7/27/2019 Numerical investigation of shock wave reflections near the head ends of rotating detonation engines - R. Zhou.pdf
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Shock Waves (2013) 23:461472
DOI 10.1007/s00193-013-0440-0
ORIGINAL ARTICLE
Numerical investigation of shock wave reflections near the headends of rotating detonation engines
R. Zhou J.-P. Wang
Received: 18 April 2012 / Revised: 16 January 2013 / Accepted: 25 February 2013 / Published online: 22 March 2013
Springer-Verlag Berlin Heidelberg 2013
Abstract The influence of various chamber geometries on
shock wave reflections near the head end of rotating det-onation engines was investigated. A hydrogen/air one-step
chemical reaction model was used. The results demonstrated
that the variation in flow field along the radial direction was
not obvious when the chamber width was small, but became
progressively more obvious as the chamber width increased.
The thrust increased linearly, and the detonation height and
the fuel-based gross specific impulse were almostconstant as
thechamber width increased. Near theheadend, shockwaves
reflected repeatedly between the inner and outer walls. Both
regular and Mach reflections were found near the head end.
The length of the Mach stem increased as the chamber length
increased. When the chamber width, chamber length and
injection parameters were the same, the larger inner radius
resulted in more shock wave reflections between the inner
and outer walls. The greater the ratio of the chamber width
to the inner radius, the weaker the shock wave reflection near
the head end. The detonation height on the outer wall and
the thrust, both increased correspondingly, while the specific
impulse was almost constant as the inner radius of the cham-
ber increased. Thenumerical shock wave reflection phenom-
ena coincided qualitatively with the experimental results.
Keywords Rotating detonation engines
Three-dimensional numerical simulation
Shock wave reflection Mach reflection
Communicated by F. Lu.
R. Zhou (B) J.-P. Wang
Department of Mechanics and Aerospace Engineering,
State Key Laboratory of Turbulence and Complex System,
College of Engineering, Peking University, Beijing 100871, China
e-mail: [email protected]
1 Introduction
Detonation is a combustion process induced by shock waves
in which energy released from the combustion results in the
shock propagation. The shock wave compresses the mixture
to initiate the detonation and provides it with self-sustaining
energy [1]. For aero-propulsion, a detonation-based engine
has a higher propulsive efficiency, wider operating ranges
from subsonic to supersonic speed, and simpler and more
compact combustor design [2]. The most common type of a
detonation-based propulsion system is the pulse detonation
engine (PDE).There are,however,several obstaclesthat need
to be overcomein PDEdevelopment [2,3]. In recentyears, an
alternative method based on detonation combustion, called
the continuously rotating detonation engine (RDE), hasbeen
underincreasedconsiderationasa viable alternative to PDEs.
In theRDE, thedetonationwavepropagates in a circumferen-
tial direction, which is perpendicular to the direction of fuel
injection. The detonation wave can continuously propagate
over a wide range of injection velocities and does not need
repetitive ignition. These characteristics can possibly greatly
reduce the difficulties in developing a detonation engine.
The basic phenomenon of the RDE has been experimen-
tally and theoretically investigatedby Voitsekhovskii [4] and
Nicholls et al. [5], followed by the work of Bykovskii et
al. [6,7]. Wolanski et al. [8] experimentally achieved rotat-
ing detonation in a coaxial combustion chamber where the
detonation velocity was close to the Chapman-Jouguet (C-J)
value. They also achieved a range of propagation stability as
a function of chamber pressure, composition, and geometry
[9]. Using numerical simulation, Zhdan et al. [10] performed
two-dimensional unsteadymodelingof rotating detonation in
an annular chamber with a hydrogenoxygen mixture, and
Davidenko et al. [11] simulated the rotating detonation with
a detailed kinetic model using a hydrogenoxygen mixture.
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462 R. Zhou, J.-P. Wang
Hishida et al. [12] numerically studied the detailed flow field
structure of the rotating detonation with a two-step chemi-
cal reaction model of an argon-diluted hydrogen and oxy-
gen mixture. Yi et al. [13] investigated the influence of vari-
ous design parameters on the propulsive performance where
the parametric variables included total pressure, total tem-
perature, injection area ratio, axial chamber length, and the
number of detonation waves. Shao et al. [14,15] comprehen-sively studied three-dimensional numerical simulations on
RDE. They demonstrated multiple cycles of rotating deto-
nation, and discussed several key issues, including the fuel
injection limit, self-ignition, thrust performance, and nozzle
effects. Zhou and Wang [16] proposed a new method for
analyzing the RDE flow-field and described its thermody-
namic properties. They analyzed the paths of flow particles
and obtained the corresponding p-v and T-s diagrams to cal-
culate the net mechanical work and thermal efficiency of a
RDE. Schwer and Kailasanath [17] examined numerically
the effect of pressure feedback into the mixture plenum in
two- and three-dimensional RDEs. Uemura et al. [18] clari-fied the detonation mechanism and dynamics of a RDE with
two- and three-dimensional simulations using compressible
Eulerequationswitha detailedchemicalreaction mechanism
of hydrogen/oxygen and hydrogen/air, especially from the
triple-point and transverse detonation points of view. They
found that at this interaction point, an unreacted gas pocket
appears and ignites periodically to generate transverse waves
at the detonationfrontand maintainsdetonationpropagation.
However, their work needs a large amount of computation,
and only a very small combustion chamber could be simu-
lated with current computer capabilities. Pan et al. [19] simu-
lated numerically the continuously rotating detonation in an
annular chamber. They found that because of the curvature
of the annular tube, the size of the cellular pattern along the
concave wall was smaller than that along the convex wall.
This implied that the detonation wave near the concave wall
was stronger than that near the divergent convex wall. How-
ever, they only analyzed one chamber size, which did not
include the comparison of flow fields for different chamber
geometries.
In the numerical investigation of RDE, most researchers
assumed that the distance between the two coaxial cylin-
ders was much smaller than their diameters and axial length,
so that the flow field could be approximated as a two-
dimensional plane without thickness along the radial direc-
tion [12]. There is little research on the difference in the flow
field along the radial direction within a three-dimensional
RDE. Schwer and Kailasanath [20] investigated the effect of
the chamber width onthe flow field for a RDE. However, they
only obtained the overall pressure distribution and compari-
son of the specific impulse for different chamber widths and
did not analyze the flow field structure along the radial direc-
tion in detail. Lee et al. [21] studied numerically the effects
of curvature on the detonation wave propagation on annu-
lar channels. The flow features, such as cell structures and
pressure variations, were investigated for different regimes
of detonation with respect to the radius of curvature. How-
ever, theirworkwasbased on the two-dimensional configura-
tion that only included the radial and circumferential direc-
tions and had no corresponding axial direction. They also
did not consider the cyclicity of the rotating detonations, andtheir physical model was not an actual RDE. Nakayama et al.
[22] studied experimentally the detonation propagation phe-
nomena in curved channels. They employed a stoichiometric
ethyleneoxygen gas mixture and five types of rectangular-
cross-section curved channels with different inner radii of
curvature. Their physical model wasa curved channel, which
was different from the RDE combustion chamber. Eude et
al. [23] described the two- and three-dimensional flow fields
of the RDE and provided a comparative analysis to demon-
strate three-dimensional effects. They found that the three-
dimensional flow had specific features due to detonation
reflection from the outer cylindrical wall. The dependenceof the three-dimensional effects on the chamber diameter
and width was investigated. They focused on a comparison
between the two- and three-dimensional results, but detailed
studies ontheeffectofchamber geometries onthewave struc-
ture near the head end are needed.
In this study, the phenomena along the radial direction
of the flow field in a coaxial annulus combustion chamber
are investigated. The structure of regular and Mach shock
wave reflection are studied. The effects of the chamber
width, length, and radius, and nozzle and inlet stagnation
pressure are investigated. Through this research, the three-
dimensional flow field structure of the RDE is better under-
stood,and the numerical results provide a basis for the expla-
nation of the experimental phenomena.
2 Numerical method and physical model
The three-dimensional Euler equations were applied with
one-step chemistry to a stoichiometric hydrogen/air mixture.
Viscosity, thermal conduction, and mass diffusion were
ignored. Generalized coordinates were used for transform-
ing the curved grids in physical space to rectangular grids in
the computational space. The governing equations in gener-
alized coordinates are
U
t+
E
+
F
+
G
= S (1)
where the dependent variable vector U, convective flux vec-
tors E, F and G, and source vector S are defined as
U =1
J[ u v w e ]T
E =
U U u + px Uv + py Uw + pz U(p + e) UT
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Numerical investigation of shock wave reflections near the head ends of RDE 463
Table 1 Model parameters for the detonation of a stoichiometric
hydrogenair mixture [24]
Model parameter Value
1 (reactant) 1.3961
2 (product) 1.1653
3 (air) 1.4
R1, J/(kg K) 395.75R2, J/(kg K) 346.2
R3, J/(kg K) 287.00
q, MJ/kg 5.4704
Ta , K 15,100
A, l/s 1.0 109
1
2
6
3
4
5
Axial InletFlow
Exhaust
Fig. 1 RDE propagation schematic structure. 1 detonation wave,
2 burnt product, 3 fresh premixed gas, 4 contact surface, 5 oblique
shock wave, 6 detonation wave propagation direction
F =
V V u + px Vv + py Vw + pz V(p + e) VT
G =
W W u + px Wv + py Ww + pz W(p + e) WT
S =1
J
0 0 0 0 0
TU = ux + vy + wz
V = ux + vy + wz
W = ux + vy + wz
The pressure p and total energy e are calculated using the
equation of state
p = RT (2)
and the energy relationship
e =p
1+ q +
1
2
u2 + v2 + w2
(3)
where is thedensity, R thegasconstant, T the temperature,
the specific heat ratio, and q the heat release per unit mass.
The mass production rate is according to Arrhenius form:
(a)
10 20 30 40 50
=4 mm, 1680 s
p (atm)
(b)
10 20 30 40 50 60
=10 mm, 1520 s
p (atm)1
(c)
10 20 30 40 50 60 70 80
=16 mm, 1440 s
p (atm) 1
Fig. 2 Pressure contours at head end when the chamber widths were
4, 10, and 16 mm
=d
dt= A exp (Ea/(RT)) (4)
where is the proportion of the mass of the reaction gas
mixture, A the pre-exponential factor, and Ta the activation
temperature. A detailed description of all the parameters can
be found in Table 1 [24]. The fifth-order MPWENO scheme
[25] was used for splitting the flux vectors, and time integra-
tion was performed using a third-order TVD Runge-Kutta
method.
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464 R. Zhou, J.-P. Wang
0.2
0.15
0.1
0.05
0
0.2
0.15
0.1
0.05
0
0.2
0.15
0.1
0.05
0
0 0.05
Azimuthal Distance (m)
AxialDistance(m)
AxialDistance(m)
AxialDistance(m)
AxialDistance(m)
AxialDistance(m)
AxialDistance(m)
Azimuthal Distance (m)
0.1 0.15
10
inner radius, =4 mm, 1680 s
mid radius, =4 mm, 1680 s
outer radius, =4 mm, 1680 s
inner radius, =10 mm, 1520 s
mid radius, =10 mm, 1520 s
outer radius, =10 mm, 1520 s
p (atm)p (atm)
p (atm)
p (atm)
p (atm)
p (atm)
20 30 40 50 60
10 20 30 4 0 50 6010 20 30 4 0 50 60
10 20 30 40 50 60 70
10 20 30 40 50 60 7010 20 30 40 50 60 70 80 90
0.2
0 0.05
Azimuthal Distance (m)
0.1 0.15 0.2
0 0.05
Azimuthal Distance (m)
(a)Chamber width 4 mm. (b)Chamber width 10 mm.
0.1 0.15 0.2
00
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0.05 0.1 0.15 0.2 0.25
Azimuthal Distance (m)
0 0.05 0.1 0.15 0.2 0.25
Azimuthal Distance (m)
0 0.05 0.1 0.15 0.2 0.25
Fig. 3 Pressure distribution on the inner, mid and outer radii as the chamber width increased
Thechemicalinduction distancewas about250mforthe
C-J detonation of the gas mixture used. As this study aimed
to investigate the RDEs macro flow field characteristics, but
not the micro cell or transverse wave structure, the average
grid size of 200m was small enough. The grid dependencywas validated in [15]. In the three-dimensional numerical
simulation, the calculated detonation propagation velocity
was close to the theoretical value and the flow fields coincide
approximately with previous results [19,20,23]. The numer-
ical convergence and grid dependency were checked using
the above comparisons.
The combustion chamber of the RDE was a coaxial cavity
with a toroidal section as shown in Fig. 1. A detonation wave
propagated circumferentially in the annular chamber while a
combustible mixture was injected from the head end, and the
burnt gas then flowed out from the downstream exit. At the
head end, there was a large number of Laval micro-nozzles
which axially injected thepremixed hydrogen/airgas into the
combustion chamber. Themass fluxof the incoming fuel wascontrolled by the relationship between the inlet stagnation
pressure and flow pressures at the head end.
The front section of the combustor was initially filled
with a quiescent, combustible gas mixture at a pressure of
0.101 MPa and temperature of 300 K. The other section
was filled with combustion products. A one-dimensional C-J
detonation wave was distributed in the front section near the
head end to initiate the detonation. In practice, the backflow
will produce an extremely dangerous explosion, so backflow
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Numerical investigation of shock wave reflections near the head ends of RDE 465
0.25
0.2
0.15
0.1
0.05
00 0.05 0.1 0.15 0.2 0.25
10 20 30 40 50 60 70
10 30 50 70 90
10 20 30 40 50 60 70 80 90 100
Azimuthal Distance (m)
0 0.05 0.1 0.15 0.2 0.25
Azimuthal Distance (m)
0 0.05 0.1 0.15 0.2 0.25Azimuthal Distance (m)
(c) Chamber width 16 mm.
AxialDistance(m)
0.25
0.2
0.15
0.1
0.05
0
AxialDistance(m)
0.25
0.2
0.15
0.1
0.05
0
AxialDistance(m)
inner radius, =16 mm, 1440 sp (atm)
mid radius, =16 mm, 1440 sp (atm)
outer radius, =16 mm, 1440 sp (atm)
Fig. 3 continued
in the experiment must be prevented using a check valve. In
the numerical simulation, a premixed stoichiometric hydro-
gen/air mixture injection condition was set according to the
local wall pressure following Laval nozzle theory. The inlet
stagnation pressure was 0 = 3MPa and the ambient pres-
sure was 0.05 MPa. The area ratio of the nozzle exit and
the nozzle throat was Aw/Athroat = 10. From the isentropic
relationship,
Aw
Athroat=
1
M
2
+ 1
1+
1
2M2
+12(1)
(5)
where M is the Mach number just in front of the head wall
and = 1.4. The solutions of(5) were M1 = 4.0 and M2 =
0.055.The threecriticalpressureswere calculated as follows:
pw1 =p0
1 +1
2M21
1
(6)
pw2 =p0
1+1
2M22
1
(7)
pw3 = pw1
1 +
2
+ 1
M21 1
(8)
The injection boundary condition was specified according to
the local gas pressure pw at the wall [15].
(1) When pw > ps: the reaction mixture cannot be injected
into the chamber. A rigid wall condition is set locally.
(2) When pw2 > pw > ps: the mass fluxes through both
throat and injection walls are subsonic.
(3) When pw3 > pw > pw2: the throat maintains choke
conditions, and the injection of the mass flux remains
constant. Shock waves develop downstream of the throat
and fresh gas is injected at subsonic velocities.
(4) When pw > pw3: the injection is not affected by the
wall pressure. The whole field downstream of the throat
is supersonic.
The above injection boundary conditions were used com-
prehensively in most of the current RDE numerical simula-
tions, and the assumed boundary conditions were the most
realistic. A rigid wall condition was used on the inner and
outer walls. Non-reflecting boundary conditions were used
at the downstream boundary.
3 Results and discussion
In this study, we focused on the effects of the chamber width,
the axial chamber length, nozzle, the inlet stagnation pres-
sure, and thechamber inner radiuson thewave structurenear
the head end and the flow field within the RDE. The cham-
berwidthalong theradial direction, thechamber lengthalong
the axial direction, and the chamber inner radius are repre-
sented by , L and Rin, respectively. The thrust F and the
specific impulse Isp arecalculated according to the followingequations:
F=
exit
w2 + p p
d A (9)
Isp =F
gm f(10)
where w is the axial velocity and m f the mass flow rate of
fuel.
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466 R. Zhou, J.-P. Wang
(a) Chamber width 4 mm.
(b)Chamber width 10 mm.
Azimuthal Distance (m)
AxialDistance(m)
0 .0 7 0 .0 75 0 .0 8 0 .0 85 0 .0 9
0
0.005
0.01
0.015
0.02
inner radius, =4 mm, 1680 s
Azimuthal Distance (m)
AxialDistance(m)
0 .0 7 0 .0 75 0 .0 8 0 .0 85 0 .0 9
0
0.005
0.01
0.015
0.02
mid radius, =4 mm, 1680 s
Azimuthal Distance (m)
AxialDistance(m)
0 .0 7 0 .0 75 0 .0 8 0 .0 85 0 .0 9
outer radius,0
0.005
0.01
0.015
0.02
=4 mm, 1680 s
Azimuthal Distance (m)
AxialDistance(m)
0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
0.005
0.01
0.015
0.02
0.025
inner radius, =10 mm, 1520 s
Azimuthal Distance (m)
AxialDistance(m)
0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
0.005
0.01
0.015
0.02
0.025
mid radius, =10 mm, 1520 s
Azimuthal Distance (m)
AxialDistance(m)
0.01 0.015 0.02 0.025 0.03 0.035 0.04
outer radius,0
0.005
0.01
0.015
0.02
0.025
=10 mm, 1520 s
Azimuthal Distance (m)
AxialDistance(m)
0.01 0.02 0.03 0.04
0
0.005
0.01
0.015
0.02
0.025
inner radius, =16 mm, 1440 s
Azimuthal Distance (m)
Axia
lDistance(m)
0.01 0.02 0.03 0.04
0
0.005
0.01
0.015
0.02
0.025
mid radius, =16 mm, 1440 s
Azimuthal Distance (m)
Axial
Distance(m)
0.01 0.02 0.03 0.04
outer radius,0
0.005
0.01
0.015
0.02
0.025
=16 mm, 1440 s
(c)Chamber width 16 mm.
Fig. 4 Enlarged pressure distribution corresponding to Fig. 3
3.1 Chamber width
The numerical simulation was performed in three different
combustion chambers whose inner radii were 3 cm, lengths
4.8 cm, and chamber widths4, 10, and 16mm.Figure2 shows
the pressure contours at the head end after the detonation
propagated in a stable manner. When the chamber width was
4 mm, there was no shock wave reflection between the inner
and outer walls, as shown in Fig. 2a. When the chamber
widths were increased to 10 or 16 mm, reflected shock waves
(wave 1) appeared on the outer wall, as shown in Fig. 2b and
c. However, shock wave1 was not reflected on the inner wall.
We extended the annulus of the combustion chamber on
a two-dimensional plane to clearly analyze the flow field
variation in the radial direction. Figure 3 shows the pressure
contours on the inner, mid, and outer radii of the combus-
tion chamber after the detonation had propagated stably in
the three chamber geometries. Figure 4 shows the enlarged
pressure distribution near the detonation front on the three
radii. The variation of the flow field in the radial direction
was not obvious when the chamber width was small such as
= 4 mm, as shown in Figs. 3a and 4a. The differences
in the flow fields were progressively more obvious when the
chamber width was increased to = 10 and = 16mm,
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Numerical investigation of shock wave reflections near the head ends of RDE 467
(a)
(b)
(c)
Azimuthal Distance (m)
AxialDistance(m)
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
0.25
0.1 0.3 0.5 0.7 0.9
inner radius, =16 mm, 1440 s
h=19.48 mm
h
Azimuthal Distance (m)
AxialDistance(m)
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
0.25
0.1 0.3 0.5 0.7 0.9
mid radius, =16 mm, 1440 s
h
h=17.19 mm
Azimuthal Distance (m)
AxialDistance(m)
0 0.05 0.1 0.15 0.2 0.25
0
0.05
0.1
0.15
0.2
0.25
0.1 0.3 0.5 0.7 0.9
outer radius, =16 mm, 1440 s
h
h=15.81 mm
Fig. 5 Reaction process parameter contours on the inner, mid, and
outer radii when the chamber width is 16 mm
as shown in Fig. 4b and c. The differences were found by
comparing the pressure distribution on the inner, mid, and
outer radii. Figure 4b and c shows that there were two strong
waves on the inner radius. One was the detonation front, and
theother thereflectedshockwave1 markedin Fig. 2. The dis-
tance between the detonation and wave 1 on the inner radius
increased as the chamber width increased. There was only
one strong detonation wave on the outer radius in the three
chambers.
Figure 5 shows that the detonation heights were 19.48,
17.19, and 15.81 mm on the inner, mid, and outer radii, when
the chamber width was 16 mm. The detonation was com-
pressed on the concave outer wall, which was stronger than
p
(atm)
0 1 2 3 4 5 60
10
20
30
40
50
60
r=3.0 cm
r=3.2 cm
r=3.4 cm
Rin
=3 cm, =4 mm
p(atm)
0 1 2 3 4 5 60
10
20
30
40
50
60
70
r=3.0 cm
r=3.5 cm
r=4.0 cm
Rin=3 cm, =10 mm
p(atm)
0 1 2 3 4 5 60
10
20
30
40
50
60
70
80
90
r=3.0 cm
r=3.8 cm
r=4.6 cm
Rin=3 cm, =16 mm
(a)
(b)
(c)
Fig. 6 Pressure variation along the circumferential direction at a point
2 mm from the head end
that expanded on the convex inner wall, consumed fuel more
rapidly, and, therefore, thedetonationheighton theouterwall
was smaller than that on the inner wall. The circumferential
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468 R. Zhou, J.-P. Wang
Chamber Width (mm)
Thrust(N),S
pecificImpulse(s)
Detonatio
nHeight(mm)
0 2 4 6 8 10 1 2 1 4 1 6 1 8 2 0
0
500
1000
1500
2000
2500
3000
3500
4000
0
5
10
15
20
Thrust
Specific impulse
Detonation height
Fig. 7 Variations of severalparametersas thechamberwidth increased
propagation velocity of the detonation on the outer wall was
higher than that on the inner wall and, therefore, the detona-tions on the different radii could maintain the same angular
velocity to propagate sustainably in the RDE chamber.
Figure 6 shows the pressure variation ata point 2 mmfrom
the head end along thecircumferential direction on the inner,
mid, andouterradii. Comparingthe maximum pressureof the
three lines in each figure, it was obvious that the largest were
on the outer wall. Itwas alsoseenthat the von Neumann spike
pressure increased with the increase in the chamber width.
The positions of the detonation front on the three radii were
the same when the chamber width was 4 mm; however, the
difference between the positions of the detonation front on
the three radii became progressively larger as the chamberwidth increased.
Figure 7 shows the influence of the chamber width on
the detonation height measured on the outer wall, the thrust,
and the specific impulse. The thrust linearly increased as the
chamber width increased because of the increase in the total
amount of combustion products exhausted from the cham-
ber exit. The detonation height on the outer wall and the
fuel-based gross specific impulse were shown to be almost
constant as the chamber width increased.
3.2 Chamber length and stagnation pressure
The next parameter of interest was the chamber length. The
width of the combustion chamber was 10 mm, and the inner
radius 3 cm. The chamber lengths were 3.6, 4.8, and 6.0 cm.
Figure 8 shows the pressure contours at the head end after
thedetonation hadpropagated stablyin the three chambers. It
was clearly seen that there were repeated shock wave reflec-
tions between the inner and outer walls near the head end.
The form of the shock wave reflection was not only limited
to regular reflections, as shown in Fig. 6a, but also Mach
10 30 50 70 90 1 10
p (atm)
L=36 mm, 1470 s
Shock WaveReflection
MachReflection
DetonationWave Front
1 0 2 0 3 0 4 0 5 0 6 0
p (atm)
L=48 mm, 1520 s
DetonationWave Front
MachReflection
Shock WaveReflection
(a)
(b)
(c)
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0
p (atm)
L=60 mm, 1410 sDetonationWave Front
MachReflection
Shock WaveReflection
Fig. 8 Pressure distribution at theheadend forchamber lengths of 3.6,
4.8 and 6.0 cm
reflections existed on the inner wall, as shown in Fig. 6b and
c. The length of the Mach stem increased as the chamber
length increased.
The detonation height on the outer wall increased slightly,
and the thrust and specific impulse were almost constant as
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Numerical investigation of shock wave reflections near the head ends of RDE 469
Chamber Length (cm)
Thrust(N),Sp
ecificImpulse(s)
Detonatio
nHeight(mm)
3 4 5 6 7
1400
1600
1800
2000
2200
0
5
10
15
20
25
30
Thrust
Specific impulse
Detonation height
Fig. 9 Detonation height, thrust, and specific impulse variations as the
chamber length increased
Axial Distance (m)
AverageAxialVelocity(m/s)
0 0.01 0.02 0.03 0.04 0.05 0.06
100
200
300
400
500
600
Length=36 mm
Length=48 mm
Length=60 mm
Fig. 10 Axial average velocity variation along the axial distance
the chamber length increased, as shown in Fig. 9. In the three
chambers, the circumferential detonation velocities were all
about 2,000 m/s on the mid radius, which was close to the C-J
value of 1,984 m/s. The detonation wave near the concave
wall was convergent and, therefore, stronger than that near
the divergent convex wall. Thus the propagating velocity of
the detonation wave of 1,750 m/s on the inner wall was lower
than the 2,300 m/s on the outer wall.
Figure 10 shows theaxial average velocity variation along
the axial direction. It is seen that the axial average velocities
were approximately the same within the 0 to 10 mm vicin-
ity (corresponding to the detonation front area) and at the
exit cross section. The axial average velocity throughout the
whole flow field decreased as the chamber length increased.
The differences between them appeared in the oblique shock
MachReflection
DetonationWave Front
ObliqueShock Wave
no nozzle,=14 mm, 1570 s
convergent divergent nozzle,=14 mm, 1520 s
DetonationWave Front
ObliqueShock Wave
Shock WaveReflection
(a)
(b)
Fig. 11 Pressure contours of the no-nozzle chamber (a) and the
convergentdivergent nozzle chamber (b)
wave andthefollowingexpansion wave area, wherethe short-
est chamber gained the highest velocity to exhaust the burnt
gas the most easily. In other words, the excessive combus-
tion chamber lengthprevented thecombustion products from
flowing out, andalsoaffectedthe wavestructure near thehead
end.
The effect of a convergentdivergent nozzle connected
to the combustion chamber was also investigated. The
convergentdivergent nozzle weakened the shock wave
reflection near the head end, and the Mach reflection
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Stagnation pressure (MPa)
DetonationHeight(mm)
Thrust(N),S
pecificImpulse(s)
1.5 2 2.5 3 3.50
5
10
15
20
25
30
0
500
1000
1500
2000
2500
3000
Detonation height
Thrust
Specific impulse
Fig. 12 Detonation height, thrust, and specific impulse variations as
the stagnation pressure increased
disappeared in the convergentdivergent nozzle chamber, as
shown in Fig. 11. The influence of the convergentdivergentnozzle on the flow field near the head end was progressively
more obvious as the chamber width increased.
The shock wave reflection was along the radial and cir-
cumferential directions, whichwere independent of the injec-
tion axial direction of the fresh gas. We simulated the flow
field when the stagnation pressure was different and found
that the wave structure was not affected by stagnation pres-
sure. Therefore, the injection parameters had no effect on the
shock wave structure. The wave structure was only affected
by the chamber geometry size. The inlet stagnation pres-
sure generally had no effect on the overall distribution of the
flow field. The distribution of the shock wave reflection nearthe head end was almost the same when the inlet stagnation
pressures were 2.0, 2.5, and 3.0 MPa. The detonation height
and the specific impulse were almost constant, and the thrust
increased linearly as the stagnation pressure increased, as
shown in Fig. 12.
3.3 Chamber radius
Another interestingaspectwas theeffectof thechamber inner
radius on the shock wave reflection near the head end. We
fixed the chamber length at 36 mm, and the chamber width
at 10 mm, while changing the chamber inner radius to 2,
3, and 4 cm. Figure 13 shows the pressure contours at the
head end of the combustion chambers after the detonation
had propagated stably in the three chambers. When the inner
radius was 2 cm, shock wave 1 was not reflected on the inner
wall. When the inner radius was 3 cm, shock wave 1 was
reflected to wave 2 by Mach reflection on the inner wall, and
shock wave2 was reflected towave 3 on the outer wall. Shock
wave 3 was not reflected on the inner wall. When the inner
radius was 4 cm, shock waves 1 and 2 were reflected on the
1 0 3 0 5 0 7 0
p (atm)
Rin=2 cm,1710 s
1
10 3 0 5 0 7 0 9 0
p (atm)
Rin=3 cm, 1470 s
12
10 20 3 0 4 0 5 0 6 0 7 0
p (atm)
Rin=4 cm, 1120 s
1234
ab
3
(a)
(c)
(b)
Fig. 13 Pressure distribution at the head end for chamber inner radii
of 2, 3, and 4 cm
inner and outer walls. Similarly, when the radius was 4 cm
shock wave 3 was again reflected to wave4 on the inner wall,
which was different from that of the radius 3 cm. The larger
inner radius resulted in much more shock wave reflections.
The greater the ratio of width to inner radius the weaker the
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Numerical investigation of shock wave reflections near the head ends of RDE 471
Chamber Inner Radius (cm)
Thrust(N),Sp
ecificImpulse(s)
DetonationHeight(mm)
1 1.5 2 2.5 3 3.5 4 4.5 50
500
1000
1500
2000
2500
3000
0
5
10
15
20
25
30
Thrust
Specific impulse
Detonation height
Fig. 14 Detonation height, thrust, and specific impulse variations as
the chamber inner radius increased
time (s)
p(atm)
500 550 600 650 700 750 800 850 9000
10
20
30
40
50
60
70
a
b
Fig. 15 Pressure history recordedat a pointon theouter wall when the
inner radius was 4 cm
shock wave reflection near the head end, as shown in Figs. 2
and 13.
When the chamber inner radius was 2 cm, the disturbance
of the specific impulse and thrust was larger than when the
chamber inner radius was 3 or 4 cm after detonation had
propagated stably. When the initial conditions, the cham-
ber widths, and lengths were the same, the detonation wave
needed a longer time to propagate stably with a decrease
in the chamber inner radius. Both the detonation height on
the outer wall and the thrustevidently increased, and thespe-
cific impulse was almostconstantas thechamber inner radius
increased, as shown in Fig. 14.
The shock wave reflection phenomenon has appeared in
many experimental results [9], but there have been few com-
putational studies to confirm this behavior. Figure 15 shows
the pressure history at a point near the head end on the outer
wall when the chamber inner radius was 4 cm. There were
two clear pressure peaks in every cycle, shown as a and
b in Fig. 15. The pressure peak a corresponded with the
detonation front a on the outer wall in Fig. 13c, and the
pressure peak b corresponded with the location of shock
wave 2 reflected onto wave three on the outer wall, as shownb in Fig.13c. The numerical shock wave reflection struc-
ture coincidesqualitatively with the experimental results [9].
Therefore, the results in this study provide a fundamental
explanation for the experimental results.
4 Conclusions
The variation in the flow field along the radial direction was
not obvious when the chamber width was small and was pro-
gressively more obvious when the chamber width increased.
The thrust increased linearly, and both the detonation heightand the fuel-based gross specific impulse were almost con-
stant as the chamber width increased.
There were repeated shock wave reflections between the
inner and outer walls near the head end. Not only were there
regular reflections but also Mach reflections when the cham-
ber length increased. The length of the Mach stem increased
as the chamber length increased. The detonation height on
the outer wall increased slightly, and the thrust and spe-
cific impulse were almost constant as the chamber length
increased.
The circumferential detonation velocity on the mid radius
was close to the C-J value. The detonation wave near the
outer wall was stronger than that near the inner wall; thus,
thepropagating velocityof detonation wave on the inner wall
was lower than that on the outer wall. Excessive combus-
tion chamber lengthprevented thecombustion products from
flowing outandalso affectedthewavestructurenear thehead
end.
The distribution of the shock wave reflection near the
head end remained constant even when the inlet stagnation
pressure was different. The detonation height and specific
impulse were almost constant, and the thrust increased lin-
early as the stagnation pressure was increased.
A larger chamber inner radius resulted in more notice-
able shock wave reflections. The greater the ratio of width
to inner radius the weaker the shock wave reflection near the
head end. Both the detonation height on the outer wall and
the thrust evidently increased, and the specific impulse was
almost constant as the chamber inner radius increased.
The numerical shock wave reflection structure coincided
qualitatively with the experimental results, and the results in
thepresent studies provide a fundamental explanation for the
experimental results.
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