Chinese Journal of Aeronautics, (2024), 37(2): 249–258

Chinese Society of Aeronautics and Astronautics

& Beihang University
Chinese Journal of Aeronautics
cja@buaa.edu.cn
www.sciencedirect.com

FULL LENGTH ARTICLE

Deflagration and detonation induced by shock wave

focusing at different Mach numbers
Zezhong YANG, Jun CHENG, Bo ZHANG *

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 5 February 2023; revised 27 February 2023; accepted 28 April 2023

Available online 28 June 2023

KEYWORDS Abstract Shock wave focusing is an effective way to create a hot spot or a high-pressure and high-
Deflagration; temperature region at a certain place, showing its unique usage in detonation initiation, which is
Detonation; beneficial for the development of detonation-based engines. The flame propagation behavior after
Hydrogen; the autoignition induced by shock wave focusing is crucial to the formation and self-sustaining of
Ignition; the detonation wave. In this study, wedge reflectors with two different angles (60° and 90°) and a
Shock wave focusing planar reflector are employed, and the Mach number of incident shock waves ranging from 2.0
to 2.8 is utilized to trigger different flame propagation modes. Dynamic pressure transducers and
the high-speed schlieren imaging system are both employed to investigate the shock-shock collision
and ignition procedure. The results reveal a total of four flame propagation modes: deflagration,
DDT (Deflagration-to-Detonation Transition), unsteady detonation, and direct detonation. The
detonation wave formed in the DDT and unsteady detonation mode is only approximately
75%
85% of the Chapman-Jouguet (C-J) speed; meanwhile, the directly induced detonation wave
speed is close to the C-J speed. Transverse waves, which are strong evidence for the existence of det-
onation waves, are discovered in experiments. The usage of wedge reflectors significantly reduces
the initial pressure difference ratio needed for direct detonation ignition. We also provide a practical
method for differentiating between detonation and deflagration modes, which involves contrasting
the speed of the reflected shock wave with the speed of the theoretically nonreactive reflected shock
wave. These findings should serve as a reference for the detonation initiation technique in advanced
detonation propulsion engines.
Ó 2023 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and
Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/
licenses/by-nc-nd/4.0/).

1. Introduction heat release rate.1,2 Detonation engines such as Pulse Detona-

tion Engines (PDEs),3,4 Rotating Detonation Engines
In recent years, the application of detonation waves in hyper- (RDEs),5 and Oblique Detonation Engines (ODEs)6,7 are all
sonic aerospace propulsion systems has been universally promising airbreathing hypersonic or supersonic propulsion
noticed for their higher thermodynamic efficiency and higher techniques. In addition, a novel shock-focusing detonation
engine configuration that used a concave reflector to ignite
* Corresponding author at: Shanghai Jiao Tong University, School the detonation wave has been proposed as an improvement
of Aeronautics and Astronautics, Shanghai 200240, China. of PDEs in recent years.8,9 Unlike the common Deflagration-
E-mail address: bozhang@sjtu.edu.cn (B. ZHANG). to-Detonation Transition (DDT) conducted in an ordinary
https://doi.org/10.1016/j.cja.2023.06.029
1000-9361 Ó 2023 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
250 Z. YANG et al.

Nomenclature

vie experimental incident shock wave velocity v2t theoretical nonreactive velocity behind incident
vit theoretical incident shock wave velocity shock wave
vre experimental reflected shock wave velocity Mai Mach number of incident shock wave
vrt theoretical reflected shock wave velocity p1 initial pressure of driven section
vCJ theoretical Chapman-Jouguet detonation velocity p4 initial pressure of driver section
v2e experimental velocity behind incident shock wave

tube or tubes with obstacles10 or jets11 to accelerate DDT, was believed to be the resource of the critical energy for igni-
where hot spots randomly form in unpredictable positions, tion.30,31 Li and Zhang32 confirmed the energy-accumulation
the shock wave focusing technique can create a certain hot effect of wedge reflectors. Schlieren images showed that it is
spot or region at the desired position and initiate direct deto- more conducive to initiate the detonation wave with a smaller
nation, showing a promising future for low-cost stable detona- angle wedge reflector at a lower speed shock wave. Yang and
tion initiation, which is crucial for detonation engine designs. Zhang33 combined both numerical and experimental methods
The important role of shock wave focusing in detonation initi- to investigate the flame propagation modes in different angle
ation can also be confirmed in various DDT processes.12,13 wedge reflectors. They found that the hot spot formed in the
As a shock wave travels into a cavity, it will converge in the 60° wedge reflector tends to have a higher intensity than the
direction of propagation and produce a locally high-pressure 90° wedge reflector, triggering a more stable detonation wave.
and high-temperature region at the center of convergence. A Although numerous studies on autoignition induced by
pioneering study of shock wave focusing was conducted by shock wave focusing have been carried out, most of them still
Meshkov.14 Several types of cylindrically concave reflectors lack quantitative analysis of the flame evolution procedure. To
were tested, and the shock wave strength was substantially the best of our knowledge, the flame velocity, which is a critical
enhanced in the shock wave focusing procedure. The three indicator of the flame mode, whether it is a deflagration wave
stages of the pressure field and the three-shock intersections or a detonation wave, has never been discussed in previous
in parabolic reflectors were experimentally studied by Sturte- shock wave focusing research. Our study combines the high-
vant and Kulkarny.15 Duong and Milton16 investigated the speed schlieren imaging technique with pressure and OH*
shock wave converging in cones ranging from 10° to 30°. Izumi emission measurement systems to explore the autoignition
et al.17 combined both experimental and computational meth- and flame propagation procedure. A hydrogen–oxygen mix-
ods to study the structure of the reflected shock wave after the ture is utilized in our shock tube experiments for its impor-
shock-shock collision in parabolic reflectors with different tance in the aerospace propulsion field.5–7 For a specific case
depths. Lodato et al.18 used a fifth-order high-resolution (Mai = 2.0, 90° wedge reflector), an additional nonreactive
method to simulate the shock wave reflected from a shallow simulation is performed to provide a better explanation of
wavy wall. It is noteworthy that they also used a nonlinear the flow field. Furthermore, two different DDT processes are
wave equation to quantitatively analyze the movement of tri- revealed. The structures of shock waves and reaction fronts
ple point and transverse waves. Wedge-like reflectors have with different reflectors are systematically discussed. The com-
been discussed by Bond,19 Eliasson,20 and Dimotakis21 et al. parison of the velocity of the reflected shock wave and
A special type of reflector that can smoothly transform a pla- Chapman-Jouguet (C-J) speed is presented and analyzed.
nar shock wave into a curved shock wave was designed by
Zhai22 and Luo23 et al. according to the Chester-Chisnell- 2. Experimental setup
Whitham (CCW) approximation. Liverts and Apazidis24
found that a focal temperature on the order of 30000 K is mea- The experiment of autoignition induced by shock wave focus-
sured in their converging-shock-wave experiments. The afore- ing is carried out in a double-diaphragm stainless steel shock
mentioned nonreactive shock wave focusing experiments and tube. As shown in Fig. 1(a), the tube is composed of three
simulations more or less indicate that the extremely high tem- parts: a 1 m-long driver section, a 1 m-long driven section,
perature and pressure point formed at the focus point of shock and a 0.1 m-long double diaphragm section. The inner cross-
waves or region after strong interactions between the shock section of the tube is a 40 mm  73 mm rectangle. Moreover,
waves. Chan25 first applied the shock wave focusing technique the shock tube is equipped with sidewall windows for schlieren
in combustible mixture ignition. The local hot spots were cap- imaging, and its size is 200 mm  73 mm. The drive section’s
able of causing strong ignition, and the blast wave could end wall is replaceable, where different-shaped reflectors can
develop into a detonation wave. Gelfand et al.26 investigated be installed. A planar reflector and two different-angled wedge
the ignition behavior in a hydrogen-air mixture, and Bartenev reflectors, a 60° wedge reflector and a 90° wedge reflector, are
et al.27 numerically studied the relationship between the Mach employed in this experiment. The apex of the wedge reflector
number of the incident shock wave and combustion mode. has been removed to create a 7 mm-wide plane, and a through
Zhang et al.28,29 systematically performed a series of experi- hole is drilled in the plane’s center to accommodate a piezo-
ments and demonstrated the existence of several ignition electric pressure transducer (PT5). Other pressure transducers
modes in shock wave focusing procedures with different reflec- (PT1–PT4) are mounted along the sidewall of the driven
tors. Energy accumulation in the shock wave focusing process
Deflagration and detonation induced by shock wave focusing 251

Fig. 1 Schematic diagram of (a) shock tube experimental system and (b) high-speed schlieren imaging system.

section. The spectral signal of combustion is obtained and on the double-diaphragm section opens, and the pressure
amplified by a Photomultiplier Tube (PMT). The optical mea- imbalance ruptures the diaphragms, generating an incident
surement window is positioned directly opposite PT4, and the shock wave with the desired intensity.
emission spectrum of the OH* radicals is isolated through a
306.5 nm bandpass filter. By comparing the arrival time of 3. Results and discussion
the pressure spike with the OH* signal, we can easily determine
the combustion state at this cross-section. PT2 and PT3 used 3.1. Validation of experimental results
here are PCB piezoelectric sensors (113B26), while PT1, PT4,
and PT5 are Kistler piezoelectric sensors (603CBA). The sig-
First, the accuracy of the experimental results has been vali-
nals from pressure transducers and PMT are recorded by a
dated. Cases 1–5 (Table 1) were tested with all three different
PicoScope 4824 with a sampling frequency of 200 kHz. A Z-
reflectors. Extra cases that have not been shown in Table 1
type schlieren system equipped with a high-speed camera
are conducted with a planar reflector to verify the agreement
(Phantom V710L) is also utilized to capture flow field changes,
between the experiment and calculation (Fig. 2(a)). The veloc-
as shown in Fig. 1(b). The camera has a frame rate of 150000
ity of the incident shock wave is obtained by comparing the
frame/s and a resolution of 320 pixel  104 pixel. The high-
location difference of the shock wave. At least 15 schlieren
speed camera can shoot continuously for 3 s in this setting,
images are used in the shock-wave velocity calculation to
which is long enough to meet the experiment’s demands.
ensure the accuracy of the calculation. For nonreactive
In this experiment, the driven gas is a stoichiometric hydro-
shock-tube experiments, as long as we know the Mach number
gen–oxygen mixture diluted with 90 vol% argon, and it is pre-
of the incident shock wave (Mai) and the initial gas state of the
mixed in a 40 L chamber for at least 24 h. Helium is chosen to
driver section and the driven section, we can obtain all the flow
be the driver gas because it is the lightest inert gas, promising a
properties in the tube according to the Rankine-Hugoniot rela-
higher Mach number for the incident shock wave. After the
tion. Because the gas after the incident shock wave has not
three sections of the tube are all vacuumed to a pressure below
been ignited in all experimental cases, the shock wave incident
0.1 kPa, the driven gas and the driver gas are filled into the cor-
process can be considered inert, which means it is also suitable
responding section at specific pressures, which are shown in
for the aforementioned theoretical calculation.
Table 1. Meanwhile, the double-diaphragm section is filled
When Mai is between 2.0 and 2.8, the experimental results
with driver gas at half the pressure between the two adjacent
agree well with the theoretical results, as shown in Fig. 2. The
sections. When all preparations are finished, the solenoid valve
experimental and theoretical results are most consistent when
252 Z. YANG et al.

Table 1 Initial gas conditions for Cases 1–5.

Case Mai Driver section Driven section p4 =p1
p4 (kPa) p1 (kPa)
1 2.0 550 60.36 9.11
2 2.2 550 44.01 12.50
3 2.4 550 32.73 16.80
4 2.6 550 24.72 22.25
5 2.8 550 18.91 29.09

Mai = 2.4. Mai is slightly lower than its theoretical value when
the pressure ratio between the driver and driven sections
(p4 =p1 ) is less than 17. This mismatch could be attributed to
the type of shock tube that is employed in the experiment.
The theoretical results are based on the one-diaphragm
assumption, whereas we adopt a double-diaphragm shock tube
in the experiment. When the solenoid valve on the double-
diaphragm section opens, the pressure of the double-
diaphragm section decreases, leading to a decrease in the pres-
sure difference between the double-diaphragm section and the
driven section and an increase in the pressure difference
between the driver section and the double-diaphragm section.
Therefore, Diaphragm 1, which is shown in Fig. 1(a), will first
be ruptured, creating a shock wave that moves into the double-
diaphragm section, increasing the pressure and temperature in
this section, bringing initial velocity to the local gas. This will
enhance the intensity of the final incident shock wave, which
will propagate in the driven section after the rupture of Dia-
phragm 2. However, the error is within the acceptable range
considering that Mai is lower than 3.0 and p4 =p1 is lower than
30 in our experiment.

3.2. Two different DDT processes with a planar reflector

Fig. 3 shows the schlieren images of the DDT process under

the condition of Mai = 2.2, and Fig. 4 shows the correspond-
ing pressure and OH* signal records. The time when the inci-
dent shock wave enters the schlieren window is set as time zero.
This is also suitable for all of the following schlieren images. At
t = 473.3 ls, a small flame kernel appears at the upper right
corner. However, this flame kernel develops at a very low
speed, which is still a relatively small flame kernel at t = 760
.0 ls, turning out to have no significant influence on the whole
flow field. Although it cannot be observed from the schlieren
images, the PMT signal shows that ignition has already started
at approximately t = 500 ls, 60 ls delayed from the time that
the reflected shock wave passed through the cross-section
where PMT and PT4 are located. At t = 586.7 ls, a strip flame
appears after the reflected shock wave and starts to expand.
The hot spot formed at t = 726.7 ls is darker than the sur-
roundings in the schlieren image, indicating a higher density
gradient at this point. It is formed by the turbulent fluctuations
behind the reflected shock wave, which is caused by the pres-
sure perturbation and wrinkling of the turbulent flame. Oran
Fig. 2 Comparison between experimental results and theoretical et al.34 considered that the ignition delay time gradient caused
results: (a) Planar reflector, (b) 60° wedge reflector, (c) 90° wedge by turbulence is the main reason for the autoignition after the
reflector. reflected shock wave. The important role of flame-turbulence
Deflagration and detonation induced by shock wave focusing 253

Fig. 3 Schlieren images of DDT process when incident Mach number is 2.2.

tion at t = 686.7 ls. Since then, a fast deflagration wave forms

from the hot spot, and it propagates toward the planar reflec-
tor. The profile of the deflagration wave at t = 773.3 ls
appears more wrinkled compared to the detonation wave at
t = 826.7 ls, and no transverse waves are observed after the
flame front, indicating that the flame acceleration is still going
on. This deflagration wave quickly goes through the DDT pro-
cess after the collision with the planar reflector, creating a
5.4 MPa peak pressure value at the end wall. Then, it travels
as a reflected blast wave toward the driver section, creating a
second peak value in PT1–PT4 and PMT. The strong coupling
of the flame front and the pressure wave indicates that the
reflected blast wave is detonation. The local velocity of the
gas after the reflected shock wave can be approximately
assumed to be zero, as the diaphragm fragment almost stands
still at the same position from 800 ls to 880 ls. Thus, the
velocity of the reflected detonation wave is 1008 ± 120.7 m/
s, measured from the schlieren images, and it is 77.2% C-J
speed (vCJ =1306 m/s).
Fig. 5 depicts another DDT mode where the Mach number
of the incident shock wave is 2.3, slightly higher than the situ-
ation in Fig. 3. At t = 280.0 ls, the reflected shock wave
forms, but no combustion wave is observed. The ignition starts
at t = 306.7 ls, creating a curved deflagration wave after the
reflected shock wave. The deflagration wave accelerates and
Fig. 4 Pressure and OH* records of planar reflector under
finally catches up with the shock wave at t = 346.7 ls. After
condition of Mai = 2.2.
the coupling of the combustion wave and the shock wave, they
form a detonation wave that keeps propagating. The trans-
in DDT is also investigated in other researches.35,36 Owing to verse waves after the newly formed combustion wave strongly
the flame-acceleration effect, the initially planar reflected support that the combustion wave is a detonation wave. More-
shock wave undergoes a slight outward curvature transforma- over, the DDT process creates a quasi-stationary density dis-
254 Z. YANG et al.

Fig. 5 Schlieren images of DDT process under condition of Mai = 2.3.

continuity at the coupling position because of the different 0 ls, the reflected bow shock waves encounter each other near
acoustic impedances before and after the shock wave. In gen- the apex of the 90° wedge reflector. The ignition starts from the
eral, Fig. 5 shows a typical chemical shock tube ignition pro- apex, but the flame is divided by the shock waves into three
cess37 where a combustion wave is formed at the end wall parts: the upper part, the middle part, and the lower part. At
after the reflected shock wave. The subsequent DDT process t = 373.3 ls, the wave forms after the reflected shock moves
that this study is focused on is often neglected in traditional at a very low velocity, and the PMT sensor fails to capture
ignition delay time measurement experiments.38 the ignition signal during the one-second sampling period.
To determine whether it is a flame front, a corresponding cold
test is conducted, by replacing the combustible driven gas with
3.3. Deflagration and detonation in wedge reflectors
a helium-argon mixture that has the same density as the driven
gas. The small flow structure at the reflector apex, caused by
Fig. 6 illustrates the deflagration with a 90° wedge reflector the complex shock wave interactions39,40 in the cold test, dissi-
under the condition of Mai = 2.0. Different from the autoigni- pates rapidly because of the pressure perturbation behind the
tion induced by the planar reflector, the incident shock wave reflected shock wave. In contrast, the wave near the reflector
reflects on the wedge, and the reflected shock waves interact apex in the combustion case keeps propagating, indicating that
with each other, partitioning out several regions. At t = 320. this is a combustion wave. Furthermore, the cold test also

Fig. 6 Schlieren images of deflagration with a 90° wedge reflector and corresponding cold test (Mai = 2.0).
Deflagration and detonation induced by shock wave focusing 255

shows that the shock wave positions in the combustible case

and the cold test are almost identical, which means that the
deflagration wave under this condition has no acceleration
effect on the reflected shock wave propagation.
The corresponding nonreactive simulation of the cold test
in Fig. 6 (t = 340.0 ls) is shown in Fig. 7 for better compre-
hension. The finite volume OpenFOAM-1041 package is
employed in our simulation. The overall accuracy of the solver
utilized here is 2nd order. The computational domain is half of
the schlieren window: 200 mm  36.5 mm, and the minimum
mesh size is 10 lm, which is fine enough for nonreactive simu-
lation. A detailed description of the simulation setup can be
found in Ref.29. The red line in Fig. 7(a) is the reflected bow
shock wave. It is deflected by the reflected shock wave because Fig. 8 Local decoupling and recoupling procedure in detonation
of the different acoustic impedances before and after the propagation with a 90° wedge reflector (Mai = 2.2).
reflected shock wave. A second small bow shock wave (blue)
also appears due to the collision between the reflected shock
wave and the side wall. Both bow shock wave tails are strongly the tendency to fail. Pantow et al.43 studied the decoupling and
disturbed by the jet formed from the end wall. The jet is also recoupling of detonation waves associated with sudden expan-
observed in the schlieren images of nonreactive experiment sion of the tube and found that confined flow conditions after
(Fig. 6, corresponding cold test, t = 373.3 ls). The accumula- the expansion are responsible for the reinitiation of the decou-
tion line18 in Figs. 7(b)–(c) shows that the upcoming stream pled detonation wave. In our research, the wedge-like reflec-
has a turnover here, dividing the local flow field into different tors act as expansion tubes for the newly formed detonation
parts, which may be the cause of the flame division. waves near the reflector apexes, which cause that the detona-
In Fig. 8, the black dashed line represents transverse shock tion waves tend to decouple. Meanwhile, the transverse waves
waves, the blue line represents the reflected shock wave, and generated from the collision of the detonation wave and the
the red line represents the combustion wave. The schematic tube wall, as well as the collision of shock waves, strongly sup-
diagrams in each frame show the coupling condition of the port the recoupling of the flame front and the reflected shock
combustion wave and shock wave. At t = 313.3 ls, the com- wave. As a result, the detonation wave experiences a decou-
bustion wave has been divided into three parts by the shock pling and recoupling procedure under the combined effects
waves. The combustion wave couples with the reflected shock of the expansion tube and transverse waves.
wave in Region 1 and Region 3 but decouples in Region 2. It The schlieren images of detonation with a 60° wedge reflec-
seems that the combustion wave propagates slower in this tor under the condition of Mai = 2.2, which is the same Mach
region. At t = 333.3 ls, the combustion wave develops into number used in Fig. 8, are shown in Fig. 9. Different from the
a curved shape but almost fully decouples from the reflected regular reflection in Fig. 6, the Mach reflection of the incident
shock wave. After 26.7 ls, due to self-acceleration, the com- shock wave makes two triple points collide with each other
bustion wave recouples with the reflected shock wave. At before the end wall. The detonation wave directly forms after
t = 386.7 ls, local decoupling appears again in the middle ignition. When the detonation wave propagates through the
of the combustion wave, but soon the reflected shock wave wedge corner, two strong transverse waves emerge from the
recouples with the flame front. The coupling between the com- corner, which will support the subsequent self-sustained deto-
bustion wave and the shock wave is highly unsteady in this nation wave propagation. At t = 466.7 ls, three transverse
case. As a detonation wave propagates from a confined chan-
nel to an unconfined space, it undergoes diffraction42 and has

Fig. 7 Nonreactive numerical streamlines and schlieren image

with 90° wedge reflector (Mai = 2.0): (a) t = 340.0 ls, (b) Fig. 9 Schlieren images of detonation with 60° wedge reflector
Enlarged drawing of Point B, (c) Enlarged drawing of Point C. (Mai = 2.2).
256 Z. YANG et al.

waves are visible after the detonation wave, indicating that the comparison between the absolute reflected shock wave velocity
reflected detonation wave is propagating in a stable mode. vre þ v2t and C-J velocity vCJ . Four different flame propagation
modes are shown in Fig. 11: Deflagration, DDT, unsteady det-
3.4. Reflected shock wave velocity and combustion mode onation, and direct detonation. The detonation waves formed
identification by DDT only have approximately 75%–86% C-J speed, while
the unsteady detonation wave speed is 84.5% C-J speed. It
Sections 3.2 and 3.3 show several cases, including deflagration, appears that the detonation becomes unstable and experiences
DDT, unsteady detonation, and direct detonation. Fig. 3, a decrease in velocity before ultimately failing. The velocity
Figs. 5–6, and Figs. 8–9 correspond to Shots 1–5 in Fig. 10 deficit of the detonation wave is usually found in near-
and Fig. 11. Fig. 10 depicts the comparison between the rela- pressure-limit detonation propagation.45 Strong boundary
tive reflected shock wave velocity and the theoretical reflected layer effects, such as small-diameter tubes and rough tube
shock wave velocity. Due to the coupling of the flame front walls,46,47 can also lead to a decrease in the detonation speed.
and shock wave, the speed of the reflected detonation wave will The onset of phenomena such as single-head spin and the reini-
be higher than that of the theoretical nonreactive reflected tiation procedure found in the aforementioned research all
shock wave. Furthermore, we also confirm that the deflagra- show that the detonation wave is traveling in a relatively unsta-
tion wave does not have a significant boost to the leading ble state. The decrease proportion of the detonation speed is
shock wave. Therefore, if the reflected shock wave velocity consistent with our findings.
vre is close to the theoretical nonreactive shock wave velocity Compared with the planar reflector, which needs
vrt , it can be considered that there is no detonation in the shock p4 =p1 ¼ 22:3 to form a detonation wave, wedge reflectors only
tube. However, one exception exists in our experiment, which need p4 =p1 ¼ 12:5, nearly reducing half of the initial pressure
is Shot 1 in Fig. 10(a). Although vre is lower than vrt , DDT ratio. The shock wave focusing technique has a great energy-
occurs after the reflected shock wave. accumulation effect, significantly reducing the pressure ratio
Because the reflected detonation wave propagates in the gas required for stable detonation initiation. Although the graphs
with initial velocity v2e , the absolute speed of the detonation of the 60° wedge reflector (Fig. 11(b)) and 90° wedge reflector
wave is vre þ v2e . However, v2e cannot be obtained from either (Fig. 11(c)) have no significant difference, where Shots 4 and 5
schlieren images or pressure signals. Section 3.1 shows that the have similar reflected shock wave speeds under the same inten-
experimental results conform well with the theoretical results. sity of the incident shock wave (Mai = 2.2), the schlieren
Hence, it is reasonable to use the theoretical velocity of the images (Fig. 8 and Fig. 9) depict that the detonation wave
shocked gas after incident shock wave v2t to substitute v2e . seems to have a more stable propagation mode (strong trans-
The C-J speed is calculated from the SDToolBox package, verse waves after the detonation wave) in the 60° wedge reflec-
which was developed by Shepherd et al.44, and the package tor, which means that the 60° wedge reflector may have a
is widely used in detonation simulations. Fig. 11 shows the better shock-wave-focusing effect.

Fig. 10 Comparison between relative reflected shock wave velocity and theoretical reflected shock wave velocity: (a) Planar reflector, (b)
60° wedge reflector, and (c) 90° wedge reflector.
Deflagration and detonation induced by shock wave focusing 257

Fig. 11 Comparison between absolute reflected shock wave velocity and C-J velocity: (a) Planar reflector, (b) 60° wedge reflector, and (c)
90° wedge reflector.

4. Conclusions Declaration of Competing Interest

In this study, the mechanism of deflagration and detonation The authors declare that they have no known competing
induced by shock wave focusing in a H2/O2/Ar mixture is sys- financial interests or personal relationships that could have
tematically explored. Schlieren images and the pressure and appeared to influence the work reported in this paper.
OH* signals under different conditions are analyzed. The accu-
racy of the experimental result is validated by comparing it Acknowledgments
with the theoretical calculation. The speed of the reflected
shock waves is utilized to analyze the flame propagation mode. The authors are grateful for the financial support from the
Additional simulation is performed for better understanding. National Natural Science Foundation of China (No.
The main conclusions are drawn as follows: 12272234), the Innovation Program of Shanghai Municipal
Education Commission, China (No. 2023KEJI05-75), and
(1) The flame initiated by the collision of the incident shock the Shanghai Science and Technology Planning Project, China
wave and the end wall generally has four different prop- (No. 22190711500).
agation modes: deflagration, DDT, unsteady detonation
and direct detonation. The detonation wave formed in References
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