Investigation of counter-rotating shock wave and wave direction control of hollow rotating detonation engine with Laval nozzle

Abstract

The counter-rotating shock wave and wave direction control of the hollow rotating detonation combustor with Laval nozzle are studied. The in-house solver BYRFoam, developed on the OpenFOAM platform, is used. The phenomenon and spatial distribution of the counter-rotating shock wave in the combustor are revealed. The result suggests that the closer the location is to the outer wall, the stronger the counter-rotating shock wave is. A method of controlling the wave direction is proposed. It's shown that the intensity of the counter-rotating shock wave is controlled by reducing the total pressure of inlet, and then the direction of the detonation wave is controlled. The process of detonation wave reversing is divided into four steps, namely, counter-rotating shock waves evolve into detonation waves, several detonation waves are extinguished, detonation waves form again, and detonation waves propagate stably. The mechanism of wave direction control is investigated. The result shows that the fluctuation of the total pressure of inlet stimulates the positive feedback interaction between the counter-rotating shock wave and the fresh gas, which causes initial detonation waves to be extinguished and the intensity of counter-rotating shock waves to become stronger and stronger, and eventually counter-rotating shock waves evolve into reverse detonation waves.

Full text

Phys. Fluids 34, 056104 (2022); https://doi.org/10.1063/5.0089207 34, 056104
© 2022 Author(s).
Investigation of counter-rotating shock wave
and wave direction control of hollow rotating
detonation engine with Laval nozzle
Cite as: Phys. Fluids 34, 056104 (2022); https://doi.org/10.1063/5.0089207
Submitted: 23 February 2022 • Accepted: 26 April 2022 • Accepted Manuscript Online: 27 April 2022 •
Published Online: 10 May 2022
Guangyao Rong (荣光耀), Miao Cheng (程杪), Zhaohua Sheng (盛兆华), et al.
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Investigation of counter-rotating shock wave and
wave direction control of hollow rotating
detonation engine with Laval nozzle
Cite as: Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207
Submitted: 23 February 2022 .Accepted: 26 April 2022 .
Published Online: 10 May 2022
Guangyao Rong (荣光耀), Miao Cheng (程杪), Zhaohua Sheng (盛兆华), Xiangyang Liu (刘向阳),
Yunzhen Zhang (张允祯), and Jianping Wang (王健平)
a)
AFFILIATIONS
Center for Combustion and Propulsion, CAPT & SKLTCS, Department of Mechanics and Engineering Sciences, College of
Engineering, Peking University, Beijing 100871, China
a)
Author to whom correspondence should be addressed: wangjp@pku.edu.cn
ABSTRACT
The counter-rotating shock wave and wave direction control of the hollow rotating detonation combustor with Laval nozzle are studied. The
in-house solver BYRFoam, developed on the OpenFOAM platform, is used. The phenomenon and spatial distribution of the counter-
rotating shock wave in the combustor are revealed. The result suggests that the closer the location is to the outer wall, the stronger the
counter-rotating shock wave is. A method of controlling the wave direction is proposed. It is shown that the intensity of the counter-rotating
shock wave is controlled by reducing the total pressure of the inlet, and then the direction of the detonation wave is controlled. The process
of detonation wave reversing is divided into four steps, namely, counter-rotating shock waves evolve into detonation waves, several detona-
tion waves are extinguished, detonation waves form again, and detonation waves propagate stably. The mechanism of wave direction control
is investigated. The result shows that the fluctuation of the total pressure of the inlet stimulates the positive feedback interaction between the
counter-rotating shock wave and the fresh gas, which causes initial detonation waves to be extinguished and the intensity of counter-rotating
shock waves to become stronger and stronger, and eventually counter-rotating shock waves evolve into reverse detonation waves.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0089207
I. INTRODUCTION
The rotating detonation engine (RDE) is one of the most promis-
ing aerospace engines, with the advantages of low entropy gain and
high thermal efficiency.
1
RDE is expected to break through the current
engine development bottleneck and gradually become an international
research hotspot. RDE uses detonation as the combustion method,
and its schematic diagram is shown in Fig. 1.
2
To date, a number of
international laboratories have conducted a series of experimental,
numerical, and theoretical studies of RDE, such as mode and
stability,
3–6
flow field details,
7–10
visualization,
11–13
multiphase
detonation,
14–16
combustor configurations,
17,18
flow field analysis
methods,
19
etc.
The mode and stability problem of RDE is one of the important
research directions in this field, because it directly determines the
operating condition of the engine. Anand et al. found four kinds of
instabilities in RDE: chaotic instability, waxing and waning instability,
mode switching, and longitudinal pulsed detonation instability.
20
They
also investigated experimentally the mechanism of longitudinal pulsed
detonation instability.
21
Smirnov et al. used different propellant com-
positions to obtain different operating modes of RDE.
22
They also
studied the effects of combustor configuration, oxygen concentration,
and inlet conditions on initiation and stability of RDE.
23
Zhang et al.
investigated the bifurcation mechanism of three-dimensional rotating
detonation waves.
24
Bennewitz et al. developed an image processing
technique and performed a two-dimensional Fourier transform on the
results to determine the mode of the rotating detonation wave.
25
Koch
and Kutz developed a modeling framework for RDE to demonstrate
the propagation of detonation waves
26
and to analyze the multiple
nonlinear dynamical behaviors of detonation waves.
27
Bluemner et al.
investigated the steady single-wave mode and the counter-rotating
wave mode of RDE by analyzing high-speed camera data and high-
frequency pressure sensor data
28
and subsequently analyzed the effect
of different injector geometries and different outlet restrictions on the
mode of RDE.
29
In experiment and numerical simulation studies of RDE, the
presence of counter-rotating shock waves in the combustor in the
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-1
Published under an exclusive license by AIP Publishing
Physics of Fluids ARTICLE scitation.org/journal/phf

opposite direction of the detonation wave propagation is often found.
These counter-rotating shock waves propagate continuously in the
combustor, colliding with the detonation wave periodically and trig-
gering instabilities, even leading to mode transitions. Bluemner et al.
used a coaxial annular cavity configuration and found two counter-
rotating waves at equal speed (2CR), two counter-rotating waves tran-
sitioning (2CRT), single wave with counter-rotating components
(SWCC), and single wave (SW) and investigated the factors influenc-
ing the mode transition of the wave.
30
The SWCC mode is the mode
with a counter-rotating shock wave in the combustor. Xia et al. investi-
gated the mode transition process from single wave to two-wave colli-
sion by numerical simulation of two-dimensional flow field, and they
concluded that the counter-rotating shock wave in the combustor
playsakeyroleintheformationofthetwo-wavecollisionmode.
31
Chacon et al. used three injection methods in a coaxial annular cavity
combustor, after which they found the secondary wave that propa-
gated in the opposite direction to the detonation wave.
32
They also
found that when the secondary wave interacts with the detonation
wave, the pressure of the detonation wave increases and the heat
release decreases, which leads to the instability in the RDE.
The operating mode parameters of RDE contain the number and
the propagation direction of detonation waves, and mechanism of
mode transition and mode control of RDE is of great value in both sci-
entific research and practical application of RDE. Yao achieved the
change in the number of detonation waves by varying the total pres-
sure of inlet and studied the process and causes of mode transition.
33
The direction change and control of the detonation wave is a very
meaningful problem. It is also the actual demand for RDE,
34,35
which
belongs to the mode transition and control problem. Zhao et al.
adjusted the sudden drop or rise of the total pressure of inlet to achieve
the extinction and reformation of the detonation wave, and they also
studied the effect of time-varying total pressure of inlet on the stability
of the RDE.
36
The evolution mechanism and control method about
the direction of the detonation wave propagation is also a very
important element in the study of the mode transition mechanism
and mode control. Stoddard and Gutmark have used barriers such as
walls and ramps in the combustor to achieve detonation wave propa-
gation in the specified direction.
37
Knowlen and Kurosaka designed
the wave generator (WG) to control the number and direction of the
detonation wave, the principle of the WG is multiple spark plugs in a
certain order of ignition.
38
They also found that if only one spark plug
is used for ignition, then the two detonation waves in opposite direc-
tions will be formed at the initial moment, and both waves will be pre-
sent throughout the experiment.
39
Kawalec found that the direction of
the detonation wave could not be controlled by using a single spark
plug, and that the use of aluminum foil next to the spark plug could
control the direction of the detonation wave in 80% of the experi-
ments, and that the use of an eccentric chamber could effectively con-
trol the direction of the detonation wave.
40
Zhao and Zhang found
that the chaotic propagation phase of the detonation wave in the RDE
is responsible for the change in direction of the detonation wave.
41
At present, the directional control of the rotating detonation
wave research is mostly focused on controlling the direction of the det-
onation wave during the initiation period, but it is rare to change the
direction of the detonation wave when the RDE works stably. When
the rotating detonation wave is stable propagation, control its inver-
sion, so that the RDE transition from one steady state to another. Such
control can make the RDE mode transition process very smooth and
reliable. We combine this very meaningful problem with the counter-
rotating shock wave problem to achieve detonation wave inversion by
controlling the counter-rotating shock wave.
So far, the research on the counter-rotating shock wave phenom-
enon in RDE is mainly focused on the configuration of coaxial annular
cavity, and the research on the hollow combustor configuration is rare.
Ye-Tao and Jian-Ping and Tang et al. first proposed the concept of
hollow combustor configuration and verified its feasibility by numeri-
calsimulation,andtheyalsoanalyzedtheflowfieldcharacteristicsof
the hollow combustor.
42,43
The hollow combustor configuration has
also been experimentally verified.
4,44–47
The advantage of hollow com-
bustor is that it can enhance the detonation wave velocity
44
and avoid
the problem of internal column ablation,
43
and the addition of the
Laval nozzle at the end of the combustor can significantly improve the
propulsion performance of the RDE.
48
However, the wave system evo-
lution of the hollow combustor is more complex than that of the coax-
ial annular cavity configuration.
49
Moreover, according to the results
we obtained, after the addition of Laval nozzle, there will be obvious
counter-rotating shock waves in the combustor, and the instability of
RDE will be enhanced again. In summary, it is significant to study the
counter-rotating shock wave phenomenon and the mode control
mechanism in the hollow combustor with Laval nozzle.
In summary, the structure of this paper is as follows. First, the
phenomenon and distribution law of counter-rotating shock waves are
analyzed. Second, the method and process of controlling the direction
of the detonation wave by controlling the counter-rotating shock
waves are proposed. Third, the mechanism of detonation wave direc-
tion control is investigated.
II. COMPUTATIONAL METHODS
Figure 2 shows the geometric configurations and mesh settings of
the computational domain of case 1 and case 2. The three-
dimensional case uses the geometry of a hollow cylinder with an axial
FIG. 1. Schematic diagram of the coaxial annular cavity RDE. Reproduced with
permission from Luan et al.,“Analytical and numerical study of the expansion effect
on the velocity deficit of rotating detonation waves,”Combust. Theory Modell. 24,
761–774 (2020). Copyright 2020 Authors, licensed under a Creative Commons
Attribution (CC BY) license.
2
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-2
Published under an exclusive license by AIP Publishing

length of L
1
¼30 mm and a radius of r
1
¼16 mm. A Laval nozzle
with an axial length of L
2
¼16.1mm is set at the end of the combus-
tor. The Laval nozzle has a throat diameter of d
1
¼22.6 mm, an outlet
diameter of d
2
¼32 mm, and a shrinkage ratio (ratio of inlet area of
the nozzle to throat area of the nozzle) of 2. A cylindrical outflow field
with a radius of r
3
¼106 mm and an axial length of L
3
¼600 mm is
set downstream of the combustor. Setting a larger area of the outflow
field is to simulate the real exit environment of the combustor. This
method can avoid the errors caused by artificially setting the outlet
boundary conditions in the previous numerical simulation. The injec-
tion method is a double-row array of injection holes, with a total of 48
injection holes of radius r
2
¼1.5 mm. The convergent nozzle boundary
condition is used for the inlet, the fuel is hydrogen/air with an equiva-
lence ratio of 1, the total temperature of inlet is 360 K, and the total pres-
sure of the inlet is different for different cases. The inlet total pressure of
case 1 is 5 atm. The inlet total pressure of case 2 is reduced from 5 to
4 atm, using the 1500 lsmomentofcase1astheinitialmoment.The
ambient back pressure is 1 atm, and the mesh of the 3D case is the poly-
hedral mesh that can adapt to the complex geometry. Seven ranges of
themeshsizearesettoensuretheaccuracyofthecalculationandto
reduce the computational effort. The total number of meshes is about
6.34 10
6
.Thenumericaltimestepsizeisabout7.810
9
s.
Referring to the error estimation method proposed by Smirnov et al.,
the total error of the cases in this paper is less than 3%.
50,51
This study uses our solver BYRFoam, developed on the open-
source computational fluid dynamics platform OpenFOAM, which
can be used to perform numerical simulations in the computational
domain of complex configurations and has been introduced and vali-
dated in previous studies.
52
According to the study by Oran et al.,
53
since the timescale is very small compared to the diffusion scale, it is
feasible to neglect the effects of viscosity, heat conduction and molecu-
lar diffusion when considering the detonation front. For this reason,
the Euler equation is currently applied to many numerical simulation
studies on detonation
54–56
and RDE.
6,10,24
In this study, the unsteady
three-dimensional Euler equations are used and coupled with chemical
reaction source terms as follows:
@q
@tþrðquÞ¼0;(1)
@qu
@tþrðquuÞþrp¼0;(2)
@qe
@tþr ðqeþpÞu½¼0;(3)
@qYi
@tþrðqYiuÞ¼ _
xi:(4)
In this paper, we use the detailed chemical reaction kinetic mech-
anism of the hydrogen/air proposed by

O Conaire et al.,
57
which con-
tains a total of 19 reactions and includes the components: H, H
2
,H
2
O,
H
2
O
2
,HO
2
,N
2
,O,O
2
, and OH.
In the following, different mesh sizes are used to solve the one-
dimensional detonation tube problem to verify the independence of
the mesh, and the detailed chemical reaction kinetic mechanism is
used for all cases.
57
The length of the detonation tube is L
6
¼0.5 m.
Eight cases are calculated with mesh sizes in the range of 0.05 to
0.2 mm. The fuel is premixed hydrogen/air with an equivalent ratio of
1, the initial pressure is 1.5 atm, and the initial temperature is 320 K.
The pressure at the hot spot on the left side of the detonation tube is
5 MPa, and the temperature is 2000 K. The results of the pressure dis-
tribution and the velocity of the detonation wave for these cases are
shown in Fig. 3.Figure 3(a) shows the pressure distribution of the
0.05, 0.1, 0.15, and 0.2 mm mesh size, and the detonation wave is
located in the local zoom. The pressure distribution curves basically
overlap, which indicates that the detonation wave can be well captured
when the mesh size is 0.05 to 0.2mm. Figure 3(b) shows the detona-
tion wave velocity of eight cases, the minimum value is 1984.8 m/s,
and the maximum value is 1985.9m/s, both of which are very close to
the C–J velocity 1985.5 m/s, which indicates that the velocity of the
detonation wave reaches mesh convergence.
The following numerical simulations of the hollow rotating deto-
nation engine with Laval nozzle at different grid sizes are performed to
verify the grid independence. The maximum grid sizes of the combus-
tor head are 0.1, 0.15, and 0.2 mm, respectively, and the calculated
results are shown in Fig. 4. There are four detonation waves and five
counter-rotating shock waves in the RDE flow field with different grid
sizes. Under different grid sizes, the flow field and wave system struc-
tures are basically the same. This indicates that the three grid sizes
capture the wave system structure exactly the same, only the difference
in resolution. This validates the reasonableness of the grid size chosen
in this paper.
The accuracy of ignoring viscosity is verified in the following.
The calculation method of Liu et al.
58
is used for the case of
FIG. 2. Geometric configuration and mesh of the computational domain. (a) Local enlarged figure. (b) Whole figure. The coordinate system is given in Fig. 2(b).
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-3
Published under an exclusive license by AIP Publishing

considering viscosity, and the calculation results are shown in Fig. 5.It
can be seen that the flow field and wave system structure of the two
cases are basically the same, and the number and structure of detona-
tion waves and counter-rotating shock waves in the flow field are the
same. This indicates that the wave system structure is captured exactly
the same for the cases with and without viscosity. This verifies the
accuracy of ignoring viscosity.
Figure 6 shows the calculation result of the outflow field, the
Mach disk structure is clear. The structure of the exhaust plume is
qualitatively consistent with the experimental and theoretical results
59
under the over-expansion condition, which shows that the additional
outflow field can truly and effectively simulate the physical conditions
at the exit of the combustor and avoid the errors caused by setting the
outlet boundary conditions.
III. RESULTS AND DISCUSSION
A. Counter-rotating shock wave in RDE
First, the origination of counter-rotating shock waves is analyzed.
Figure 7 shows the pressure gradient nephogram of the flow field
(r ¼14.25mm). At the moment of 806 ls, there are four detonation
waves in the flow field, which are indicated by blue arrow 1. In addi-
tion, there are some weak shock waves in the flow field with the oppo-
site direction of the detonation wave, such as the yellow arrow 2.
Zhang et al. suggest that some weak shock waves are generated in the
flow field due to the intensity and height oscillation of the detonation
wave, which makes the flow field very complicated.
60
At the moment
of 814 ls, the shock wave represented by the yellow arrow 2 collides
with the detonation wave and its intensity gets increased, as shown by
the yellow arrow 3. After a period of evolution, the intensity of the ini-
tial weak shock wave keeps increasing, and eventually, the weak shock
wave becomes the stronger counter-rotating shock wave, as shown by
yellow arrow 4. Finally, five stronger counter-rotating shock waves are
formed in the flow field, and they continue to interact with the four
detonation waves in the flow field and remain in the flow field. This is
the formation process of counter-rotating shock waves.
Figure 8 shows the wave system structure of the flow field at dif-
ferent locations in the detonation combustor of case 1 to analyze the
interaction between the detonation wave and the counter-rotating
shock wave and the variation of the intensity of the counter-rotating
shock wave with radial position. The total pressure of inlet of case 1 is
5 atm. The circles in the figure represent the locations of the taken
cross sections. The method of distinguishing detonation wave and
counter-rotating shock wave is as follows: First, the pressure gradient
of the detonation wave and oblique shock wave is significantly higher
than that of the counter-rotating shock wave. The line of the detona-
tion wave and oblique shock wave is very deep in the pressure gradient
nephogram, but the line of the counter-rotating shock wave is shallow.
FIG. 3. Pressure distribution and detonation wave velocity of the one-dimensional detonation tube (different mesh sizes, t ¼150 ls). (a) One-dimensional detonation tube
pressure distribution. (b) The velocity of the one-dimensional detonation wave.
FIG. 4. Grid independence verification of the hollow rotating detonation engine. (a)
Pressure nephogram, the maximum sizes of the combustor head grid from the left
to the right figure are 0.1, 0.15, and 0.2 mm, respectively. (b) Temperature nepho-
gram, the maximum sizes of the combustor head grid from the left to the right figure
are 0.1, 0.15, and 0.2 mm, respectively.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-4
Published under an exclusive license by AIP Publishing

Second, the angle between the oblique shock wave and the inlet wall is
obviously smaller than the angle between the counter-rotating shock
wave and the inlet wall. Third, the pressure gradient of both the deto-
nation wave surface and the region behind the detonation wave is
high, but the pressure gradient behind the counter-rotating shock
wave is significantly lower. It is shown in the pressure gradient nepho-
gram that there is a dark region behind the detonation wave, but there
is no obvious dark region behind the counter-rotating shock wave.
The light blue circle (radius 16 mm) in Fig. 8(g) shows the radial
position of the outer wall of the combustor, and the wave system struc-
ture corresponds to Fig. 8(a). The figure uses cyan arrows to indicate
the location and direction of detonation waves and oblique shock
waves. Red arrows indicate the location and direction of counter-
rotating shock waves. Cyan circle 1 indicates the high-pressure region
behind detonation waves and oblique shock waves. Red circle 2 indi-
cates the high-pressure region behind the counter-rotating shock
waves. Light green box 3 indicates the high-pressure region formed by
the collision of detonation waves and counter-rotating shock waves.
Detonation wave and counter-rotating shock wave in the Laval nozzle
contraction section are reflected and form two reflected shock waves.
Orange box 4 indicates the high-pressure area behind the detonation
wave, counter-rotating shock wave, and two reflected shock waves.
This indicates that the strength of the counter-rotating shock wave is
great.
The light green circles in Fig. 8(g) (radii of 14.25 and 11 mm,
respectively) show the radial locations of the outer and inner injection
holes of the combustor, and the wave system structures correspond to
Figs. 8(b) and 8(c), respectively. The wave system structure in Fig. 8(b)
is the same as that in Fig. 8(a). The intensity of the counter-rotating
shock wave in Fig. 8(c) is reduced. The high-pressure region behind
the counter-rotating shock wave is not obvious, even if the high-
pressure region formed by the collision of the counter-rotating shock
wave and the detonation wave is also weak. The counter-rotating
shock wave does not produce a strong reflected shock wave in the con-
striction section of the Laval nozzle. The red box 5 is used to indicate
the post-wave high-pressure region generated by the oblique shock
wave and reflected shock wave. The dark blue circles (radii of 9.25, 8,
and 6 mm, respectively) in Fig. 8(g) are the radial locations of the
FIG. 5. Comparison of hollow rotating
detonation engine with and without vis-
cosity. (a) Pressure nephogram, the left
figure is the result of ignoring viscosity,
and the right figure is the result of con-
sidering viscosity. (b) Temperature
nephogram, the left figure is the result
of ignoring viscosity, and the right figure
is the result of considering viscosity.
FIG. 6. Temperature nephogram of the outflow field outside the combustor.
FIG. 7. Schematic diagram of the counter-rotating shock waves originate. (a) 806 ls.
(b) 814 ls. (c) 1500 ls.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-5
Published under an exclusive license by AIP Publishing

internal non-inlet of the combustor, and the wave system structures
correspond to Figs. 8(d)–8(f), respectively. It can be seen that the
counter-rotating shock waves are almost non-existent in the region of
internal non-intake. In summary, the counter-rotating shock wave
interacts with the detonation wave and triggers the instability of the
flow field, and the intensity of the counter-rotating shock wave
decreases with the decrease in the radius of the location.
It should be noted that the peak pressure at the counter-rotating
shock wave surface is much smaller than that at the detonation sur-
face. Figure 9 visualizes the results of the strength evaluation of the
detonation waves and counter-rotating shock waves. Figure 9(a) shows
thepressurenephogram(r¼14.25 mm) with cyan arrow 2 indicating
the detonation wave and red arrow 3 indicating the counter-rotating
shock wave. Figure 9(b) shows the pressure distribution curve at yel-
low line 1. Blue circle 4 indicates the peak pressure of the detonation
wave surface, and red circle 5 indicates the peak pressure of the
counter-rotating shock wave surface. It can be seen that the peak pres-
sure of the counter-rotating shock wave surface is small, while the
peak pressure of the detonation wave surface is large. This shows that
the strength of the counter-rotating shock wave is much smaller than
that of the detonation wave. Therefore, when the inlet conditions do
not change, the interactions between the detonation waves and the
counter-rotating shock waves do not lead to a change in the number
and direction of the detonation wave.
B. The method and process of detonation wave
direction control
Figure 10 shows the pressure distribution at the inlet wall from
the head of the combustor, demonstrating the evolution of the wave
system in case 2 where the detonation wave changes direction and
finally tends to propagate steadily, by reducing the total pressure of
inlet from 5 to 4 atm at 1500 ls, with an applied disturbance of 20% of
the original total pressure of inlet. At 1500 ls, there are four counter-
clockwise rotating detonation waves (D1–D4) and five clockwise rotat-
ing counter-rotating shock waves (S1–S5) in the flow field. One
thousand five hundred to 1548 ls is the first stage of the detonation
wave changing direction. From 1506 to 1518 ls, the intensity of the
detonation wave gradually decreases. At 1524 ls, the detonation waves
FIG. 8. The flow field in the detonation combustor—detonation wave and counter-rotating shock wave interaction. (a)–(f) Wave system structure of the flow field for cylindrical
cross sections of different radii. The red region is the fresh gas, and the blue region represents the region where the pressure is higher than the total pressure of inlet.
(g) Three-dimensional temperature nephogram of the combustor. The circles show the positions of multiple cross sections.
FIG. 9. Strength evaluation of the detonation waves and counter-rotating shock
waves. (a) Pressure nephogram. (b) Pressure distribution curve at yellow line 1.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-6
Published under an exclusive license by AIP Publishing

D3 and D4 are extinguished, degenerated into shock waves S6 and S7,
and counter-rotating shock waves S2 to S4 are transformed into the
detonation waves D5 to D7. At 1530 ls, the detonation wave D1 is
extinguished, degenerated into shock wave S8, and counter-rotating
shock wave S5 is transformed into the detonation wave D8. At
1536 ls, the detonation wave D2 is extinguished, degenerated into
shock wave S9, and counter-rotating shock wave S1 is transformed
into the detonation wave D9. After the abovementioned process, the
intensity of the original four detonation waves decreases gradually
until detonation waves are extinguished and transformed into shock
waves, and the intensity of five counter-rotating shock waves increases
gradually until counter-rotating shock waves transform into detona-
tion waves. From 1536 to 1548 ls, the intensity of four shock waves
formed by the extinction of the original detonation waves gradually
decreases until the shock waves disappear, and there are only five deto-
nation waves in the flow field with the direction opposite to that of the
original detonation waves.
Figure 11 shows the pressure distribution at the inlet wall from
the head of the combustor, demonstrating the evolution of the wave
system in case 2 where the detonation wave tends to propagate
steadily. One thousand five hundred forty-eight to 1596 lsisthesec-
ond stage of the detonation wave change direction. In this process, det-
onation waves D5, D6, and D9 gradually extinguished, which is due to
the flow field of fresh gas cannot maintain the propagation of five det-
onation waves, there are only two detonation waves (D7 and D8) in
the flow field. One thousand five hundred ninety-six to 1634 lsisthe
third stage of the detonation wave change direction. In this process,
due to the number of detonation waves in the flow field being less, the
mass of fresh gas accumulated in front of the detonation wave starts to
increase. Shock waves formed when the detonation wave is extin-
guished in the second stage encounter enough fresh gas to form deto-
nation waves D10 and D11 again. Then, in 1634 ls, there are four
detonation waves in the flow field. One thousand six hundred thirty-
four to 1866 ls is the fourth stage of the detonation wave change direc-
tion. In this process, the four detonation waves in the flow field are
self-adjusting and, eventually, they are almost the same spacing and
stable propagation, but their direction is opposite to the direction of
detonation waves at 1500 ls. At this point, the detonation wave direc-
tion conversion and stability are completed. These are the four stages
in which detonation waves change direction and eventually stabilize.
At the 2000 ls moment, counter-rotating shock waves form
again. Figure 12 shows the pressure nephogram and pressure gradient
nephogram (r ¼14.25 mm) at the 2000 ls moment, where D7, D8,
D10, and D11 are the four detonation waves. Five new counter-
rotating shock waves are formed in the flow field, which are denoted
by S10 to S14. The origin and formation process of counter-rotating
shock waves are similar to what is described in Fig. 7. The above
results show that the coexistence mode of the detonation waves and
counter-rotating shock waves will eventually be formed under the inlet
conditions of both case 1 and case 2.
The above phenomena and processes of detonation wave rever-
sal in case 2 are very similar to the experimental results of
Canteins.
61
Figure 13 shows the pressure signal obtained by
Canteins. They used high-frequency pressure sensors in their experi-
ments, showing the phenomenon and process of detonation wave
reversal. They found that the intensity of the clockwise propagating
detonation wave gradually decreases and eventually becomes a shock
wave, and the intensity of the counter-rotating shock wave gradually
increases and eventually evolves into a counterclockwise detonation
wave, achieving a change in the direction of the detonation wave.
The above results are consistent with the phenomenon and process
of case 2. The comparison with the experimental data verifies the
accuracy of the phenomenon obtained in this study. Then, we will
explain in detail the reasons for the change of direction of the deto-
nation wave in the RDE and proposes the mechanism to control the
direction of the detonation wave.
FIG. 10. Pressure distribution at the inlet wall from the combustor head: evolution
of the wave system as the detonation wave changes direction. The solid arrows
indicate the direction of the detonation wave, the dashed arrows indicate the direc-
tion of the shock wave, the yellow arrows represent counterclockwise, the orange
arrows represent clockwise, and the position of the shock wave is marked by a cir-
cle. (a) 1500 ls. (b) 1506 ls. (c) 1512 ls. (d) 1518 ls. (e) 1524 ls. (f) 1530 ls.
(g) 1536 ls. (h) 1542 ls. (i) 1548 ls.
FIG. 11. Pressure distribution at the inlet wall from the combustor head: evolution
of the wave system as the detonation eventually tends to propagate steadily. The
meaning of the markers is the same as in Fig. 10. (a) 1596 ls. (b) 1634 ls. (c)
1866 ls.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-7
Published under an exclusive license by AIP Publishing

C. The mechanism of detonation wave
direction control
Figure 14 shows the pressure–time curve of the detonation wave,
frequency scatter diagram, spatiotemporal distribution of pressure and
spatiotemporal distribution of fresh gas height. The above data are
analyzed to investigate the evolution mechanism of the wave system in
which the detonation wave changes direction and eventually tends to
propagate steadily. The procedure for graphing the spatiotemporal dis-
tribution of pressure is as follows. First, the pressure value of the
expanded cylindrical section is integrated along the axial direction and
divided by the circumference of the cylindrical section, and the inte-
gration range is 0 to 4.2 mm. Then, the obtained one-dimensional set
of pressure data are expressed in shades of color. Finally, the color bar
corresponding to each time step is arranged in time order to obtain
the spatiotemporal distribution of pressure with the horizontal axis of
time and the vertical axis of angle. The dark line in the graph repre-
sents the motion trajectory of the detonation wave, and the light line
represents the motion trajectory of the counter-rotating shock wave.
The spatiotemporal distribution of pressure clearly shows the positions
of detonation waves and counter-rotating shock waves at different
times, as well as their trajectories. Thus, the evolution of the wave sys-
tem in the flow field and the wave propagation characteristics can be
understood clearly. The plotting steps of the spatiotemporal distribu-
tion of fresh gas height are as follows: first, extract the fresh gas height
values of the expanded cylindrical cross section, after which the
obtained one-dimensional set of height data are represented by the
shades of color, and finally the color bars corresponding to each time
step are arranged in chronological order to obtain the spatiotemporal
distribution of fresh gas height with the time on the horizontal axis
FIG. 13. Pressure time profile during the
change of direction of the detonation wave
in the experiment of RDE. Reproduced with
permission from G. Canteins, Etude de la
D
etonation Continue Rotative-Application 
a
la Propulsion (Universit
e de Poitiers, 2006).
Copyright 2006 Author, licensed under a
Creative Commons Attribution (CC BY)
license.
61
FIG. 12. Pressure nephogram and pres-
sure gradient nephogram at 2000 ls
moment. (a) Pressure nephogram of the
inlet wall surface of the combustor head.
The legend is the same as that of Fig. 10.
(b) Pressure gradient nephogram of the
flow field. The red arrows indicate the
direction of the detonation wave, the yel-
low arrows indicate the direction of the
counter-rotating shock wave, and the blue
ellipse circle indicates one of the counter-
rotating shock waves.
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Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-8
Published under an exclusive license by AIP Publishing

and the angle on the vertical axis. The white areas in the figure repre-
sent the areas where fresh gas is present. The spatiotemporal distribu-
tion of fresh gas heights clearly shows the distribution of fresh gas at
different moments and the variation of the incoming fresh gas flow
with time.
Figure 14(a) shows the pressure time profile and frequency scat-
ter of the detonation wave, and the position of the pressure probe is
marked by the solid blue line in Fig. 14(b). It can be seen that after the
decrease in the total pressure of inlet at 1500 ls, the peak of the
pressure–time curve and the frequency scatter of the detonation wave
become unstable, which is the instability of the flow field caused by the
change of direction of the detonation wave. After 1675 ls, i.e., after
the detonation wave has changed direction, the frequency scatter of
the detonation wave gradually stabilizes, while the peak of the pressure
time curve fluctuates, which is caused by the interaction between the
counter-rotating shock wave and the detonation wave. Figure 14(b)
shows the spatiotemporal distribution of the pressure of the detona-
tionwave,usingthebluedashedlineandthereddashedlinetoiden-
tify the detonation wave and the counter-rotating shock wave,
respectively. Near 1500 ls, the slope of the line represented by the det-
onation wave is positive, indicating that the detonation wave propa-
gates counterclockwise. The slope of the line represented by the
counter-rotating shock wave is negative, indicating that the counter-
rotating shock wave propagates clockwise. Red circle 1 reflects the phe-
nomenon of gradual deepening of the lines represented by five
counter-rotating shock waves, which indicates that the intensity of the
counter-rotating shock waves gradually increases and eventually trans-
forms into detonation waves. This corresponds to the first stage of the
change of the detonation wave direction.
Around 1550 ls, the line representing the detonation wave trans-
formed by the counter-rotating shock wave gradually becomes lighter,
indicating that the intensity of the detonation wave is gradually weak-
ening, and some of the detonation wave is extinguished, which corre-
sponds to the second stage of the detonation wave changing direction.
Then near 1625 ls, four detonation waves are formed again, marked
by the light blue circle 2, which corresponds to the third stage of the
change of direction of the detonation wave. Near 1625 ls, the lines
representing the detonation waves are not parallel and not equally
spaced in the spatiotemporal distribution of the pressure, and the fre-
quency scatter in Fig. 14(a) fluctuates. The above results indicate that
the spacing of the detonation waves is not equal. Then, the spacing of
the detonation waves gradually converges to the same by self-
adjustment. In the vicinity of 1866 ls, the frequency scatter almost no
fluctuations, which indicates that the spacings of the detonation wave
are basically the same, which corresponds to the fourth stage of the
detonation wave change direction. It is worth noting that in the pro-
cess of completing the change of the detonation wave direction to sta-
bilization, the counter-rotating shock wave gradually generated again,
the dark blue circle 3 identifies the coexistence of the detonation wave
and counter-rotating shock wave state. This indicates that after the
propagation characteristic of the detonation wave change, the counter-
rotating shock wave will still reappear.
Figure 14(c) shows the spatiotemporal distribution of the fresh
gas height, and the green triangle 4 identifies the change process of the
fresh gas layer after the change of the total pressure of inlet. It can be
seen that after the decrease in the total pressure of inlet, the interrup-
tion of the wide white line representing the fresh gas layer becomes
larger and larger, and the fresh gas layer becomes more and more
FIG. 14. Pressure time curves, frequency scatter diagrams, spatiotemporal distribution of pressure, spatiotemporal distribution of fresh gas height. (a) Pressure time profile
and frequency scatter diagram of the detonation wave. (b) Spatio-temporal distribution of the pressure of the detonation wave. The blue solid line marks the position of the
pressure probe. (c) Spatio-temporal distribution of the fresh gas height. (a)–(c) Share a common time axis. The radius r of the cylindrical coordinate system is 14.25 mm.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-9
Published under an exclusive license by AIP Publishing

irregular, which indicates that the fresh gas is increasingly blocked by
the counter-rotating shock wave. The reason for this phenomenon will
be analyzed in Fig. 15. Cyan box 5 identifies a large influx of fresh gas.
The reason for this phenomenon is that when multiple counter-
rotating shock waves transform into the detonation wave, the lack of
fresh gas leads to the extinction of some detonation waves. Then, the
blocking effect on the fresh gas is reduced and a large amount of fresh
gas flows into the combustor. These fresh gases allow the re-formation
of the extinguished detonation wave, which explains the mechanism
analyzed above that the detonation wave converted from counter-
rotating shock wave can be regenerated after extinction. The orange
circle 6 marks the break in the wide white line representing the fresh
gas, which proves the re-formation of the counter-rotating shock
wave.
Figure 15 shows the pressure gradient nephogram
(r ¼14.25mm) of the flow field, and the fresh gas layer is marked with
the blue line in the diagram. The counter-rotating shock wave must
enter the fresh gas before colliding with the detonation wave, and the
red boxes 1 to 3 are local enlargements of the phenomenon of the
counter-rotating shock wave entering the fresh gas. As the counter-
rotating shock wave into the fresh gas process is the shock wave from
the high temperature and low density gas into the low temperature
and high density gas, according to the law of motion of the shock
wave through the interface of different media,
62
the discriminative for-
mula is shown as
c1q1<c2q2;(5)
ðc1þ1Þq1<ðc2þ1Þq2;(6)
where c1and q1are the specific heat ratio and density of the high tem-
perature low density gas, c2and q2are the specific heat ratio and den-
sity of the low temperature high density gas, respectively. This
condition can be inferred that the counter-rotating shock wave will be
reflected after passing through the interface between the product and
the fresh gas, forming a reflected shock wave. As can be seen in the fig-
ure, after the counter-rotating shock wave enters the fresh gas, a trans-
mitted shock wave and a reflected shock wave are formed. The
transmitted shock waves are marked by yellow arrows a1 to a3, the
positions of the reflected shock wave are marked by cyan oval circles,
and the propagation directions of the reflected shock wave are marked
by dark blue arrows b1 to b3. The reflected shock wave generated at
the fresh gas interface increases the post-wave pressure of the counter-
rotating shock wave, which in turn causes the intensity of the counter-
rotating shock wave to increase as it enters the fresh gas. As in the case
of the counter-rotating shock wave a2 in red box 2, the part of a2 in
the fresh gas has a greater intensity than the part of a2 in the detona-
tion product.
When the total pressure of inlet is reduced, the blocking effect of
the fresh gas by the counter-rotating shock wave is enhanced, which in
turn leads to a more irregular fresh gas layer. Then, when the counter-
rotating shock wave enters the fresh gas, its interaction with the inter-
face between the product and the fresh gas will be more intense. This
results in a higher intensity of the reflected shock wave and a higher
intensity of the counter-rotating shock wave, which in turn will pro-
duce a greater blocking effect on the fresh gas. This process repeats
itself over and over again, creating a positive feedback type of interac-
tion between the counter-rotating shock waves and the fresh gas,
which is the reason for the increasing blocking effect of the counter-
rotating shock wave on the fresh gas analyzed above. As shown in Fig.
15,at1518ls, the irregularity of the fresh gas layer is greater than that
at 1510 ls, and the intensity of the reflected shock wave b3 at 1518 ls
is also greater than that of the reflected shock waves b1 and b2 at
1510 ls. As the interaction between the counter-rotating shock wave
and fresh gas is positive feedback, that is, after the total pressure of
inlet is reduced, the fresh gas is blocked more and more strongly, the
intensity of the counter-rotating shock wave is stronger and stronger,
and eventually counter-rotating shock waves will evolve into detona-
tionwavestoachieveachangeinthedirectionofdetonationwaves.
The following is a summary of the above analysis of the phenom-
enon of the detonation wave changing direction and eventually tend-
ing to stable propagation, and a summary of the proposed mechanism
of the above phenomenon. As shown in Fig. 16, when the total pres-
sure p0of inlet decreases, the blocking effect of the counter-rotating
shock wave on the fresh gas will be enhanced, the concavity of the
fresh gas increases, the fresh gas layer will become more irregular.
Thus, the next counter-rotating shock wave will be disturbed more
strongly by the boundary between the fresh gas and the high tempera-
ture product when it enters the fresh gas. This leads to an increase in
the pressure PRof the reflected shock wave generated at the boundary
due to reflection, an increase in the post-wave pressure of the counter-
rotating shock wave, and an increase in the counter-rotating shock
wave pressure PS. Later, stronger counter-rotating shock waves collide
with the fresh gas, causing the blocking effect on the fresh gas to be
further enhanced. This forms a complete cycle characterizing the inter-
action between the counter-rotating shock wave and the fresh gas.
This cycle is a positive feedback regulation, which means that each var-
iable in the cycle varies monotonically as the cycle continues, indicat-
ing that at some point, the modalities of the system must change
because some variables exceed the threshold of the system. When the
intensity of the blocking effect of the fresh gas by the counter-rotating
shock wave is too large, original detonation waves due to the lack of
fresh gas in front of the wave are extinguished. When counter-rotating
shock waves are too strong and collide with the fresh gas, they will
transform into detonation waves. The combination of the above two
results makes the direction of the detonation wave change. The above
is the mechanism of the change in the direction of the detonation
wave, which is the mechanism of the first stage mentioned above.
FIG. 15. Pressure gradient nephogram of the flow field, with the fresh gas layer
identified by the blue line in the figure. (a) 1510 ls. (b) 1518 ls.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-10
Published under an exclusive license by AIP Publishing

As shown in Fig. 17, after more counter-rotating shock waves are
transformed into detonation waves, some of the detonation waves are
extinguished and transformed into shock waves because there is not
enough fresh gas in the combustor to support all of the detonation waves.
This will reduce the pressure at the head of the combustor, a large influx
of fresh gas into the combustor, then shock waves formed by detonation
waves extinguished will be transformed into detonation waves again.
Then, the distances between the detonation waves will be self-regulating
FIG. 16. Mechanistic diagram of the change in the direction of the detonation wave. Where the light blue box represents the cause and effect, and the dark blue box repre-
sents the interaction process between the shock wave and fresh gas.
FIG. 17. Mechanistic diagram of stabilization of the detonation wave after changing its direction. The light blue boxes represent the characteristics of the flow field wave system
at each evolutionary stage, and the dark blue boxes represent intermediate processes and phenomena. The characteristic phenomenon boxes identify the very typical phe-
nomena analyzed above.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-11
Published under an exclusive license by AIP Publishing

and tend to the same. After the flow field stabilizes, due to the reemer-
gence of counter-rotating shock waves, detonation waves and counter-
rotating shock waves continue to collide. The above is the mechanism of
stabilization of the detonation wave after changing direction.
IV. CONCLUSION
The object of this paper is a hollow combustor with Laval nozzle
and an array of injection holes for the inlet. The physical conditions at
the exit of the combustor are realistically simulated by using the out-
flow field of a larger area. The phenomenon and distribution law of
the counter-rotating shock wave in the combustor are studied, the
method and process of detonation wave direction control are pro-
posed, and the mechanism of detonation wave direction control is
investigated. The main conclusions are as follows:
1. The sources of the counter-rotating shock waves are some weak
shock waves in the flow field with the opposite direction of the
detonation wave. The interaction between the counter-rotating
shock wave and the detonation wave will trigger the instability of
the flow field, and the intensity of the counter-rotating shock
wave decreases with the decrease of the radius of its position.
The peak pressure at the counter-rotating shock wave surface is
much smaller than that at the detonation surface.
2. The strength of the counter-rotating shock wave is controlled by
reducing the total pressure of the inlet, which, in turn, enables the
control of the direction of the detonation wave. The process of
detonation wave redirection is that counter-rotating shock waves
evolve into detonation waves, several detonation waves are extin-
guished, detonation waves form again, and then detonation waves
propagate stably.
3. The mechanism of controlling the direction of the detonation
wave is that the change in the inlet conditions break the balance
of the detonation wave and counter-rotating shock wave. This
stimulates the positive feedback interaction between the counter-
rotating shock wave and fresh gas, and makes the intensity of the
counter-rotating shock wave increase. Eventually, original deto-
nation waves are extinguished, and counter-rotating shock waves
evolve into reverse detonation waves. After the change of direc-
tion of the detonation wave, the counter-rotating shock wave
will be formed again.
ACKNOWLEDGMENTS
This research is sponsored by the National Natural Science
Foundation of China (Grant Nos. 91741202 and 52076003).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 34, 056104 (2022); doi: 10.1063/5.0089207 34, 056104-13
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