Numerical Simulation of the Transition to Detonation in a Hydrogen–Air Mixture Due to Shock Wave Focusing on a 90-Degree Wedge ^†

Abstract

This study numerically explores the initiation of detonation through shock wave reflection and focusing on a 90-degree wedge in varying mixtures of hydrogen–air. The simulations were conducted using the ddtFoam code, an integral part of the OpenFOAM open-source Computational Fluid Dynamics (CFD) package of density-based code for solving the unsteady, compressible Navier–Stokes equations. The simulation results unveil three potential outcomes in the corner post-reflection: deflagrative ignition in the corner, deflagrative ignition with intermediate transient phases leading to a delayed transition to detonation in the trailing combustion zone close to the apex of the wedge, and ignition with an immediate transition to detonation, resulting in the formation of a detonation wave in the corner tip. In the experimental investigation, the transition velocity for the stoichiometric mixture stood at approximately 719 m/s. In contrast, the numerical simulation indicated a transition velocity of 664 m/s for the same stoichiometric mixture, reflecting a 5.5% decrease in velocity. Such an underestimation level of 5–8% by the simulation results was observed for mixtures of 25–45% H₂ in air.

1. Introduction

Hydrogen has gained significant interest across diverse industries as a sustainable and environmentally friendly alternative to conventional energy sources. However, a significant safety concern in hydrogen production, transportation, storage, and consumption facilities arises from the potential for unforeseen explosions. In the event of a hydrogen leak in an enclosed space, flammable mixtures may form, posing a serious risk of unintended deflagration upon the slightest ignition source. The severity of the consequences depends on the propagation speed of the deflagration. The hazardous potential of detonation, particularly during the unsteady transition process, far surpasses that of deflagration. Consequently, substantial research has been dedicated to examining the characteristics of accelerating deflagrations and identifying conditions leading to successful deflagration-to-detonation transition (DDT) in industrial settings, aiming to address safety concerns [1].

Research on DDT has an extensive development history, encompassing various methods aimed at accelerating flame propagation by enhancing turbulence. These methods include Shchelkin spirals, perforated plates, multiple barriers, and intricate geometries. Both experimental [2,3,4,5,6,7,8,9,10] and numerical research [3,4,5,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25] highlight the significance of channel geometry, obstruction ratios, reflecting walls, and particularly corner structures that concentrate shock waves, creating localized regions of high pressure and temperature, conducive to initiating detonation in combustible mixtures.

Various investigations have explored shock focusing, employing numerical simulations and experimental approaches. Gelfand et al. [10] experimentally investigated detonation and deflagration using different focusing elements like wedges, semi-cylinders, and parabolas. They observed so-called mild and strong ignition inside the reflector cavity, with the latter resulting in direct detonation initiation inside and the former resulting in deflagration only. The study also found a correlation between detonation initiation distance and the incident shock wave’s Mach number. Bartenev et al. [16] investigated combustion initiation via shock focusing on various reflectors, aiming to identify the critical incident shock wave (ISW) Mach number, mixture parameters, and reflector geometries for different combustion regimes. Their simulations showed that detonation initiation is primarily driven by energy release in the gas dynamics focus of the reflector cavity after triple-point collision, with critical ISW Mach numbers being key triggers.

Wintenberger et al. [26] carried out numerical simulations involving planar parabolic reflectors, where they varied the incident shock wave Mach number and depth of the reflector. Their findings provided insights into pressure amplification in specific reflection types, indicating a small, compressed region between merging triple points and the reflected shock. Buraczewski and Shepherd [6] expanded our knowledge through experiments on detonation initiation via shock wave focusing in a parabolic reflector, identifying various combustion scenarios, such as no reaction, flame, and transition to detonation, based on the reflector depth, incident wave strength, and gas composition. They found direct detonation initiation for mixtures with minimal dilution and in deeper reflectors, suggesting further research directions, including examining the initial pressure influence and exploring different reflector shapes.

The work by Smirnov et al. [18,21,27] studied detonation initiation via shock wave focusing inside cones and wedges using numerical simulations and experiments. They validated a 3D transient model for reacting gas, and they identified various flow scenarios based on incident shock wave intensity, including shock wave reflection with or without combustion, detonation wave formation, and transient regimes with deflagration-to-detonation transitions. Zhang et al. [28] explored shock wave focusing using different concave profiles, identifying weak and strong ignition modes dependent on the incident shock wave velocity, with critical values between 740 and 780 m/s for various reflectors in a stoichiometric methane–oxygen–argon mixture. They observed a variation in ignition delay time and maximum pressure among reflectors, noting shorter delays and higher pressures in hemispherical reflectors in the weak ignition mode, but the reverse in the strong ignition mode. Li and Zhang [29] researched different ignition modes and detonation initiation in methane-based mixtures using wedge reflectors due to shock focusing. They identified three ignition modes: peak local ignition mode (PLIM), boundary ignition mode (BIM), and strong ignition mode (SIM). In PLIM, flame originates at the apex of the reflector, with slow propagation and a concentrated flame area. In BIM, flames start at the apex and rapidly move along the tube wall, then slowly spread to the central unburned area. SIM involves oblique shock waves generating detonation via their interaction after reflection from wedged walls. Yang and Zhang [30] investigated ignition modes and detonation initiation in wedge reflectors via shock wave focusing using numerical simulations and experiments. They identified three ignition modes—deflagration, quasi-detonation, and direct detonation—with increasing shock wave intensity, emphasizing the importance of new hot spot formation and transverse waves in sustaining detonation, with Richtmyer–Meshkov instability playing a crucial role.

Finally, Rudy [31,32] presents experimental observations of ignition modes by shock wave focusing in a 90-degree wedge using varying mixtures of H₂ in air under an initial pressure of 1 bar, aiming to identify the critical conditions for detonation transition. Three ignition modes were identified: weak ignition followed by deflagration with an ignition delay time (IDT) exceeding ~1 µs, strong ignition with an instant transition to detonation, and deflagrative ignition with a delayed transition to detonation. Detonation initiation required specific shock wave velocities, with the lowest observed at ~715 m/s for stoichiometric mixtures. The study revealed a strong correlation between IDT and shock wave velocity and demonstrated that detonation is triggered only when a specific velocity threshold is met.

The present study aimed to extends the understanding of the transition to detonation in the specific context of a 90-degree wedge for different concentrations of hydrogen in air mixtures at an initial pressure of 1 bar, mirroring the experimental conditions set by Rudy [31], utilizing numerical simulations employing the ddtFoam code, which provides the capability to explore these phenomena in a controlled and systematic manner, allowing for the isolation and analysis of specific variables that contribute to DDT. Additionally, this paper serves as an extended version of our work published for the 2023 International Conference on Hydrogen Safety [33], expanding upon the findings with a more in-depth analysis and further discussion of key phenomena.

The selection of a 90-degree wedge geometric configuration enables the effective focusing of shock waves, creating localized high-pressure and high-temperature conditions favorable to detonation initiation. This study sought to identify critical parameters, such as the concentration of hydrogen and incident shock wave velocity, which influence the ignition modes and the transition to detonation.

The insights gained from this study have practical implications for enhancing safety in industrial applications where hydrogen is used or produced. By understanding the conditions under which detonation can be initiated and the mechanisms that govern this transition, it is possible to develop strategies to mitigate the risks associated with hydrogen explosions.

2. Model Description

A 2D numerical representation of the experiments conducted by Rudy [31] is illustrated in Figure 1. The computational domain represents a cross-sectional view of the final segment of a rectangular shock tube with dimensions 0.11 × 0.11 × 2 m, used in the experiments. Within the tube, a customized end section was integrated, creating a cavity with a 90° opening angle. The tube was filled with a hydrogen–air mixture. The numerical domain adhered to a conventional shock tube configuration, encompassing high-pressure (8–15 bar, 298 K) and low-pressure (1 bar, 298 K) sections, generating a planar shock wave directed toward the wedge at the tube’s terminus with a specific velocity. The shock wave reflected from the wedge, and the associated focusing processes were analyzed. Before the simulation initiation, the gases were at rest, and the shock wave was instigated at the simulation outset, possessing a leading velocity ranging from approximately 600 to 900 m/s. In Rudy’s experiments [31], the leading shock wave was generated through ignition and flame acceleration in the initial tube segment. The subsequent part of the tube was employed to stabilize the leading shock wave.

The numerical simulations focused on only half of the tube cross-section, introducing symmetry conditions at the axial plane to reduce the computational cost and maintain accuracy. The mesh comprises hexahedral and prism shapes, partitioned into two segments: one structured and orthogonal, and the other unstructured. The mesh encompasses approximately 50,300 cells, with cell sizes ranging from a minimum of 0.3 mm to a maximum of 1.6 mm. The finest mesh resolution is concentrated around the wedge walls, ensuring an accurate representation of the critical region near the apex. Figure 2a illustrates the geometry of the computational domain and the overall mesh structure, while Figure 2b provides an enlarged view of the mesh around the wedge apex. This level of mesh resolution surpasses what Ettner’s study [34,35,36] established as the minimum required to accurately predict global phenomena, such as flame speeds and pressure loads, without the need to resolve all intricate mechanisms on smaller scales.

A total of 60 simulations were executed, encompassing various hydrogen in air concentrations ranging from 15% to 50%, corresponding to an equivalence ratio of 0.42 to 2.91. The initial temperature and pressure were set to 293 K and 101,325 Pa, respectively, while the walls were assumed to be non-slip and adiabatic. To replicate the experimental setup, the numerical model incorporates a set of temperature and pressure sensors (PS1–PS5). These sensors were positioned at locations similar to those in the experimental setup but adjusted to match the scaled length of the simulated domain (

Table 1. Sensor position (m).

	PS1	PS2	PS3	PS4	PS5
Experiments	0.461	0.941	1.261	1.581	1.932
Simulation	1.394	1.523	1.652	1.797	1.928

). This adjustment was necessary because the computational domain is scaled differently from the experimental setup, and placing sensors at the exact experimental positions would have resulted in some sensors being positioned outside the computational domain. The scaled positioning ensured that all sensors remained within the computational domain while preserving the ability to compare simulation results with experimental data accurately.

3. Numerical Methodology

The ddtFoam solver, part of the OpenFOAM^® toolbox (version 2.1) developed by Ettner et al. [34,35,36], was used for the numerical simulations. This solver is specifically designed to simulate complex phenomena such as flame acceleration, detonation propagation, and deflagration-to-detonation transition (DDT) within a single framework. It solves equations that describe the behavior of compressible fluids and uses advanced methods to accurately capture shock wave dynamics while minimizing numerical errors. By utilizing the unsteady, compressible Navier–Stokes equations with the HLLC (Harten–Lax–van Leer–Contact) scheme for shock capturing, ddtFoam ensures the accurate determination of convective fluxes, resulting in precise shock propagation speeds and reduced dissipation at discontinuities compared to standard schemes like the PISO (Pressure-Implicit with Splitting of Operators) scheme [37]. The solver displays the computational efficiency by describing both deflagration and detonation using a reaction progress variable. Combustion is modeled with the Weller gradient combustion model [38], and the autoignition delay time using O’Conaire’s reaction scheme mechanism [39]. To optimize computational resources, the solver utilizes a pre-calculated table of ignition delay times (IDTs) generated with Cantera software (version 2.4). During simulation, the solver references this IDT table and compares it with the simulated fluid residence time in specific conditions. If the residence time exceeds the pre-calculated IDT value, the solver activates the second source term in the reaction progress transport equation, significantly reducing the numerical cost of the simulation. The solver is also versatile, capable of handling complex geometries and coarse computational grids while maintaining reliability.

Obtaining the numerical solution involves solving the Navier–Stokes equations, along with energy and species conservation equations, employing the finite-volume method on a computational grid. For turbulence modeling, the k-ω SST (K-omega Shear Stress Transport) model is used. This model combines the strengths of two widely used turbulence approaches, making it suitable for accurately capturing flow behavior near walls and in free-stream regions.

It is important to note that the simulations are conducted under idealized conditions, which simplify heat transfer effects and boundary layer dynamics and exclude three-dimensional effects such as localized flow instabilities. These simplifications allow the study to focus on key phenomena like shock propagation and ignition dynamics while maintaining computational efficiency. Further details of the solver setup, mathematical models, and numerical methods are described in the works by Ettner et al. [34,35,36].

4. Results and Discussion

The numerical outcomes were processed in a similar manner to the experiments, using the Time of Arrival method to extract shock wave velocity profiles along the tube from pressure sensor data recorded. Three ignition modes were observed, similar to the experimental results [18,29,30,31]. The first mode observed was the Deflagrative Ignition Mode (DIM), featured deflagrative ignition in the corner, similar to the “weak” ignition mode, occurring at lower velocities in [10]. The second mode, Delayed Detonation Ignition Mode (DDIM), involved deflagrative ignition with a transition to detonation delayed by 301–647 µs after reflection. The third, Immediate Transition to Detonation Mode (ITDM), similar to ‘strong’ ignition as described in [10], encompassed a direct transition to detonation in the wedge, with an ignition delay time lower than 12 µs.

In the Deflagrative Ignition Mode, the flame initiation process is distinguished by its slow propagation and concentration at the apex of the wedge reflector, as depicted in Figure 3. Another characteristic of DIM is a single pressure peak recorded by sensor PS5, followed by a decay to a relative constant pressure post-reflection. Example pressure readings are presented in Figure 4, where the pressure profiles obtained from the simulation are compared to the experimental data. The comparison in Figure 4 reveals a strong agreement in the general behavior of the shock wave and flame dynamics between the simulation and the experiments, with both profiles following a similar pattern and reaching nearly identical pressure levels. However, two notable observations emerge from this comparison. First, a small discrepancy in the timing of the post-shock pressure can be observed across sensors PS1–PS4, which can be attributed to differences in shock velocities and the adjusted sensor positions in the simulation domain relative to the experimental setup. Despite this temporal offset, the simulation achieves the same pressure levels as the experiment, ~0.3 MPa. Second, post-reflection, the simulation shows the pressure stabilizing at 0.9–1 MPa, whereas in the experimental data, the pressure falls to 0.4–0.5 MPa. This disparity may arise from additional energy dissipation mechanisms in the experiments, such as heat losses to the walls, non-ideal effects in the shock-tube setup, or unaccounted mixing phenomena in the experimental environment. This mode highlights the initial stages of ignition and subsequent propagation, providing insights into the early dynamics of deflagration before any transition to detonation occurs.

The Delayed Detonation Ignition Mode, illustrated in Figure 5, begins similarly to DIM, with a flame kernel forming at the apex of the wedge and propagating slowly at first. However, at 1.45 ms, a distinct flame kernel emerges at approximately 1.8 mm downstream from the apex of the wedge, marking the onset of a detonation wave propagating in multiple directions. The pressure profile for DDIM, as captured by sensor PS5 and shown in Figure 6, exhibited two main pressure peaks. The first peak, associated with the initial flame kernel and shock focusing, reaches 4.4 MPa in the simulation, which is nearly doubled in the experimental data. Approximately 0.6–0.7 ms later, both the simulation and experimental data exhibit a second peak of significantly higher magnitude, indicating the transition to detonation. In the simulation, this second peak reaches 8.8 MPa, while the experimental counterpart is nearly twice as high.

The comparison in Figure 6 further demonstrates that the general behavior in cases of DDIM is consistent between the simulation and experiments, as both display a pattern of two primary peaks. Across sensors PS1–PS4, the post-shock pressure initially reaches the same level of 0.3–0.4 MPa in both simulation and experiments. Notably, in the simulation, the second peak at PS5 coincides with an elevated pressure reading at PS4, aligning with the findings by Buraczewski and Shepherd [6] that the transition to detonation occurs in the volume between PS4 and PS5. However, in the experimental data, no evidence of comparable pressure levels at PS4 is observed during the second peak. This discrepancy, again, may result from additional energy dissipation mechanisms in the experiments, for instance, unaccounted mixing phenomena, which are minimized in the idealized conditions of the simulation.

In the Immediate Transition to Detonation Mode (Figure 7), the incident shock wave initially travels at 715 m/s before reflecting upon colliding with the inner wall of the wedge. This interaction ignites the mixture at the apex of the wedge at 0.855 ms, leading to a coupled propagation of the flame and the reflected shock wave. The coupled propagation process exhibits a velocity of 1830 m/s relative to the tube walls. However, considering that the detonation wave propagates within a gas already in motion, the axial velocity of the shocked gas is calculated as 396 m/s using Gaseq software (version 0.79). This results in a coupled wave propagation velocity of 2226 m/s, which exceeds the ideal Chapman–Jouguet (C–J) detonation velocity of 2077 m/s for this mixture, indicating the presence of an overdriven detonation. This behavior is characteristic of ITDM and highlights the intense energy release during this process. The pressure profiles recorded by sensor PS5 (Figure 8) show a steeper pressure increase and higher maximal values than those observed in DIM, with the peak pressure reaching 10.5 MPa compared to 4.5 MPa in DIM.

The comparison presented in Figure 8 confirms that the general behavior observed in the ITDM case aligns with experimental findings, as both follow similar patterns in the pressure profiles. While the experimental and simulated results show agreement in the timing and magnitude of the primary peak, the post-detonation pressure profiles highlight a notable difference. In the simulation, pressure stabilizes around 3 MPa after detonation, while in the experiments, it gradually decays over time. This disparity is consistent with observations from the other ignition modes and can be attributed to additional energy dissipation mechanisms in the experimental setup.

Figure 9 depicts the correlation between hydrogen concentration and the velocity of the leading shock wave, both in absolute terms (left) and relative to the speed of sound in the reactants (center) and products (right), providing insights into the transition to detonation across various mixture compositions. Remarkably, the transition limit exhibits a characteristic U-shaped pattern, consistent with experimental observations. Typically, the lowest velocity values are observed near stoichiometric concentration. A comparative analysis between experimental and numerical results reveals a slight difference. For instance, the transition velocity for a 30% H₂-in-air mixture was around 719 m/s in the experiments, whereas the numerical simulation yielded a slightly lower value of 664 m/s. This discrepancy indicates a 5.5% decrease in transition velocity compared to the experimental findings obtained by Rudy [31]. While the difference between the numerical and experimental data remained within 5–8% for mixtures containing 25% to 45% hydrogen in air, it became more significant for leaner and richer mixtures, specifically H₂ concentrations below 25% and above 45%.

The summary graph in Figure 10 presents the maximum pressures recorded by sensor PS5, shedding light on the critical pressure thresholds associated with detonation initiation across different hydrogen concentrations in air mixtures. The limit line delineates the range within which direct detonation initiation occurs, representing an average value between the lowest recorded pressure of the detonation mode and the highest from the deflagration mode. Notably, the pressure required for direct detonation initiation is constrained within a narrow range of 7.9–9 MPa for 25–50% H₂ in air, as inferred from experimental data. However, a prominent deviation from this threshold was observed in numerical simulations, with differences ranging from 10% to 36% compared to the experimental limits, particularly pronounced for mixtures containing 35–40% hydrogen.

Figure 11 provides valuable insights into the relationship between ignition delay time and shock wave velocity, revealing the crucial role of the shock velocity in detonation onset. The findings underscore that higher shock velocities lead to higher pressure and temperature levels within the wedge tip, resulting in shorter ignition delay times and an augmented likelihood of transition to detonation. Generally, there is a critical shock wave velocity beyond which detonation occurrence becomes more probable. While the influence of shock velocity on IDT is visible for mixtures of 20–50%, a notable deviation is observed when compared to experimental data, with numerical outcomes yielding higher IDT values. Despite this disparity, the simulations demonstrate reliability, as both experimental and numerical IDT values fall within the order of microseconds.

5. Conclusions

This study rigorously evaluated the predictive capability of the ddtFoam code by replicating experiments conducted by Rudy [31] and analyzing the transition to detonation limits induced by shock focusing. The numerical outcomes revealed three distinct scenarios post-shock wave focusing: deflagrative ignition in the corner, deflagrative ignition with intermediate transient phases leading to a delayed transition to detonation in the trailing combustion zone approximately 1.8 mm from the apex of the wedge, and ignition with an immediate transition to detonation, resulting in the formation of a detonation wave in the corner tip.

The pressure profiles from Figure 4, Figure 6 and Figure 8 reveal strong agreement in the general behavior of the shock wave and flame dynamics between the simulations and the experiments. Both approaches exhibit similar patterns, achieving comparable pressure levels across sensors. This agreement demonstrates the capability of the numerical model to replicate the key features of deflagrative and detonative ignition processes. Discrepancies in the timing of pressure peaks observed in the results can largely be attributed to differences in shock velocities and the adjusted sensor positions in the simulation domain relative to the experimental setup. Despite these temporal offsets, the simulations reliably reproduce pressure levels recorded in the experiments, affirming the validity and robustness of the numerical approach.

Notably, a 5–8% underestimation in shock velocities was observed in the numerical simulations compared to the experimental results, with greater variances—up to 17%—for leaner and richer mixtures (H₂ < 25% and H₂ > 45% in air). Several factors could contribute to this discrepancy. Differences in chemical kinetics models used in the simulations versus those governing the actual combustion processes in the experiments may influence the predicted reaction rates and, consequently, the shock wave velocity. Additionally, minor differences in sensor alignment or positioning between the experimental setup and the scaled simulation domain can affect the timing and velocity measurements, introducing small but cumulative errors. Variability in experimental conditions, such as inhomogeneities in mixture composition, temperature gradients, or turbulence, could also lead to deviations that are not fully captured in the idealized numerical model. Addressing these factors in future studies by refining the chemical kinetics model, improving sensor calibration in simulations, and exploring additional boundary condition sensitivities would provide deeper insights into the potential for reducing these discrepancies and enhancing the accuracy of numerical predictions.

Analysis of the impact of shock velocity on IDT revealed a disparity compared to experimental results, with numerical outcomes showing higher IDT levels across mixtures containing 20–50% hydrogen in air. Despite this deviation, the simulations demonstrated accuracy in replicating IDT, aligning within the order of microseconds. Similarly, the simulations accurately captured trends in maximum pressure recordings, although with a notable quantitative underestimation (10–36%) in the pressure required for direct detonation initiation, particularly for mixtures containing 35–40% hydrogen in air. The overestimation of IDT suggests that the numerical model provides a conservative estimate of the time available for detonation mitigation strategies, such as quenching or venting. Additionally, the underestimation of the required pressure for direct detonation initiation introduces a built-in safety margin, reducing the risk of underpredicting hazardous scenarios.

The observed disparities between the numerical simulations and experimental findings warrant a more profound exploration of underlying factors. While this study successfully replicated key ignition modes and transition phenomena observed in experiments by Rudy [31], discrepancies in shock velocity and pressure thresholds for detonation initiation necessitate a closer examination. Potential factors contributing to these disparities include differences in shock wave dynamics between simulations and experiments, which could arise from variations in shock tube geometries, reflector designs, or the initial conditions of shock wave generation; the fidelity of shock wave propagation and combustion dynamics; and experimental conditions such as ambient temperature, mixture preparation, and ignition sources, which introduce variability that is not fully captured in the simulations. Additionally, real-world deviations such as boundary layer dynamics, ambient temperature gradients, wall heat losses, and unmodeled flow instabilities could significantly affect detonation behavior and are not accounted for in the idealized numerical setup. Acknowledging these limitations, further investigation into these factors within the simulation setup may provide insights into the potential for refining the model to enhance its quantitative agreement with experimental data. Nonetheless, the observed underestimation in pressure and shock velocity provides a safety margin when using ddtFoam to assess the transition to detonation under controlled conditions.

The findings of this study have direct implications for the design and safety of hydrogen systems in industrial and transportation applications. By accurately predicting ignition modes and detonation thresholds, the ddtFoam model offers a reliable tool for assessing detonation risks in hydrogen storage and transport systems. The conservative estimates of ignition delay times provide a valuable safety margin for designing mitigation strategies, such as quenching systems, venting mechanisms, and barrier placement. Additionally, the insights gained into shock wave dynamics and the influence of key parameters, such as hydrogen concentration and shock velocity, can inform guidelines for operating conditions to minimize detonation risks. These results contribute to advancing safety standards and improving the reliability of hydrogen technologies in real-world scenarios.