Chinese

Journal of
Aeronautics

Chinese Journal of Aeronautics 23(2010) 647-652

www.elsevier.com/locate/cja

Continuous Detonation Engine and Effects of Different Types of

Nozzle on Its Propulsion Performance
Shao Yetao, Liu Meng, Wang Jianping*
State Key Laboratory of Turbulence and Complex System, College of Engineering, Peking University, Beijing 100871, China
Received 11 January 2010; accepted 1 August 2010

Abstract
The rotating propagation of a continuous detonation engine (CDE) with different types of nozzles is investigated in
three-dimensional numerical simulation using a one-step chemical reaction model. Flux terms are solved by the so-called
monotonicity-preserving weighted essentially non-oscillatory (MPWENO) scheme. The simulated flow field agrees well with the
previous experimental results. Once the initial transient effects die down, the detonation wave maintains continuous oscillatory
propagation in the annular chamber as long as fuel is continuously injected. Using a numerical flow field, the propulsion performance of a CDE is computed for four types of nozzles, namely the constant-area nozzle, Laval nozzle, diverging nozzle and
converging nozzle. The gross specific impulse of the CDE ranges 1 540-1 750 s and the mass flux per square meter ranges
313-330 kg/(m2s) for different nozzles. Among these four types of nozzles, Laval nozzle performs the best, and these parameters
are 1 800 N, 1 750 s and 313 kg/(m2s). A nozzle can greatly improve the propulsion performance.
Keywords: continuous detonation engine; propulsion performance; nozzle effects; Laval nozzle; hypersonic

1. Introduction1
After one hundred years of research and advancements, it has been steadily becoming more difficult to
achieve further improvements in conventional constant-pressure combustion engines in terms of high
efficiency. Urgency now surrounds the development of
new-concept propulsion systems to meet demands for
higher velocity and higher efficiency. Taking advantage of the nearly isochoric combustion process, detonation is inherently seen as a way to achieve higher
thermodynamic efficiency than usual deflagrationbased propulsion devices such as gas turbine engine
and ramjet engine. In particular, detonation allows
more intense and steadier combustion, therefore
comparatively small combustors are capable of creating enormous thrust. And also, it is simple and light,
and does not require turbo-pumps or compressors.
Such benefits have gained worldwide interest in research into detonation engines.
Now, how to design a detonation engine to create
*Corresponding author. Tel.: +86-10-82529038.
E-mail address: wangjp@pku.edu.cn
Foundation item: Aeronautical Science Foundation of China
(2008ZH71006)
1000-9361/$ - see front matter 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/S1000-9361(09)60266-1

thrust becomes a challenge. The pulse detonation engines (PDEs)[1-2] have been widely investigated during
the past three decades. This has shown without doubt
that PDEs have many advantages. Nevertheless, there
are several hurdles that need to be overcome in PDE
research. For one, it is difficult to achieve high operational frequency resulting consequently in low mass
flux at a PDE. For another, high ignition energies are
needed for each pulse cycle making ignition difficult.
As a result, a reliably-functioning detonation engine
has not been developed until now. Developing a detonation engine operating without need of periodic ignition and continuously injecting fuel would greatly reduce difficulties in designs of, in particular, detonation
combustion aerospace thrusters. In this article, the recently investigated continuous detonation engine
(CDE) (also known as rotating detonation engine
(RDE)) is simulated, which is expected to meet the
above demands.
In a CDE, detonation wave (DW) propagates in
azimuthal direction which is perpendicular to but not
against fuel injection direction. As DW propagating
direction and fuel injection direction are independent,
DW can continuously propagate in a wide range of
injection velocity from low subsonic to hypersonic[3]
and need not multi-time ignition naturally. These
characteristics would greatly reduce the difficulties in
the design of a detonation engine.

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Shao Yetao et al. / Chinese Journal of Aeronautics 23(2010) 647-652

The basic concept behind a CDE was first proposed

by B. V. Voitsekhovskii[4]. Experimentally, he achieved
short-lived continuous detonation fuelled by ethylene
or acetylene. In recent years, CDE has been extensively studied both theoretically and experimentally,
by F. A. Bykovskii, et al.[5-7]. J. Kindracki, et al.[8] have
experimentally obtained very promising thrust performances from a rocket-type CDE. F. Falempin, et
al.[9] have performed some preliminary tests to evaluate the capability of C/SiC composite materials to sustain the high temperature of about 1 000-2 000 K at the
head part of the combustor. In numerical aspect, S. A.
Zhdan, et al.[10] have made a wave structure analysis
on a two-dimensional plane domain. M. Hishida, et
al.[11] have performed a detailed analysis of a CDE.
They have discovered Kelvin-Helmholtz (K-H) instabilities and cell structures in CDE. Y. T. Shao, et al.[12]
and X. H. Jiang, et al.[13] have done three-dimensional
simulations to analyze CDEs flow field and wave
structure for a constant area tube, respectively.
Up to now, numerical works on CDEs mainly focus
on the DW structure. In this article, detailed analyses
of the propulsion performance of a CDE with four
different types of nozzles, constant-area nozzle, Laval
nozzle, diverging nozzle and converging nozzle, have
been carried out. Constant-area tube studies[10-12] have
shown that the detonation product flows out of the
combustor exit at high temperature, wasting a lot of
energy. Like a rocket engine, a continuous combustion
engine with a nozzle may efficiently expand the product to create more thrust. In addition, previous performance studies have mainly computed the gross specific impulse, and hence, they have not considered the
enormous injection momentum that may provide
higher propulsion performance. With this background,
CDEs with four types of nozzles are numerically
simulated to investigate their propulsion performance.
2. Computation Scheme
2.1. Governing equation
A one-step chemical reaction model is used in this
simulation. Three-dimensional Euler equations in generalized coordinates are used as governing equations
as follows:
wU wE wF wG
(1)
S



wt w[ wK w]
where the dependent variable vector U, convective
flux vectors E, F and G, and source vector S are defined as
U = [ U U u U v U w e UE ]T
S = [0 0 0 0 0 UZ E ]T

E = [ UU

UUu  p[ x

UUv  p[ y

UUw  p[ z

F = [ UV

U ( p  e) UU E ]
UVu  pK x UVv  pK y

UVw  pK z

V ( p  e) UE ]

G = [ UW

No.6

UWu  p] x

UWv  p] y

UWw  p] z

W ( p  e) UW E ]

u[ x  v[ y  w[ z

uK x  vK y  wK z

u] x  v] y  w] z

The pressure p and total energy e are calculated using

equations of state:
p
e

U RT

p
1
1
1
 EU q  U u 2  U v 2  U w2
J 1
2
2
2

(2)
(3)

where U is density, R gas constant, T temperature, J specific heat ratio and q heat release per unit mass.
The mass production rate is

Z E

dE
dt

 AE exp( Ea /( RT ))

(4)

where E is the mass proportion of reaction mixture gas

(E =1 represents a fresh gas mixture, whereas E =0 indicates the detonation products are in equilibrium), A
preexponential factor and Ea activation energy per unit
mass. A detailed description of all the parameters can be
found in Ref.[14]. Flux terms are solved by using the
five-order so-called monotonicity preserving weighted
essentially non-oscillatory scheme (MPWENO)[15], and
the time integration is performed by using the
third-order total variation diminishing (TVD) RungeKutta method. Grid numbers are 31 (radial direction) u
800 (azimuthal direction) u301 (axial direction), with
mesh size of about 0.4 mm.
2.2. Grid dependency
The chemical induction distance is about 250 Pm for
the Chapman-Jouguet (CJ) detonation of the present
gas mixture. To simulate this problem, a grid size of
less than 25 Pm may be needed, however, it is too
computationally extensive to carry out three-dimensional simulation of a practical-size configuration with
detailed reaction model. As this article aims to investigate CDEs main propagation characteristics and its
propulsive performance but not detailed wave structure, a one-step reaction model with average grid size
of 0.4 mm and a maximum Courant number 0.3 is
used.
First, the thermochemical model and computation
scheme have been validated for some simple
one-dimensional test cases with grid size of 0.1 mm,
0.2 mm and 0.4 mm, respectively. Stoichiometric
H2/air mixture at 400 K and 0.18 MPa is directly ignited by hot spot of 2 000 K and 5 MPa. After propagation of 160 Ps, except the van-Neumann spike pressure, the wave structure almost coincides together for
these three grid size cases, shown in Fig.1. At CJ point
the pressure and temperature are 2.5 MPa and 3 050 K

Shao Yetao et al. / Chinese Journal of Aeronautics 23(2010) 647-652

No.6

which are close to theory value. The computed propagation velocity is 1 980 m/s, which is close to CJ value
of 1 984 m/s too. Furthermore, for three-dimensional
simulation of contant-nozzle tube case, another fine
grid system with average grid size of 0.2 mm is computed to check the grid dependency. Fig.2 shows the
comparison of the pressure contours computed based
on these two grid systems from 400 to 850 Ps of about
18 000 iteration steps. It can be seen that the flow
fields approximately coincide with these two grid systems. The above comparison proves the numerical
convergence and grid dependency.

cross-sections of the four nozzles are shown in Fig.4.

The front parts of the combustors have the same configurations. The Laval nozzle and diverging nozzle
have the same exit area. The exit width of the converging nozzle is 0.62 times of the combustor width. The
geometrical parameters are designed according to the
ratio of the pressure of the steady state detonation
product to the environment pressure.

Fig.3

Fig.1

649

Reaction process parameter E contour at four instantaneousness during continuous propagation of a CDE
with a Laval nozzle from 1 390 to 1 480 s.

One-dimensional detonation wave propagation cases

with three grid sizes respectively.

Fig.2 Comparison of instantaneous pressure contour based

on coarse and fine grid systems from 400 Ps to 850 s.

2.3. Physical model

To begin with, an overview of the continuous
propagation of a continuous detonation engine is presented in Fig.3. The dark regions indicate newly injected combustible fresh gas mixture and the remainder represents detonation products. Fresh gas is continuously injected into the detonation chamber from
the left, and detonation products are ejected to the right
to provide thrust. The annular gap between the two
coaxial cylinders constitutes the combustor. The inner
radius of the constant-area part is 40 mm and the outer
radius is 53 mm. Schematic diagrams of the axial

Fig.4

Cross-sections of CDE configurations studied.

The front section of the combustor is initially injected with quiescent, combustible gas mixture at
pressure p = 0.103 MPa and temperature T = 300 K.
The other part is filled with air. In the experiment[8],
the DW is ignited by a branching pre-detonation tube
that is connected tangentially to the outer wall of the
combustor. In numerical simulation, for simple,
pre-detonation ignition is substituted by a section of
classical one-dimensional CJ detonation. After ignition,
the DW continuously propagates azimuthally around the
combustor. From 1 390 Ps to 1 480 Ps, the DW has

Shao Yetao et al. / Chinese Journal of Aeronautics 23(2010) 647-652

650

propagated 9/16 round. Till to 1 480 Ps, the detonation

wave has propagated nine rounds and propagates at
quasi- steady state continuously. A premixed
stoichiometric H2/air mixture injection condition is set
according to the local wall pressure following Laval
tube theory. The inlet stagnation pressure is p0 = 2 MPa
and the environment pressure pf is 0.05 MPa. The
injection hole exit area normalized by the hole throat
area is Aw/Athroat = 10.
From the isentropic relation, there is the following
expression:
J 1

Aw
Athroat

1 2 J 1
2(J 1)
Ma 2
1 

Ma J  1
2

(5)

where Ma is the Mach number just in front of the head

wall and J =1.4. The injection boundary condition is
specified according to the local environment pressure
pw just near the wall.
(1) When pw>2 MPa: the reaction mixture could not
be injected into the chamber. A rigid wall condition is
set locally.
(2) When 1.994 MPa<pw<2 MPa: mass fluxes
through both throat and injection walls are subsonic.
(3) When 0.244 MPa<pw<1.994 MPa: the throat
maintains choke conditions, and the mass flux of injection remains constant. Shock waves develop downstream of the throat and fresh gas is injected at subsonic velocities.
(4) When pw<0.244 MPa, the injection is not affected by the wall pressure. Whole field downstream
of the throat is supersonic.
The side wall boundary conditions correspond to
adiabatic, slipping and noncatalytic fluid flow. In real
engine, the injected combustible gas would be heated
by the high temperature wall[9] which would be helpful
for fuel and air mixing. This mechanism has not been
considered here. Conditions on the outflow boundary
correspond to a non-reflecting surface[16].
3. Presentation of Results

with the DW that extends to the exit to compress the

former cycle detonation product. The numerical results
for the above wave structures qualitatively agree with
experimental results[5]. At the head wall, it is seen from
the pressure contours that reflecting shock waves
repeatedly reflect between the inner wall and outer
wall and finally become a sound wave. Because of the
curvature of the cylinders, the DW front on the inner
expansion wall goes ahead of the wave front on the
outer wall by about 3. Conversely, the maximum
pressure on the outer compression wall is about
13 MPa, which is nearly twice the pressure of 7 MPa
on the inner wall.

Fig.5

Pressure contour at 1 500 s.

Fig.6 shows the Mach number contours in Laval

nozzle combustor. It is seen that the flow is mainly
subsonic in front of the throat. As the detonation
product flows through the diverging section, it accelerates to a high Mach number greater than 2. The
maximum Mach number is 2.5 for the axial velocity of
2 100 m/s at the exit. Fig.7 shows the instantaneous
pressure history at a point near the head (radius =
48 mm, z = 1 mm) for the period of 0-1 500 s. It is
seen that once the initial transient effects die down, the
DW maintains continuous oscillatory propagation in
the annular chamber as long as fuel is continuously

3.1. Flow field structure

Fig.5 shows the pressure contours at 1 500 s when
the DW has propagated more than nine rounds. The
DW can continuously propagate in this quasi-steady
state for a long time. The DW maintains an acute angle
to the head wall so that it moves against the injection
flow direction, thereby avoiding being blown downstream. Now the CJ velocity is divided into two velocity components, i.e., axial component and azimuthal
component. The axial component velocity has the
same value with the injection velocity in front of the
detonation wave but opposite direction. The azimuthal
component velocity is the DW rotating velocity. According to the triangle relation, the azimuthal component of the DW velocity is lower than the classical CJ
velocity. There is an oblique shock wave associated

No.6

Fig.6

Mach number contours at 1 500 s.

Shao Yetao et al. / Chinese Journal of Aeronautics 23(2010) 647-652

No.6

651

injected. According to Fig.7, in the period of 3361 486 s, the DW has propagated through seven
rounds, and the DW rotating velocity is thus
VR = 7u2Sr/'t =1 841 m/s. This value is about 5% less
than the classical CJ velocity of about 1 984 m/s. This
deficit is mainly caused by the incline of the DW discussed at the beginning of this section. And also,
unlike the case for the CJ theory model, one side of the
DW is in soft contact with the detonation product. This
condition would also reduce the detonation velocity a
little.

Fig.8

Thrust history of four CDEs with different nozzles.

Fig.9 shows the fuel-based gross specific I spg history. The gross specific impulse ranges 1 540-1 750 s
for the four types of nozzles. The Laval nozzle has the
best performance of about 1 750 s. All previous articles[10-12] use the criterion of the gross specific impulse
to determine the gross propulsive performance.
Clearly, higher I spg could be achieved with greater

Fig.7

Pressure history at a point near head of combustor

(r = 46 mm, z = 1 cm) for period of 0-1 500 s.

injection momentum according to this criterion. In this

article, the net specific criterion is used to compare the
momentum increase due to continuous detonation
combustion.

3.2. Propulsive performance

The gross thrust F, the fuel mass flow rate per
 , the gross specific impulse I spg not
square meter m
including the momentum of the incoming gas, and the
net specific impulse I spn are calculated according to the
following equations.
(6)
(7)
(8)
Fig.9

(9)
where w is the axial velocity, m f mass flow rate fuel,
and Ahead the cut area at the tube head. Fig.8 shows
the thrust history of the models for the period
0-1 500 s. After the flow field becomes steady, the
models continuously create enormous, almost steady
thrust. The CDE with the diverging nozzle creates
about 1 900 N thrust, which is considerable for such a
small combustor. The CDE with the constant-area nozzle outputs the least thrust of about 1 600 N.

Gross specific impulse history for period of

0-1 500 s.

Fig.10 shows that the net specific impulses specifically produced by the four nozzles vary from 60 s to
630 s. For these cases, a Laval nozzle has the best efficiency over the other three nozzles.
The average mass flux m values are shown in
Fig.11. The mass flux per square meter ranges
313-330 kg/(m2s) for the different nozzles. It is seen
that these values vary little, and the average mass flow
rate is about 320 kg/(m2s), which is significantly favorable value. This demonstrates the mass flow rate
advantage of a CDE.

Shao Yetao et al. / Chinese Journal of Aeronautics 23(2010) 647-652

652

[2]

[3]

[4]
[5]
[6]
Fig.10

Net specific impulse history for period 0-1 000 s.

[7]
[8]

[9]
[10]

[11]
Fig.11

Average mass flow rates per square meter for four

types of nozzles.

4. Conclusions

The typical flow field structure of a CDE is numerically obtained and found to agree well with the results
of previous experiments. The key propulsion performance parameters, namely thrust, specific impulse
and the mass flow rate, are exhaustively analyzed for
the CDE. Different effects of different kinds of nozzles
on the CDE performance are compared. We find that
the Laval nozzle has the best performance of 1 800 N
thrust, 1 750 s gross specific impulse and 313 kg/
(m2s) mass rate. The Laval nozzle has some advantages over other three nozzles.
As CDE concept is an essentially innocative combustor design concept of propulsion system, it is easy
to substitute conventional aviation systems combustor
to achieve higher performance. Once proven to be effective, it would be a good candidate for the new generation aerospace propulsion system.
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Biography:
Shao Yetao Born in 1981, he received B.S. degree from
Jilin University in 2005 and Ph.D. degree from Peking
University in 2010 respectively. His main research interests
are CFD and detonation propulsion.
E-mail: shaoyt@163.com