhydene.2021.12.026

20. Ciccarelli, G., Dorofeev, S.: Flame acceleration and transition to detonation in ducts. Prog. Energy Combust. Sci. **34**(4), 499-550 (2008). https://doi.org/10.1016/j.pecs.2007.11.002

21. Lee, J., Knystautas, R., Yoshikawa, N.: Photochemical initiation of gaseous detonations. Acta Astronaut. **5**(11), 971-982 (1978). https://doi.org/10.1016/0094-5765(78)90003-6

22. Yoshikawa, N.: Coherent shock wave amplification in photochemical initiation of gaseous detonations. Ph.D. thesis, McGill University (1978)

23. Vasil'ev, A.A.: Cell size as the main geometric parameter of a multifront detonation wave. J. of Propulsion and Power **22**(6), 1245-1260 (2006). https://doi.org/10.2514/1.20348

24. Mitrovanov, V., Soloukhin, R.: The diffraction of multifront detonation waves. Sov. Phys. Dokl. **9**, 1055-1058 (1964)

25. Moen, I. O., Thomas, G. O., Bjerketvedt, D., Thibault, P. A.: Influence of cellular regularity on the behavior of gaseous detonations. In: Leyer, J.-C., Soloukhin, R.I., Bowen, J.R. (eds.) Dynamics of Explosions. Progr. in Astro. and Aero, AIAA, vol. 104, pp. 220-243 (1986). https://doi.org/10.2514/5.9781600865800.0220.0243

26. Kuznetsov, M., Alekseev, V., Dorofeev, S.: Comparison of critical conditions for DDT in regular and irregular cellular detonation systems. Shock Waves **10**, 217-223 (2000). https://doi.org/10.1007/s001930050009

27. Dorofeev, S., Sidorov, V., Kuznetsov, M., Matsukov, I., Alekseev, V.: Effect of scale on the onset of detonations. Shock Waves **10**, 137-149 (2000). https://doi.org/10.1007/s001930050187

28. Plaster, M.J., McClenagan, R., Benz, F.J., Shepherd, J.E., Lee, J.H.S.: Detonation of cryogenic gaseous hydrogen-oxygen mixtures. In: Leyer, J.-C., Borisov, A.A., Kuhl, A.L., Sirignano, W.A. (eds.) Dynamics of Detonations and Explosions: Detonations. Progr. in Astro. and Aero, AIAA, vol. 133, pp. 37-55 (1991). https://doi.org/10.2514/5.9781600866067.0037.0055
29. Guirao, C., Knystautas, R., Lee, J., Benedick, W., Berman, M.: Hydrogen-air detonations. Symp. (Int.) Combust. **19**(1), 583–590 (1982). https://doi.org/10.1016/S0082-0784(82)80232-4

30. Rodriguez, V., Monnier, V., Vidal, P., Zitoun, R.: Non-dimensionalized distances and limits for the transition of deflation to detonation. Shock Waves **32**, 417–425 (2022). https://doi.org/10.1007/s00193-022-01088-0

31. Monnier, V., Rodriguez, V., Vidal, P., Zitoun, R.: An analysis of three-dimensional patterns of experimental detonation cells. Combust. Flame **245**, 112310 (2022). https://doi.org/10.1016/j.combustflame.2022.112310

32. Monnier, V., Rodriguez, V., Vidal, P., Zitoun, R.: Three-dimensional dynamics of detonation cells in linearly diverging channels: experimental analysis of the cross-sectional shape and a detonation-shock dynamics interpretation. Exp. Fluids **65**, 154 (2024). https://doi.org/10.1007/s00348-024-03893-y

33. Boulal, S.: Comportements dynamiques de la détonation dans les écoulements des compositions gazeuses non-uniformes. Ph.D. thesis, École Nationale Supérieure de Mécanique et d'Aérotechnique (2017) (in French)

34. Ciccarelli, G., Boccio, J.: Detonation wave propagation through a single orifice plate in a circular tube. Symp. (Int.) Combust. **27**(2), 2233–2239 (1998). https://doi.org/10.1016/S0082-0784(98)80072-6

35. Vasil'ev, A.A.: Diffraction of multifront detonation. Comb. Expl. and Shock Waves **24**, 92–99 (1988). https://doi.org/10.1007/s00193-012-0409-4

36. Pantow, E., Fischer, M., Kratzel, T.: Decoupling and recoupling of detonation waves associated with sudden expansion. Shock Waves **6**, 131–137 (1996). https://doi.org/10.1007/BF02510993

37. Sun, H., Kawasaki, A., Itouyama, N., Matsuoka, K., Kasahara, J.: Experimental study on detonation-diffraction reflection point distances in hydrogen and gaseous hydrocarbon reactive systems. Combust. Flame **245**, 112329 (2022). https://doi.org/10.1016/j.combustflame.2022.112329

38. Sorin, R., Zitoun, R., Khasainov, B., Desbordes, D.: Detonation diffraction through different geometries. Shock Waves **19**, 11–23 (2009). https://doi.org/10.1007/s00193-008-0179-1

39. Polley, N.L., Egbert, M.Q., Petersen, E.L.: Methods of re-initiation and critical conditions for a planar detonation transforming to a cylindrical detonation within a confined volume. Combust. Flame **160**(1), 212–221 (2013). https://doi.org/10.1016/j.combustflame.2012.09.017

40. Yousefi-Asli, V., Lau-Chapdelaine, S., Ciccarelli, G.: Novel weak detonation initiation from normal shock reflection in square cross-section shock tubes. Combust. Flame **266**, 113502 (2024). https://doi.org/10.1016/j.combustflame.2024.113502

41. Peraldi, O., Knystautas, R., Lee, J.: Criteria for transition to detonation in tubes. Symp. (Int.) Combust. **21**(1), 1629–1637 (1988). https://doi.org/10.1016/S0082-0784(88)80396-5

42. Poroshyna, Y., Loiseau, J., Lau-Chapdelaine, S.S.-M., Ciccarelli, G.: 3D effects of detonation re-initiation after diffraction at a back-facing step. Proc. Combust. Inst. **40**(1), 105325 (2024). https://doi.org/10.1016/j.proci.2024.105325

**Publisher's Note** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.