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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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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)


0.1 0.15

10

inner radius, =4 mm, 1680 s

mid radius, =4 mm, 1680 s

outer radius, =4 mm, 1680 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


0.1 0.15 0.2

0 0.05


(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


0 0.05 0.1 0.15 0.2 0.25


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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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


0 0.05 0.1 0.15 0.2 0.25


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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(a) Chamber width 4 mm.

(b)Chamber width 10 mm.


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



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



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


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



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



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


AxialDistance(m)

0.01 0.02 0.03 0.04

0

0.005

0.01

0.015

0.02

0.025



Axia

lDistance(m)

0.01 0.02 0.03 0.04

0

0.005

0.01

0.015

0.02

0.025



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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(a)

(b)

(c)


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


h=19.48 mm

h


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


h

h=17.19 mm


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


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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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


MachReflection


(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


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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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


ObliqueShock Wave

no nozzle,=14 mm, 1570 s

convergent divergent nozzle,=14 mm, 1520 s


ObliqueShock Wave


(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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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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