JOURNAL OF PROPULSION AND POWER
Vol. 39, No. 5, September–October 2023
Analysis on Propulsive Performance of Hollow Rotating Detonation
Engine with Laval Nozzle
Yunzhen Zhang,∗ John Z. Ma,† Kevin Wu,∗ Miao Cheng,∗ Zhaohua Sheng,∗ Guangyao Rong,∗
Dawen Shen,∗ and Jianping Wang‡
Peking University, Beijing 100871, People’s Republic of China
and
Shujie Zhang§
Beijing Institute of Astronautical Systems Engineering, 100076 Beijing, People’s Republic of China
Downloaded by NC State University on September 8, 2023 | http://arc.aiaa.org | DOI: 10.2514/1.B38830
https://doi.org/10.2514/1.B38830
In the present study, an experimental performance analysis of hollow rotating detonation engines (RDEs)
with Laval nozzles is carried out for the first time. Experiments of a hollow rotating detonation engine with a
Laval nozzle were performed with a modular RDE at a backpressure condition of 1 atm. Two configurations
with area ratios of the outlet throat to the inlet of Ath ∕Ain 5.3 and 2.7 have been tested with gaseous
methane/oxygen as propellants. Three normalized metrics, usually used for evaluating the performance of
conventional rocket engines, are introduced to analyze the performance deficit between the measured value
of an RDE and the ideal value of an isobaric-combustion-based engine. These metrics allow for assessing the
entire engine and each component separately. The metric analysis suggests a small outlet-to-inlet area ratio
(Ath ∕Ain ) is detrimental to the propulsive performance. To explain the mechanism, a gas-stratification
flowfield model is further proposed. It is found that the unchoked region in the combustible gas layer, which
is caused by unchoked injection on the injecting plate, is responsible for the performance deficit of the
combustion chamber. This model is then validated by one-dimensional numerical simulations and
experimental data. In addition, we also focus on the global performance, including the gross thrust, the
specific impulse, and the utilization of the supplied stagnation pressure. The result implies a tradeoff space
when choosing an appropriate Ath ∕Ain .
p
Ma
_
m
N
n
Q
Q_
R
T
t
u
Yi
z
γ
εe
η
ρ
φ
_
ω
Nomenclature
A
Cf
c
D
DCJ
Dw
E
Es
F
g
H
Hs
hi
hi;s
h0i
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Isp
L
=
=
area, mm2
thrust coefficient
characteristic velocity, m/s
diameter, mm
Chapman–Jouguet speed of the detonation wave, m/s
the speed of the detonation wave, m/s
total energy, J/kg
total sensible energy, J/kg
thrust, N
gravity, m∕s2
total enthalpy, J/kg
total sensible enthalpy, J/kg
specific enthalpy of the ith species, J/kg
specific sensible enthalpy of the ith species, J/kg
specific standard enthalpy of formation of the ith species, J/kg
the specific impulse, s
the total inlet area in the flowfield model
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
pressure, bar
Mach number
mass flow rate, kg/s
number of species
number of detonation waves
the heat release in the whole region, J∕m2
heat release rate, J∕s∕m3
the molar gas constant, J∕mol ⋅ K
temperature, K
time, s
velocity, m/s
the mass fraction of the ith species
position of the one-dimensional detonation tube, m
specific heat ratio
nozzle expansion area ratio
normalized metric
density, kg∕m3
equivalent ratio
production rate of the ith species, kg∕s∕m3
Subscripts
a
b
c
cr
e
exp
f
i
id
in
ox
pl
th
unch
0
3, 4
Received 17 March 2022; revision received 22 March 2023; accepted for
publication 23 April 2023; published online 15 June 2023. Copyright © 2023
by the American Institute of Aeronautics and Astronautics, Inc. All rights
reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3876 to
initiate your request. See also AIAA Rights and Permissions www.aiaa.org/
randp.
*Ph.D. Student, Center for Combustion and Propulsion, CAPT (Center for
Applied Physics and Technology) and SKLTCS (State Key Laboratory for
Turbulence and Complex System), Department of Mechanics and Engineering Sciences, College of Engineering.
†
Asistant Research Fellow, Center for Combustion and Propulsion, CAPT
and SKLTCS, Department of Mechanics and Engineering Sciences, College
of Engineering.
‡
Professor, Center for Combustion and Propulsion, CAPT and SKLTCS,
Department of Mechanics and Engineering Sciences, College of Engineering;
wangjp@pku.edu.cn (Corresponding Author).
§
Engineer.
765
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
ambient
backstate
combustion chamber
critical choked state
exit section of the nozzle
experimental value
fuel (methane)
the ith species
ideal value
injectors
oxidizer (oxygen)
propellant plenums
nozzle throat
unchoked
total state
capillary tube attenuated pressure ports
766
ZHANG ET AL.
I.
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T
Introduction
HE detonation propulsion device, which is also known as the
pressure gain device, has received increasing attention in the last
few decades due to its higher thermodynamic efficiency as compared
to conventional Brayton-cycle-based propulsion systems. The
thought of applying detonation to propulsion systems was first
proposed by Zel’dovich [1]. Since then, attempts have been carried
out, such as pulse detonation engines [2], oblique detonation engines
[3], and rotating detonation engines (RDEs) [4,5]. Nowadays, the
RDE is regarded as a more feasible solution for further practical
applications on account of its inherent advantages such as higher
thrust density and wider operating range of Mach number.
A detonation wave is a shock wave coupled with a following
reacting region that restrengthens the front shock wave in turn. Early
attempts to achieve rotating detonation waves were conducted by
Voitsekhovskii [6] and Nicholls et al. [7] in the 1950s and 1960s.
After several decades, Bykovskii and Mitrofanov [8] achieved steady
rotating detonations again in 1980. They carried out a series of
experiments afterward [9] and proposed guidelines to get steady
rotating detonations by taking into account the relationship between
the cell size and the combustor scale [10]. Their inspiring works have
led to the resurging interest in RDEs ever since [4,5].
The combustion chambers of most existing RDEs are usually
annular chambers with an inner cylinder. For rocket applications,
the hollow RDE with no inner cylinder is also an attractive alternative due to its lighter structure weight, similar structure to existing
rocket engines, and the potential to avoid high-pressure and hightemperature damage of the chamber wall [11,12]. Tang et al. [11]
verified the feasibility of this configuration numerically. They
achieved detonation waves in both annular RDEs and hollow RDEs;
they found out that the pressure peaks in the annular chamber are
higher than those in the hollow chamber and that some fresh propellants will diffuse into the inner region because of the lack of the inner
cylinder, which may lead to deflagration in the center region. Anand
and Gutmark [13] proposed that this needs to be rectified in future
iterations of hollow RDEs. Lin et al. [14], Anand et al. [15], Wang and
Le [16], Zhang et al. [17], and Peng et al. [18] managed to obtain
rotating detonation in hollow RDEs experimentally. Some common
features were observed that the wave speeds of the detonation waves
in hollow chambers are overall higher than those in annular ones, and
that the channel pressures in the hollow RDEs are higher than those in
annular ones when there is no nozzle attached. There has also been
some research on the propulsive performance of the hollow RDE. Lin
et al. [19] measured the thrust of a nozzleless hollow RDE and found
that it is relatively small as compared with annular ones. Kawasaki
et al. [20] performed experiments on a nozzleless RDE with inner
cylinders with diameters of 0 (no inner cylinder), 9, 15, 23, and
31 mm; and they found that the performance will be reduced as the
inner cylinder decreases. Goto et al. [21] and Yokoo et al. [22] grew
particular interest in hollow RDEs of small scales. They performed
experiments on nozzleless hollow RDEs with diameters of20 and
24 mm, and they attained relatively high performances. Therefore, it
is believed that the lower channel pressure and performance in these
studies are attributed to the lack of confinement of the inner cylinder
when there is no nozzle attached. For applications on the booster and
upper stages, a nozzle with a constriction is usually necessary to
extract more work from the burned gas [23]. Yao et al. [24] and Sun
et al. [25] numerically studied the propulsive performance of hollow
RDEs with Laval nozzles. Zhang et al. [26] carried out experiments in
hollow RDEs equipped with Laval nozzles and analyzed the stability
of the RDE operation, but they failed to provide discussions about the
propulsive performance. It can be seen that there has been a conspicuous lack of reporting of the analyses on the propulsive performances of hollow RDEs with nozzles.
The propulsive performance of RDEs has always been a topic of
major interest in the research of RDEs. Frolov et al. [27] demonstrated that the specific impulse of the detonation mode is 6–7%
higher than that of the combustion mode in the same combustor.
Mizener and Lu [28] and Kaemming et al. [29] developed models
to evaluate the performance theoretically and investigated the
influences of different factors. Zhang et al. [30] proposed an equivalent expansion model and provided a new perspective to account for
the pressure gain property of RDEs. Furthermore, a nozzle is usually
necessary to achieve better performance. Fotia et al. [31] conducted
RDE experiments with four types of exit forms, and they subsequently investigated the influence of nozzle design parameters
including the expansion area ratio, the half-angle of the tail cone,
and the level of nozzle truncation systematically [32]. Bennewitz
et al. [33] and Goto et al. [34] also performed experiments with
different nozzle constriction ratios and evaluated the propulsive
performance. Bach et al. [35] studied the pressure gain property of
the RDE by changing the outlet area ratio along with other parameters, and they proposed an empirical model to describe the pressure
gain. Harroun et al. [36] conducted experiments and numerical
simulations of RDEs with two aerospike nozzle geometries and
analyzed the influence of the plume pressure distribution on the
performance. Liu et al. [37] investigated the propulsive performance
of optimized aerospike nozzles in RDEs numerically and compared it
to the ones with a flat ramp surface. Stechmann et al. [38] established
a model to assess the influence of aerospike and bell nozzles on
performance. Rankin et al. [39] reviewed performance studies of
RDEs and concluded that an area constriction is beneficial for thrust
gain in RDEs. However, to our best knowledge, there has hitherto
been no experimental research on the effects of the Laval nozzle on
the performance of hollow RDEs.
Due to the high frequency and the inherent unsteady property of
the RDE flowfield, there are difficulties in evaluating the performance of RDEs. To date, several approaches and metrics have been
proposed and widely used. Stechmann [40] suggested that the performances of detonation-based combustors should be compared with
theoretical values rather than measured values of isobaric-combustion-based combustors. Following this idea, he proposed an approach
to calculate ideal values under the same testing condition. Kaemming
and Paxson [41] defined a metric called the equivalent available
pressure based on thrust measurements, and they used it to determine
the pressure gain of the whole RDE setup. Both of these methods
have been widely used in research on the performance of RDEs
[42–44]. In the present study, the performance assessment framework of conventional rocket engines is introduced to the study of
RDEs systematically for the first time. Based on the calculation of
ideal performance using the approach proposed by Stechmann [40],
three normalized metrics are adopted to determine the performance
deficit. These metrics allow the performance evaluation of the
entire engine and each component, including the combustion chamber and nozzle. In addition, metrics based on the works of Fotia et al.
[32,45] are used to evaluate the utilization of the supplied stagnation pressure.
The present study aims to explore the effects of the Laval nozzle on
the propulsive performance of hollow RDEs. Experiments of hollow
RDEs with designed Laval nozzles fueled by gaseous methane (CH4 )
and gaseous oxygen (O2 ) have been conducted at ambient exit
conditions. Two throat sizes have been tested. We will first analyze
the propulsive performance deficit with the normalized metrics.
Then, to explain the mechanism of the performance loss, a new
flowfield model based on gas stratification is proposed. Finally, the
focus will be placed on the consideration of global performance,
including the gross thrust, the specific impulse, and the utilization of
the supplied stagnation pressure.
II.
Experimental Apparatus and Performance Metrics
A. Engine Hardware Configurations
Figure 1 is a schematic diagram of the hollow RDE equipped with
the Laval nozzle used in this study. The injecting scheme is kept the
same as in previous studies by Ma et al. [46]. The fuel and the oxidizer
are injected into the combustion chamber coaxially, which is shown
in Fig. 1, through an array of 90 coaxial injecting tubes, which are
evenly distributed along the circumference. The injectors are chosen
as fuel-centered injectors according to the pre-experiment. The total
injecting areas for the fuel and oxidizer are 70.7 and 203.6 mm2 ,
respectively. The fuel and oxidizer are injected into the chamber from
767
ZHANG ET AL.
Fig. 1
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Table 1
Part
Injectors
Combustion chamber
Laval nozzle
Schematic diagram of the hollow RDE and the supply system.
The RDE geometry properties
Geometry measured
Fuel injecting area Af
Oxygen injecting area Aox
Total injecting area Ain
Chamber diameter Dc
Throat diameter Dth
Exit diameter De
Throat to injector area ratio,
Ath ∕Ain
Expansion area ratio
εe Ae ∕Ath
Properties
70.7 mm2
203.6 mm2
274.3 mm2
120 mm
43 mm 30.8 mm
55 mm 40.8 mm
5.3
2.7
1.64
1.76
are suitable for the propagation of detonation waves. Other detailed
dimensions can be found in Table 1.
In the present study, the predetonator is used to initiate detonation
waves; this is shown in Fig. 1. The predetonator is a straight steel tube
tangentially attached to the outer wall of the combustion chamber.
The schematic is shown in Fig. 1. The inner diameter of the predetonator is 5 mm, and the length is 100 mm. The tangential installation
of the predetonator has been widely used by RDE researchers to
generate symmetric detonation waves in the combustion chamber.
The predetonator is located at the end of the combustion chamber and
the entrance of the converging–diverging nozzle to minimize the
effects of the predetonator on the detonation waves.
B. Measuring Instruments
two separated plenums to ensure there is no risk of backfire and
explosion. The dimensions of the injectors are listed in Table 1. The
outer wall of the chamber and the thrust wall (injector faceplate) are
made of C101 copper, which has high thermal conductivity. The
diameter of the chamber is Dc 120 mm. A designed Laval nozzle
is mounted on the chamber. The nozzle is designed according to
Refs. [23,47]. The nozzle consists of a contraction section, a throat,
and an expansion section. The arc radius of the contraction surface is
chosen to be 2.5 times the radius of the combustion chamber to
weaken reflected oblique shock waves. A straight section is set at
the throat to steady the flow and the pressure. The length of the
straight section is chosen to be half the throat diameter. As for the
divergence section, the profile type of the nozzle is chosen to be a
conical nozzle. Plenty of reported works have indicated that the area
ratio of the outlet throat to the inlet is a key parameter for RDE
performance. It is found that the area ratio of the outlet throat to the
inlet has a great effect on the pressure gain of RDEs [35,43]. Here,
Ath ∕Ain (which represents the outlet throat to the inlet area ratio) is
used to refer to different nozzles. According to the pre-experiments,
the area ratios (Ath ∕Ain ) in this study are chosen as 2.7 and 5.3, which
Fig. 2
Piezoresistive pressure transducers are used to measure the pressure of the fuel and oxidizer plenums. These measurements provide
the total pressure of the propellants before injection because the
propellants in the plenums are practically stationary. To measure
the low-frequency average pressure, several capillary tube attenuated
pressure (CTAP) probes are employed, as illustrated in Fig. 2. The
CTAP probes have been extensively employed in RDE experiments
to determine the static pressure [44,48]. The CTAP probe has a
length of 1 m and an inner diameter of 1 mm according to Fotia
et al.’s research [48]. Piezoresistive pressure transducers are
installed on the CTAP tubes to obtain pressure signals. Three CTAP
instruments are positioned on the combustion chamber to measure
pressures p3 , p4 , and pth at different ports on the chamber wall. Note
that p3 is the pressure just after injecting, p4 is the pressure at the
end of the combustion chamber and the entry of the nozzle, and pth
is the pressure at the nozzle throat. The total pressure of the burned
gas can then be estimated by the following isentropic relation when
the throat reaches a choked condition:
pc pth 1 γ−1
Ma2th
2
The hollow RDE lying on the thrust stand and the exhaust plume.
γ∕γ−1
pth
γ1
2
γ∕γ−1
(1)
Downloaded by NC State University on September 8, 2023 | http://arc.aiaa.org | DOI: 10.2514/1.B38830
768
ZHANG ET AL.
where γ is chosen to be 1.22. However, it should be noted that this
can only be considered as an estimation of the chamber pressure due
to the unsteadiness of the flow. This approach to measuring the total
pressure in the RDE has been adopted by Bach et al. [44] in their
experiments. Apart from this, they also measured the total pressure
of the exhausted gas via Kiel tubes and compared the results. It is
found that the difference is fairly small. Figure 3 is a plot of the
estimated chamber pressure pc and the measured pressure at the
entrance of the Laval nozzle p4 . It can be seen that these two fit each
other very well. This indicates that the burned gas is nearly stagnant
before entering the nozzle and that the pressure estimated from
Eq. (1) represents the total pressure of the combustion chamber.
A high-speed camera is placed against the RDE to capture the
video data of the rotating detonation waves in the chamber. The highspeed camera used in this study is a Phantom-v611. The frame size is
128 × 128, and the frame rate is 48;000 s−1 . The RDE is bedded on
the moving part of the thrust stand, and the thrust is measured with a
load cell. In our experiments, gaseous methane and oxygen are
chosen as the fuel and oxidizer. The feeding mass flow rate is
measured by two Coriolis flowmeters that directly measure the mass
flow rate of the supplied propellants and ensure accuracy. The total
mass flow rate of the present experiments varies between 100 and
300 g/s, and the equivalence ratio varies between 0.5 and 1.2. A
National Instruments acquisition system that allows multichannel
acquisition is used to acquire low-frequency variables like the thrust,
pressure, and mass flow rate measurement. The uncertainties of the
measurement are evaluated using the approach in Ref. [49]. The Atype uncertainty and B-type uncertainty are considered together to
determine the total uncertainties of the variables directly measured.
The uncertainty of the mass flow rate is determined by the standard
deviation of the fluctuation within the average duration, along with
the accuracy of the Coriolis flowmeters and the uncertainties in the
data acquisition systems. The total uncertainty of the mass flow rate is
found to be up to 0.29% with a 95% confidence interval. The
uncertainty of the pressure measured is calculated in the same way
and is found to be up to 5% with a 95% confidence interval. The
uncertainty of the thrust is calculated by taking into account the
calibration and the uncertainty caused by the placement posture of
the test objects, which is estimated to be up to a 5 deg deviation from
the normal direction of the thrust sensor. The value is up to 1.1% with
a 95% confidence interval. The uncertainties of the dependent variables are calculated by the propagation of uncertainty. Therefore, the
uncertainty of the equivalent ratio is estimated to be up to 1.14% with
a 95% confidence interval. The uncertainty of the specific impulse is
up to 1.96% with a 95% confidence interval. The uncertainty of the
chamber pressure derived through Eq. (1) is also calculated by the
uncertainty propagation. Herein, the uncertainty of γ is set as
0.1∕1.22 8.2%. By doing this, the uncertainty of the chamber
pressure is calculated as up to 5% with a 95% confidence interval.
In the present study, the experimental data are plotted with error bars
to show the uncertainties in Figures 3, 7–11, and 17–19. However, in
some figures, error bars are not visible due to their smaller sizes in
comparison with the markers.
Fig. 3
The estimated chamber pressure pc versus the measured p4 .
C. Normalized Metrics for Rocket Performance
There are three important performance metrics for designing a
conventional rocket engine. They are the specific impulse I sp , which
measures the overall performance of the entire engine; the characteristic velocity c, which only measures the performance of the propellants and the combustion chamber; and the thrust coefficient Cf ,
which only measures the performance of the nozzle [23]. These three
metrics are defined as
I sp F
1
_
mg
g
2γRT c
pe
1−
γ−1
pc
γ−1∕γ
c pe − pa Ae
pc
g
Ath
(2)
c Cf F
pc Ath
p
γRT
pc Ath
_
m
γ 2∕γ 1γ1∕γ−1
2γ 2
2
γ−1 γ1
γ1∕γ−1
pe − pa Ae
pc
Ath
1−
pe
pc
(3)
γ−1∕γ
(4)
The first versions of Eqs. (2–4) allow the determination of the
metrics from measured values. The second versions of Eqs. (2–4) give
the ideal values. The relationship between these metrics is given by
I sp c Cf
g
(5)
In the current work, several normalized performance metrics are
used to compare the measured performance with the ideal performance
of the isobaric combustion and the analysis of the performance deficit.
First of all, to obtain the ideal performance, the method that Stechmann
[40] proposed is adopted, which is based on the NASA CEA [50] code.
For the input condition, the propellant composition, the equivalent
ratio, and the initial temperature are set according to the testing
conditions. The chamber pressure pc is used as the input value of
the first iteration, and then a characteristic velocity c is obtained. This
process is iterated until the following relation [Eq. (6)] is satisfied:
_ exp m
pc;id Ath
cid
(6)
Applying the ideal chamber pressure pc;id and the ideal characteristic
velocity cid to the second version of Eq. (4), we then get the ideal
thrust coefficient Cf;id . Subsequently, the ideal specific impulse I sp;id
can be calculated from Eq. (5).
Normalizing the measured specific impulse and the characteristic
velocity with the ideal values, the characteristic specific impulse ηIsp
and c efficiency ηc can then be obtained. However, the nozzle
efficiency ηCf cannot be obtained by directly normalizing the measured thrust coefficient Cf;exp with Cf;id calculated from Eq. (5). It is
worth noting that Cf;id may not accurately represent the ideal thrust
coefficient because it is calculated based on the ideal chamber
pressure instead of the measured chamber pressure in the experiments. In fact, as indicated by Eq. (4), there are two ways to calculate
the ideal value of the thrust coefficient. One is to apply the ideal
chamber pressure pc;id to Eq. (4), namely, the mass-flow-rate-based
ideal thrust coefficient. Apparently, the ideal thrust coefficient Cf;id
calculated from Eq. (5) is the mass-flow-rate-based ideal thrust
coefficient. The other is to apply the measured chamber pressure
pc;exp to Eq. (4), namely, the chamber-pressure-based ideal thrust
0 . As is known, the nozzle efficiency η
coefficient Cf;id
Cf should be
obtained by normalizing the measured thrust coefficient Cf;exp with
the ideal thrust coefficient calculated under the same condition of the
chamber pressure; here, it is the chamber-pressure-based ideal thrust
0
.
coefficient Cf;id
769
ZHANG ET AL.
In rocket engine design, the c efficiency ηc is usually used to
represent the degree of completion of chemical energy released in the
combustion chamber; and the nozzle efficiency ηCf is usually used to
evaluate the nozzle design [23]. The definitions of these normalized
metrics are
ηIsp Isp;exp
;
Isp;id
ηc cexp pc;exp
;
cid
pc;id
ηCf Cf;exp
0
Cf;id
(7)
These normalized metrics allow the performance deficit analysis of
the whole engine and each component. This enables us to gain a better
understanding of how well the engine is functioning and identify
specific areas where improvements can be made.
III.
Results and Discussions
Downloaded by NC State University on September 8, 2023 | http://arc.aiaa.org | DOI: 10.2514/1.B38830
A. The Existence of Detonation Waves
In this study, we judge whether the detonation waves are formed in
the combustion chamber mainly from high-speed video. Figure 4 is
an example of the high-speed video results of test 7 in Table 2. In
Fig. 4, two rotating high-brightness regions are visible. It is hard to
determine whether these observed waves are detonation waves
directly. This is because the typical propagation of detonation waves
occurs along the outer wall of the combustion chamber, where they
interact with the unburned mixture and generate a pressure wave.
However, due to the presence of a nozzle with a small throat covering
the combustion chamber, the propagation of the pressure wave may
not be observed directly. The high-brightness region can be attributed
to either the detonation wave or the rapid deflagration.
To get more information on the propagation of the high-brightness
regions, a ring area (denoted by blue circles in Fig. 4) was extracted
from the high-speed video images and was processed with the
method that Bennewitz et al. [51] proposed so that spatial-temporal
distribution was obtained, which is shown in Fig. 5a. The spatialtemporal distribution can visually present the propagation characteristic. Bright and dark stripes are visible in the figure, which indicate
that the high-brightness region propagates steadily at a certain speed.
And, it is noted that two waves are rotating in the combustion
chamber. Thus, the propagation speed within this time interval is
calculated as 1774.1 m/s, which is 75.4% of the Chapman–Jouguet
(CJ) velocity at the same testing condition. To determine the propagation speed during the entire hot-fire period, we arrange a probe on
the images and extract the light intensity history, which is shown in
Figs. 5a and 5b. Then, we perform a fast Fourier transform (FFT) and
short-time Fourier transform (STFT) with the light intensity history;
and the results are shown in Figs. 5c and 5d. It can be seen that
the frequency domain is neat and there is a strong main frequency
of 9358.9 Hz. Thereafter, the propagation speed is calculated as
1764.1 m/s, which is consistent with the previous instantaneous
result. Therefore, it can be concluded that the high-brightness regions
Table 2
A summary of detonation
wave speeds
Test Ath ∕Ain
n
Dw
DCJ
ηCJ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
1
2
2
1
1
2
2
1
2
1
2
1
1
2
2
2
1
1
2
1
1
1
1
1
2
1
1
1
1773.4
1538.1
1669
2425.2
2499.6
1673.8
1769.9
2553.2
1543.7
1908.6
1728.1
2562.3
2617.9
1802
1712.4
1576.7
1863.8
1941.1
1726.1
2528.7
2561.8
2585.5
2500.4
2443.9
1473.5
2288.6
2427.6
2298.7
1936.8
2072.7
2259.6
2378.4
2362.5
2257.1
2352.9
2474.1
2035
2059.9
2298.9
2461.2
2527.4
2378.3
2254.5
2030.1
1980.5
2088.6
2266
2387.4
2524.6
2496.7
2393.6
2299.2
1921.9
2208.5
2348.2
2517.1
0.916
0.742
0.739
1.020
1.058
0.742
0.752
1.032
0.759
0.927
0.752
1.041
1.036
0.758
0.760
0.777
0.941
0.929
0.762
1.059
1.015
1.036
1.045
1.063
0.767
1.036
1.034
0.913
Table 3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
5.3
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
A summary of the mass flux and wave number of
experiments from other research groups
Bigler and Bennewitz
et al. [33,51]
Stechmann et al. [40]
Goto et al. [52]
The present study
Mass flux in the combustion
chamber, Kg∕s∕m2
Wave number
distribution
80–1000
2–10
1500–2300
40–150
10–100
1–9
1–2
1–2
are steady detonation waves because the fast deflagration cannot
reach such a high speed.
In this way, we determined the wave speed and the operating mode
of all the tests. The results are summarized in Table 2. In this study, all
of these tests operated mostly in single-wave mode or double-wave
mode, and no collision phenomenon was observed. It should be noted
that the diameter to determine the wave speed is chosen as the
chamber diameter. However, there is a possibility that the detonation
waves travel at a smaller radial position [17]. This might be the reason
for those wave speeds larger than the CJ speed in Table 2. Also, as can
be seen in Table 2, the wave numbers in this study are generally one or
two. In other studies adopting the same propellants [40,51], the wave
numbers can be pretty large and are distributed between 1 and 10.
The reason for the difference might lie in the mass flux. The mass
flux of these studies is on a level of 100–2000 kg∕s∕m2 , whereas the
mass flux of the current experiments is distributed on a level of
10–100 kg∕s∕m2 . A higher mass flux may widely expand the upper
limit of the wave numbers, and this might account for fewer
wave numbers in the current study. However, it should be noted
that this is a very preliminary hypothesis that requires further
detailed investigation.
B. Analysis of the Performance Deficit
Fig. 4
The high-speed video of test 7.
Figure 6a shows the curves of the measured thrust, pressures, and
mass flow rates, during a hot-fire test. The reason why p4 is bigger
than p3 might be due to the unsteadiness of the flow or a macroscopic
flow near the injector. Figure 6b is a spatiotemporal distribution of the
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Fig. 5 The a) spatial-temporal distribution, b) light intensity history, c) amplified light intensity history, d) FFT of the light intensity signal, and e) STFT of
the light intensity signal of test 7.
detonation waves of the same hot-fire test. The detonation speed is
1908.6 m∕s with a single-wave mode, which is 92.6% of the theoretical CJ velocity under the same testing condition. For each ignition, the oxidizer is first supplied into the chamber and the fuel supply
begins immediately after it. When the propellant supply becomes
steady, the engine is ignited by a predetonator. When the flow rates of
the propellants supplied to the combustion chamber stabilize, an
electric sparking plug ignites the combustible mixture, generating a
detonation wave in the predetonator. This wave then propagates into
the combustion chamber, creating rotating detonation waves within
it. The propellants used to ignite these waves are gaseous methane
and gaseous oxygen: the same ones used by the engine itself. The
gray dashed lines in Fig. 6 denote the periods that we adopted to get
the average thrust, pressure, and mass flow rate of cold flow and the
hot-fire test. The period ahead of the ignition indicates the averaging
duration of the cold flow, and the period after the ignition indicates
the averaging duration of the hot-fire test.
The measured specific impulses versus the mass flux at the exit
throat are shown in Fig. 7. The dashed lines represent the measured
specific impulses, whereas the solid lines represent the results of the
ideal specific impulse calculated from the method introduced in
Sec. II.C. Different colors denote different equivalent ratios. The
reason why mass flux is chosen as the horizontal axis rather than
the chamber pressure is that the measured specific impulses and the
ideal values are related to each other at the same mass flux instead of
the chamber pressure. In fact, if the experimental chamber pressure is
0
in CEA,
chosen as the input condition to calculate the ideal value I sp;id
the obtained ratio of I sp;ex ∕I 0 sp;id cannot evaluate the performance of
the entire engine but only of the nozzle, which can be illustrated by
the following derivation:
I sp;ex Fex ∕mex g Fex ∕pc;ex Ath Cf;ex
0
0 ηCf
0
0
I sp;id
Cf;id
Fid ∕mex g
Fid ∕pc;ex Ath (8)
where F’id denotes the ideal thrust calculated from the measured
0
chamber pressure pc;ex . As can be expected, the ratio Isp , I sp;ex ∕I sp;id
is equal to the actual nozzle efficiency ηCf , which evaluates the
performance of the nozzle as discussed in Sec. II.C. This can also
account for the difference between the ideal I sp values calculated by
two different methods in Ref. [42].
In Fig. 7, it can be observed that the specific impulse increases as
the mass flux increases, which agrees with the trend of the ideal
values. Within the configuration of Ath ∕Ain 5.3, the highest specific impulse reaches around 130 s with an equivalent ratio of 1.2 and
a mass flux of 170 kg∕m2 ⋅ s. There is a proclivity that the specific
impulse becomes higher as the equivalent ratio increases from 0.5 to
1.2, which agrees with the ideal values and the results of other
reported research [33,40,42] using the same propellants but in annular chambers. It should be noted that at low mass flux, the specific
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Fig. 8 The characteristic specific impulse ηIsp of the hollow RDE with
Ath ∕Ain 5.3 and 2.7.
deficit quantitatively, the characteristic specific impulse ηIsp is first
taken into consideration, of which the result is shown in Fig. 8. The
unfilled markers represent the results of the Ath ∕Ain 5.3 case,
whereas the solid markers represent the results of the Ath ∕Ain 2.7 case. It can be seen that the performances of both configurations
are relatively low as compared with the ideal value of ηIsp 1. It is
also obvious in the figure that the overall characteristic specific
impulses ηIsp of the Ath ∕Ain 5.3 case are higher than those of the
Ath ∕Ain 2.7 case. The large deficit of I sp indicated poor usage of
the nozzle or the propellants and is conducive to the goal of larger
thrust. To further look into the reason for the deficit in I sp , a detailed
performance analysis for the nozzle and combustion chamber should
be carried out.
As the description of the thrust coefficient Cf [which is shown in
Eq. (4)] indicates,
Fig. 6 Representations of a) typical temporal curves of thrust, pressures, and mass flow rates; and b) spatiotemporal distribution of detonation waves of test 10. Mass flow rates of O2 and CH4 are
192.6 0.28 g∕s and 23.27 0.03 g∕s. Plenum pressures of O2 and
CH4 are 3.34 0.02 bar and 3.47 0.02 bar, respectively. Thrust is
203.1 1.0 N.
The thrust coefficient may be thought of as representing the
amplification of thrust due to the gas expansion in the
supersonic nozzle as compared to the thrust that would be
exerted if the chamber pressure acted over the throat area
only [23].
Thus, the chamber pressure has a significant effect on the value of
the thrust coefficient. The chamber pressures of both configurations
are shown in Fig. 9. It is exhibited that the chamber pressure of the
Ath ∕Ain 2.7 case is overall higher than the Ath ∕Ain 5.3 case at
the same mass flow rate. This is attributed to the smaller nozzle throat
of the Ath ∕Ain 2.7 case.
It is known that a high chamber pressure usually indicates a high
theoretical performance, which will be further discussed in detail in
Sec. III.C. However, considering that this section is about analyzing
the performance deficit, and the chamber pressures of different
configurations are not on the same level, we will therefore discuss
the normalized metrics, ηCf and ηc , which can determine the
Fig. 7 The specific impulse of Ath ∕Ain 5.3 ratio varying with the
throat mass flux.
impulses are close to and even exceed the ideal value. This might be
attributed to the choked assumption in CEA, which implies there are
no valid ideal values of performance metrics for the unchoked tests.
Thus, the following discussion will only concentrate on the tests
where the nozzle is choked, which means the data points presented in
the following figures will not include the unchoked tests.
As can be seen in Fig. 7, there is a significant performance deficit
between the measured performances and ideal values. To evaluate the
Fig. 9 The chamber pressure versus the mass flow rate.
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Fig. 10 Representations of a) nozzle efficiency ηCf and b) c efficiency ηc of the hollow RDE with Ath ∕Ain 5.3 and 2.7.
performance deficit but not the Cf and c themselves. The nozzle
efficiency ηCf and the c* efficiency ηc are plotted in Fig. 10. The
highest values of ηCf and ηc are 0.91 and 0.84. As for the nozzle
efficiency ηCf , the data are plotted over the chamber pressure, and it is
shown that almost all results with Ath ∕Ain 2.7 are lower than those
with Ath ∕Ain 5.3. Nozzle efficiency is usually used to evaluate the
level of the nozzle profile design. The closer its value to the unit, the
closer the nozzle can make the obtained thrust to the ideal value,
which means the better the nozzle design. It is a parameter that is only
related to the nozzle. Hence, the different performance deficits can
only result from the nozzle design. From Table 1, it is noticed the
nozzle expansion area ratios of these two configurations are different,
and the values are εe 1.64 and 1.76 for Ath ∕Ain 5.3 and 2.7,
respectively. According to Fotia et al. [32], the performance of the
RDEs is sensitive to the expansion ratio. Therefore, it is assumed that
it is the difference in the nozzle design that may account for the result.
The different expansion area ratios lead to different nozzle profiles,
and may subsequently lead to the differences in flow separation,
friction, heat transfer, etc., which have a significant influence on the
performance. However, the exact mechanism is not clear. To further
look into the mechanism, numerical simulations and more carefully
designed experiments are needed.
As for the c efficiency, the values are distribute between 0.6 and
0.9. The average c efficiency is ηc , withav 0.71 for Ath ∕Ain 2.7, which is lower than 0.79 for Ath ∕Ain 5.3. According to Eq. (3),
it can be seen that the characteristic c is sensitive to the gas property γ
and the combustion temperature T, which are both related to the
propellant, the combustion process, and heat release inside the combustion chamber. Thus, parameter c is generally used as a figure of
merit when comparing different propellant combinations for combustion chamber performance [23]. The c efficiency can be used to
express the degree of completion of the chemical energy releases in
the combustion chamber. Apparently, the lower c -efficiency value
of the Ath ∕Ain 2.7 case means a reduced degree of energy release.
This means the degree of energy release in the Ath ∕Ain 2.7 case is
somehow lower than that in the Ath ∕Ain 5.3 case.
To uncover the reason behind the difference in c* efficiencies
observed between the two configurations, as explained earlier, it is
appropriate to direct our attention toward the factors that may impact
the combustion process taking place within the chamber. Unlike the
volume combustion on which traditional engines are based, the
detonation waves can be seen as a process of surface burning. And,
in RDEs, the detonation waves are usually rotating at the injecting
end of the combustion chamber where the injected fresh propellants
are distributed. The distribution of fresh propellants therefore has a
great influence on the propagation and heat release of detonation
waves. One of the factors that will significantly affect the propellant
distribution is the injection process. Figure 11 shows the pressure
ratio of the plenums to the bottom of the combustion chamber:
ppl ∕p3 . It reveals that within the Ath ∕Ain 5.3 case, the injection
pressure ratios of both the fuel and oxidizer are either around or above
the critical choked pressure ratio, which is estimated by
ppl ∕p3 cr γ1
2
γ∕γ−1
(9)
with γ 1.3. Meanwhile, within the Ath ∕Ain 2.7 case, all the
injection ratios of the oxidizer are lower than the critical value; and
only a few cases exceed the critical choked ratio, which means the
choking condition was not met in the injecting process in most tests.
The reason also lies in the smaller throat:
Choked:
p A
_ p0
m
RT 0
γ
2
γ1
Fig. 11 The injection pressure ratio, ppl ∕p3 , of the hollow RDE with Ath ∕Ain 5.3 and 2.7.
γ1∕γ−1
(10)
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ZHANG ET AL.
Fig. 12 A simplified model of the engine.
Unchoked:
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p A
_ p0
m
RT 0
2γ
γ−1
p0
pb
2∕γ
−
p0
pb
γ1γ
(11)
Here is the mechanism of the unchoked injection. The engine
can be simplified as the model shown in Fig. 12 with reference to
Ref. [43]. And, the mass flow rate of the flow can be determined by
Eqs. (10) and (11). A represents the section areas, such as the injecting
area Ain , and the throat area Ath . Subscript 0 denotes the stagnation
state, and subscript b denotes the backstate of the relevant section.
Because only the tests where the nozzle was choked are considered
here, the mass flow rate at the nozzle throat can be estimated
from Eq. (10).
It can be seen that at the same mass flow rate, a small nozzle throat
will lead to a high chamber pressure. This is the reason for the higher
chamber pressure in the Ath ∕Ain 2.7 case shown in Fig. 9. From
Figs. 3 and 6, it can be seen that the discrepancy between p3 and pc is
not large. Thus, the rising of the chamber pressure will lead to a
pressure ratio, ppl ∕p3 , lower than the critical value if the plenum
pressure does not change. According to Eq. (11), a lower pressure
ratio, ppl ∕p3 , will result in a reduction of the mass flow rate. Hence,
the plenum pressure has to rise to maintain the mass flow rate. We
then assume that the plenum pressure will rise to a value to make the
pressure ratio, ppl ∕p3 , higher than the critical value so that the
injectors are still choked. According to Eq. (10), because the injecting
areas are the same in these two cases, the mass flow rate will rise
because of the rising plenum pressure. This is against the assumption
of the constant mass flow rate. Therefore, the choking condition
cannot be met at the injection section anymore within the Ath ∕Ain 2.7 case. It is worth noting that the unchoked injection might bring
some loss in performance, and it may cause the occurrence of a
backflow event, which is dangerous for RDE operation when structure optimization has not been conducted.
This analysis can be done for the isentropic flow without any
problems. Because the flow in RDEs is periodic, this can only be
considered a trend estimation. Even so, it can still provide some
insightful information.
C. A Gas-Stratification Flowfield Model
To provide more insight into the mechanism behind the larger
deficit in c efficiency, a gas-stratification flowfield model was
proposed here. As is known, behind the detonation wave there is a
region of high pressure and high temperature, which blocks the
injection and therefore reduces the effective injection area [52].
The blocked injectors will experience a response time before recovering from the blockage. The disparity in the plenum pressures of the
fuel and oxidizer will lead to a difference in response time [33,53],
and may therefore lead to the stratification of fresh gas, which might
have some effects on the performance. Hence, to assess the influence
of the response time on the performance, a flowfield model considering the gas stratification, which is shown in Fig. 13, is proposed
to explain the increased performance deficit of the combustion
chamber.
In Fig. 13, the curve represents the pressure distribution on the
inlet wall resulting from the detonation wave. This is also the
backpressure for injectors. The pressure profile is exponential
profile that is commonly used to simulate detonation pressure
profile. The red and blue solid lines represent the plenum pressures
of the fuel and oxidizer: pf;pl and pox;pl . The red and blue dashed
lines represent the critical backpressure of choked injection: pf;cr
Fig. 13 A gas-stratification flowfield model
and pox;cr . Ignoring the aerodynamic response time of the orifice
caused by the time-varied downstream pressure, the injection process
can then be divided into three states by two characteristic pressure
ratios [the unit and ppl ∕p3 cr determined by Eq. (9)]. The three states
are the blocked state for ppl ∕p3 < 1, where propellants are not able to
enter the chamber; unchoked state for 1 < ppl ∕p3 < ppl ∕p3 cr ; and
the choked state for ppl ∕p3 > ppl ∕p3 cr. Herein, the choked state is
considered as the stable injecting state because disturbances from the
downstream cannot affect the upstream injection.
The fresh gas layer can then be divided into three regions: the
buffer region (in Ref. [54]), the unchoked region, and the choked
region. The buffer region is due to the difference in the time it takes to
recover from blocking, which has been verified experimentally by
Chacon and Gamba [54]. There is only one side of the propellants
(fuel or oxidizer) in the buffer region. The unchoked region is due to
the difference in the time it takes to return to the choked state. In this
region, the equivalent ratio of the combustible gas ahead of the
detonation waves will deviate from the designed condition and vary
in a wide range because of the difference in the injecting flow rate and
the interaction of disturbances with injectors, which will reduce the
energy release of the propellants (as will be validated later). Unlike
the two regions described earlier in this paper, in the choked region,
the equivalent ratio is evenly distributed owing to choked injection. It
can be seen that the buffer region and the unchoked region are
responsible for the detriment to the performance.
Here, a parameter is defined to describe the influence of the
injection pressure ratio, ppl ∕p3 , namely, the unchoked area ratio:
ηunch Δxunch
L
(12)
where Δxunch is the unchoked region on the inlet wall. It includes the
area of the buffer region, and the unchoked region on the inlet wall L
denotes the total inlet region. Within Δxunch , the backpressure is
lower than the higher one of the plenum pressures and higher than
the lower one of the critical pressure, min pf;cr ; pox;cr < p <
max pf;pl ; pox;pl ; so, at least one side of the propellants is injected
into the combustion chamber at an unchoked state. As shown in
Fig. 14, a small injection ratio, ppl ∕p3 , usually means a large
unchoked area ratio, ηunch , which will adversely affect the performance. It is worth noting that the aforementioned flowfield model
makes no special assumption for the wave’s form, and so the model is
also suitable for shock waves and other forms of disturbances. Moreover, an unchoked injection will also lead to unstable operations like
mode switching [55] and even extinguish–reignition events [46],
which may also affect the energy release [53].
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Fig. 16 The heat release as a function of the mass fraction amplitude of
CH4 .
Fig. 14 The change of the unchoked area Δxunch at different injection
pressure ratios.
To validate and quantitatively evaluate the effect of the nonuniform
distribution of the equivalent ratio on the total energy release, a onedimensional (1-D) simulation is performed. Our in-house solver,
BYRFoam, developed on the OpenFOAM® platform, is used, which
has been validated in previous work [56–58]. The physical model
used in the 1-D simulation is a detonation tube that is 1 m in length.
The combustible gas is the premixed mixture of CH4 ∕O2 , which is
kept the same as the propellants used in the experiment. The chemical
kinetic model is the CH4 ∕O2 detailed chemical reaction mechanism,
which has been used for simulations of detonation waves [59,60] and
rotating detonation engines [61]. The mesh size is 0.1 mm. The time
step is controlled by the Courant number. The maximum Courant
number is set as 0.15 in these simulations. The boundary conditions
of the left and right ends are both wall conditions.
The initiated flowfield is shown in Fig. 15. The field is full of fresh
combustible CH4 ∕O2 gas at the beginning. The initial pressure and
temperature are set as 1.013 bar and 300 K, respectively. The detonation wave is ignited by a hot spot at the left end. The pressure and
the temperature within the hot spot are set as 35 bar and 3000 K,
respectively. Region 1 is filled with combustible gas with an evenly
distributed equivalent ratio to reduce the influence of the ignition.
The mass fraction of CH4 in region 2 is distributed sinusoidally while
keeping the average equivalent ratio in region 2 the same as that in
region 1. Here, here only the heat release in region 2 Q is discussed. A
series of simulations with different amplitudes and different average
equivalent ratios was carried out to evaluate the impact of the unevenness degree of the equivalent ratio distribution, that is, the amplitude
of CH4 mass fraction, B in region 2, and the results of Q are
summarized in Fig. 16. The way to calculate the total heat release
Q is shown in the Appendix. As can be seen, there is a proclivity for
the heat release to decrease as the fraction amplitude increases. This
result verifies the hypothesis that a poor equivalent ratio distribution due to the unchoked injection will result in a severe deficit of
the performance, and therefore the aforementioned mechanism.
According to the gas-stratification flowfield model proposed earlier in this paper, the smaller the pressure difference between the fuel
and oxidizer plenums, the less performance loss; i.e., a higher c
efficiency ηc will occur. Therefore, we further verified our experimental data and found that they were in agreement with the results
predicted by the flowfield model. The result is shown in Fig. 17. As
shown in the figure, with a mass flow rate of the same level, a pressure
ratio closer to the unit results in a higher ηc , which indicates less
performance loss. This phenomenon in the experiment also serves as
a further validation of the gas-stratification flowfield model.
The gas-stratification flowfield model indicates that a low injection
pressure ratio should be avoided. However, the low injection pressure
ratio has its own advantage. It is worth noting that the main reason for
studying RDEs is that rotating detonation is a new type of combustion
that can achieve pressure gain; and our aim is to break the performance bottleneck of conventional engines. For rocket engines, we are
counting on reducing the requirement for the exit pressure of the
turbopump via the application of RDEs, thus simplifying the turbopump system and improving the overall performance of the integrated engine system. For a ramjet or scramjet, we hope to get a
higher total pressure recovery coefficient through the pressure gain
capacity of rotating detonation waves [62]. In this respect, the injection pressure loss should be reduced as much as possible, which is
contrary to the previous conclusion. It can be seen that there is a
tradeoff space between the performance deficit and the injection
pressure ratio when choosing the area ratio, Ath ∕Ain , for RDEs, which
will be discussed in detail in the following subsection.
It should be noted that the gas-stratification flowfield model only
considers the influence of circumferential distribution. However, as
indicated by previous works [24,63], the flowfield structure behind
the detonation waves is quite complex. To investigate the effect of the
detailed flow structure on the performance, further numerical simulation should be performed.
D. Global Performance Consideration
Fig. 15 The initial set of the mass fractions of CH4 and O2 in the 1-D
simulation.
The preceding discussion is mainly about performance deficits.
But in practical engineering applications of the rocket engine, what
we are more concerned with is the actual obtained thrust and specific
ZHANG ET AL.
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Fig. 17 The c efficiency ηc under different pressure ratios between the fuel and oxidizer plenums, pf ∕pox . The color bar indicates the pressure ratio of
pf ∕pox .
Fig. 18 The thrust F and specific impulse Isp of the RDE as a function of the equivalent ratio φ. The oxygen mass flow rates of these tests are all distributed
around 200 g∕s within 190.8 ∼ 208.4 g∕s, of which the average value is 199.2 g∕s.
impulse. There is still a possibility that the actually obtained thrust or
the specific impulse might still be enhanced by a smaller throat, even
with a larger performance deficit. Therefore, it is necessary to discuss
global performance.
Figure 18 shows the variation of the gross thrust and the specific
impulse with the equivalent ratio at a set oxygen mass flow rate. Note
that the oxygen mass flow rates of these tests are all distributed
around 200 g∕s within 190.8 ∼ 208.4 g∕s, and the average value is
199.2 g∕s. It can be seen that the RDE with the throat ratio of
Ath ∕Ain 2.7 achieves a higher gross thrust and specific impulse,
which are consistent with the ideal values that are presented in Fig. 18
with solid lines. However, at a high fuel mass flow rate, the measured
gross thrusts of the two configurations become close, and the specific
impulse of the smaller ratio of Ath ∕Ain 2.7 becomes even lower
than that of the larger ratio of Ath ∕Ain 5.3. This might be due to the
higher performance loss discussed earlier in this paper.
The increase in the gross thrust and specific impulse is actually due
to the increase in chamber pressure. However, the increased chamber
pressure will put additional demands on the supplied stagnation
pressure, which will result in the complexity of the supply system.
Therefore, it is necessary to consider the engine as a whole and
evaluate the performance of the whole system. Herein, two metrics
developed from the works of Fotia et al. [32,45] are adopted to assess
the usage of the supplied stagnation pressure:
F
;
pox Aox
I sp pa
pox
the injection. In Fig. 19, the chamber with the ratio of Ath ∕Ain 2.7
shows much worse usage of the supplied stagnation pressure, which
is in contrast with the results of the gross thrust and specific impulse
discussed earlier in this paper. This means that although we can get
higher thrust and specific impulse by increasing the chamber pressure
with a reduced ratio of Ath ∕Ain , the chamber with the reduced ratio
of Ath ∕Ain may exhibit a lower efficiency in transforming the
increased supplied stagnation pressure, which is also corresponding
to increased potential energy, into the thrust obtained.
(13)
The former one of F∕pox Aox evaluates the usage of the upstream
pressure in the oxygen plenum, and the latter one of I sp pa ∕pox
evaluates the usage of both the supplied pressure and propellants. It
is noteworthy that both metrics have included the pressure loss during
Fig. 19 The F∕Aox pox and Isp pa ∕pox of the RDE as functions of the
equivalent ratio φ. The oxygen mass flow rates of these tests are all
distributed around 200 g∕s within 190.8 ∼ 208.4 g∕s, of which the average value is 199.2 g∕s.
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776
ZHANG ET AL.
The preceding discussion implies that there is a tradeoff
between the small and large area ratios of Ath ∕Ain . As mentioned
earlier in this paper, a small Ath ∕Ain ratio is typically associated
with a low injection pressure ratio and high chamber pressure at
the same mass flow rate: both of which have advantages and
disadvantages. As for the high chamber pressure, it usually corresponds to a higher theoretical propulsive performance, as shown
in Fig. 18. However, this also imposes additional requirements on
the pressure supply system upstream, leading to increased complexity in the integrated engine system. As for the low injection
ratio, it corresponds to minimal pressure loss during injection,
which is a current research focus in the RDE community. However, a low injection pressure ratio can significantly harm chamber
performance and lead to a large performance deficit. In summary,
a small Ath ∕Ain ratio can bring benefits such as high theoretical
propulsive performance and minimal pressure loss during injection, but it may also result in a significant performance deficit and
increased system complexity. Selecting the appropriate Ath ∕Ain
ratio can be challenging. Therefore, the RDE community’s primary goal in the next stage should be to minimize the performance
deficit while maintaining a small Ath ∕Ain ratio to use the high
theoretical performance effectively.
Based on the findings of this study, it is recommended that the
design procedure for practical engineering applications and the system design of RDEs should follow the steps outlined in the following.
First, a performance target, such as thrust or specific impulse, that
aligns with the vehicle’s function, working conditions, and background should be proposed. Throughout the design process, a systematic design methodology should be used to consider all relevant
factors, including pressure loss, performance deficit, stability of
detonation, etc. This will aid in determining an appropriate area ratio
of Ath ∕Ain , as well as making other essential decisions. Moreover, it is
important to bear in mind that the ultimate goal is to surpass the
performance of conventional engines. Therefore, in order to achieve
the final aim, some sacrifices may need to be made, which could be
deemed acceptable.
IV.
Conclusions
Experimental analysis of the effects of the Laval nozzle on the
propulsive performance of a hollow RDE has been carried out for the
first time. Experiments using CH4 and O2 were performed on a
modular RDE. Pressure and thrust measurements were taken within
two configurations. The outlet throat to inlet area ratios of Ath ∕Ain are
set to be 5.3 and 2.7. It was found that the scales of the Laval nozzle
have great impacts on the propulsive performance. The leading
findings are listed as follows:
1) The performance assessment framework of conventional rocket
engines is introduced to the study of RDEs systematically, which
provides a way to compare the new-concept propulsion system with
developed traditional rocket engines. Three normalized metrics are
adopted to evaluate the performance deficit of the entire engine,
nozzle, and combustion chamber separately. The maximum nozzle
efficiency ηCf and c efficiency ηc reached 0.91 and 0.84 in the
current experiments, respectively.
2) It is found that the characteristic specific impulse ηIsp , the nozzle
efficiency ηCf , and the c efficiency ηc of Ath ∕Ain 2.7 are lower
than those of Ath ∕Ain 5.3. The deteriorated nozzle efficiency ηCf
might be due to the nozzle design, whereas the lower c efficiency
could be explained by the gas-stratification flowfield model proposed
in this study.
3) A gas-stratification flowfield model is proposed in this study to
account for the deficit of c efficiency. According to the model,
the fresh layer is divided into three regions: the buffer region, the
unchoked region, and the choked region. It was found that the
unchoked region, in which the equivalent ratio of propellants distributes unevenly, reduces the chemical energy release, and is hence
responsible for the performance deficit. With the small area ratio of
Ath ∕Ain 2.7, the injection pressure ratio of ppl ∕p3 is lower than the
critical ratio of ppl ∕p3 cr ; as a result, a large unchoked injection area
is created that leads to a larger performance deficit. This model is
validated against 1-D numerical simulations and experimental data.
The results agree well.
4) From the perspective of global performance, it is suggested that
a small Ath ∕Ain ratio will theoretically enhance the thrust and specific
impulse, but it will have less efficient use of the supplied stagnation
pressure due to the higher performance loss.
This study reveals the main tradeoff when designing a RDE. A
small Ath ∕Ain ratio can bring benefits like high theoretical propulsive
performance and small pressure loss during injection, but it will also
lead to complexity of the integrated system and a large performance
deficit, which also imply a low usage of the supplied pressure. To
balance the influence of different aspects, a systematical design
methodology should be applied in practical applications. The work
that needs to be done in the next stage is to minimize the injection
pressure loss with extra considerations on performance loss. It is
worth noting that although this paper concentrates on the hollow
RDE, the design idea developed in this study can also be applied to
other configurations.
The present study also sheds insight onto comparing the performance of detonation-based and deflagration-based engines directly.
This is a reminder that there might be a possibility to further introduce
the design methods of conventional rocket engines into the optimization of RDEs in the future.
Appendix: Calculation of the Heat Release
The way to calculate the heat release of the whole region is
illustrated here.
The definition of the heat release rate is
N
Q_ −
_i
h0i ω
(A1)
i1
where Q_ is the heat release rate, h0i is the specific standard enthalpy of
_ is the production rate of the ith
formation of the ith species, and ω
species.
The derivation begins with the energy equation and the species
equation from the Euler equations:
∂ρE
∇ ⋅ ρHu 0
∂t
(A2)
∂ρY i
_i
∇ ⋅ ρY i u ω
∂t
(A3)
where ρ is the density, E is the total energy, H is the total enthalpy, u is
velocity, and Y i is the mass fraction of the ith species. The total
energy E and total enthalpy H can be written as
N
Y i hi H
i1
u2
2
N
Y i h0i hs;i i1
N
N
Y i h0i i1
u2
2
Y i hs;i i1
u2
2
N
Y i h0i H s
i1
and
(A4)
ZHANG ET AL.
EH−
p
ρ
N
N
Y i h0i i1
Y i hs;i i1
[4]
u2 p
−
2 ρ
N
[5]
Y i h0i Es
(A5)
i1
where hi is the specific enthalpy of the ith species, and hi;s is the
specific sensible enthalpy of the ith species. Hs is the total sensible
enthalpy, and Es is the total sensible energy. Therefore, the energy
equation [Eq. (A2)] can be written as
N
∂ρEs
∂ρY i ∇ ⋅ ρY i u
h0i
∇ ⋅ ρuHs −
∂t
∂t
i1
[6]
[7]
[8]
N
Downloaded by NC State University on September 8, 2023 | http://arc.aiaa.org | DOI: 10.2514/1.B38830
−
h0i ω_ i
[9]
i1
Q_
(A6)
This is the form of energy equation adopted in the computational fluid
dynamics code. It can be seen that Q_ is the source term of the energy
equation.
The integral form of Eq. (A6) is
∂
∂t
[10]
[11]
V
ρEs dV V
∇ ⋅ ρuHs dV V
Q_ dV
(A7)
According to the Gauss formula, this equation can be written as
∂
∂t
V
ρEs dV S
ρuH s ⋅ dS V
Q_ dV
(A8)
Apply this equation to the calculating region in the simulation. It
should be noted that the boundary conditions of the left and right ends
are both wall conditions, where u is zero. Therefore,
∂
∂t
l
l
ρEs dz 0
Q_ dz
[12]
(A9)
[13]
[14]
0
Integrating over time, we get
t
Q
0
l
l
Q_ dz dt 0
ρEs dz
0
[15]
t
(A10)
0
Q is the heat release in the whole region when the fresh gas is
completely consumed by the detonation wave. This is the way we
calculate the heat release in the simulation.
[16]
[17]
Acknowledgments
This study was sponsored by the National Natural Science Foundation of China (grant numbers 91741202 and 52076003). The
authors would like to acknowledge Xiangyu Chen for his remarkable
assistance with the experiments.
[18]
[19]
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T. Lee
Associate Editor