The present numerical simulations were performed within the open-source computational framework OpenFOAM-6, employing the detonationFoam solver originally developed by Sun et al. (2023) on the basis of the parent solver rhoCentralFoam (Greenshields et al., 2010). This open-source solver has been extensively utilized for modeling compressible, multicomponent reactive flows and gaseous detonations (Sun et al., 2022). Structurally, detonationFoam integrates two main modules: a Euler-equation solver (detonationEulerFoam) and a Navier–Stokes-equation solver (detonationNSFoam). Depending on the adopted diffusion formulation, the latter is further divided into detonationNSFoam_mixtureAverage and detonationNSFoam_Sutherland. In the present work, the Sutherland-based version was employed because it offers greater computational efficiency while producing results that remain consistent with those from the mixture-averaged formulation. ∂ ρ ∂ t + ∇ · ρ V = 0 ∂ ∂ t ρ V + ∇ · ρ V V = − ∇ P + ∇ · τ ∂ ρ ∂ t ρ E + ∇ · ρ E + P V = − ∇ · q + ∇ · V · τ ∂ ∂ t ρ Y k + ∇ · ρ V + V k ′ Y k = ω k k = 1 , ⋯ , N S − 1 (1)

The prototype rhoCentralFoam serves as a canonical foundation for many subsequent detonation and combustion solvers—such as RYrhoCentralFoam (Zhang et al., 2020) and rhoCentralRfFoam (Gutiérrez Marcantoni et al., 2017)—both widely applied in supersonic combustion research. The governing equations implemented in this solver can be expressed in conservative form, as given in Equation 1, where ρ denotes the mixture density, V = (u, v, w) is the velocity vector, P is the static pressure, and Y _k, V k ′ , and ω _k represent, respectively, the mass fraction, diffusion velocity, and production rate of the kth species. The total number of chemical species is denoted by N _s. The viscous stress tensor τ is determined according to the Stokes hypothesis, while the total energy E and heat flux q follow their corresponding conservation equations together with the ideal gas law for the reactive mixture. The associated constitutive relations and thermodynamic closure are summarized in Equation 2. τ = μ ∇ V + ∇ V T − 2 3 I ∇ · V P = ρ R T W ― q = − λ ∇ T + ρ ∑ k = 1 N s Y k h k V k ′ E = − P ρ + ∑ k = 1 N s Y k h k (2)

In these equations, μ is the mixture’s dynamic viscosity, I the identity tensor, R = 8.314 J mol^-1 K⁻¹ the universal gas constant, W ¯ the mean molar mass, λ the thermal conductivity, and h _k the specific enthalpy of the kth species. Thermophysical properties are evaluated from the JANAF polynomial database, where the specific heat at constant pressure (C _p) and enthalpy (h) are obtained through the JANAF functional forms (Burcat and Ruscic, 2005). The elementary reaction rate constant (K) follows the Arrhenius relation (Lee, 2008), parameterized by the pre-exponential factor A, temperature exponent β, and activation energy E _a. These relations are given in Equation 3. C p R = ∑ n = 0 4 a n T n h R T = ∑ n = 0 4 a n T n n + 1 + a n , 5 T K = A T β e − E a R T (3) where a _n is the auxiliary input parameter provided by the thermodynamic files, E _a is the activation energy, A is the pre-exponential factor, and β is the temperature exponent. For numerical treatment, the Strang splitting technique (Strang, 1968) is used to decouple convection and reaction processes. The second-order MUSCL scheme (Hussaini et al., 2012)reconstructs face-centered variables from cell-centered data, and the HLLC-P Riemann solver (Xie et al., 2019) computes the convective fluxes. Chemical kinetics are integrated using the Seulex ODE solver (Yang et al., 2019), while the explicit first-order Euler method advances the solution in time. The adaptive time-step control ensures that the Courant number remains below 0.02, resulting in a temporal resolution Δ t – ≈ 10^–12 – 10^–10 s. To describe chemical reactions, a reduced hydrogen–oxygen mechanism containing 11 species and 31 elementary reactions (Hong et al., 2011) is adopted. This mechanism has been validated over a broad pressure range (100 kPa – 10 MPa) and is therefore well suited for the present detonation simulations.

The computational model is shown in Figure 1. The double-bend duct consists of an inlet, a cavity, and an outlet. The inlet duct has dimensions of 20 mm × 100 mm, while the cavity height is fixed 120 mm and the cavity width is set to 20mm, 60mm, 100mm and 160 mm to represent the subcritical mode, critical mode, and supercritical mode of detonation propagation, respectively. The total streamwise length of the computational domain is 400 mm. The computational domain is two-dimensional, and this choice of dimensionality helps reduce computational cost and optimize resource utilization. The entire computational domain is filled with a premixed hydrogen-air mixture with a molar ratio of 2:1:3.76. The initial pressure P ₀ is 6670 Pa and the initial temperature T ₀ is 300 K. At the left end of the pre-detonator, there are three hot spots, each with a pressure of 1 MPa and a temperature of 2000 K. Both the inlet and outlet boundaries are treated with outflow conditions, whereas the top wall, bottom wall, upstream cavity wall, and downstream cavity wall are modeled as adiabatic, no-slip boundaries.

To verify the reliability of the numerical simulations and assess grid convergence, detonation propagation in the pre-detonator was simulated using grid resolutions of Δx = 50 μm, Δx = 25 μm, and Δx = 12.5 μm. Figure 2a presents the temporal pressure histories recorded by several pressure probes for the Δx = 25 μm case. The computed detonation velocity V _D = 1973 m/s, which agrees well with the Chapman–Jouguet (CJ) detonation velocity V _CJ = 1975 m/s predicted by CJ theory (McBride and Gordon, 1992), demonstrating the reliability of the present numerical model. Figure 2b compares the pressure profiles obtained under different grid resolutions. As shown, the grid with Δx = 50 μm fails to adequately resolve the rapid transient evolution and peak pressure during triple points collisions. Noticeable discrepancies in peak pressures are observed when compared with the Δx = 25 μm and Δx = 12.5 μm grids. In contrast, the peak pressures predicted by Δx = 25 μm and Δx = 12.5 μm are nearly identical, indicating that grid convergence has been achieved. Therefore, a three-level refinement strategy is adopted to ensure sufficient accuracy while reducing computational cost and overall runtime. Based on the grid-convergence results, the resolution of Δx = 25 μm is considered adequate to capture the essential detonation features and is consequently employed in all simulations throughout this study.

Figure 3 presents the cellular structure of the detonation wave obtained in the present simulations. Owing to the intense shear layers generated at triple points, the associated wall traces form the characteristic diamond-shaped patterns that define cellular detonations. The resulting cell geometry and the inferred cell spacing are consistent with the experimental measurements reported in Ref (Shepherd, 2009), indicating that the current numerical framework captures the essential multi-wave dynamics governing cellular motion. Under conditions where the post-shock flow remains free of pronounced downstream perturbations, the cellular pattern exhibits quasi-periodic repeatability, and both the cell size and the overall front topology remain approximately stationary over successive cycles, as shown in Figure 3.

To compare the evolution of detonation cellular structures as the detonation wave propagates through cavities of different widths, numerical soot foils were constructed using the maximum pressure in history, as shown in Figure 4. For a cavity width of w = 20 mm (Figure 4a), the detonation wave entering from the inlet duct undergoes strong reflection at the downstream wall. Owing to the narrow cavity, part of the reflected wave propagates upstream into the inlet duct and obscures the cellular tracks there, while the remaining portion undergoes multiple reflections within the cavity, maintaining high pressure and leaving a dark region on the soot-foil. After entering the outlet duct, well-defined cellular structures emerge immediately, with a darker coloration than that of the initial detonation, indicating that the detonation strengthened after passing through the narrow cavity. When the cavity width increases to w = 60 mm (Figure 4b), the detonation weakens during reflection and no longer induces reverse flow into the inlet duct. The major high-pressure region becomes concentrated along the downstream wall, where a cellular structure develops, reflecting complex triple points merging and wave-system evolution. Meanwhile, an expansion wave induced low-pressure region with lighter coloration appears at the cavity center. The cell vanishes immediately upon entering the outlet duct, indicating detonation failure. The residual high-temperature products subsequently ignite the premixed gas in the outlet duct, causing flame acceleration along the wall. After multiple reflections, the flame re-couples with the shock front, and cellular structures reappear. With a further increase in cavity width to w = 100 mm (Figure 4c), expansion waves continue to act during diffraction, leading to the disappearance of most high-pressure regions. Upon entering the outlet duct, the flame becomes decoupled from the leading shock, preventing the formation of distinct cellular tracks. When the cavity width reaches w = 160 mm (Figure 4d), the fully developed diffraction process causes the remaining high-pressure regions to vanish entirely. This indicates not only complete flame–shock decoupling but also a substantial reduction in shock pressure due to expansion waves, thereby resulting in detonation quenching.

Comparison of the numerical soot-foil images for different cavity sizes reveals the following: in narrow cavities, multiple reflections occur, and the localized hot spots generated by reflected waves help sustain the detonation, enabling its transmission into the outlet duct. As the cavity width increases, the prolonged influence of expansion waves generated during diffraction gradually weakens the reflected shock at the downstream wall, eventually leading to detonation extinction. The soot-foil image for w = 60 mm further demonstrates that complex triple points dynamics arise during diffraction, and because soot-foil images record only the historical peak pressure, they cannot fully resolve the transient evolution. Therefore, a more detailed analysis of detonation evolution within the cavity and the outlet duct is needed.

To elucidate the evolution of detonation wave structures within double-bend ducts of varying cavity widths, numerical schlieren during detonation propagation are presented in Figure 5. When the cavity width is w = 20 mm (Figure 5a), the detonation diffracts upon exiting the inlet duct (Pintgen and Shepherd, 2009). The expansion fan propagates upward along the leading shock front, causing the decoupling between the shock and the high-density-gradient reaction zone (Figure 5(a₁)). Subsequently, the detonation front reflects from the downstream cavity wall, producing a high-density region near the upper wall, while a Mach reflection forms along the downstream cavity wall (Figure 5(a₂)). Part of the reflected wave travels downward and merges with the leading shock, and another part reflects from the upstream cavity wall. The resulting localized hot spot further strengthens the detonation, creating an extended region of elevated density along the front (Figure 5(a₃)–(a₄)). The wave eventually undergoes a second diffraction event near the outlet duct (Figure 5(a₅)). When the cavity width increases to w = 60 mm, the prolonged diffraction process leads to near extinction of the detonation before reaching the downstream wall. Only a weak reflection appears in the upper-right corner of the cavity (Figure 5(b₁)–(b₂)). After reflection, however, the detonation front propagating along the cavity wall rapidly pursues and overtakes the decayed leading shock, setting up a stable triple-wave configuration (Figure 5(b₃)–(b₄)). The Mach stem then continues to move along the downstream wall toward the outlet duct, while the triple point drifts toward the upstream wall, continuously igniting the unburned mixture in the cavity (Figure 5(b₅)).

For larger cavities (Figures 5c,d), the detonation is quenched before reaching the downstream wall (Figure 5(c₁), (d₁)). Although the reflection from the downstream wall at w = 100 mm can still generate a high-density shock front, its large separation from the expansion front causes the reflected wave to decay again during the catch-up process, and the flow undergoes another diffraction near the outlet duct (Figure 5(c₁)–(c₅)). At w = 160 mm, the increased transverse dimension allows the detonation front moving in the +x direction and the shock front moving in the–y direction to strike the downstream and bottom cavity walls nearly simultaneously. Both reflections then propagate toward the outlet duct (Figure 5(d₁)–(d₃)). Ultimately, the reflection along the downstream wall reaches the outlet first and diffracts, while colliding with the reflection propagating along the bottom wall (Figure 5(d₄)–(d₅)).

To elucidate how cavity width governs the attenuation, reinforcement, and possible recovery of the detonation front during diffraction, Figure 6 presents pressure contours for different cavity widths. For w = 20 mm, the detonation wave reflects from the downstream cavity wall (Figure 6(a₁) – (a₂)) and subsequently undergoes alternating reflections between the upstream and downstream walls. These repeated reflections continuously strengthen the shock, allowing the leading front to maintain a high-pressure state (Figure 6(a₃) – (a₄)) until diffraction occurs at the outlet duct. When the cavity width increases to w = 60 mm, the reflected detonation fails to reach the upstream cavity wall due to the larger transverse scale (Figure 6(b₁)–(b₂)). Nevertheless, the wave propagating along the downstream wall establishes a stable, self-sustained triple-wave configuration, enabling the leading shock to maintain a persistent high-pressure profile (Figure 6(b₃)–(b₅)).

For w = 100 mm, the detonation is completely quenched before reaching the downstream wall. Although the reflected shock still forms a localized high-pressure spot near the wall (Figure 6(c₁)–(c₂)), the prolonged decoupling leaves the flame front far behind the leading shock, preventing re-initiation. The high-pressure spot then travels along the downstream wall, followed by the flame front, which reflects from the wall and attempts to catch up with the leading shock. During this pursuit, however, the flame is continuously weakened by the opposing expansion waves, causing the pressure to drop (Figure 6(c₃)–(c₄)). When the cavity width is further increased to w = 160 mm, the rarefaction region behind the expansion wave becomes fully developed. As a result, the reflected shock from the downstream wall is unable to generate a significant high-pressure region (Figure 6(d₁)–(d₃)). Two reflected shocks instead propagate separately along the bottom wall and the downstream cavity wall toward the outlet duct (Figure 6(d₄)–(d₅)).

To illustrate the evolution of flame–shock coupling, the temperature fields for different cavity widths are shown in Figure 7. For w = 20 mm, the detonation entering the cavity generates a decoupled region near the inlet duct due to the expansion induced by diffraction (Figure 7(a₁)), which is consistent with the canonical diffraction structure reported by Pintgen and Shepherd (Pintgen and Shepherd, 2009). After reflecting from the downstream cavity wall, the resulting strong reflected detonation wave propagates downward and gradually merges with the previously decoupled leading shock (Figure 7(a₂)–(a₃)), until a second diffraction occurs at the outlet duct (Figure 7(a₄)–(a₅)). For w = 60 mm, a pronounced flame–shock decoupling is observed when the detonation reflects from the downstream cavity wall (Figure 7(b₁)–(b₂)). The reflected wave, however, rapidly encounters the flame front and behaves as a localized hot spot, thereby re-initiating the detonation (Li et al., 2021) (Figure 7(b₃)). The re-established detonation then propagates along the downstream wall, overtakes the leading shock, and undergoes diffraction again at the outlet duct (Figure 7(b₄)–(b₅)).

For w = 100 mm, the detonation becomes fully decoupled prior to reaching the downstream wall (Figure 7(c₁)). Such complete decoupling is characteristic of near-critical or failing diffraction, in which the rarefaction field suppresses transverse-wave support and substantially reduces the likelihood of prompt re-ignition (Jun et al., 2024). The high-pressure region produced by the reflected shock does not intersect the flame front and therefore lacks sufficient energy to trigger re-initiation (Figure 7(c₂)). Only after the flame front reaches the downstream wall and reflects does the combination of its elevated temperature and the increased pressure behind the reflected shock produce a new hot spot that propagates along the wall toward the outlet duct (Figure 7(c₃)–(c₄)). At this stage, however, the hot spot is generated after the leading shock has already undergone substantial rarefaction; consequently, the newly formed detonation kernel is not adequately supported while traversing the expansion field and is forced to decouple again (Jun et al., 2024), ultimately failing to re-initiate (Figure 7(c₅)). For w = 160 mm, the decoupled shock front arrives at both the bottom wall and the downstream cavity wall nearly simultaneously (Figure 7(d₁)–(d₃)). Under the sustained influence of the expansion wave, the flame front remains far behind the shock. Even when the two reflected shocks collide near the outlet duct, the flame front has still not reached the downstream wall and thus cannot couple with the shock to form a re-initiation hot spot (Figure 7(d₄)–(d₅)).

To quantitatively assess the influence of cavity width on detonation propagation, temperature probes were placed near the downstream cavity wall and the upper wall, and the resulting temperature-time histories are plotted in Figure 8. As shown, at t ≈ 80 μs the detonation front in the w = 20 mm case has already reached the downstream wall. Owing to the confinement imposed by the narrow cavity, the wave reflected from the wall immediately encounters the hot products residing near the inlet duct, producing a brief stagnation-like interaction. This process manifests as a short-lived quasi-plateau in the temperature trace. As the reflected detonation travels downward, the hot front retreats from the monitoring location and the local temperature decreases; subsequently, continued post-shock heat release drives a sustained temperature rise as the mixture approaches complete reaction. For w = 60 mm, a hot spot is generated by detonation reflection at the downstream wall. In the enlarged cavity, however, the fully reacted gases behind the wave are decelerated by the expansion field, and the high-temperature zone does not persist at the downstream wall. Consequently, only a transient temperature peak is observed, followed by a decay as the wave system moves away. Once reactions in the post-shock region are completed, the temperature recovers to a level comparable to that in the w = 20 mm case.

For w = 100 mm, the detonation becomes fully decoupled before reaching the downstream wall. The leading shock therefore reflects first, yielding a relatively low temperature peak, and then propagates along the downstream wall toward the outlet duct. Because no ignition kernel with sufficient strength is established, the downstream wall region remains filled with compressed but largely unreacted mixture. When the delayed high-temperature products subsequently impinge on the downstream wall, these gases ignite and the temperature increases. Nevertheless, the absence of successful re-initiation implies a lower overall burning rate than in the w = 20 mm and w = 60 mm cases, and complete consumption is not achieved until t ≳ 300 μs. When the cavity width is further increased to w = 160 mm, the detonation remains fully decoupled. The reacted products are strongly decelerated by the long-lived expansion field, and the reflected shock compresses and effectively traps the high-temperature gases within the cavity rather than promoting near-wall re-ignition. As a result, the temperature in the vicinity of the cavity wall no longer exhibits a sustained rise.

Analysis of the diffraction processes at different cavity widths reveals that detonation re-establishment within the cavity is governed by a common mechanism: the leading shock forms a high-pressure spot upon reflecting from a wall, and this high-pressure region couples with the flame front to generate a new hot spot, thereby enabling re-initiation. For w = 20 mm, the very short diffraction time results in only partial decoupling, and the narrow cavity forces the detonation to undergo multiple reflections between the walls. These repeated reflections continuously produce high-pressure spots that help sustain the detonation strength. Meanwhile, the interaction between reflected waves and the flame front enhances mixing in the post-shock region, ensuring complete reaction and raising the temperature further. Such strong back-and-forth reflections also drive the detonation into an overdriven state, accelerating the wave and maintaining full coupling between the flame front and the leading shock. For w = 60 mm, the increased cavity width causes the detonation to approach near extinction as it reaches the downstream wall. However, the reflected shock immediately encounters the adjacent flame front, forming a hot spot that successfully re-initiates the detonation. The subsequently arriving flame front supplies energy to this hot spot, and during its pursuit of the leading shock, a stable triple-wave structure forms, enabling the detonation to transition into a self-sustained propagation mode.

When the cavity width increases to w = 100 mm, the shock front and flame front become completely decoupled. Before the flame front reaches the downstream wall, the high-pressure spot generated by shock reflection has already moved far away. Although reflection of the flame front from the wall can still generate a hot spot, this ignition kernel is rapidly quenched by the strong rarefaction field during propagation, preventing successful re-initiation within the cavity. For sufficiently large cavities, such as w = 160 mm, the detonation is entirely extinguished. When the reflected shock traveling upstream encounters the flame front, the leading shock has already existed through the outlet duct, making re-initiation impossible.

To clarify how cavity width regulates the re-ignition dynamics and the subsequent evolution of triple point structures as the detonation enters the outlet duct, Figure 9 presents the corresponding temperature fields for different cavity widths. According to previous studies, whether re-initiation occurs is typically governed by two coupled requirements: (i) whether the shock–wall interaction can deposit sufficient energy to form a viable detonation kernel, and (ii) whether the associated transverse-wave system can survive the post-diffraction expansion process (Yang et al., 2022; Hu et al., 2024).

For w = 20 mm, the detonation undergoes diffraction at the entrance of the outlet duct. After reflecting from the bottom wall, it generates a strong transverse wave that, together with the leading shock front, forms a triple point. As this triple point propagates toward the upper wall, it rapidly consumes the unreacted pockets produced during diffraction (Figure 9(a₁)–(a₂)), resulting in complete re-ignition after reflection from the upper wall (Figure 9(a₃)–(a₄)). When the cavity width increases to w = 60 mm, unreacted pockets persist near the upstream cavity wall. Following reflection at the bottom wall, part of the detonation front propagates upward and ignites these pockets, while another portion forms a localized hot spot that moves toward the outlet duct (Figure 9(b₁)–(b₂)). Because combustion is concentrated near the upstream cavity wall at this stage, the hot spot attenuates as it attempts to catch up with the leading expansion front (Figure 9(b₃)). After the hot spot extinguishes, large Kelvin–Helmholtz (K–H) vortices develop near the flame-tip region; these vortical structures enhance mixing and facilitate ignition of the unreacted gas behind the wave (Li et al., 2021). Meanwhile, the reflected detonation—sustained by the energy release from the unreacted pockets—retains sufficient strength to reflect again from the upstream wall and subsequently propagate into the outlet duct, where it pursues the expansion-wave front (Figure 9(b₄)).

As the cavity width further increases to w = 100 mm, the hot spot extinguishes before reaching the outlet duct, and the leading shock reflected from the bottom wall fails to generate a new ignition kernel (Figure 9(c₁)). Collisions between the reflected shock and the flame front impede further flame propagation toward the bottom wall, producing extensive unreacted pockets along the bottom boundary (Figure 9(c₂)–(c₃)). As a result, the flame remains confined within the cavity, whereas only expansion-induced pressure disturbances enter the outlet duct. Because these pressure waves carry insufficient thermal energy to ignite the mixture, detonation re-initiation ultimately fails (Figure 9(c₄)). For sufficiently large cavity widths (w = 160 mm), the suppressing influence of the reflected shock becomes more pronounced. No flame region develops near the downstream wall (Figure 9(d₁)), and the flame severely weakened by strong diffraction—largely loses its propagation speed. After colliding with the reflected shock, the front flame is driven backward by a strong reverse flow and becomes nearly stationary within the cavity. This reverse flow acts as an “air cushion,” effectively isolating the flame front from the walls. Flame propagation then proceeds primarily through K–H-instability-induced roll-up along the flame boundary, which gradually consumes the unreacted pockets (Figure 9(d₂)–(d₃)). Meanwhile, the shock wave entering the outlet duct continues to propagate independently; since the flame cannot recouple with the shock front, detonation re-initiation does not occur.

To quantitatively compare re-ignition behavior in the outlet duct, a temperature monitoring line was placed along the duct centerline, and the flame position was extracted to obtain flame-speed histories, as shown in Figure 10. For w = 20 mm, diffraction occurs near t = 140 μs, during which expansion-wave effects cause temporary flame–shock decoupling and reduce the flame speed to approximately 1000 m/s. After reflection from the bottom wall, re-ignition occurs rapidly, and within approximately 50 μs, the flame speed reaches the theoretical Chapman–Jouguet (CJ) velocity V _CJ. This indicates complete recoupling of the flame and shock fronts. The strengthened shock induced by wall reflection produces a mild overdriven condition, with an overdrive factor of approximately f = 1.2. For w = 60mm, because re-ignition also occurs at the downstream cavity wall, the detonation reaches the outlet duct with a speed near V _CJ. Between t = 160μs and 280 μs, as the detonation front propagates toward the upstream wall, the flame in the outlet duct remains in a deflagration-like mode with an average velocity of approximately 1200 m/s. After t = 280 μs, the reflected detonation catches up with the flame front, triggering another re-initiation event, and the flame speed then increases toward the CJ velocity. For w = 100 mm and w = 160mm, the detonation has fully quenched before reaching the outlet duct. The flame speed fluctuates near 1200 m/s, and no successful re-initiation occurs within the outlet duct.

A comparison of the detonation evolution in the outlet duct with the corresponding flame-speed curves reveals the following. For w = 20 mm, the detonation has already been strengthened by multiple reflections within the narrow cavity before entering the outlet duct. The compact geometry keeps a high-temperature, high-pressure environment next to the outlet, supporting the triple-wave structure after reflection and enabling rapid re-initiation. For w = 60 mm, unreacted regions exist within the cavity, and the detonation, after reflecting from the bottom wall, propagates upstream to ignite the remaining unreacted mixture. The triple-wave structure formed in the outlet duct is initially separated from the high-temperature flame region inside the cavity and thus does not re-initiate at once. Only after the flame reflects again from the upstream cavity wall does it continue to chase the leading shock and eventually recouple, completing re-initiation.

For w = 100 mm and w = 160 mm, prolonged flame–shock decoupling significantly reduces flame speed. After reflecting from the walls, the weakened shock front suppresses flame propagation, effectively forming an “air cushion” between the flame front and the wall. The shock entering the outlet duct continues to propagate independently, while the flame advances only slowly via Kelvin–Helmholtz instability induced mixing, further distancing itself from the shock. As a result, the flame and shock front do not recouple, and detonation re-initiation does not occur.

Industrial explosion scenarios typically involve variations in both transition-cavity dimensions and explosion intensity. Changes in the lateral scale of the outlet duct and in the initiation energy level can simultaneously alter the post-diffraction expansion history, the survivability of the transverse-wave system, and the local strengthening induced by wall reflections. Motivated by these considerations, two sets of comparative cases were conducted based on the baseline configuration. First, with the cavity width fixed at w = 60 mm, two outlet-duct widths were examined, w _o = 40 mm and w _o = 60 mm. Second, with w = 100 mm held constant, cases with hot-spot initiation pressures of P _i = 1 MPa and P _i = 3 MPa were simulated. Figure 11 shows the corresponding evolution of the temperature fields for these cases.

When the outlet width is increased to w _o = 40 mm, the hot spot enters the outlet duct earlier (Figure 11(a₂)) and undergoes diffraction. Because a stable triple point does not develop, the decoupled gases are not sufficiently ignited while the wave system propagates toward the upstream cavity wall, leaving unreacted pockets (Figure 11(a₃)). Later reflections from the upstream cavity wall and the bottom wall generate a localized hot spot, which ignites these pockets. After a sustained reflected reinforcement is set up, the re-energized wave system penetrates the outlet duct and pursues the attenuated leading shock (Figure 11(a₄)). When the outlet width is increased to w _o = 60 mm, the hot spot and the leading shock do not couple promptly at the downstream cavity wall (Figure 11(b₁)). Instead, they meet inside the outlet duct and experience diffraction almost simultaneously (Figure 11(b₂)). Owing to the larger outlet cross-section, the expansion wave more effectively reduces the post-shock pressure and temperature, thereby weakening the transverse wave formed after reflection. After successive reflections from the bottom wall and the upstream cavity wall, the reflected wave system can still catch up with the leading shock; however, a longer recovery distance/time is required, and the reflection-induced hot spot is correspondingly weaker (Figure 11(b₃)–(b₄)).

For the w = 100 mm cavity, increasing the hot-spot initiation pressure to P _i = 1 MPa yields a markedly more stable coupling between the shock and the reaction zone, and the decoupled region formed upon entering the cavity is substantially reduced (Figure 11(c₁)). Reflection at the downstream cavity wall produces a sufficiently concentrated local strengthening that successfully triggers a hot spot. As this hot spot travels along the downstream wall, a relatively stable triple point is kept (Figure 11(c₂)–(c₃)). The wave system then enters the outlet duct; following reflection from the bottom wall, it rapidly catches up with the leading shock and recouples, achieving re-initiation and recovering a self-sustained detonation (Figure 11(c₄)). When the initiation pressure is further increased to P _i = 3 MPa, the higher-energy hot spot elevates the wavefront temperature and further reduces the decoupled region compared to the P _i = 1 MPa case (Figure 11d1). Nevertheless, the overall propagation sequence and the evolution of the key structures in both the cavity and the outlet duct remain essentially unchanged (Figure 11(d₂)–(d₄)).

These results indicate that, under the present conditions, increasing P _i primarily raises the detonation energy level and thus makes the wave more resistant to quenching, but it does not alter the characteristic propagation scenario in the double-bend geometry: diffraction-reflection-induced hot-spot formation-re-diffraction-re-reflection and catch-up. Diffraction-induced decoupling, governed by the cavity scale, persists; so, re-initiation continues to rely on localized hot spots constructed by wall reflections.

To compare the influence of outlet width on the pursuit of the leading shock and the ensuing re-initiation, Figure 12 presents the pressure evolution in the w = 60 mm configuration after the detonation reflects from the bottom wall and enters the outlet duct. When w _o = 20 mm, reflection occurs at once upon entering the outlet: one branch of the wave system generates a high-pressure region near the downstream cavity wall, while other forms a localized hot spot near the upper wall of the outlet duct. Together, these features set up a stable triple point within the outlet duct (Figure 12(a₁)). The transverse wave is continually reinforced through wall reflections, which accelerates the flame and appears as pronounced oscillations in the flame-speed history (Figure 12(a₂)). Between t = 273 μs and t = 298 μs, the flame front completes its catch-up to the leading shock and recouples, thereby achieving detonation re-initiation (Figure 12(a₃)–(a₄)). When w _o is increased to 40 mm, the enlarged outlet section prolongs the diffraction process, and the associated expansion wave causes an earlier decay of the local pressure. As a result, it becomes difficult to set up a high-pressure shock structure near the outlet comparable to that in the w _o = 20 mm case (Figure 12(b₁)). Although the transverse wave can still be strengthened gradually through successive reflections (Figure 12(b₂)), the resulting triple points stays weaker overall. Eventually, recoupling with the leading shock occurs at approximately t ≈ 292 μs (Figure 12(b₃)), after which the wave system advances toward the outlet (Figure 12(b₄)). This behaviour shows that the detonation is not completely quenched for w_o = 40 mm; rather, re-initiation is characterized by a weaker triple point and a delayed recoupling time. For w_o = 60 mm, the expansion effect is further intensified and the pressure in the flow field decreases accordingly (Figure 12(c₁)). In addition, the wider outlet reduces the frequency of transverse wave interactions with the walls and diminishes the cumulative reinforcement, making it difficult to form a triple point structure of sufficient strength (Figure 12(c₂)). Ultimately, the reflected shock and the leading shock tend to merge into a weak pressure-wave system (Figure 12(c₃)–(c₄)), and detonation re-initiation does not occur.

To quantify the effect of outlet width on re-initiation, a monitoring line was placed along the outlet-duct centerline, and the flame position was extracted to obtain flame-speed–time histories (Figure 13). For w _o = 20 mm, periodic velocity peaks are observed over 180–260 μs, reflecting repeated wall-reflection reinforcement. Near t ≈ 280 μs, re-initiation occurs and the flame speed rises abruptly, stabilizing at approximately 1,600 m/s, which indicates recovery to a self-sustained detonation. When w _o = 40 mm, the larger length scale increases the characteristic reflection time, leading to less frequent peaks. The extended diffraction history also weakens the triple point, so the flame-speed peak is generally lower than those for w _o = 20 mm. Although a catch-up recoupling event occurs at t ≈ 290 μs, producing a sustained increase in flame speed, the post-recoupling level remains noticeably below that in the w _o = 20 mm case. In the w _o = 60 mm configuration, the prolonged diffraction and expansion-induced depressurization prevent the formation of a sufficiently strong triple point. Consequently, the flame never recoupling with an effective triple point; the propagation speed remains well below that of the other two outlet widths, and re-initiation fails.

By comparing the effects of outlet width and hot-spot initiation pressure on detonation propagation, the following conclusions can be drawn. The outlet width (w _o) primarily modulates whether a triple point can be established and sustained by altering the post-diffraction expansion/attenuation time scale and “reinforcement efficiency” associated with transverse-wave reflections. As w _o increases, the hot spot is more likely to diffract prematurely near the downstream cavity wall, which accelerates the decay of local pressure and temperature and consequently weakens the reflection-induced triple point strength. In parallel, a larger transverse scale reduces the effective frequency of transverse-wave interactions with the walls, making cumulative strengthening through repeated reflections less attainable and thereby raising the threshold for re-initiation (Hu et al., 2024).

In contrast, the hot-spot initiation pressure (P _i) mainly governs the coupling strength at cavity entry. A higher P _i promotes tighter coupling between the leading shock and the reaction zone and reduces the extent of the decoupled region, ensuring that the wave system still retains sufficient intensity to trigger a hot spot upon reaching the downstream cavity wall. Nevertheless, geometry-induced diffraction decoupling does not vanish with increasing P _i. A substantial decoupled zone persists within the double-bend cavity; therefore, elevating the initiation overpressure alone is insufficient to enable “spontaneous” re-ignition inside the cavity. Instead, detonation re-establishment remains controlled by localized hot spots generated through wall reflections and the subsequent catch-up/recoupling process in the outlet section (Tang et al., 2025; Li et al., 2021).