x. he and a. r. karagozian- reactive flow phenomena in pulse detonation engines

8/3/2019 X. He and A. R. Karagozian- Reactive Flow Phenomena in Pulse Detonation Engines

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AIAA 2003-1171

Reactive Flow Phenomena inPulse Detonation Engines

X. He and A. R. Karagozian

UCLA

Los Angeles, CA

41st AIAA Aerospace Sciences

Meeting and Exhibit

69 January 2003Reno, Nevada

For permission to copy or to republish, contact the American Institute of Aeronautics and Astronautics,

1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344.


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AIAA20031171

REACTIVE FLOW PHENOMENA IN

PULSE DETONATION ENGINES

X. He and A. R. Karagozian

Department of Mechanical and Aerospace EngineeringUniversity of California, Los Angeles, CA 90095-1597

Abstract

This paper describes one- and two-dimensional

numerical simulations, with simplified as well as

full reaction kinetics, of a single cycle pulse deto-

nation engine (PDE). Focus of the present studies

is on 1) the presence of a nozzle extension at the

end of the tube, and its effect on performance pa-

rameters as well as noise characteristics, 2) criticalspark ignition energies associated with the initia-

tion of a detonation in the PDE tube, and 3) quan-

tification of performance parameters associated with

full kinetics simulations of the PDE and compari-

son of these data sets with available experimentaldata. The present simulations demonstrate the abil-

ity to predict PDE reactive flow phenomena and as-

sociated performance and noise characteristics, and

hence have promise as a predictive tool for the evo-

lution of future PDE designs.

Introduction and Background

The Pulse Detonation Wave Engine (often calledthe Pulse Detonation Engine or PDE) is a device

which allows periodic ignition, propagation, and

transmission of detonation waves within a detona-

tion tube, with associated reflections of expansion

and compression waves which can act in periodic

fashion to produce thrust1,2

. A summary of the rel-

evant gasdynamics within the PDE tube is shown in

Figure 1. The figure indicates ignition and propaga-

tion of the detonation out of the PDE tube (Figures

1a-c), reflection of an expansion fan into the tube(Figures 1de), reflection of the expansion fan from

Graduate ResearcherProfessor; Associate Fellow, AIAA. Corresponding author

([email protected]).Copyright (c) 2003 by X. He Published by the American

Institute of Aeronautics and Astronautics, Inc., with permis-sion.

the thrust wall (Figures 1ef), allowing reactants to

be drawn into the tube, and propagation of the ex-

pansion fan out of the tube (Figures 1gh), with si-

multaneous reflection of a compressive disturbance

into the tube (Figures 1hij), which reflects from

the thrust wall, ignites the fresh reactants, and re-

initiates the cycle. Because the PDE concept holdspromise for high thrust density in a constant vol-

ume device requiring little or no rotating machinery,a number of groups have been exploring PDEs for

propulsion applications. This exploration has built

on fundamental PDE work over several decades14

,so that modern experimental diagnostic as well as

computational methods may be used to bring about

significant advances in the state of the art2,46

.

Performance parameters commonly used to char-

acterize the pulse detonation engine include the im-

pulse, I, typically defined as

I A

0

ptw(t)dt (1)

where A is the area of the thrust wall and ptw(t)

is the time-dependent pressure differential at the

thrust wall. The impulse is usually scaled to pro-

duce the engines specific impulse, Isp,

Isp I

V g(2)

where is the initial mass of the reactive gas mixture

in the tube, V is the detonation tube volume (includ-

ing the nozzle volume, if containing reactants), and

g is the earths gravitational acceleration. As an

alternative performance parameter, the fuel-basedspecific impulse, Isp,f, is often used:

Isp,f IspYf

(3)

where Yf is the fuel mass fraction present within the

premixed reactants in the tube. Both Isp and Isp,f

1

American Institute of Aeronautics and Astronautics


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AIAA20031171

are often used to compare performance among dif-

ferent PDE configurations and also to compare PDE

performance with that of alternative engine cycles7

.

Overviews of past and ongoing numerical simu-

lations of PDEs are described in recent articles byKailasanath

2,8,9

. Other recent simulations have fo-

cused on various flow and geometrical features ofthe PDE, including the effects of nozzles placed

downstream of the detonation tube. Cambier and

Tegner4

, for example, find that the presence of the

nozzle can have a significant effect on the impulse

of a single cycle of the PDE. In 2D axisymmetric

simulations, which employ a second order TVD (to-

tal variation diminishing) scheme, they find that in-

creasing the ratio of the nozzle exit area to the tube

area can produce monotonic increases in impulse I.

Increasing the nozzle exit area causes a dropoff in Ispuntil Aexit/Atube reaches 4.0, since the nozzle volume

increases; for area ratios larger than this value, Ispis seen to increase, suggesting that the impulse is

increasing faster than the nozzle volume increases,

per equation (2).

In recent PDE experiments with nozzle exten-

sions, Johnson10

observes a decrease in the fuel

specific impulse for converging-diverging nozzles ascompared with straight nozzles or converging noz-

zles. Similarly, tests as well as modeling by Cooper

and Shepherd6

suggest that the relative impulse of a

PDE tube with a flared nozzle is lower than that for a

straight nozzle at a given fill fraction (or percentageof the PDE tube initially filled with reactants). It is

of interest to understand the mechanisms whereby

nozzles can increase or decrease PDE performance

and what the associated changes in engine noise lev-

els may be.

Another issue of interest with respect to the suc-

cessful performance of the pulse detonation engine

is quantification of the required energy input needed

to ignite and sustain a propagating detonation wave

from the closed end of the tube. Thermally initi-

ated detonations via the deflagration-to-detonation

transition (DDT) have been examined over manyyears

1113

. When a mixture of reactants is ignited

by a bulk power deposition of limited duration, a

sequence of events is initiated which eventually re-

sults in a sudden power burst or explosion in the

explosion, accelerating the flame front and leading

to formation of a propagating detonation. Since the

PDE tube is designed to operate in a cyclical man-

ner, it is of interest to quantify the required energy

input to be able to repetitively initiate a detonation

front.

Prior computational studies by our group pertain-

ing to detonation phenomena in general14

and the

pulse detonation engine in particular15

involve bothone- and two-dimensional simulations, employing

the essentially non-oscillatory or ENO scheme1618

for spatial integration. An examination of the one-

dimensional overdriven detonation14

suggests spe-

cific requirements for spatial resolution of the det-onation front to be able to obtain accurate wave

speeds, peak pressures, and frequencies of detona-

tion oscillation. These ideas are incorporated into

1D and 2D simulations of the single cycle PDE15

with single step reaction kinetics. The studies sug-gest that useful performance and noise related esti-

mates may be obtained even from one-dimensionalcomputations of the pulse detonation wave engine

with simplified reaction kinetics.

The present study focused on using these high

order numerical schemes to study the behavior of

the pulse detonation engine, using simplified as well

as complex reaction kinetics. Special attention was

paid to the PDEs geometrical, flow, and reaction

characteristics and their influence on performance

parameters as well as noise generation. NASAs in-

terest in the PDE for advanced vehicle propulsion19

is incumbent upon the ability of the engine to ef-

ficiently generate thrust without having to pay asignificant penalty in engine noise. While specific

geometries for PDE nozzle extensions may have the

effect of reducing the relative Isp, as suggested in

recent experiments10

, there may be benefits associ-

ated with noise reduction.

Problem Formulation and Numerical Methodology

The equations of mass, momentum, energy, and

species conservation were solved in both one and two

spatial dimensions, assuming inviscid flow. Single

step reaction kinetics for CH4 O2 and H2 O2,

as outlined in detail in He and Karagozian

15

, as wellas full reaction kinetics for mixtures of H2 O2,H2O2Ar, and H2O2N2 were employed. Both

straight PDE tubes and tubes with nozzle extensions

were explored. In the 1D simulations, the compu-

tational domain consisted primarily of the detona-

tion tube or tube and nozzle (containing at least

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600 grid points), with only a few grid points ex-

tending beyond the tube end in order to capture

the external pressure. In the 2D simulations, the

air external to the detonation tube was assumed to

be uniformly at atmospheric pressure, and the com-

putational domain extended well downstream of the

end of the tube, in general at least one and one halftube lengths downstream and at least two tube di-

ameters away from the detonation tube in the di-

mension perpendicular to the axial dimension. The

effects of employing a 1D pressure relaxation length

(PRL), as done by Kailasanath and Patnaik9

, were

explored in our prior PDE study15

, but for most of

the conditions examined, a relaxation length was not

needed in order to obtain equivalent results between

1D and 2D simulations.

In cases where alternative nozzle geometries were

considered, a locally 1D flow approximation was em-

ployed to represent nozzle shapes of slowly vary-ing cross-sectional areas A(x). In this quasi-one-dimensional case, with a single step reaction, for

example, the governing equations reduce to the fol-

lowing form:

tU +

xF =

1

A

dA

dx( H F) + S (4)

where the vectors containing conserved variables,

flux terms, and source terms are:

U =

u

E

Y

F =

u

u2 + p

(E+ p)u

uY

(5)

H =

0p

0

0

S =

00

0

KY eTA

T

(6)

Here E may be written

E =p

1+

u2 + v2

2+ qY (7)

where represents density, p is the static pressure,u is the x-component of the velocity vector, and

is the ratio of specific heats. q is a heat release

parameter which characterizes the amount of en-

ergy released during the reaction, and TA represents

the activation temperature. Y is the reactant mass

fraction, which varies from 0 to 1, while K is the

reaction-rate multiplier for the reaction source term.

Through equation (4) it became possible, in an ap-

proximate way, to represent the effects of nozzle ge-

ometry in 1D PDE simulations.

Four different nozzle extension shapes were ex-plored here; these are shown in Figure 2. The nozzle

shapes included a fifth order polynomial (configura-tion 1), a flared divergent section (configuration 2), a

nozzle section with a constant conical divergence an-

gle (configuration 3), and a straight tube (configura-

tion 4). In the simulations of PDEs with nozzles, the

straight portion of the PDE tube, of length L, was

assumed to be initially filled with reactants, while

the nozzle section, of length Ln, was filled with inert

gas (for a single step reaction, effectively products).

In the simulations of straight PDE tubes without a

nozzle, the tube was assumed to be initially filled

completely with reactants. Unless otherwise stated,

the straight tube lengths L used in the present com-putations were 1 m, and the nozzle lengths Ln were

also 1 m.

For the simulations involving complex reaction ki-

netics, the equations (5) - (7) were replaced by rela-

tions for the straight PDE tube (with A(x) constant)

but with N 1 species equations for the N speciesinvolved in the reactions. Full kinetics simulations of

the combustion reactions for H2O2, H2O2Ar,

and H2 O2 N2 (representing hydrogen-air) were

considered here; the latter mechanism contained 23

elementary reactions and was part of the CHEMKINII library

20.

As in He and Karagozian15

, the present study used

the Weighted Essentially Non-Oscillatory (WENO)

method21

, a derivative of the ENO method1618

for

spatial interpolation of the system of governingequations. The WENO scheme was fifth order ac-

curate in smooth regions and third order accurate

in the vicinity of discontinuities. The ENO/WENO

schemes were tested on a variety of problems, in-

cluding shock tubes with open ends, analogous to

the exit of the PDE15

, and that of the classical

one-dimensional, overdriven, pulsating detonation

14

.For the single step kinetics simulations, the third

order total variation diminishing (TVD) Runge-

Kutta method was used for time discretization. For

full kinetics simulations, the method of operator

splitting22

was used, whereby the system of govern-

ing equations (including N1 species equations) was

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split into two separate equations, one which only

included the advection-diffusion terms (solved via

WENO) and one which only included the reaction

rate source terms. A stiff ODE solver, DVODE (a

variation of VODE23

) was employed for the solution

of the rate equations; thermodynamic parameters

and rate constants were obtained via the CHEMKINII subroutine

20

.

A computational spark adjacent to the thrust

wall was used to initiate the detonation at the startof the PDE cycle. This narrow, high pressure, high

temperature region (3 grid cells in width) was able to

initiate a propagating shock and ignite the reactants;

the flame front then caught up with the shock, form-

ing a detonation. As suggested by prior studies1113

,

however, such thermal initiation of detonation de-

pends very strongly on the initial rate of deposi-

tion of energy in the reactants. This concept was

explored in the present studies by altering the ini-tial temperature and pressure in the computational

spark to be able to determine minimum input en-

ergy densities leading to detonation initiation.

In addition to the standard performance parame-

ters used to characterize the PDE (I, Isp, and Isp,f),

the sound pressure level (SPL) at various locations

within and external to the detonation tube was also

computed. As done previously15

, these noise levelswere estimated by examining the Fourier transform

of the time-dependent pressure measured at various

locations within the computational domain. The

SPL was then computed based on peak pressuresin the Fourier spectrum. In most cases these peaksoccurred at the PDE cycle frequency.

Results

An example of the temporal evolution of the pres-

sure distribution along the centerline of a straight

PDE tube, over a single cycle, is shown in Figure

3 for a 2D axisymmetric configuration with a sin-

gle step methane-oxygen reaction. Here the initia-

tion and propagation of the detonation wave through

the tube (Figures 3ab) and the exit of the shockfrom the tube and reflection of the expansion fan

from the exhaust back into the tube (Figures 3cd)

are clear. Our prior studies15

demonstrate that a

1D simulation of this same PDE tube quantitatively

yields a very similar pressure field evolution to that

of the 2D simulation, even without inclusion of a

1D pressure relaxation length. 1D simulations of

the PDE tube do not precisely replicate the pres-

sure and Mach number evolution at the tube end,

with or without a PRL. Yet the evolution of the

tubes interior pressure without a PRL is, in most

cases previously explored15

, sufficiently close to that

obtained from the 2D simulations so as to producesimilar PDE performance estimates. This is shown,

for example, in Figure 4, which compares the spe-

cific impulse for 2D simulations with that for 1D

simulations, with and without inclusion of a PRL.

Time-series pressure data at specific locationswere used to estimate the noise generated at vari-

ous points in the flowfield over a single PDE cycle.

Estimates of the sound pressure level were made us-

ing both 1D and 2D simulations of the straight PDE

tube with a single step CH4 O2 reaction. Sincethe 1D simulations only resolved the flow within the

PDE tube, comparisons were made only for inte-rior and tube exit noise levels. In all locations for

this case we observed the peak in pressure to appear

close to the frequency associated with the period of

the PDE cycle, roughly 330 Hz. The noise levels at

various tube locations are quantified in Table 1.

Location 2D SPL 1D SPL

Thrust Wall 212 dB 212 dB

Mid-tube 211 dB 211 dB

Tube end 202 dB 203 dB

Table 1. Computed sound pressure level (SPL) at vari-

ous locations within the tube (thrust wall, center of tube,

and tube end). Results are computed from both 2D and

1D simulations of the CH4 O2 reaction.

Consistent with the evolution of the pressure fieldand the performance parameters (e.g., Figure 4), the

noise levels were nearly the same for 1D as for 2D

simulations. The magnitudes of the noise levels were

close to those quantified in PDE experiments24,25

.

The influence of the presence of the nozzle is

shown in Figure 5, where the straight tube (nozzle

configuration 4) had the same length as the tubeswith the divergent nozzles. Again, a CH4 O2 sin-

gle step reaction was used in this set of simulations.

Interestingly, the straight tube was observed to pro-

duce the highest values ofI, Isp, and Isp,f, while the

conical nozzle (configuration 3) produced the low-

est values. These findings were generally consistent

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with the observations of Johnson10

, i.e., that Isp,fdecreased with inclusion of a divergent exit nozzle.

Our results were also consistent with those of Cam-

bier and Tegner4

, in that Isp decreased for nozzle-

to-tube area ratios of 4.0 (examined here).

The fact that the straight tube produced the high-

est thrust (resulting from the highest sustained pres-sure at the thrust wall) has interesting implications

for PDE noise estimates. Figure 6 shows the results

of taking the Fourier transform of the time depen-

dent pressure within (Figure 6ab) and at the end

(Figure 6c) of the tube/nozzle, for the four different

nozzle configurations explored here. For example, in

the middle of the PDE tube, the straight nozzle caseproduced the smallest pressure perturbation at the

PDE cycle frequency, yet at the end of the straight

nozzle, the pressure and hence the SPL were both

larger than those for the other nozzles, albeit at

a higher harmonic (667 Hz) of the PDE cycle fre-quency (about 333 Hz). This behavior is reflected in

Table 2 for SPL values at various locations.

Location Config. 1 SPL Config. 4 SPL

Thrust wall 212 dB 210 dB

Mid-tube 211 dB 208 dB

Tube end 205 dB 202 dB

Nozzle end 188 dB 208 dB

Table 2. Computed sound pressure level at various lo-

cations within the tube (thrust wall, center of tube, PDE

tube end) and at the nozzle exit for two different noz-

zle configurations (see Figure 2). Results are computed

from quasi 1D simulations of the CH4 O2 reaction.

The above behavior likely resulted from the weak-

ened downstream-propagating shock that formed in

the divergent nozzles as compared with that forthe straight nozzle. The lower pressure and SPL

in the upstream portions of the straight nozzle

(as compared with the divergent nozzles) possibly

could have resulted from stronger reflected expan-

sion waves that occurred at the contact surface be-

tween reactants and air at the start of the noz-

zle. While there were clear tradeoffs between perfor-mance and nozzle exit noise conditions, it is unclear

why the pressure perturbation of the higher har-

monic (660 Hz) in the straight nozzle was so much

larger than for the divergent nozzles. This and other

noise related issues require further exploration in the

future.

The full kinetics simulations of the reactant-filled,

straight PDE tube without a nozzle allowed a more

detailed examination of the detonation ignition and

propagation process to be made, in addition to more

quantitative comparisons with experimental data.

Figure 7 displays the evolution of the 2D pressure

field associated with the PDE tube and its sur-roundings, for a full H2 O2 reaction. As seen in

prior 2D simulations of the PDE but with simplified

kinetics15

, the propagation of the detonation out of

the tube resulted in the propagation of a vortical

structure coincident with the shock and simultane-ous reflection of an expansion fan back into the tube.

Increased complexity in the wave structures as com-

pared with that for simplified reaction kinetics was

observed, especially in the propagating shock/vortex

structure downstream of the tube exit.The influence of the initial pressure and tempera-

ture (and resulting energy deposition) on initiationof a detonation wave was studied using these full

kinetics simulations. Figure 8 shows the centerline

pressure distribution for a 1D, full kinetics simula-

tion of an H2O2Ar mixture, for different initial

temperatures and pressures in the 3 grid cell-wide

spark adjacent to the thrust wall. Critical combi-

nations of temperature and pressure were observed

to be necessary for the classical ZND detonation

structure to evolve; if the initial energy deposition

was too small, a weak shock front did not ignite the

mixture and thus did not transition to a detonation,

as seen by the solid lines in Figures 8ab. Tables 3and 4, for H2 O2 Ar and H2 O2 N2 reac-

tions, respectively, quantify the conditions that were

required for ignition of a detonation.

Temp. Press. Energy (erg/cm2) Deton.?

500K 3 atm 2.02 105 No

1000K 3 atm 8.06 105 No

1500K 3 atm 9.95 105 Yes

2000K 3 atm 11.0 105 Yes

1500K 5 atm 15.3 105 Yes

1500K 2.5 atm 8.63 105 Yes

1500K 2.0 atm 7.3 105 No

Table 3. Initial temperatures, pressures, and input

energies for a computational spark used to ignite a

H2 O2 Ar mixture, and determination of the possi-

bility of detonation ignition.

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Temp. Press. Energy (erg/cm2) Deton.?

800K 1 atm 4.54 105 No

900K 1 atm 4.89 105 No

1000K 1 atm 5.18 105 Yes

1200K 1 atm 5.64 105 Yes

Table 4. Initial temperatures, pressures, and input

energies for a computational spark used to ignite a

H2 O2 N2 mixture, and determination of the possi-

bility of detonation ignition.

As expected, the critical input energies for ignition

of a detonation were found to be different for these

different reactions. In the case of H2 O2 N2,

a critical energy deposition per unit area of about

5105 erg/cm2 was required for detonation, whereas

for the case of H2 O2 Ar, this critical value rose

to about 8.5 105 erg/cm2.

The full kinetics simulations also allowed quanti-tative comparisons to be made between performance

parameters from the present simulations and those

obtained by experiment (for a single cycle PDE) or

analysis. Table 5 below shows the current estima-

tions ofIsp for the PDE for H2O2 and H2O2N2(hydrogen-air) reactions, as compared with the anal-

ysis and experiments described in Wintenberger26

and the experiments of Zitoun27

and Schauer28

.

Study Isp, H2-air Isp, H2 O2

Present 128.5 s 240 s

Wintenberger26 123.7 s 173 s

CIT expts.26

200 s

Zitoun expts.27

149 s 226 s

Schauer expts.28

113 s

Table 5. Comparison of specific impulse for single cycle

PDE between the present simulations and corresponding

experiments and modeling efforts, as noted.

While the present simulations appeared quantita-

tively to replicate the experimentally observed per-

formance parameters reasonably well, detailed com-

parisons of the pressure field evolution and noise es-timates require further examination and are the sub-

ject of continued studies.

Conclusions

High resolution numerical simulations of pulse

detonation engine phenomena revealed useful infor-

mation that may be used in future PDE designs.

Simulations of the effects of the presence of a di-

vergent nozzle downstream of the PDE tube sug-gested that, while performance parameters such as

Isp may decrease with increasing nozzle exit area,

the noise generation at the nozzle exit may actu-

ally be reduced, and hence these tradeoffs may be

explored through simplified reaction kinetics stud-

ies. Simulations of PDE evolution with full chemi-

cal kinetics suggested that specific minimum energy

densities were required to enable the initiation of a

detonation, and hence to sustain the PDE cycle. Fi-

nally, it was observed that full kinetics simulations

were able to capture quantitatively the physical phe-

nomena and corresponding performance parametersfor the PDE. Future studies will continue with this

quantitative comparison as well as noise generation

issues relevant to the PDE.

Acknowledgments

This work has been supported at UCLA by NASA

Dryden Flight Research Center under Grant NCC4-

153, with Dr. Trong Bui and Dave Lux as technical

monitors, and by the Office of Naval Research un-

der Grant ONR N00014-97-1-0027, with Dr. Wen

Masters as technical monitor.

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[24] Schauer, F., private communication.

[25] Perkins, H. D., private communication.

[26] Wintenberger, E., Austin, J. M., Cooper, M.,

Jackson, S., and Shepherd, J. E., An Analyt-

ical Model for the Impulse of a Single Cycle

Pulse Detonation Engine, AIAA Paper 2001-3811, July, 2001.

[27] Zitoun, R. and Desbordes, D., Propulsive Per-

formances of Pulsed Detonations, Combustion

Science and Technology, Vol. 144, pp. 93-114,

1999.

[28] Schauer, F., Stutrud, J., and Bradley, R., Det-

onation Initiation Studies and Performance Re-

sults for Pulsed Detonation Engines, AIAA

Paper no. 2001-1129, 2001.

a)

)

c)

reactants

reactantsproducts

products

detonation

products

reflected expansion wave

products

expansion wave

d)

e)

detonation front

propagating detonation

f)

g)

h)

i)

j)

reactants

enterproducts

reactants

reactants

reactants

reactants

reflected expansion wave

expansion wave

reflected compression wave

compression wave

shock/detonation reflection

Fig. 1: The generic Pulse Detonation Engine (PDE)

cycle.

8



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AIAA20031171

0 0.5 1 1.5

X (m)

0

0.1

0.2

0.3

0.4

0.5

Radius(m)

nozzle 1nozzle 2

nozzle 3nozzle 4

Fig. 2: Different nozzle geometries explores in

present computations. These include straight tubes

(configuration 4), flared divergent sections (config-uration 2), divergent sections with inflection (con-

figuration 1), and a nozzle section with a constant

divergence angle (configuration 3). In all case the

reactants are assumed to lie initially upstream of

nozzle, in the constant area tube, while the nozzleitself contains inert gas.

X (m)

Pressure(atm)

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

(a)

X (m)

Pressure(atm)

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

(b)

X (m)

Pressure(atm)

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

(c)

X (m)

Pressure(atm)

0 0.5 1 1.5 2 2.50

5

10

15

20

25

30

(d)

Fig. 3: Evolution of the centerline pressure for a

straight 2D axisymmetric PDE of 1 m length, at

times (a) 0.06 ms, (b) 0.15 ms, (c) 0.49 ms, and (d)

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0 0.001 0.002 0.003 0.004

Time (sec.)

0

50

100

150

200

250

300

350

SpecificImpulse(sec.)

2D simulation1D without pressure relaxation1D with pressure relaxation

Fig. 4: Comparisons of time-dependent specific im-

pulse for both 1D and 2D axisymmetric simulationsof the PDE tube with a CH4 O2 reaction, takenfrom He and Karagozian

15

. 1D simulations explored

the use of a pressure relaxation length l = 0.5L.

Here the 1D simulations incorporated a computa-

tional spark consisting of a pressure of 10 atm and

a temperature of 3000K in order to match the initial

conditions for the 2D simulation.

0 0.001 0.002 0.003

Time (sec.)

0

500

1000

1500

2000

2500

Impulseperunit

area(ps.s

)

nozzle 1nozzle 2nozzle 3nozzle 4

0 0.001 0.002 0.003

Time (sec.)

0

50

100

150

200

250

Isp(sec.)

0 0.001 0.002 0.003

Time (sec.)

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

Ispf(sec.)

Fig. 5: Comparisons of time-dependent performance

parameters computed from 1D simulations using dif-

ferent nozzle geometries. Results shown are for im-

pulse I, specific impulse Isp, and fuel specific impulseIsp,f. The reaction of methane and oxygen was sim-

ulated.

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0 2500 5000 7500 10000

Frequency (HZ)

0

1

2

3

4

5

6

7

[P]/P

0


(a)

0 2500 5000 7500 10000

Frequency (HZ)

0

0.5

1

1.5

2

2.5

3

3.5

4

[P]/P0


(b)

0 2500 5000 7500 10000

Frequency (HZ)

0

1

2

3

[P]/P0


(c)

Fig. 6: Comparisons of pressure spectra: (a) in the

middle of the detonation tube, (b) at the end of the

straight part of the detonation tube, and (c) at the

end of the nozzle. Results are shown for different

nozzle configurations (see Fig. 2).

50 100 150 200 250 300 350

X (cm)

-50

0

50

100

150

200

Y(cm)

(a)

50 100 150 200 250 300 350

X (cm)

-50

0

50

100

150

200

Y(cm)

(b)

50 100 150 200 250 300 350

X (cm)

-50

0

50

100

150

200

Y(cm)

(c)

Fig. 7: Temporal evolution of the 2D planar pres-sure field within and external to the PDE over one

cycle, with pressure given in units of dyn/cm2. Data

shown are at times corresponding to (a) 0.15 ms, (b)

0.47 ms, and (c) 1.34 ms. A H2 O2 reaction was

simulated here with full chemical kinetics.

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0 10 20 30 40

X (cm)

0

2E+06

4E+06

6E+06

8E+06

1E+07

1.2E+07

1.4E+07

1.6E+07

1.8E+07

Pressure(dyn/cm

2)

Ps = 2.0 AtmPs = 2.5 AtmPs = 3.0 AtmPs = 5.0 Atm

(a)

0 10 20 30 40

X (cm)

0

2E+06

4E+06

6E+06

8E+06

1E+07

1.2E+07

1.4E+07

1.6E+07

1.8E+07

Pressure(dyn/cm

2)

Ts = 500KTs = 1000KTs = 1500KTs = 2000K

(b)

Fig. 8: Centerline pressure distribution for the H2

O2 reaction with full kinetics, for different compu-

tational spark conditions: (a) fixed temperature

1500K and variable pressure, and (b) fixed pressure

3 atm and variable temperature, each at time 0.2

msec.

12


x. he and a. r. karagozian- reactive flow phenomena in pulse detonation engines

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