JOURNAL OF PROPULSION AND POWER

Vol 19 No 4 JulyndashAugust 2003

System Performance and Thermodynamic Cycle Analysisof Airbreathing Pulse Detonation Engines

Yuhui Wucurren Fuhua Madagger and Vigor YangDagger

Pennsylvania State University University Park Pennsylvania 16802

A modular approach to the study of system performance and thermodynamic cycle ef ciency of airbreathingpulse detonation engines (PDEs) is described Each module represents a speci c component of the engine and itsdynamic behavior is formulated using conservation laws in either one or two spatial dimensions A frameworkis established for assessing quantitatively the in uence of all known processes on engine dynamics Various lossmechanisms limiting the PDE performance are identi ed As a speci c example a supersonic PDE for high-altitudeapplicationsis studied comprehensivelyThe effects of chamber con gurationand operating sequence on the enginepropulsive ef ciency are examined The results demonstrate the existence of an optimum cycle frequency and valveclose-up time for achieving maximumperformance in terms of thrust and speci c impulse Furthermore a chokedconvergentndashdivergent nozzle is required to render the PDE competitive with other airbreathingpropulsionsystemssuch as gas-turbine and ramjet engines

NomenclatureA = cross-sectionalarea of detonation tubeC p = constant-pressurespeci c heatD = diameter of detonation tubeDe = diameter of nozzle exitDt = diameter of nozzle throatF = thrustFsp = speci c thrust (air-based) F= Pma

f = ratio of fuel to air mass ow rateg = gravitational accelerationh = ight altitudeI = impulseIsp = speci c impulse (fuel-based) F= Pm f gL = length of detonation tubeLnozzle = length of nozzle sectionM = Mach numberMR = molar ratio of nitrogen to oxygenPm = mass ow ratep = pressurepp = plateau pressure in single-pulse detonation studyq = heat addition per unit mass of airQq = nondimensionalheat addition q=C p1T0

s = entropyT = temperatureu = velocityreg = half conical angle of nozzlemacr = ratio of nozzle length to detonation tube lengthdeg = ratio of speci c heatacuteth = thermodynamic cycle ef ciencyiquest = time periodiquestD = residence time of detonation wave L=uD

Aacute = stoichiometric ratioAtilde = cycle static-temperatureratio T1=T0

Received 23 October 2002 revision received 7 March 2003 acceptedfor publication 7 March 2003 Copyright cdeg 2003 by the authors Publishedby the American Institute of Aeronautics and Astronautics Inc with per-mission Copies of this paper may be made for personal or internal useon condition that the copier pay the $1000 per-copy fee to the CopyrightClearance Center Inc 222 Rosewood Drive Danvers MA 01923 includethe code 0748-465803 $1000 in correspondence with the CCC

currenPostdoctoral research associate Department of Mechanical Engineeringyxw119psuedu

daggerPhD Candidate Department of Mechanical Engineering mafuhuapsuedu

DaggerDistinguished Professor Department of Mechanical Engineeringvigorpsuedu Fellow AIAA

Subscripts

a = airclose = time duration during which valve is closedcycle = pulse detonation engine operation cycleD = detonation wavef = fueli = preconditionedstatepurge = purging stagere ll = re lling stage0 = freestream condition1 = fresh reactant upstream of detonation wave front2 = combustion product downstream of detonation

wave front3 = ow property after isentropic expansion

I Introduction

P ULSE detonation engines (PDEs) have recently been rec-ognized as a promising propulsion technology that offers

advantages in thermodynamic cycle ef ciency hardware simplic-ity operation scalability and reliability1 The potential for self-aspirating operation is highly attractive from the perspectives ofef ciency and operation Studies of PDEs have been conductedfor several decades The earliest experimental investigation maybe traced back to Hoffman2 Nicholls et al3 later performed a se-ries of single-pulse detonation experiments with hydrogenoxygenand acetyleneoxygen mixtures Because a low-energy spark igni-tor was used in their experiments and no de agration-to-detonation(DDT) augmentation device was utilized it is not clear whetherfull detonationwaves were realized Signi cant progress was madeby Krzycki4 at the US Naval Ordinance Test Station demonstrat-ing the use of propaneair mixtures for a pulse detonation deviceThe tube had an internal diameter of 1 in (254 cm) and a lengthof 6 ft (1829 cm) Cycle frequency of up to 55 Hz was achievedusing a high-energy spark discharge Krzycki concluded that thisintermittentdetonationdevice was not promising for propulsionap-plicationsdue to the low speci c impulse associatedwith the limitedcycle rates attained

Exploratory research on detonation as an alternative reactionmechanism for airbreathing and rocket propulsion was terminatedin the late 1960s due to the lack of fundingand was not resumed un-til the 1980s Helman et al5 carried out a series of experimentswithethyleneoxygen and ethyleneair mixtures at the US Naval Post-graduate School Both single- and multicycle modes were studiedA predetonatorusingethyleneoxygenwas employedto enhancetheDDT process in the main tube lled with an ethyleneair mixtureThe pressure recordings however suggested that full detonation

556

WU MA AND YANG 557

Table 1 Survey of single-pulse single-tube experimental investigations of PDEs

Propellant Reference Con gurations Isp s impulse

H2 O2 Hinkey et al6 L D unknown idD 51 cm fully lled Mixture-based1 atm 298 K (1995) spark ignitor of 17 J DDT enhanced by Shchelkin spiral 240 sa 185 sb

Aacute D 10 Sterling et al7 (1996) L D 1759 cm id D 22 cm fully lled spark ignitor NALitchford8 (2001) L D 90 cm id D 5 cm partially and fully lled Aacute unknown Peak thrust D 1201 N

spark plug of 011 J DDT enhanced by Shchelkin spiral (in fully lled case)Shepherd et al9 L D 800 cm id D 28 cm fully lled diluted with Ar or N2 NA

(2002) predetonator (direct initiation using exploding wire)H2 air Hinkey et al6 L D unknown idD 51 cm fully lled Fuel-based

1 atm 298 K (1995) spark ignitor of 17 J DDT enhanced by Shchelkin spiral 1000 sa

Aacute D 10 predetonator lled with H2 O2 mixtures 1200 sb

Meyer et al10 L D 914 cm idD 51 cm spark ignitor with DDT NA(2002) enhanced by Shchelkin spiral extended cavity plus

Shchelkin spiral and coannulusC2H4 O2 Sanders et al11 L D 135 cm id D 38 cm partially and fully lled Total impulseD

1 atm 298 K (2000) spark ignitor 3000 N cent sm2 a 792 sAacute D 10 Daniau et al12 Exploding wire source of 30 J id D 50 cm Mixture-based

(2000) L D 65 cm no nozzle fully lled 200 sc

L D 65 cm straight nozzles gt200 sc

L D 10 cm diverging nozzles 257ndash340 sc

Sinibaldi et al13 L D 120 cm id D 127 cm Aacute D 02ndash10 NA(2001) spark ignitor with DDT enhanced by Shchelkin spiral

Falempin et al14 L D unknown idD unknown Mixture-based(2001) ignition source unknown 200 s

C2H4 air Broda et al15 L D 1829 cm id D 34 cm Aacute D 11ndash13 fully lled NA1 atm (1999) spark ignitor of 35 J DDT enhanced by obstacles298 K Sanders et al11 L D 135 cm id D 38 cm Aacute D 13 fully lled NA

(2000) spark ignitor with DDT enhanced by Shchelkin spiralWatts et al16 Square tube W D 45 cm L D 165 cm Aacute D 12 fully lled NA

(2000) spark plug of 25 J DDT enhanced by obstaclesSinibaldi et al17 L D 120 cm id D 127 cm Aacute D 10 predetonator NA

(2000) spark ignitor with unknown energyC2H4 O2N2 Sinibaldi et al17 L D 1905 cm id D 57 cm Aacute D 06ndash20 fully lled NA

p and T unknown (2000) ignitor of 033ndash831 JO2 N2 volumetric ratio 100 and 75

C2H4 O2N2 Cooper et al18 L D 1016 cm L=D D 13 id D 76 cm Aacute D 10 fully lled Mixture-based 170 sc

30ndash100 kPa 298 K (2002) spark ignitor N2 dilution 0ndash75 (by volume) at p D 100 kPaC3H8 O2 Sinibaldi et al13 L D 120 cm id D 127 cm Aacute D 04ndash10 NA

1 atm 298 K (2001) predetonator spark ignitorHarris et al19 L D 250 cm id D 505 cm Aacute D 10 fully lled Total impulseD

(2001) spark plug of 005 J for DDT initiation 10 N cent sc(MR D 0)exploding wire ignitor of 203ndash502 J for direct initiation 80 N cent sc(MR D 10)

C3H8 O2N2 Cooper et al18 Tube fully lled N2 dilution 0ndash75 (by volume) Mixture-based at p D 100 kPa05ndash10 atm 298 K (2002) idD 76 cm L D 609 cm L=D D 8 various spiral pitches without N2 dilutionAacute D 10 idD 38 cm L D 150 cm L=D D 40 spiral pitch D 11 cm 150 sc 172 scL=D D 8

130 sc L=D D 40C3H8 O2N2 Lieberman et al20 L D 100 cm id D 75 cm N2 dilution 20ndash40 (by volume) Mixture-based

1 atm T unknown (2002) driver section C3H8O2 (Aacute D 1 at p D 1 4 atm 152 sc (20 N2 dilution)Aacute D 10 L D 14 cm id D 3 cm spark plug of 30 mJ 144 sc (40 N2 dilution)

aBased on time history of pressure at thrust wall bBased on time history of force measured by load cell cMaximum impulse measured from ballistic pendulumdisplacement

was not achieved as revealed by the presence of signi cant com-pressionwaves precedingthe pressureriseof the reporteddetonationwaves

Much effort has been applied to the study of various aspects ofPDEs since the mid-1990s Tables 1 and 2 summarize the exper-imental work performed to date6iexcl32 In single-pulse experimentsonly detonation ignitionpropagationand attenuationwere investi-gated at preconditionedstates Total impulse was obtainedbased onmeasured chamber pressureandor force histories for a limited timeperiodIn practicenegativethrustmay appeardue to the low-energylevel of the gases in detonationtubes during the blowdown purgingand re lling processesin a multicyclemode As a result systemper-formance obtained from single-pulseexperimentsusually exceededthat in a real engine with multicycle operation In contrast multi-cycle experiments involved all necessary PDE operation processesand thus provided more direct simulation However much impor-tant information required to characterizethe system dynamics suchas air ow rate and purgingre lling data was not measured in mostexperiments The details for thrust measurements were not clearlyde ned either renderingassessmentof the performanceof differentsystems a dif cult task

In parallel to experimental investigations attempts were madeboth theoretically and numerically to estimate the performance ofPDEs Talley and Coy33 employed a lumped-parameter analysis todetermine the theoretical limit of PDE performanceby approximat-ing detonation chamber dynamics with an ideal constant-volumeprocess The blowdown time was assumed to be much longer thanthe characteristic wave transit times in the chamber Tew34 andKent eld35 estimated the PDE performance using an ideal-cycleassumption Detonation was approximated as either a polytropicor a constant-volume heat addition process Heiser and Pratt36 uti-lized a more realistic Zeldovich von Neumann and Doring (ZND)model to simulate the detonation process in their PDE cycle analy-sis Wintenbergeret al37 developeda semi-analyticalmodel for theimpulse of a single-pulse detonation tube by means of dimensionalanalysis and empirical observations In addition to these theoreticalmodels several numerical analyses based on one-dimensional3839

and two-dimensional40iexcl47 approaches were carried out One majorde ciency of one-dimensionalanalyses is that the boundary condi-tion at the detonation tube exit can not be correctly speci ed be-cause it depends on the local ow evolution in the downstreamam-bient regime46 Multidimensional simulations with computational

558 WU MA AND YANG

Table 2 Survey of multicycle experimental investigation of PDEs

Propellants Reference Con gurations Isp s and Thrust N

H2 O2 Sterling et al7 Single tube f D 33 Hz id D 22 cm L D 152 cm NA1 atm (1996) fully lled H2 and O2 adopted as buffer gases spark ignitor298 K Stuessy and Single tube f D 10ndash12 Hz fully lled air used as buffer gas NAAacute D 10 Wilson21 cylindrical tube with idD 76 cm L D 533 cm

(1996) annular tube with id D 25 cm odD 76 cm L D 533 cmannular tube with diverging conical nozzle

H2 air Aarnio et al22 Single tube f D 5 Hz id D 51 cm L D 1219 cm Aacute D 10 Fuel-based1 atm (1996) partially lled air used as buffer gas 1333 sa

298 K spark ignitor of 17 J 1116 sb

Hinkey et al23 Twin tube f D 10 Hz common air inlet manifold NA(1997) id D 51 cm L D 914 cm Aacute D 07ndash13

predetonator lled by H2 O2 mixtures spark ignitor of 15 JSchauer et al24 Multitube (1 2 and 4) f D 05ndash100 Hz NA

(1999) id D 51 cm L D 914 cm and id D 89 cm L D 914 cmair used as buffer gasspark ignitor with DDT enhanced by Shchelkin spiral

Schauer et al25 f D 14ndash40 Hz L D 914 cm id D 51 cm Aacute D 04ndash285 Fuel-based for Aacute D 10 f D 16 Hz(2000) spark ignitor with DDT enhanced by Shchelkin spiral and 35 s ignition delay

air used as buffer gas with 50 tube lling ratio 7100 sa (30 lling length)various ignition delays and tube lling fractions 4200 sa (90 lling length)

McManus et al26 Single tube f D 10ndash35 Hz conical converging nozzle F D 1112 Nb ( f D 10 Hz)(2001) id D 476 cm L D 254 cm Aacute D 06 F D 5338 Nb ( f D 35 Hz)

spark ignitor of 002 J at tube exitFrankey et al27 Single tube f D 11ndash21 Hz converging nozzle F D 222 Na f D 21 Hz)

(2002) id D 508 cm L D 18288 cm Aacute D 10spark ignitor with DDT enhanced by Shchelkin spiralair used as buffer gas

C2H4O2 Sterling et al7 f D 100 Hz L D 508 cm id D 22 cm 13 tube lled NA1 atm 298 K (1996) spark ignitor with DDT enhanced by unspeci ed deviceAacute D 10 Falempin et al14 Single tube f D 80 Hz id D 50 cm L D 50ndash426 cm NA

(2001) ignition source not mentionedC2H4air Broda et al15 f D 8ndash10 Hz L D 1829 cm id D 34 cm Aacute D 11 NA

1 atm (1999) spark ignitor of 4ndash8 J DDT enhanced by obstacles298 K Watts et al16 Single tube f D 10 Hz Aacute D 12 NA

(2000) spark plug of 25 J DDT enhanced by obstaclesbuffer air injected for 5ndash10 ms in each cycle

Brophy et al28 Single tube f D 80 Hz id D 40 cm L D 25 cm Aacute D 10ndash18 Total impulse(2002) spark ignitor with unknown energy 048 N cent sa (Aacute D 10)

Shimo et al29 f D 15 Hz id D 51 cm L D 822 cm Aacute D 09 60 lling ratio NA(2002) spark ignitor with DDT enhanced by Shchelkin spirals

JP-10O2 g Brophy et al30 f D 5 Hz L D 152 305 762 cm id D 38 cm Aacute D 07ndash17 NAp and T unknown (1998) ignitor of 14 J ignition delay 40 50 and 70 ms

1 atm Brophy and Single tube f D 10 Hz id D 381 cm L D 29 cm Aacute D 09ndash12 NA292 K Netzer31 (1999) ignitor of 05 J at locations x D 0ndash2D from head end

1 atm Brophy et al28 Single tube f D 30 Hz id D 40 cm L D 25 cm Aacute D 10ndash18 Total impulse280 K (2002) spark ignitor with unknown energy 059 N cent sa (Aacute D 10)

C3H8O2 Farinaccio et al32 f D 10 15 20 25 Hz L D 40 cm id D 81 cm Aacute D 10 Mixture-based 71 sb f D 15 Hz)p and T unknown (2002) fully or partially lled N2 purged for 13-cycle period 71 sb f D 20 Hz fully lled)

aBased on time history of pressure at thrust wall bBased on time history of force measured by load cell

domains including both detonation tubes and ambient ows arethus required to describe the system dynamics faithfullyespeciallyin the near eld of the tube exit where the ow is intrinsicallymultidimensional To date only single-pulse operations have beentreated using two-dimensional analyses Multicycle simulationshave been limited to one-dimensionalmodels whose results are ap-parently questionable Most of the analyses developed so far haveconsideredonly detonationchamber dynamicsnot a complete PDEsystem

The current work attempts to establish a global analysis to de-termine the overall system performance and thermodynamic cy-cle ef ciency of airbreathing PDEs Figure 1 shows schemati-cally the con guration under consideration It includes a coaxialsupersonic inlet with mixed compression a multitube detonationchamber and a nozzle A rotary valve is placed in front of thecombustor entrance to distribute the air ow evenly into the indi-vidual tubes The following section describes the development ofan ideal PDE thermodynamic cycle analysis to assess the theoret-ical limit of system performance A modular approach is then uti-lized to analyze the engine dynamics numerically As a speci cexample an engine operating at a ight altitude of 93 km and afreestream Mach number of 21 is considered The effects of var-

ious operating parameters on the engine propulsion ef ciency arestudied systematically Finally the in uence of nozzle design isexamined

II Thermodynamic Cycle AnalysisAn ideal thermodynamic cycle analysis is presented in this sec-

tion to estimate the theoretical limit of the performance of an air-breathingPDE The work extendsthe approachof Heiserand Pratt36

for perfect gases with constant properties to accommodate prop-erty variations across the detonation wave front Figure 2 showsthe temperaturendashentropy diagram of an ideal PDE cycle The cor-responding Humphrey (constant-volumecombustion) and Brayton(constant-pressure combustion) cycles are included for compari-son The process from point 0 to point 1 is an adiabatic isentropiccompressionprocess in the course of which the ow temperature israisedfromits freestreamvalueT0 to thatat thecombustorentranceT1 The path from point 1 to point 2 corresponds to the detonationprocess Here the ZND model is adopted that is the dashed linefrom point 1 to point 1a corresponds to the entropy increase causedby shockcompressionand the solid line from point 1a to point 2 rep-resents subsequent heat release due to chemical reactions Point 2

WU MA AND YANG 559

Table 3 Variations of ow properties for stoichiometricH2air mixturea

s1a ndashs1 s2ndashs1Cycle p1a =p1 p2=p1 T1a=T1 T2=T1 kJkg cent K kJkg cent K deg2

Ideal PDE 1550 547 982 104 184 116Humphrey NA 796 NA 917 NA 189 117Brayton NA 100 NA 794 NA 228 118

aT1 D 300 K p1 D 1 atm and deg1 D 14

Fig 1 Supersonic airbreathing PDE

Fig 2 Temperaturendashentropy diagram of ideal PDE Humphrey andBrayton cycles

is known as the ChapmanndashJouguet (CndashJ) point where the chemicalsystem reaches an equilibriumstate and the ow relative to the deto-nation wave front is sonic For a given initial condition and reactantcomposition the detonation wave velocity and CndashJ properties canbe easilydeterminedby means of a chemicalequilibriumanalysis48

Table 3 summarizes the ow properties at various stations for a sto-ichiometric hydrogenair mixture initially at 1 atm and 300 K Notethat the ldquohumpedrdquo nature of the heat additionprocess from point 1ato point 2 re ects the well-known phenomenonof the existenceof amaximumtemperaturewhen heat is added to a steadysubsonic owThe process from point 2 to point 3 involves isentropic expansionwith the pressure of the burned gases decreasing to the freestreamcondition p3D p0 The cycle is closed by an imaginary constantstatic-pressureprocess in which the abundantheat is removed to thesurroundings from the exhaust ow

Following the conventional de nition the thermodynamic cycleef ciency acuteth is expressed as

acuteth D 1 iexcl qreject=q (1)

where q and qreject are the amount of heat added during the processfrom point 1a to point 2 and the amount of heat removed to the am-bient during the process from point 3 to point 0 respectivelyAfterextendingthe analysisof Heiser and Pratt36 to includepropertyvari-ations across the detonation wave front a closed-form expressionof acuteth is obtained as follows49

acutethPDE D 1 iexcl

(deg1 iexcl 1deg2 iexcl 1

sup3deg2

deg1

acute21

M2D

sup31 C deg1 M2

D

1 C deg2

acutedeg2 C 1=deg2

pound Atilde 1iexcl[deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)

(2)

where deg1 and deg2 are the speci c-heat ratios of the unburned andburned gases separated by the detonation wave front respectivelyThe Mach number of the detonation wave relative to the unburnedgas MD can be calculated using the following equation for xedheat addition

M2D D

microdeg 2

2 iexcl deg1

deg 21 iexcl deg1

C deg 22 iexcl 1

deg1 iexcl 1

QqAtilde

para

C

smicrodeg 2

2 iexcl deg1

deg 21 iexcl deg1

Cdeg 2

2 iexcl 1

deg1 iexcl 1

QqAtilde

para2

iexcldeg 2

2

deg 21

(3)

In comparison the thermodynamic ef ciencies of the Humphreyand Brayton cycles are

acutethHumphrey D 1 iexcl

(deg2

deg1

sup3deg1 iexcl 1

deg2 iexcl 1

acutedeg2 iexcl 1=deg2sup3

deg1QqAtilde

C 1

acute1=deg2

pound Atilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)(4)

acutethBrayton D 1 iexclraquomicrosup3

QqAtilde

C 1

acuteAtilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1deg1=deg2 iexcl 1

paraiquestQqfrac14

(5)

With the thermodynamic cycle ef ciency available the thrust Fcan be obtained by a control-volumeanalysis

F D Pma C Pm f u3 iexcl Pmau0 frac14 Pma

iexclpu2

0 C 2acutethq iexcl u0

cent(6)

where u0 is the freestream velocity Pma the cycle-averagedair mass ow rate delivered to the engine through the inlet and Pm f the cycle-averaged fuel mass ow rate Note that the preceding analysis isbased on the assumptions that every uid particle experiences thesame processes sequentially and that the effects of purging and by-pass air are ignored The fuel-based speci c impulse can then beobtained as follows

Isp DF

Pm f gD

pu2

0 C 2acutethq iexcl u0

f g(7)

Figures 3 and 4 show a typical result of cycle ef ciencyand speci c impulse for a stoichiometric hydrogenair system

Fig 3 Thermodyanmiccycle ef ciencies of idealPDE Humphrey andBraytoncycles as function of static temperature ratio Atilde for stoichiomet-ric H2 air system

560 WU MA AND YANG

Fig 4 Speci c impulses of ideal PDE Humphrey and Brayton cy-cles as function of static temperature ratio Atilde for stoichiometric H2 airsystem

The freestream velocity and temperature are u0 D 636 ms andT0 D 228 K respectively The nondimensional heat addition q is2247The system performanceincreaseswith increasingstatic tem-peratureratioThe PDE offers the bestperformanceamong the threecycles especiallywhen the static temperatureratio Atilde is smaller than3 This may be because for a given amount of heat addition theMach number of the detonation wave increases with decreasing T1

(or Atilde ) as indicated by Eq (3) The shock-compression effect be-comes more signi cant for a lower T1 leading to a higher increasein the temperature and pressure of the unburned gases before com-bustion The Isp of an ideal PDE reaches 5263 s when T1 D 428 Kthat is Atilde D 1877

III System Performance AnalysisThis section deals with the developmentof a system performance

analysis for airbreathing PDEs as shown schematically in Fig 1The study is based on a modular approach Each module repre-sents a speci c component of the engine and its dynamic behav-ior is formulated using complete conservationequationsThe workinvolves the following three components 1) supersonic inlet dy-namics 2) detonation chamber dynamics and system performanceand 3) effect of nozzle con guration The effects of fuel supply airdistribution and inlet isolator are ignored for simplicity They canhowever be straightforwardlyincluded as submodels in the overallengine performance analysis

A Supersonic Inlet DynamicsThe inlet and its interaction with combustor represent a crucial

aspect in the development of any airbreathing engine includingPDEs The inlet is designed to capture and supply stable air ow at arate demanded by the combustor and to maintain high pressure re-covery and stability margin at various engine operating conditionsThe overallvehicleperformancedependsgreatlyon the energy leveland ow quality of the incoming air A small loss in inlet ef ciencymay translate to a substantial penalty in engine thrust Moreoverany change in the inlet ow structure may modify the downstreamcombustioncharacteristicsand subsequentlylead to undesirablebe-haviors such as ame blowoff and ashback Thus matching inletbehavior to engine requirements is of fundamental importance todesigners50

In addition to its primary function of supplyingair an inlet has adetermining in uence on the dynamics of the entire system throughits intrinsicunsteadinessand interactionswith thecombustioncham-berTypicallypressurewaves areproducedin thecombustioncham-ber and propagate upstream to interact with the inlet ow througha manifold where mixing of air and fuel occurs The resultant owoscillations in the inlet diffuser then either propagate downstreamin the form of acoustic waves or are convected downstream withthe mean ow in the form of vorticity and entropy waves andfurther reinforce the unsteady motions in the combustor A feed-back loop is thus established between the inlet and combustor

Fig 5 Supersonic inlet with mixed compression

Fig 6 Mach number contours with different back pressures understeady-state conditions

The situation is much more complicated in a supersonic PDE dueto the shock-waveboundary-layer interaction and shockacoustic-wave interaction51 [also Oh J Y Ma F H Hsieh S Y andYang V ldquoInteractionsBetween Shock Waves and Acoustic Wavesin a Supersonic Inlet Diffuserrdquo (to be submitted for publication)]The engine dynamics exhibits features that are qualitatively differ-ent from those predicted by treating the inlet and combustor as twoseparate entities

The inlet analysis is based on the axisymmetric Favre-averagedconservationequationsTurbulenceclosure is achievedusing a two-layer model of Rodi calibrated for supersonic ows with shockwaves52 The governingequationsare solved numerically by meansof a density-based nite volume methodology The temporal dis-cretization is obtained using a four-stage RungendashKutta integrationmethod and the spatial discretization employs an upwind total-variation-diminishing scheme developed by Harten53 Speci c de-tails of the numerical algorithm can be found in Ref 54

Figure 5 shows the con guration treated in the present study amixed-compression inlet optimized for a ight altitude of 93 kmand a freestream Mach number of 21 (Ref 50) Figure 6 presentsthe Mach-numbercontoursat two differentbackpressures(pb D 21and 22 atm) which are carefully chosen such that the engine oper-ates at a supercritical condition to provide a suf cient shock stabil-ity margin Under these conditions the two leading conical shocksgenerated by the double-cone centerbody compress the air ow ex-ternallymerge slightlyabove thecowl lip and forma strongobliqueshock which extends into the external-ow region In addition ashockstemming from the cowl inner surface continuesdownstreamhitting and re ecting from both the cowl and centerbodywalls and nally leading to a terminal normal shock The ow undergoes asequence of compression and expansion waves and becomes sub-sonic after passingthroughthe normal shock located in thedivergentsection of the diffuser The inlet recovers a high percentage of thefreestream total pressure by decelerating the air ow through theshock train The pressure recovery coef cients for the two cases are84 and 88 respectively and the Mach numbers immediately infront of the terminal shocks are 142 and 132 respectively

The response of the inlet shock system to downstream distur-bances has also been studied by imposing periodic pressureoscilla-tions at theexit planeA wide rangeof uctuationfrequencyandam-plitude were considered Important phenomena of concern includetemporal and spatial variationsof mass ow rate pressure recoveryand ow distributionas well as shock displacementIn general theacoustic response of the inlet ow increases with increasing ampli-tude of the imposed oscillation but decreases with frequencyAlsoincluded as part of the result is the acoustic impedance function atthe inlet exit a parameter that can be effectively used to character-ize the inletcombustor couplingA more detaileddiscussionon thissubject will be given in subsequent work

WU MA AND YANG 561

B Detonation Chamber DynamicsThe detonationchamberdynamicsis formulatedbasedon thecon-

servation laws for a multicomponentchemically reacting system intwo-dimensionalcoordinatesDiffusive transport is neglected in thecurrent study because of its minor role in determining detonationdynamics and system performance The governing equations andtheir associatedboundary conditionsare solved using a recently de-velopedspacendashtime conservation-elementsolution-elementmethodthat circumvents the de ciencies of existing numerical methodsfor treating detonation waves and shock discontinuities55iexcl59 Theresultant computer code is further parallelized using the messagepassing interface library with domain decomposition to improve itsef ciency

Both simple global and detailed chemical kinetics models areutilized4959 The former involvesonly one progressvariable to char-acterize the chemical reaction rate and assumes constant proper-ties Because of its computational ef ciency and reasonable accu-racy in determining the PDE propulsive performance the model isimplemented in the present work The associated thermochemicalparameters are optimized by comparing the calculated detonationwave properties with those from the NASA chemical equilibriumanalysis48 The relative errors are less than 5 in terms of the det-onation velocity and the CndashJ pressure and temperature

As part of the model validation effort a series of single-pulsecalculations were conducted for a straight tube of 60 cm in lengthinitially lled with a stoichiometric mixture of hydrogen and air atpreconditionedpressure pi and temperature Ti A driver-gas regionspanning 02 mm near the head end with a temperature of 2000 Kand a pressure of 30 atm was employed to initiate the detonationwave directly Four different numerical grids with the sizes of 0201 005 and 0025 mm were used to check the solution accuracyin terms of grid independenceAll of the calculatedpressurepro lescollapsed onto a single curve with the CndashJ propertiesmatching theanalytical values exactly As a result the 02-mm grid was chosenfor the entire study to alleviate the computational burden For asingle-pulse operation the head-end pressure remains at a plateauvalue pp that is p3 in Refs 37 and 46 for certain period soonafter the detonation initiation and then decays gradually to a levellower than the ambient state3746 The impulse can be determinedby integrating temporally the force exerted on the head end fromt D 0 to the instant when the head-endpressure reaches the ambientvalue The contribution to the impulse from the ignition source isestimated to be less than 05 Figure 7 shows the impulse per unitcross-sectionalarea as a function of the plateau pressure pp and thedetonation residence time iquestD (de ned as the tube length L dividedby the detonation wave velocity u D ie iquestD acute L=u D Results canbe correlated well in the following form

I=A D 41pp iexcl pi iquestD (8)

This expression is quite similar to those obtained from the semi-analytical analysis of Wintenberger et al37 and the experimentalwork of Falempin et al14 The constants of proportionality differslightly in the various studies for example 413 in Ref 37 for the

Fig 7 Generalized impulse curve for single-pulse detonation instraight tube with stoichiometric H2air mixture

current conditions suggesting dependenceon the details of experi-mentalprocedureandoperatingconditionsThe generalizedformulaproposed by Kailasanath60 based on his numerical simulations forhydrogenair ethyleneoxygen and propaneoxygen mixtures has alarger constant of proportionality of 465 One factor contributingto this disparity may be differencesin ignition source The width ofthe detonation initiation region is 20 mm in Ref 60 as opposed to02 mm in the present simulationsNonethelessthe precedingpara-metric study demonstrates the capacity and delity of the presentapproach for the PDE performance analysis

The validatedanalysis is then employed to study the performanceof airbreathing PDEs As a speci c example the ight conditioninvolving an altitude of 93 km and a Mach number of 21 is consid-ered The freestream static pressure and temperature are 029 atmand 228K respectivelycorrespondingto a totalpressureof 265atmand a total temperature of 428 K The total pressure and tempera-ture at the combustor entranceobtained from the inlet ow analysisdescribed in the preceding section are 212 atm and 428 K respec-tively The detonation tube is 60 cm long and has a diameter of16 cm which is similar to the dimensions of contemporary ramjetcombustors for air defense applicationsAs a rst approach only asingle straight tube is considered to provide direct insight into thechamber dynamics without complications arising from the nozzleThe interactions among the tubes are also ignored The valve atthe tube entrance is assumed to be either fully open or fully closedThe operationsequenceis thuscontrolledby three time periodsthevalveclose-upperiodiquestclose duringwhich the valve is closedand thetube undergoesdetonationinitiationand propagationand blowdownprocesses the purging period iquestpurge during which a small amountof cold air is injected to prevent preignition of fresh reactants andthe re lling period iquestre ll during which the combustible mixture isdelivered to the chamber The sum of these three periods is equal tothe operationcycleperiodiquestcycle that is iquestcycle D iquestclose C iquestpurge C iquestre llThe purgingperiodshouldbeminimizedto reduceperformancelossA small value of 01 ms is used throughoutall of the calculationsinthe present study

The boundaryconditionsat thehead endof thedetonationtubearespeci ed according to the engine operationDuring the purgingandre lling stages the total pressureand total temperatureare obtainedfrom the inlet analysis The axial velocity is extrapolated from theinterior points and the reactant mass fraction is treated as an inputparameterWhen the valve is closed the head end is simply modeledas a rigid wall

A series of analysesare conductedover a wide range of operationparameters The baseline case has iquestcycle of 3 ms and iquestclose of 24 msThe tube is initially lled with a stoichiometrichydrogenairmixtureat ambient pressure and temperature It takes about ve cycles toreach steady cyclic operation Figure 8 shows the xndasht diagram forthe rst cycle of operation obtained by tracing the characteristiclines of the ow eld along the centerline of the tube The timehistories of the ow properties at the head end are also presentedThe detonation wave is directly initiated by a hot driver gas andpropagates downstream at the CndashJ velocity toward the unburnedmixture(region1) It then inducesTaylorexpansionwaves (region2)to satisfy the stationaryconditionat the head end causinga uniformregion (region 3) with constant-ow properties in the upstream

The detonationwave reaches the reactantair interfaceat the tubeexit at t D 0305 ms (point A) which deviates slightly from thefollowing analytical prediction by 06 due to the effect of theexternally imposed ignition source

iquestD D L=u D D 06 m=1956 ms D 0307 ms (9)

The wave then degenerates to a nonreactive shock that is theprimary shock wave proceeding farther downstream into the ex-ternal region followed by a contact surface separating the ambientair and combustion products A sonic region is gradually formednear the tube exit due to the local ow expansion as evidenced bythe clustered characteristic lines in the xndasht diagram Downstreamof the sonic region the ow is expanded to become supersonic and nally leads to the formationof a secondaryshock to match with thesubsonic ow behind the primary shockThis secondaryshockwave

562 WU MA AND YANG

Fig 8 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms 1 = unburned region 2 = Taylor expansion waves 3 = stationary region 4 = non-simple wave regionand 5 = simple wave region

Fig 9 Snapshots of pressure density and their gradients elds at t = 07 ms

moves farther downstreammeeting with expansionwaves originat-ing from the primary shockwaveThese complicated ow structurescan be also observed in Fig 9 which shows the instantaneouspres-sure and density and their gradient elds at t D 07 ms Many salientfeatures are clearly shown including the expansion fans vorticesand rolled-upslip lines that are developedas the shockdiffractsoverthe edge of the tube exit

As the detonation wave catches the reactantair interface and theresultant primary shock wave travels outside the tube a series ofexpansionwaves is generatedwhich propagate upstream resulting

in a nonsimple wave region (region 4) when interactingwith the in-coming Taylor waves A simple wave region (region 5) is recoveredafter passing through the Taylor waves The rst expansion wavereaches the head end at t D 0935 ms (point B) which can be deter-mined by consideringthe interactionbetween the expansionand theTaylor waves and the sound speed in region 3 A similarity solutionhas been derivedby Wintenbergeret al37 to predict this time instantanalytically

t D L=uD C regL=c3 (10)

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

WU MA AND YANG 557

Table 1 Survey of single-pulse single-tube experimental investigations of PDEs

Propellant Reference Con gurations Isp s impulse

H2 O2 Hinkey et al6 L D unknown idD 51 cm fully lled Mixture-based1 atm 298 K (1995) spark ignitor of 17 J DDT enhanced by Shchelkin spiral 240 sa 185 sb

Aacute D 10 Sterling et al7 (1996) L D 1759 cm id D 22 cm fully lled spark ignitor NALitchford8 (2001) L D 90 cm id D 5 cm partially and fully lled Aacute unknown Peak thrust D 1201 N

spark plug of 011 J DDT enhanced by Shchelkin spiral (in fully lled case)Shepherd et al9 L D 800 cm id D 28 cm fully lled diluted with Ar or N2 NA

(2002) predetonator (direct initiation using exploding wire)H2 air Hinkey et al6 L D unknown idD 51 cm fully lled Fuel-based

1 atm 298 K (1995) spark ignitor of 17 J DDT enhanced by Shchelkin spiral 1000 sa

Aacute D 10 predetonator lled with H2 O2 mixtures 1200 sb

Meyer et al10 L D 914 cm idD 51 cm spark ignitor with DDT NA(2002) enhanced by Shchelkin spiral extended cavity plus

Shchelkin spiral and coannulusC2H4 O2 Sanders et al11 L D 135 cm id D 38 cm partially and fully lled Total impulseD

1 atm 298 K (2000) spark ignitor 3000 N cent sm2 a 792 sAacute D 10 Daniau et al12 Exploding wire source of 30 J id D 50 cm Mixture-based

(2000) L D 65 cm no nozzle fully lled 200 sc

L D 65 cm straight nozzles gt200 sc

L D 10 cm diverging nozzles 257ndash340 sc

Sinibaldi et al13 L D 120 cm id D 127 cm Aacute D 02ndash10 NA(2001) spark ignitor with DDT enhanced by Shchelkin spiral

Falempin et al14 L D unknown idD unknown Mixture-based(2001) ignition source unknown 200 s

C2H4 air Broda et al15 L D 1829 cm id D 34 cm Aacute D 11ndash13 fully lled NA1 atm (1999) spark ignitor of 35 J DDT enhanced by obstacles298 K Sanders et al11 L D 135 cm id D 38 cm Aacute D 13 fully lled NA

(2000) spark ignitor with DDT enhanced by Shchelkin spiralWatts et al16 Square tube W D 45 cm L D 165 cm Aacute D 12 fully lled NA

(2000) spark plug of 25 J DDT enhanced by obstaclesSinibaldi et al17 L D 120 cm id D 127 cm Aacute D 10 predetonator NA

(2000) spark ignitor with unknown energyC2H4 O2N2 Sinibaldi et al17 L D 1905 cm id D 57 cm Aacute D 06ndash20 fully lled NA

p and T unknown (2000) ignitor of 033ndash831 JO2 N2 volumetric ratio 100 and 75

C2H4 O2N2 Cooper et al18 L D 1016 cm L=D D 13 id D 76 cm Aacute D 10 fully lled Mixture-based 170 sc

30ndash100 kPa 298 K (2002) spark ignitor N2 dilution 0ndash75 (by volume) at p D 100 kPaC3H8 O2 Sinibaldi et al13 L D 120 cm id D 127 cm Aacute D 04ndash10 NA

1 atm 298 K (2001) predetonator spark ignitorHarris et al19 L D 250 cm id D 505 cm Aacute D 10 fully lled Total impulseD

(2001) spark plug of 005 J for DDT initiation 10 N cent sc(MR D 0)exploding wire ignitor of 203ndash502 J for direct initiation 80 N cent sc(MR D 10)

C3H8 O2N2 Cooper et al18 Tube fully lled N2 dilution 0ndash75 (by volume) Mixture-based at p D 100 kPa05ndash10 atm 298 K (2002) idD 76 cm L D 609 cm L=D D 8 various spiral pitches without N2 dilutionAacute D 10 idD 38 cm L D 150 cm L=D D 40 spiral pitch D 11 cm 150 sc 172 scL=D D 8

130 sc L=D D 40C3H8 O2N2 Lieberman et al20 L D 100 cm id D 75 cm N2 dilution 20ndash40 (by volume) Mixture-based

1 atm T unknown (2002) driver section C3H8O2 (Aacute D 1 at p D 1 4 atm 152 sc (20 N2 dilution)Aacute D 10 L D 14 cm id D 3 cm spark plug of 30 mJ 144 sc (40 N2 dilution)

aBased on time history of pressure at thrust wall bBased on time history of force measured by load cell cMaximum impulse measured from ballistic pendulumdisplacement

was not achieved as revealed by the presence of signi cant com-pressionwaves precedingthe pressureriseof the reporteddetonationwaves

Much effort has been applied to the study of various aspects ofPDEs since the mid-1990s Tables 1 and 2 summarize the exper-imental work performed to date6iexcl32 In single-pulse experimentsonly detonation ignitionpropagationand attenuationwere investi-gated at preconditionedstates Total impulse was obtainedbased onmeasured chamber pressureandor force histories for a limited timeperiodIn practicenegativethrustmay appeardue to the low-energylevel of the gases in detonationtubes during the blowdown purgingand re lling processesin a multicyclemode As a result systemper-formance obtained from single-pulseexperimentsusually exceededthat in a real engine with multicycle operation In contrast multi-cycle experiments involved all necessary PDE operation processesand thus provided more direct simulation However much impor-tant information required to characterizethe system dynamics suchas air ow rate and purgingre lling data was not measured in mostexperiments The details for thrust measurements were not clearlyde ned either renderingassessmentof the performanceof differentsystems a dif cult task

In parallel to experimental investigations attempts were madeboth theoretically and numerically to estimate the performance ofPDEs Talley and Coy33 employed a lumped-parameter analysis todetermine the theoretical limit of PDE performanceby approximat-ing detonation chamber dynamics with an ideal constant-volumeprocess The blowdown time was assumed to be much longer thanthe characteristic wave transit times in the chamber Tew34 andKent eld35 estimated the PDE performance using an ideal-cycleassumption Detonation was approximated as either a polytropicor a constant-volume heat addition process Heiser and Pratt36 uti-lized a more realistic Zeldovich von Neumann and Doring (ZND)model to simulate the detonation process in their PDE cycle analy-sis Wintenbergeret al37 developeda semi-analyticalmodel for theimpulse of a single-pulse detonation tube by means of dimensionalanalysis and empirical observations In addition to these theoreticalmodels several numerical analyses based on one-dimensional3839

and two-dimensional40iexcl47 approaches were carried out One majorde ciency of one-dimensionalanalyses is that the boundary condi-tion at the detonation tube exit can not be correctly speci ed be-cause it depends on the local ow evolution in the downstreamam-bient regime46 Multidimensional simulations with computational

558 WU MA AND YANG

Table 2 Survey of multicycle experimental investigation of PDEs

Propellants Reference Con gurations Isp s and Thrust N

H2 O2 Sterling et al7 Single tube f D 33 Hz id D 22 cm L D 152 cm NA1 atm (1996) fully lled H2 and O2 adopted as buffer gases spark ignitor298 K Stuessy and Single tube f D 10ndash12 Hz fully lled air used as buffer gas NAAacute D 10 Wilson21 cylindrical tube with idD 76 cm L D 533 cm

(1996) annular tube with id D 25 cm odD 76 cm L D 533 cmannular tube with diverging conical nozzle

H2 air Aarnio et al22 Single tube f D 5 Hz id D 51 cm L D 1219 cm Aacute D 10 Fuel-based1 atm (1996) partially lled air used as buffer gas 1333 sa

298 K spark ignitor of 17 J 1116 sb

Hinkey et al23 Twin tube f D 10 Hz common air inlet manifold NA(1997) id D 51 cm L D 914 cm Aacute D 07ndash13

predetonator lled by H2 O2 mixtures spark ignitor of 15 JSchauer et al24 Multitube (1 2 and 4) f D 05ndash100 Hz NA

(1999) id D 51 cm L D 914 cm and id D 89 cm L D 914 cmair used as buffer gasspark ignitor with DDT enhanced by Shchelkin spiral

Schauer et al25 f D 14ndash40 Hz L D 914 cm id D 51 cm Aacute D 04ndash285 Fuel-based for Aacute D 10 f D 16 Hz(2000) spark ignitor with DDT enhanced by Shchelkin spiral and 35 s ignition delay

air used as buffer gas with 50 tube lling ratio 7100 sa (30 lling length)various ignition delays and tube lling fractions 4200 sa (90 lling length)

McManus et al26 Single tube f D 10ndash35 Hz conical converging nozzle F D 1112 Nb ( f D 10 Hz)(2001) id D 476 cm L D 254 cm Aacute D 06 F D 5338 Nb ( f D 35 Hz)

spark ignitor of 002 J at tube exitFrankey et al27 Single tube f D 11ndash21 Hz converging nozzle F D 222 Na f D 21 Hz)

(2002) id D 508 cm L D 18288 cm Aacute D 10spark ignitor with DDT enhanced by Shchelkin spiralair used as buffer gas

C2H4O2 Sterling et al7 f D 100 Hz L D 508 cm id D 22 cm 13 tube lled NA1 atm 298 K (1996) spark ignitor with DDT enhanced by unspeci ed deviceAacute D 10 Falempin et al14 Single tube f D 80 Hz id D 50 cm L D 50ndash426 cm NA

(2001) ignition source not mentionedC2H4air Broda et al15 f D 8ndash10 Hz L D 1829 cm id D 34 cm Aacute D 11 NA

1 atm (1999) spark ignitor of 4ndash8 J DDT enhanced by obstacles298 K Watts et al16 Single tube f D 10 Hz Aacute D 12 NA

(2000) spark plug of 25 J DDT enhanced by obstaclesbuffer air injected for 5ndash10 ms in each cycle

Brophy et al28 Single tube f D 80 Hz id D 40 cm L D 25 cm Aacute D 10ndash18 Total impulse(2002) spark ignitor with unknown energy 048 N cent sa (Aacute D 10)

Shimo et al29 f D 15 Hz id D 51 cm L D 822 cm Aacute D 09 60 lling ratio NA(2002) spark ignitor with DDT enhanced by Shchelkin spirals

JP-10O2 g Brophy et al30 f D 5 Hz L D 152 305 762 cm id D 38 cm Aacute D 07ndash17 NAp and T unknown (1998) ignitor of 14 J ignition delay 40 50 and 70 ms

1 atm Brophy and Single tube f D 10 Hz id D 381 cm L D 29 cm Aacute D 09ndash12 NA292 K Netzer31 (1999) ignitor of 05 J at locations x D 0ndash2D from head end

1 atm Brophy et al28 Single tube f D 30 Hz id D 40 cm L D 25 cm Aacute D 10ndash18 Total impulse280 K (2002) spark ignitor with unknown energy 059 N cent sa (Aacute D 10)

C3H8O2 Farinaccio et al32 f D 10 15 20 25 Hz L D 40 cm id D 81 cm Aacute D 10 Mixture-based 71 sb f D 15 Hz)p and T unknown (2002) fully or partially lled N2 purged for 13-cycle period 71 sb f D 20 Hz fully lled)

aBased on time history of pressure at thrust wall bBased on time history of force measured by load cell

domains including both detonation tubes and ambient ows arethus required to describe the system dynamics faithfullyespeciallyin the near eld of the tube exit where the ow is intrinsicallymultidimensional To date only single-pulse operations have beentreated using two-dimensional analyses Multicycle simulationshave been limited to one-dimensionalmodels whose results are ap-parently questionable Most of the analyses developed so far haveconsideredonly detonationchamber dynamicsnot a complete PDEsystem

The current work attempts to establish a global analysis to de-termine the overall system performance and thermodynamic cy-cle ef ciency of airbreathing PDEs Figure 1 shows schemati-cally the con guration under consideration It includes a coaxialsupersonic inlet with mixed compression a multitube detonationchamber and a nozzle A rotary valve is placed in front of thecombustor entrance to distribute the air ow evenly into the indi-vidual tubes The following section describes the development ofan ideal PDE thermodynamic cycle analysis to assess the theoret-ical limit of system performance A modular approach is then uti-lized to analyze the engine dynamics numerically As a speci cexample an engine operating at a ight altitude of 93 km and afreestream Mach number of 21 is considered The effects of var-

ious operating parameters on the engine propulsion ef ciency arestudied systematically Finally the in uence of nozzle design isexamined

II Thermodynamic Cycle AnalysisAn ideal thermodynamic cycle analysis is presented in this sec-

tion to estimate the theoretical limit of the performance of an air-breathingPDE The work extendsthe approachof Heiserand Pratt36

for perfect gases with constant properties to accommodate prop-erty variations across the detonation wave front Figure 2 showsthe temperaturendashentropy diagram of an ideal PDE cycle The cor-responding Humphrey (constant-volumecombustion) and Brayton(constant-pressure combustion) cycles are included for compari-son The process from point 0 to point 1 is an adiabatic isentropiccompressionprocess in the course of which the ow temperature israisedfromits freestreamvalueT0 to thatat thecombustorentranceT1 The path from point 1 to point 2 corresponds to the detonationprocess Here the ZND model is adopted that is the dashed linefrom point 1 to point 1a corresponds to the entropy increase causedby shockcompressionand the solid line from point 1a to point 2 rep-resents subsequent heat release due to chemical reactions Point 2

WU MA AND YANG 559

Table 3 Variations of ow properties for stoichiometricH2air mixturea

s1a ndashs1 s2ndashs1Cycle p1a =p1 p2=p1 T1a=T1 T2=T1 kJkg cent K kJkg cent K deg2

Ideal PDE 1550 547 982 104 184 116Humphrey NA 796 NA 917 NA 189 117Brayton NA 100 NA 794 NA 228 118

aT1 D 300 K p1 D 1 atm and deg1 D 14

Fig 1 Supersonic airbreathing PDE

Fig 2 Temperaturendashentropy diagram of ideal PDE Humphrey andBrayton cycles

is known as the ChapmanndashJouguet (CndashJ) point where the chemicalsystem reaches an equilibriumstate and the ow relative to the deto-nation wave front is sonic For a given initial condition and reactantcomposition the detonation wave velocity and CndashJ properties canbe easilydeterminedby means of a chemicalequilibriumanalysis48

Table 3 summarizes the ow properties at various stations for a sto-ichiometric hydrogenair mixture initially at 1 atm and 300 K Notethat the ldquohumpedrdquo nature of the heat additionprocess from point 1ato point 2 re ects the well-known phenomenonof the existenceof amaximumtemperaturewhen heat is added to a steadysubsonic owThe process from point 2 to point 3 involves isentropic expansionwith the pressure of the burned gases decreasing to the freestreamcondition p3D p0 The cycle is closed by an imaginary constantstatic-pressureprocess in which the abundantheat is removed to thesurroundings from the exhaust ow

Following the conventional de nition the thermodynamic cycleef ciency acuteth is expressed as

acuteth D 1 iexcl qreject=q (1)

where q and qreject are the amount of heat added during the processfrom point 1a to point 2 and the amount of heat removed to the am-bient during the process from point 3 to point 0 respectivelyAfterextendingthe analysisof Heiser and Pratt36 to includepropertyvari-ations across the detonation wave front a closed-form expressionof acuteth is obtained as follows49

acutethPDE D 1 iexcl

(deg1 iexcl 1deg2 iexcl 1

sup3deg2

deg1

acute21

M2D

sup31 C deg1 M2

D

1 C deg2

acutedeg2 C 1=deg2

pound Atilde 1iexcl[deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)

(2)

where deg1 and deg2 are the speci c-heat ratios of the unburned andburned gases separated by the detonation wave front respectivelyThe Mach number of the detonation wave relative to the unburnedgas MD can be calculated using the following equation for xedheat addition

M2D D

microdeg 2

2 iexcl deg1

deg 21 iexcl deg1

C deg 22 iexcl 1

deg1 iexcl 1

QqAtilde

para

C

smicrodeg 2

2 iexcl deg1

deg 21 iexcl deg1

Cdeg 2

2 iexcl 1

deg1 iexcl 1

QqAtilde

para2

iexcldeg 2

2

deg 21

(3)

In comparison the thermodynamic ef ciencies of the Humphreyand Brayton cycles are

acutethHumphrey D 1 iexcl

(deg2

deg1

sup3deg1 iexcl 1

deg2 iexcl 1

acutedeg2 iexcl 1=deg2sup3

deg1QqAtilde

C 1

acute1=deg2

pound Atilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)(4)

acutethBrayton D 1 iexclraquomicrosup3

QqAtilde

C 1

acuteAtilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1deg1=deg2 iexcl 1

paraiquestQqfrac14

(5)

With the thermodynamic cycle ef ciency available the thrust Fcan be obtained by a control-volumeanalysis

F D Pma C Pm f u3 iexcl Pmau0 frac14 Pma

iexclpu2

0 C 2acutethq iexcl u0

cent(6)

where u0 is the freestream velocity Pma the cycle-averagedair mass ow rate delivered to the engine through the inlet and Pm f the cycle-averaged fuel mass ow rate Note that the preceding analysis isbased on the assumptions that every uid particle experiences thesame processes sequentially and that the effects of purging and by-pass air are ignored The fuel-based speci c impulse can then beobtained as follows

Isp DF

Pm f gD

pu2

0 C 2acutethq iexcl u0

f g(7)

Figures 3 and 4 show a typical result of cycle ef ciencyand speci c impulse for a stoichiometric hydrogenair system

Fig 3 Thermodyanmiccycle ef ciencies of idealPDE Humphrey andBraytoncycles as function of static temperature ratio Atilde for stoichiomet-ric H2 air system

560 WU MA AND YANG

Fig 4 Speci c impulses of ideal PDE Humphrey and Brayton cy-cles as function of static temperature ratio Atilde for stoichiometric H2 airsystem

The freestream velocity and temperature are u0 D 636 ms andT0 D 228 K respectively The nondimensional heat addition q is2247The system performanceincreaseswith increasingstatic tem-peratureratioThe PDE offers the bestperformanceamong the threecycles especiallywhen the static temperatureratio Atilde is smaller than3 This may be because for a given amount of heat addition theMach number of the detonation wave increases with decreasing T1

(or Atilde ) as indicated by Eq (3) The shock-compression effect be-comes more signi cant for a lower T1 leading to a higher increasein the temperature and pressure of the unburned gases before com-bustion The Isp of an ideal PDE reaches 5263 s when T1 D 428 Kthat is Atilde D 1877

III System Performance AnalysisThis section deals with the developmentof a system performance

analysis for airbreathing PDEs as shown schematically in Fig 1The study is based on a modular approach Each module repre-sents a speci c component of the engine and its dynamic behav-ior is formulated using complete conservationequationsThe workinvolves the following three components 1) supersonic inlet dy-namics 2) detonation chamber dynamics and system performanceand 3) effect of nozzle con guration The effects of fuel supply airdistribution and inlet isolator are ignored for simplicity They canhowever be straightforwardlyincluded as submodels in the overallengine performance analysis

A Supersonic Inlet DynamicsThe inlet and its interaction with combustor represent a crucial

aspect in the development of any airbreathing engine includingPDEs The inlet is designed to capture and supply stable air ow at arate demanded by the combustor and to maintain high pressure re-covery and stability margin at various engine operating conditionsThe overallvehicleperformancedependsgreatlyon the energy leveland ow quality of the incoming air A small loss in inlet ef ciencymay translate to a substantial penalty in engine thrust Moreoverany change in the inlet ow structure may modify the downstreamcombustioncharacteristicsand subsequentlylead to undesirablebe-haviors such as ame blowoff and ashback Thus matching inletbehavior to engine requirements is of fundamental importance todesigners50

In addition to its primary function of supplyingair an inlet has adetermining in uence on the dynamics of the entire system throughits intrinsicunsteadinessand interactionswith thecombustioncham-berTypicallypressurewaves areproducedin thecombustioncham-ber and propagate upstream to interact with the inlet ow througha manifold where mixing of air and fuel occurs The resultant owoscillations in the inlet diffuser then either propagate downstreamin the form of acoustic waves or are convected downstream withthe mean ow in the form of vorticity and entropy waves andfurther reinforce the unsteady motions in the combustor A feed-back loop is thus established between the inlet and combustor

Fig 5 Supersonic inlet with mixed compression

Fig 6 Mach number contours with different back pressures understeady-state conditions

The situation is much more complicated in a supersonic PDE dueto the shock-waveboundary-layer interaction and shockacoustic-wave interaction51 [also Oh J Y Ma F H Hsieh S Y andYang V ldquoInteractionsBetween Shock Waves and Acoustic Wavesin a Supersonic Inlet Diffuserrdquo (to be submitted for publication)]The engine dynamics exhibits features that are qualitatively differ-ent from those predicted by treating the inlet and combustor as twoseparate entities

The inlet analysis is based on the axisymmetric Favre-averagedconservationequationsTurbulenceclosure is achievedusing a two-layer model of Rodi calibrated for supersonic ows with shockwaves52 The governingequationsare solved numerically by meansof a density-based nite volume methodology The temporal dis-cretization is obtained using a four-stage RungendashKutta integrationmethod and the spatial discretization employs an upwind total-variation-diminishing scheme developed by Harten53 Speci c de-tails of the numerical algorithm can be found in Ref 54

Figure 5 shows the con guration treated in the present study amixed-compression inlet optimized for a ight altitude of 93 kmand a freestream Mach number of 21 (Ref 50) Figure 6 presentsthe Mach-numbercontoursat two differentbackpressures(pb D 21and 22 atm) which are carefully chosen such that the engine oper-ates at a supercritical condition to provide a suf cient shock stabil-ity margin Under these conditions the two leading conical shocksgenerated by the double-cone centerbody compress the air ow ex-ternallymerge slightlyabove thecowl lip and forma strongobliqueshock which extends into the external-ow region In addition ashockstemming from the cowl inner surface continuesdownstreamhitting and re ecting from both the cowl and centerbodywalls and nally leading to a terminal normal shock The ow undergoes asequence of compression and expansion waves and becomes sub-sonic after passingthroughthe normal shock located in thedivergentsection of the diffuser The inlet recovers a high percentage of thefreestream total pressure by decelerating the air ow through theshock train The pressure recovery coef cients for the two cases are84 and 88 respectively and the Mach numbers immediately infront of the terminal shocks are 142 and 132 respectively

The response of the inlet shock system to downstream distur-bances has also been studied by imposing periodic pressureoscilla-tions at theexit planeA wide rangeof uctuationfrequencyandam-plitude were considered Important phenomena of concern includetemporal and spatial variationsof mass ow rate pressure recoveryand ow distributionas well as shock displacementIn general theacoustic response of the inlet ow increases with increasing ampli-tude of the imposed oscillation but decreases with frequencyAlsoincluded as part of the result is the acoustic impedance function atthe inlet exit a parameter that can be effectively used to character-ize the inletcombustor couplingA more detaileddiscussionon thissubject will be given in subsequent work

WU MA AND YANG 561

B Detonation Chamber DynamicsThe detonationchamberdynamicsis formulatedbasedon thecon-

servation laws for a multicomponentchemically reacting system intwo-dimensionalcoordinatesDiffusive transport is neglected in thecurrent study because of its minor role in determining detonationdynamics and system performance The governing equations andtheir associatedboundary conditionsare solved using a recently de-velopedspacendashtime conservation-elementsolution-elementmethodthat circumvents the de ciencies of existing numerical methodsfor treating detonation waves and shock discontinuities55iexcl59 Theresultant computer code is further parallelized using the messagepassing interface library with domain decomposition to improve itsef ciency

Both simple global and detailed chemical kinetics models areutilized4959 The former involvesonly one progressvariable to char-acterize the chemical reaction rate and assumes constant proper-ties Because of its computational ef ciency and reasonable accu-racy in determining the PDE propulsive performance the model isimplemented in the present work The associated thermochemicalparameters are optimized by comparing the calculated detonationwave properties with those from the NASA chemical equilibriumanalysis48 The relative errors are less than 5 in terms of the det-onation velocity and the CndashJ pressure and temperature

As part of the model validation effort a series of single-pulsecalculations were conducted for a straight tube of 60 cm in lengthinitially lled with a stoichiometric mixture of hydrogen and air atpreconditionedpressure pi and temperature Ti A driver-gas regionspanning 02 mm near the head end with a temperature of 2000 Kand a pressure of 30 atm was employed to initiate the detonationwave directly Four different numerical grids with the sizes of 0201 005 and 0025 mm were used to check the solution accuracyin terms of grid independenceAll of the calculatedpressurepro lescollapsed onto a single curve with the CndashJ propertiesmatching theanalytical values exactly As a result the 02-mm grid was chosenfor the entire study to alleviate the computational burden For asingle-pulse operation the head-end pressure remains at a plateauvalue pp that is p3 in Refs 37 and 46 for certain period soonafter the detonation initiation and then decays gradually to a levellower than the ambient state3746 The impulse can be determinedby integrating temporally the force exerted on the head end fromt D 0 to the instant when the head-endpressure reaches the ambientvalue The contribution to the impulse from the ignition source isestimated to be less than 05 Figure 7 shows the impulse per unitcross-sectionalarea as a function of the plateau pressure pp and thedetonation residence time iquestD (de ned as the tube length L dividedby the detonation wave velocity u D ie iquestD acute L=u D Results canbe correlated well in the following form

I=A D 41pp iexcl pi iquestD (8)

This expression is quite similar to those obtained from the semi-analytical analysis of Wintenberger et al37 and the experimentalwork of Falempin et al14 The constants of proportionality differslightly in the various studies for example 413 in Ref 37 for the

Fig 7 Generalized impulse curve for single-pulse detonation instraight tube with stoichiometric H2air mixture

current conditions suggesting dependenceon the details of experi-mentalprocedureandoperatingconditionsThe generalizedformulaproposed by Kailasanath60 based on his numerical simulations forhydrogenair ethyleneoxygen and propaneoxygen mixtures has alarger constant of proportionality of 465 One factor contributingto this disparity may be differencesin ignition source The width ofthe detonation initiation region is 20 mm in Ref 60 as opposed to02 mm in the present simulationsNonethelessthe precedingpara-metric study demonstrates the capacity and delity of the presentapproach for the PDE performance analysis

The validatedanalysis is then employed to study the performanceof airbreathing PDEs As a speci c example the ight conditioninvolving an altitude of 93 km and a Mach number of 21 is consid-ered The freestream static pressure and temperature are 029 atmand 228K respectivelycorrespondingto a totalpressureof 265atmand a total temperature of 428 K The total pressure and tempera-ture at the combustor entranceobtained from the inlet ow analysisdescribed in the preceding section are 212 atm and 428 K respec-tively The detonation tube is 60 cm long and has a diameter of16 cm which is similar to the dimensions of contemporary ramjetcombustors for air defense applicationsAs a rst approach only asingle straight tube is considered to provide direct insight into thechamber dynamics without complications arising from the nozzleThe interactions among the tubes are also ignored The valve atthe tube entrance is assumed to be either fully open or fully closedThe operationsequenceis thuscontrolledby three time periodsthevalveclose-upperiodiquestclose duringwhich the valve is closedand thetube undergoesdetonationinitiationand propagationand blowdownprocesses the purging period iquestpurge during which a small amountof cold air is injected to prevent preignition of fresh reactants andthe re lling period iquestre ll during which the combustible mixture isdelivered to the chamber The sum of these three periods is equal tothe operationcycleperiodiquestcycle that is iquestcycle D iquestclose C iquestpurge C iquestre llThe purgingperiodshouldbeminimizedto reduceperformancelossA small value of 01 ms is used throughoutall of the calculationsinthe present study

The boundaryconditionsat thehead endof thedetonationtubearespeci ed according to the engine operationDuring the purgingandre lling stages the total pressureand total temperatureare obtainedfrom the inlet analysis The axial velocity is extrapolated from theinterior points and the reactant mass fraction is treated as an inputparameterWhen the valve is closed the head end is simply modeledas a rigid wall

A series of analysesare conductedover a wide range of operationparameters The baseline case has iquestcycle of 3 ms and iquestclose of 24 msThe tube is initially lled with a stoichiometrichydrogenairmixtureat ambient pressure and temperature It takes about ve cycles toreach steady cyclic operation Figure 8 shows the xndasht diagram forthe rst cycle of operation obtained by tracing the characteristiclines of the ow eld along the centerline of the tube The timehistories of the ow properties at the head end are also presentedThe detonation wave is directly initiated by a hot driver gas andpropagates downstream at the CndashJ velocity toward the unburnedmixture(region1) It then inducesTaylorexpansionwaves (region2)to satisfy the stationaryconditionat the head end causinga uniformregion (region 3) with constant-ow properties in the upstream

The detonationwave reaches the reactantair interfaceat the tubeexit at t D 0305 ms (point A) which deviates slightly from thefollowing analytical prediction by 06 due to the effect of theexternally imposed ignition source

iquestD D L=u D D 06 m=1956 ms D 0307 ms (9)

The wave then degenerates to a nonreactive shock that is theprimary shock wave proceeding farther downstream into the ex-ternal region followed by a contact surface separating the ambientair and combustion products A sonic region is gradually formednear the tube exit due to the local ow expansion as evidenced bythe clustered characteristic lines in the xndasht diagram Downstreamof the sonic region the ow is expanded to become supersonic and nally leads to the formationof a secondaryshock to match with thesubsonic ow behind the primary shockThis secondaryshockwave

562 WU MA AND YANG

Fig 8 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms 1 = unburned region 2 = Taylor expansion waves 3 = stationary region 4 = non-simple wave regionand 5 = simple wave region

Fig 9 Snapshots of pressure density and their gradients elds at t = 07 ms

moves farther downstreammeeting with expansionwaves originat-ing from the primary shockwaveThese complicated ow structurescan be also observed in Fig 9 which shows the instantaneouspres-sure and density and their gradient elds at t D 07 ms Many salientfeatures are clearly shown including the expansion fans vorticesand rolled-upslip lines that are developedas the shockdiffractsoverthe edge of the tube exit

As the detonation wave catches the reactantair interface and theresultant primary shock wave travels outside the tube a series ofexpansionwaves is generatedwhich propagate upstream resulting

in a nonsimple wave region (region 4) when interactingwith the in-coming Taylor waves A simple wave region (region 5) is recoveredafter passing through the Taylor waves The rst expansion wavereaches the head end at t D 0935 ms (point B) which can be deter-mined by consideringthe interactionbetween the expansionand theTaylor waves and the sound speed in region 3 A similarity solutionhas been derivedby Wintenbergeret al37 to predict this time instantanalytically

t D L=uD C regL=c3 (10)

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

558 WU MA AND YANG

Table 2 Survey of multicycle experimental investigation of PDEs

Propellants Reference Con gurations Isp s and Thrust N

H2 O2 Sterling et al7 Single tube f D 33 Hz id D 22 cm L D 152 cm NA1 atm (1996) fully lled H2 and O2 adopted as buffer gases spark ignitor298 K Stuessy and Single tube f D 10ndash12 Hz fully lled air used as buffer gas NAAacute D 10 Wilson21 cylindrical tube with idD 76 cm L D 533 cm

(1996) annular tube with id D 25 cm odD 76 cm L D 533 cmannular tube with diverging conical nozzle

H2 air Aarnio et al22 Single tube f D 5 Hz id D 51 cm L D 1219 cm Aacute D 10 Fuel-based1 atm (1996) partially lled air used as buffer gas 1333 sa

298 K spark ignitor of 17 J 1116 sb

Hinkey et al23 Twin tube f D 10 Hz common air inlet manifold NA(1997) id D 51 cm L D 914 cm Aacute D 07ndash13

predetonator lled by H2 O2 mixtures spark ignitor of 15 JSchauer et al24 Multitube (1 2 and 4) f D 05ndash100 Hz NA

(1999) id D 51 cm L D 914 cm and id D 89 cm L D 914 cmair used as buffer gasspark ignitor with DDT enhanced by Shchelkin spiral

Schauer et al25 f D 14ndash40 Hz L D 914 cm id D 51 cm Aacute D 04ndash285 Fuel-based for Aacute D 10 f D 16 Hz(2000) spark ignitor with DDT enhanced by Shchelkin spiral and 35 s ignition delay

air used as buffer gas with 50 tube lling ratio 7100 sa (30 lling length)various ignition delays and tube lling fractions 4200 sa (90 lling length)

McManus et al26 Single tube f D 10ndash35 Hz conical converging nozzle F D 1112 Nb ( f D 10 Hz)(2001) id D 476 cm L D 254 cm Aacute D 06 F D 5338 Nb ( f D 35 Hz)

spark ignitor of 002 J at tube exitFrankey et al27 Single tube f D 11ndash21 Hz converging nozzle F D 222 Na f D 21 Hz)

(2002) id D 508 cm L D 18288 cm Aacute D 10spark ignitor with DDT enhanced by Shchelkin spiralair used as buffer gas

C2H4O2 Sterling et al7 f D 100 Hz L D 508 cm id D 22 cm 13 tube lled NA1 atm 298 K (1996) spark ignitor with DDT enhanced by unspeci ed deviceAacute D 10 Falempin et al14 Single tube f D 80 Hz id D 50 cm L D 50ndash426 cm NA

(2001) ignition source not mentionedC2H4air Broda et al15 f D 8ndash10 Hz L D 1829 cm id D 34 cm Aacute D 11 NA

1 atm (1999) spark ignitor of 4ndash8 J DDT enhanced by obstacles298 K Watts et al16 Single tube f D 10 Hz Aacute D 12 NA

(2000) spark plug of 25 J DDT enhanced by obstaclesbuffer air injected for 5ndash10 ms in each cycle

Brophy et al28 Single tube f D 80 Hz id D 40 cm L D 25 cm Aacute D 10ndash18 Total impulse(2002) spark ignitor with unknown energy 048 N cent sa (Aacute D 10)

Shimo et al29 f D 15 Hz id D 51 cm L D 822 cm Aacute D 09 60 lling ratio NA(2002) spark ignitor with DDT enhanced by Shchelkin spirals

JP-10O2 g Brophy et al30 f D 5 Hz L D 152 305 762 cm id D 38 cm Aacute D 07ndash17 NAp and T unknown (1998) ignitor of 14 J ignition delay 40 50 and 70 ms

1 atm Brophy and Single tube f D 10 Hz id D 381 cm L D 29 cm Aacute D 09ndash12 NA292 K Netzer31 (1999) ignitor of 05 J at locations x D 0ndash2D from head end

1 atm Brophy et al28 Single tube f D 30 Hz id D 40 cm L D 25 cm Aacute D 10ndash18 Total impulse280 K (2002) spark ignitor with unknown energy 059 N cent sa (Aacute D 10)

C3H8O2 Farinaccio et al32 f D 10 15 20 25 Hz L D 40 cm id D 81 cm Aacute D 10 Mixture-based 71 sb f D 15 Hz)p and T unknown (2002) fully or partially lled N2 purged for 13-cycle period 71 sb f D 20 Hz fully lled)

aBased on time history of pressure at thrust wall bBased on time history of force measured by load cell

domains including both detonation tubes and ambient ows arethus required to describe the system dynamics faithfullyespeciallyin the near eld of the tube exit where the ow is intrinsicallymultidimensional To date only single-pulse operations have beentreated using two-dimensional analyses Multicycle simulationshave been limited to one-dimensionalmodels whose results are ap-parently questionable Most of the analyses developed so far haveconsideredonly detonationchamber dynamicsnot a complete PDEsystem

The current work attempts to establish a global analysis to de-termine the overall system performance and thermodynamic cy-cle ef ciency of airbreathing PDEs Figure 1 shows schemati-cally the con guration under consideration It includes a coaxialsupersonic inlet with mixed compression a multitube detonationchamber and a nozzle A rotary valve is placed in front of thecombustor entrance to distribute the air ow evenly into the indi-vidual tubes The following section describes the development ofan ideal PDE thermodynamic cycle analysis to assess the theoret-ical limit of system performance A modular approach is then uti-lized to analyze the engine dynamics numerically As a speci cexample an engine operating at a ight altitude of 93 km and afreestream Mach number of 21 is considered The effects of var-

ious operating parameters on the engine propulsion ef ciency arestudied systematically Finally the in uence of nozzle design isexamined

II Thermodynamic Cycle AnalysisAn ideal thermodynamic cycle analysis is presented in this sec-

tion to estimate the theoretical limit of the performance of an air-breathingPDE The work extendsthe approachof Heiserand Pratt36

for perfect gases with constant properties to accommodate prop-erty variations across the detonation wave front Figure 2 showsthe temperaturendashentropy diagram of an ideal PDE cycle The cor-responding Humphrey (constant-volumecombustion) and Brayton(constant-pressure combustion) cycles are included for compari-son The process from point 0 to point 1 is an adiabatic isentropiccompressionprocess in the course of which the ow temperature israisedfromits freestreamvalueT0 to thatat thecombustorentranceT1 The path from point 1 to point 2 corresponds to the detonationprocess Here the ZND model is adopted that is the dashed linefrom point 1 to point 1a corresponds to the entropy increase causedby shockcompressionand the solid line from point 1a to point 2 rep-resents subsequent heat release due to chemical reactions Point 2

WU MA AND YANG 559

Table 3 Variations of ow properties for stoichiometricH2air mixturea

s1a ndashs1 s2ndashs1Cycle p1a =p1 p2=p1 T1a=T1 T2=T1 kJkg cent K kJkg cent K deg2

Ideal PDE 1550 547 982 104 184 116Humphrey NA 796 NA 917 NA 189 117Brayton NA 100 NA 794 NA 228 118

aT1 D 300 K p1 D 1 atm and deg1 D 14

Fig 1 Supersonic airbreathing PDE

Fig 2 Temperaturendashentropy diagram of ideal PDE Humphrey andBrayton cycles

is known as the ChapmanndashJouguet (CndashJ) point where the chemicalsystem reaches an equilibriumstate and the ow relative to the deto-nation wave front is sonic For a given initial condition and reactantcomposition the detonation wave velocity and CndashJ properties canbe easilydeterminedby means of a chemicalequilibriumanalysis48

Table 3 summarizes the ow properties at various stations for a sto-ichiometric hydrogenair mixture initially at 1 atm and 300 K Notethat the ldquohumpedrdquo nature of the heat additionprocess from point 1ato point 2 re ects the well-known phenomenonof the existenceof amaximumtemperaturewhen heat is added to a steadysubsonic owThe process from point 2 to point 3 involves isentropic expansionwith the pressure of the burned gases decreasing to the freestreamcondition p3D p0 The cycle is closed by an imaginary constantstatic-pressureprocess in which the abundantheat is removed to thesurroundings from the exhaust ow

Following the conventional de nition the thermodynamic cycleef ciency acuteth is expressed as

acuteth D 1 iexcl qreject=q (1)

where q and qreject are the amount of heat added during the processfrom point 1a to point 2 and the amount of heat removed to the am-bient during the process from point 3 to point 0 respectivelyAfterextendingthe analysisof Heiser and Pratt36 to includepropertyvari-ations across the detonation wave front a closed-form expressionof acuteth is obtained as follows49

acutethPDE D 1 iexcl

(deg1 iexcl 1deg2 iexcl 1

sup3deg2

deg1

acute21

M2D

sup31 C deg1 M2

D

1 C deg2

acutedeg2 C 1=deg2

pound Atilde 1iexcl[deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)

(2)

where deg1 and deg2 are the speci c-heat ratios of the unburned andburned gases separated by the detonation wave front respectivelyThe Mach number of the detonation wave relative to the unburnedgas MD can be calculated using the following equation for xedheat addition

M2D D

microdeg 2

2 iexcl deg1

deg 21 iexcl deg1

C deg 22 iexcl 1

deg1 iexcl 1

QqAtilde

para

C

smicrodeg 2

2 iexcl deg1

deg 21 iexcl deg1

Cdeg 2

2 iexcl 1

deg1 iexcl 1

QqAtilde

para2

iexcldeg 2

2

deg 21

(3)

In comparison the thermodynamic ef ciencies of the Humphreyand Brayton cycles are

acutethHumphrey D 1 iexcl

(deg2

deg1

sup3deg1 iexcl 1

deg2 iexcl 1

acutedeg2 iexcl 1=deg2sup3

deg1QqAtilde

C 1

acute1=deg2

pound Atilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)(4)

acutethBrayton D 1 iexclraquomicrosup3

QqAtilde

C 1

acuteAtilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1deg1=deg2 iexcl 1

paraiquestQqfrac14

(5)

With the thermodynamic cycle ef ciency available the thrust Fcan be obtained by a control-volumeanalysis

F D Pma C Pm f u3 iexcl Pmau0 frac14 Pma

iexclpu2

0 C 2acutethq iexcl u0

cent(6)

where u0 is the freestream velocity Pma the cycle-averagedair mass ow rate delivered to the engine through the inlet and Pm f the cycle-averaged fuel mass ow rate Note that the preceding analysis isbased on the assumptions that every uid particle experiences thesame processes sequentially and that the effects of purging and by-pass air are ignored The fuel-based speci c impulse can then beobtained as follows

Isp DF

Pm f gD

pu2

0 C 2acutethq iexcl u0

f g(7)

Figures 3 and 4 show a typical result of cycle ef ciencyand speci c impulse for a stoichiometric hydrogenair system

Fig 3 Thermodyanmiccycle ef ciencies of idealPDE Humphrey andBraytoncycles as function of static temperature ratio Atilde for stoichiomet-ric H2 air system

560 WU MA AND YANG

Fig 4 Speci c impulses of ideal PDE Humphrey and Brayton cy-cles as function of static temperature ratio Atilde for stoichiometric H2 airsystem

The freestream velocity and temperature are u0 D 636 ms andT0 D 228 K respectively The nondimensional heat addition q is2247The system performanceincreaseswith increasingstatic tem-peratureratioThe PDE offers the bestperformanceamong the threecycles especiallywhen the static temperatureratio Atilde is smaller than3 This may be because for a given amount of heat addition theMach number of the detonation wave increases with decreasing T1

(or Atilde ) as indicated by Eq (3) The shock-compression effect be-comes more signi cant for a lower T1 leading to a higher increasein the temperature and pressure of the unburned gases before com-bustion The Isp of an ideal PDE reaches 5263 s when T1 D 428 Kthat is Atilde D 1877

III System Performance AnalysisThis section deals with the developmentof a system performance

analysis for airbreathing PDEs as shown schematically in Fig 1The study is based on a modular approach Each module repre-sents a speci c component of the engine and its dynamic behav-ior is formulated using complete conservationequationsThe workinvolves the following three components 1) supersonic inlet dy-namics 2) detonation chamber dynamics and system performanceand 3) effect of nozzle con guration The effects of fuel supply airdistribution and inlet isolator are ignored for simplicity They canhowever be straightforwardlyincluded as submodels in the overallengine performance analysis

A Supersonic Inlet DynamicsThe inlet and its interaction with combustor represent a crucial

aspect in the development of any airbreathing engine includingPDEs The inlet is designed to capture and supply stable air ow at arate demanded by the combustor and to maintain high pressure re-covery and stability margin at various engine operating conditionsThe overallvehicleperformancedependsgreatlyon the energy leveland ow quality of the incoming air A small loss in inlet ef ciencymay translate to a substantial penalty in engine thrust Moreoverany change in the inlet ow structure may modify the downstreamcombustioncharacteristicsand subsequentlylead to undesirablebe-haviors such as ame blowoff and ashback Thus matching inletbehavior to engine requirements is of fundamental importance todesigners50

In addition to its primary function of supplyingair an inlet has adetermining in uence on the dynamics of the entire system throughits intrinsicunsteadinessand interactionswith thecombustioncham-berTypicallypressurewaves areproducedin thecombustioncham-ber and propagate upstream to interact with the inlet ow througha manifold where mixing of air and fuel occurs The resultant owoscillations in the inlet diffuser then either propagate downstreamin the form of acoustic waves or are convected downstream withthe mean ow in the form of vorticity and entropy waves andfurther reinforce the unsteady motions in the combustor A feed-back loop is thus established between the inlet and combustor

Fig 5 Supersonic inlet with mixed compression

Fig 6 Mach number contours with different back pressures understeady-state conditions

The situation is much more complicated in a supersonic PDE dueto the shock-waveboundary-layer interaction and shockacoustic-wave interaction51 [also Oh J Y Ma F H Hsieh S Y andYang V ldquoInteractionsBetween Shock Waves and Acoustic Wavesin a Supersonic Inlet Diffuserrdquo (to be submitted for publication)]The engine dynamics exhibits features that are qualitatively differ-ent from those predicted by treating the inlet and combustor as twoseparate entities

The inlet analysis is based on the axisymmetric Favre-averagedconservationequationsTurbulenceclosure is achievedusing a two-layer model of Rodi calibrated for supersonic ows with shockwaves52 The governingequationsare solved numerically by meansof a density-based nite volume methodology The temporal dis-cretization is obtained using a four-stage RungendashKutta integrationmethod and the spatial discretization employs an upwind total-variation-diminishing scheme developed by Harten53 Speci c de-tails of the numerical algorithm can be found in Ref 54

Figure 5 shows the con guration treated in the present study amixed-compression inlet optimized for a ight altitude of 93 kmand a freestream Mach number of 21 (Ref 50) Figure 6 presentsthe Mach-numbercontoursat two differentbackpressures(pb D 21and 22 atm) which are carefully chosen such that the engine oper-ates at a supercritical condition to provide a suf cient shock stabil-ity margin Under these conditions the two leading conical shocksgenerated by the double-cone centerbody compress the air ow ex-ternallymerge slightlyabove thecowl lip and forma strongobliqueshock which extends into the external-ow region In addition ashockstemming from the cowl inner surface continuesdownstreamhitting and re ecting from both the cowl and centerbodywalls and nally leading to a terminal normal shock The ow undergoes asequence of compression and expansion waves and becomes sub-sonic after passingthroughthe normal shock located in thedivergentsection of the diffuser The inlet recovers a high percentage of thefreestream total pressure by decelerating the air ow through theshock train The pressure recovery coef cients for the two cases are84 and 88 respectively and the Mach numbers immediately infront of the terminal shocks are 142 and 132 respectively

The response of the inlet shock system to downstream distur-bances has also been studied by imposing periodic pressureoscilla-tions at theexit planeA wide rangeof uctuationfrequencyandam-plitude were considered Important phenomena of concern includetemporal and spatial variationsof mass ow rate pressure recoveryand ow distributionas well as shock displacementIn general theacoustic response of the inlet ow increases with increasing ampli-tude of the imposed oscillation but decreases with frequencyAlsoincluded as part of the result is the acoustic impedance function atthe inlet exit a parameter that can be effectively used to character-ize the inletcombustor couplingA more detaileddiscussionon thissubject will be given in subsequent work

WU MA AND YANG 561

B Detonation Chamber DynamicsThe detonationchamberdynamicsis formulatedbasedon thecon-

servation laws for a multicomponentchemically reacting system intwo-dimensionalcoordinatesDiffusive transport is neglected in thecurrent study because of its minor role in determining detonationdynamics and system performance The governing equations andtheir associatedboundary conditionsare solved using a recently de-velopedspacendashtime conservation-elementsolution-elementmethodthat circumvents the de ciencies of existing numerical methodsfor treating detonation waves and shock discontinuities55iexcl59 Theresultant computer code is further parallelized using the messagepassing interface library with domain decomposition to improve itsef ciency

Both simple global and detailed chemical kinetics models areutilized4959 The former involvesonly one progressvariable to char-acterize the chemical reaction rate and assumes constant proper-ties Because of its computational ef ciency and reasonable accu-racy in determining the PDE propulsive performance the model isimplemented in the present work The associated thermochemicalparameters are optimized by comparing the calculated detonationwave properties with those from the NASA chemical equilibriumanalysis48 The relative errors are less than 5 in terms of the det-onation velocity and the CndashJ pressure and temperature

As part of the model validation effort a series of single-pulsecalculations were conducted for a straight tube of 60 cm in lengthinitially lled with a stoichiometric mixture of hydrogen and air atpreconditionedpressure pi and temperature Ti A driver-gas regionspanning 02 mm near the head end with a temperature of 2000 Kand a pressure of 30 atm was employed to initiate the detonationwave directly Four different numerical grids with the sizes of 0201 005 and 0025 mm were used to check the solution accuracyin terms of grid independenceAll of the calculatedpressurepro lescollapsed onto a single curve with the CndashJ propertiesmatching theanalytical values exactly As a result the 02-mm grid was chosenfor the entire study to alleviate the computational burden For asingle-pulse operation the head-end pressure remains at a plateauvalue pp that is p3 in Refs 37 and 46 for certain period soonafter the detonation initiation and then decays gradually to a levellower than the ambient state3746 The impulse can be determinedby integrating temporally the force exerted on the head end fromt D 0 to the instant when the head-endpressure reaches the ambientvalue The contribution to the impulse from the ignition source isestimated to be less than 05 Figure 7 shows the impulse per unitcross-sectionalarea as a function of the plateau pressure pp and thedetonation residence time iquestD (de ned as the tube length L dividedby the detonation wave velocity u D ie iquestD acute L=u D Results canbe correlated well in the following form

I=A D 41pp iexcl pi iquestD (8)

This expression is quite similar to those obtained from the semi-analytical analysis of Wintenberger et al37 and the experimentalwork of Falempin et al14 The constants of proportionality differslightly in the various studies for example 413 in Ref 37 for the

Fig 7 Generalized impulse curve for single-pulse detonation instraight tube with stoichiometric H2air mixture

current conditions suggesting dependenceon the details of experi-mentalprocedureandoperatingconditionsThe generalizedformulaproposed by Kailasanath60 based on his numerical simulations forhydrogenair ethyleneoxygen and propaneoxygen mixtures has alarger constant of proportionality of 465 One factor contributingto this disparity may be differencesin ignition source The width ofthe detonation initiation region is 20 mm in Ref 60 as opposed to02 mm in the present simulationsNonethelessthe precedingpara-metric study demonstrates the capacity and delity of the presentapproach for the PDE performance analysis

The validatedanalysis is then employed to study the performanceof airbreathing PDEs As a speci c example the ight conditioninvolving an altitude of 93 km and a Mach number of 21 is consid-ered The freestream static pressure and temperature are 029 atmand 228K respectivelycorrespondingto a totalpressureof 265atmand a total temperature of 428 K The total pressure and tempera-ture at the combustor entranceobtained from the inlet ow analysisdescribed in the preceding section are 212 atm and 428 K respec-tively The detonation tube is 60 cm long and has a diameter of16 cm which is similar to the dimensions of contemporary ramjetcombustors for air defense applicationsAs a rst approach only asingle straight tube is considered to provide direct insight into thechamber dynamics without complications arising from the nozzleThe interactions among the tubes are also ignored The valve atthe tube entrance is assumed to be either fully open or fully closedThe operationsequenceis thuscontrolledby three time periodsthevalveclose-upperiodiquestclose duringwhich the valve is closedand thetube undergoesdetonationinitiationand propagationand blowdownprocesses the purging period iquestpurge during which a small amountof cold air is injected to prevent preignition of fresh reactants andthe re lling period iquestre ll during which the combustible mixture isdelivered to the chamber The sum of these three periods is equal tothe operationcycleperiodiquestcycle that is iquestcycle D iquestclose C iquestpurge C iquestre llThe purgingperiodshouldbeminimizedto reduceperformancelossA small value of 01 ms is used throughoutall of the calculationsinthe present study

The boundaryconditionsat thehead endof thedetonationtubearespeci ed according to the engine operationDuring the purgingandre lling stages the total pressureand total temperatureare obtainedfrom the inlet analysis The axial velocity is extrapolated from theinterior points and the reactant mass fraction is treated as an inputparameterWhen the valve is closed the head end is simply modeledas a rigid wall

A series of analysesare conductedover a wide range of operationparameters The baseline case has iquestcycle of 3 ms and iquestclose of 24 msThe tube is initially lled with a stoichiometrichydrogenairmixtureat ambient pressure and temperature It takes about ve cycles toreach steady cyclic operation Figure 8 shows the xndasht diagram forthe rst cycle of operation obtained by tracing the characteristiclines of the ow eld along the centerline of the tube The timehistories of the ow properties at the head end are also presentedThe detonation wave is directly initiated by a hot driver gas andpropagates downstream at the CndashJ velocity toward the unburnedmixture(region1) It then inducesTaylorexpansionwaves (region2)to satisfy the stationaryconditionat the head end causinga uniformregion (region 3) with constant-ow properties in the upstream

The detonationwave reaches the reactantair interfaceat the tubeexit at t D 0305 ms (point A) which deviates slightly from thefollowing analytical prediction by 06 due to the effect of theexternally imposed ignition source

iquestD D L=u D D 06 m=1956 ms D 0307 ms (9)

The wave then degenerates to a nonreactive shock that is theprimary shock wave proceeding farther downstream into the ex-ternal region followed by a contact surface separating the ambientair and combustion products A sonic region is gradually formednear the tube exit due to the local ow expansion as evidenced bythe clustered characteristic lines in the xndasht diagram Downstreamof the sonic region the ow is expanded to become supersonic and nally leads to the formationof a secondaryshock to match with thesubsonic ow behind the primary shockThis secondaryshockwave

562 WU MA AND YANG

Fig 8 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms 1 = unburned region 2 = Taylor expansion waves 3 = stationary region 4 = non-simple wave regionand 5 = simple wave region

Fig 9 Snapshots of pressure density and their gradients elds at t = 07 ms

moves farther downstreammeeting with expansionwaves originat-ing from the primary shockwaveThese complicated ow structurescan be also observed in Fig 9 which shows the instantaneouspres-sure and density and their gradient elds at t D 07 ms Many salientfeatures are clearly shown including the expansion fans vorticesand rolled-upslip lines that are developedas the shockdiffractsoverthe edge of the tube exit

As the detonation wave catches the reactantair interface and theresultant primary shock wave travels outside the tube a series ofexpansionwaves is generatedwhich propagate upstream resulting

in a nonsimple wave region (region 4) when interactingwith the in-coming Taylor waves A simple wave region (region 5) is recoveredafter passing through the Taylor waves The rst expansion wavereaches the head end at t D 0935 ms (point B) which can be deter-mined by consideringthe interactionbetween the expansionand theTaylor waves and the sound speed in region 3 A similarity solutionhas been derivedby Wintenbergeret al37 to predict this time instantanalytically

t D L=uD C regL=c3 (10)

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

WU MA AND YANG 559

Table 3 Variations of ow properties for stoichiometricH2air mixturea

s1a ndashs1 s2ndashs1Cycle p1a =p1 p2=p1 T1a=T1 T2=T1 kJkg cent K kJkg cent K deg2

Ideal PDE 1550 547 982 104 184 116Humphrey NA 796 NA 917 NA 189 117Brayton NA 100 NA 794 NA 228 118

aT1 D 300 K p1 D 1 atm and deg1 D 14

Fig 1 Supersonic airbreathing PDE

Fig 2 Temperaturendashentropy diagram of ideal PDE Humphrey andBrayton cycles

is known as the ChapmanndashJouguet (CndashJ) point where the chemicalsystem reaches an equilibriumstate and the ow relative to the deto-nation wave front is sonic For a given initial condition and reactantcomposition the detonation wave velocity and CndashJ properties canbe easilydeterminedby means of a chemicalequilibriumanalysis48

Table 3 summarizes the ow properties at various stations for a sto-ichiometric hydrogenair mixture initially at 1 atm and 300 K Notethat the ldquohumpedrdquo nature of the heat additionprocess from point 1ato point 2 re ects the well-known phenomenonof the existenceof amaximumtemperaturewhen heat is added to a steadysubsonic owThe process from point 2 to point 3 involves isentropic expansionwith the pressure of the burned gases decreasing to the freestreamcondition p3D p0 The cycle is closed by an imaginary constantstatic-pressureprocess in which the abundantheat is removed to thesurroundings from the exhaust ow

Following the conventional de nition the thermodynamic cycleef ciency acuteth is expressed as

acuteth D 1 iexcl qreject=q (1)

where q and qreject are the amount of heat added during the processfrom point 1a to point 2 and the amount of heat removed to the am-bient during the process from point 3 to point 0 respectivelyAfterextendingthe analysisof Heiser and Pratt36 to includepropertyvari-ations across the detonation wave front a closed-form expressionof acuteth is obtained as follows49

acutethPDE D 1 iexcl

(deg1 iexcl 1deg2 iexcl 1

sup3deg2

deg1

acute21

M2D

sup31 C deg1 M2

D

1 C deg2

acutedeg2 C 1=deg2

pound Atilde 1iexcl[deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)

(2)

where deg1 and deg2 are the speci c-heat ratios of the unburned andburned gases separated by the detonation wave front respectivelyThe Mach number of the detonation wave relative to the unburnedgas MD can be calculated using the following equation for xedheat addition

M2D D

microdeg 2

2 iexcl deg1

deg 21 iexcl deg1

C deg 22 iexcl 1

deg1 iexcl 1

QqAtilde

para

C

smicrodeg 2

2 iexcl deg1

deg 21 iexcl deg1

Cdeg 2

2 iexcl 1

deg1 iexcl 1

QqAtilde

para2

iexcldeg 2

2

deg 21

(3)

In comparison the thermodynamic ef ciencies of the Humphreyand Brayton cycles are

acutethHumphrey D 1 iexcl

(deg2

deg1

sup3deg1 iexcl 1

deg2 iexcl 1

acutedeg2 iexcl 1=deg2sup3

deg1QqAtilde

C 1

acute1=deg2

pound Atilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1]deg1=deg2 iexcl 1

Qq

)(4)

acutethBrayton D 1 iexclraquomicrosup3

QqAtilde

C 1

acuteAtilde 1 iexcl [deg2 iexcl 1=deg1 iexcl 1deg1=deg2 iexcl 1

paraiquestQqfrac14

(5)

With the thermodynamic cycle ef ciency available the thrust Fcan be obtained by a control-volumeanalysis

F D Pma C Pm f u3 iexcl Pmau0 frac14 Pma

iexclpu2

0 C 2acutethq iexcl u0

cent(6)

where u0 is the freestream velocity Pma the cycle-averagedair mass ow rate delivered to the engine through the inlet and Pm f the cycle-averaged fuel mass ow rate Note that the preceding analysis isbased on the assumptions that every uid particle experiences thesame processes sequentially and that the effects of purging and by-pass air are ignored The fuel-based speci c impulse can then beobtained as follows

Isp DF

Pm f gD

pu2

0 C 2acutethq iexcl u0

f g(7)

Figures 3 and 4 show a typical result of cycle ef ciencyand speci c impulse for a stoichiometric hydrogenair system

Fig 3 Thermodyanmiccycle ef ciencies of idealPDE Humphrey andBraytoncycles as function of static temperature ratio Atilde for stoichiomet-ric H2 air system

560 WU MA AND YANG

Fig 4 Speci c impulses of ideal PDE Humphrey and Brayton cy-cles as function of static temperature ratio Atilde for stoichiometric H2 airsystem

The freestream velocity and temperature are u0 D 636 ms andT0 D 228 K respectively The nondimensional heat addition q is2247The system performanceincreaseswith increasingstatic tem-peratureratioThe PDE offers the bestperformanceamong the threecycles especiallywhen the static temperatureratio Atilde is smaller than3 This may be because for a given amount of heat addition theMach number of the detonation wave increases with decreasing T1

(or Atilde ) as indicated by Eq (3) The shock-compression effect be-comes more signi cant for a lower T1 leading to a higher increasein the temperature and pressure of the unburned gases before com-bustion The Isp of an ideal PDE reaches 5263 s when T1 D 428 Kthat is Atilde D 1877

III System Performance AnalysisThis section deals with the developmentof a system performance

analysis for airbreathing PDEs as shown schematically in Fig 1The study is based on a modular approach Each module repre-sents a speci c component of the engine and its dynamic behav-ior is formulated using complete conservationequationsThe workinvolves the following three components 1) supersonic inlet dy-namics 2) detonation chamber dynamics and system performanceand 3) effect of nozzle con guration The effects of fuel supply airdistribution and inlet isolator are ignored for simplicity They canhowever be straightforwardlyincluded as submodels in the overallengine performance analysis

A Supersonic Inlet DynamicsThe inlet and its interaction with combustor represent a crucial

aspect in the development of any airbreathing engine includingPDEs The inlet is designed to capture and supply stable air ow at arate demanded by the combustor and to maintain high pressure re-covery and stability margin at various engine operating conditionsThe overallvehicleperformancedependsgreatlyon the energy leveland ow quality of the incoming air A small loss in inlet ef ciencymay translate to a substantial penalty in engine thrust Moreoverany change in the inlet ow structure may modify the downstreamcombustioncharacteristicsand subsequentlylead to undesirablebe-haviors such as ame blowoff and ashback Thus matching inletbehavior to engine requirements is of fundamental importance todesigners50

In addition to its primary function of supplyingair an inlet has adetermining in uence on the dynamics of the entire system throughits intrinsicunsteadinessand interactionswith thecombustioncham-berTypicallypressurewaves areproducedin thecombustioncham-ber and propagate upstream to interact with the inlet ow througha manifold where mixing of air and fuel occurs The resultant owoscillations in the inlet diffuser then either propagate downstreamin the form of acoustic waves or are convected downstream withthe mean ow in the form of vorticity and entropy waves andfurther reinforce the unsteady motions in the combustor A feed-back loop is thus established between the inlet and combustor

Fig 5 Supersonic inlet with mixed compression

Fig 6 Mach number contours with different back pressures understeady-state conditions

The situation is much more complicated in a supersonic PDE dueto the shock-waveboundary-layer interaction and shockacoustic-wave interaction51 [also Oh J Y Ma F H Hsieh S Y andYang V ldquoInteractionsBetween Shock Waves and Acoustic Wavesin a Supersonic Inlet Diffuserrdquo (to be submitted for publication)]The engine dynamics exhibits features that are qualitatively differ-ent from those predicted by treating the inlet and combustor as twoseparate entities

The inlet analysis is based on the axisymmetric Favre-averagedconservationequationsTurbulenceclosure is achievedusing a two-layer model of Rodi calibrated for supersonic ows with shockwaves52 The governingequationsare solved numerically by meansof a density-based nite volume methodology The temporal dis-cretization is obtained using a four-stage RungendashKutta integrationmethod and the spatial discretization employs an upwind total-variation-diminishing scheme developed by Harten53 Speci c de-tails of the numerical algorithm can be found in Ref 54

Figure 5 shows the con guration treated in the present study amixed-compression inlet optimized for a ight altitude of 93 kmand a freestream Mach number of 21 (Ref 50) Figure 6 presentsthe Mach-numbercontoursat two differentbackpressures(pb D 21and 22 atm) which are carefully chosen such that the engine oper-ates at a supercritical condition to provide a suf cient shock stabil-ity margin Under these conditions the two leading conical shocksgenerated by the double-cone centerbody compress the air ow ex-ternallymerge slightlyabove thecowl lip and forma strongobliqueshock which extends into the external-ow region In addition ashockstemming from the cowl inner surface continuesdownstreamhitting and re ecting from both the cowl and centerbodywalls and nally leading to a terminal normal shock The ow undergoes asequence of compression and expansion waves and becomes sub-sonic after passingthroughthe normal shock located in thedivergentsection of the diffuser The inlet recovers a high percentage of thefreestream total pressure by decelerating the air ow through theshock train The pressure recovery coef cients for the two cases are84 and 88 respectively and the Mach numbers immediately infront of the terminal shocks are 142 and 132 respectively

The response of the inlet shock system to downstream distur-bances has also been studied by imposing periodic pressureoscilla-tions at theexit planeA wide rangeof uctuationfrequencyandam-plitude were considered Important phenomena of concern includetemporal and spatial variationsof mass ow rate pressure recoveryand ow distributionas well as shock displacementIn general theacoustic response of the inlet ow increases with increasing ampli-tude of the imposed oscillation but decreases with frequencyAlsoincluded as part of the result is the acoustic impedance function atthe inlet exit a parameter that can be effectively used to character-ize the inletcombustor couplingA more detaileddiscussionon thissubject will be given in subsequent work

WU MA AND YANG 561

B Detonation Chamber DynamicsThe detonationchamberdynamicsis formulatedbasedon thecon-

servation laws for a multicomponentchemically reacting system intwo-dimensionalcoordinatesDiffusive transport is neglected in thecurrent study because of its minor role in determining detonationdynamics and system performance The governing equations andtheir associatedboundary conditionsare solved using a recently de-velopedspacendashtime conservation-elementsolution-elementmethodthat circumvents the de ciencies of existing numerical methodsfor treating detonation waves and shock discontinuities55iexcl59 Theresultant computer code is further parallelized using the messagepassing interface library with domain decomposition to improve itsef ciency

Both simple global and detailed chemical kinetics models areutilized4959 The former involvesonly one progressvariable to char-acterize the chemical reaction rate and assumes constant proper-ties Because of its computational ef ciency and reasonable accu-racy in determining the PDE propulsive performance the model isimplemented in the present work The associated thermochemicalparameters are optimized by comparing the calculated detonationwave properties with those from the NASA chemical equilibriumanalysis48 The relative errors are less than 5 in terms of the det-onation velocity and the CndashJ pressure and temperature

As part of the model validation effort a series of single-pulsecalculations were conducted for a straight tube of 60 cm in lengthinitially lled with a stoichiometric mixture of hydrogen and air atpreconditionedpressure pi and temperature Ti A driver-gas regionspanning 02 mm near the head end with a temperature of 2000 Kand a pressure of 30 atm was employed to initiate the detonationwave directly Four different numerical grids with the sizes of 0201 005 and 0025 mm were used to check the solution accuracyin terms of grid independenceAll of the calculatedpressurepro lescollapsed onto a single curve with the CndashJ propertiesmatching theanalytical values exactly As a result the 02-mm grid was chosenfor the entire study to alleviate the computational burden For asingle-pulse operation the head-end pressure remains at a plateauvalue pp that is p3 in Refs 37 and 46 for certain period soonafter the detonation initiation and then decays gradually to a levellower than the ambient state3746 The impulse can be determinedby integrating temporally the force exerted on the head end fromt D 0 to the instant when the head-endpressure reaches the ambientvalue The contribution to the impulse from the ignition source isestimated to be less than 05 Figure 7 shows the impulse per unitcross-sectionalarea as a function of the plateau pressure pp and thedetonation residence time iquestD (de ned as the tube length L dividedby the detonation wave velocity u D ie iquestD acute L=u D Results canbe correlated well in the following form

I=A D 41pp iexcl pi iquestD (8)

This expression is quite similar to those obtained from the semi-analytical analysis of Wintenberger et al37 and the experimentalwork of Falempin et al14 The constants of proportionality differslightly in the various studies for example 413 in Ref 37 for the

Fig 7 Generalized impulse curve for single-pulse detonation instraight tube with stoichiometric H2air mixture

current conditions suggesting dependenceon the details of experi-mentalprocedureandoperatingconditionsThe generalizedformulaproposed by Kailasanath60 based on his numerical simulations forhydrogenair ethyleneoxygen and propaneoxygen mixtures has alarger constant of proportionality of 465 One factor contributingto this disparity may be differencesin ignition source The width ofthe detonation initiation region is 20 mm in Ref 60 as opposed to02 mm in the present simulationsNonethelessthe precedingpara-metric study demonstrates the capacity and delity of the presentapproach for the PDE performance analysis

The validatedanalysis is then employed to study the performanceof airbreathing PDEs As a speci c example the ight conditioninvolving an altitude of 93 km and a Mach number of 21 is consid-ered The freestream static pressure and temperature are 029 atmand 228K respectivelycorrespondingto a totalpressureof 265atmand a total temperature of 428 K The total pressure and tempera-ture at the combustor entranceobtained from the inlet ow analysisdescribed in the preceding section are 212 atm and 428 K respec-tively The detonation tube is 60 cm long and has a diameter of16 cm which is similar to the dimensions of contemporary ramjetcombustors for air defense applicationsAs a rst approach only asingle straight tube is considered to provide direct insight into thechamber dynamics without complications arising from the nozzleThe interactions among the tubes are also ignored The valve atthe tube entrance is assumed to be either fully open or fully closedThe operationsequenceis thuscontrolledby three time periodsthevalveclose-upperiodiquestclose duringwhich the valve is closedand thetube undergoesdetonationinitiationand propagationand blowdownprocesses the purging period iquestpurge during which a small amountof cold air is injected to prevent preignition of fresh reactants andthe re lling period iquestre ll during which the combustible mixture isdelivered to the chamber The sum of these three periods is equal tothe operationcycleperiodiquestcycle that is iquestcycle D iquestclose C iquestpurge C iquestre llThe purgingperiodshouldbeminimizedto reduceperformancelossA small value of 01 ms is used throughoutall of the calculationsinthe present study

The boundaryconditionsat thehead endof thedetonationtubearespeci ed according to the engine operationDuring the purgingandre lling stages the total pressureand total temperatureare obtainedfrom the inlet analysis The axial velocity is extrapolated from theinterior points and the reactant mass fraction is treated as an inputparameterWhen the valve is closed the head end is simply modeledas a rigid wall

A series of analysesare conductedover a wide range of operationparameters The baseline case has iquestcycle of 3 ms and iquestclose of 24 msThe tube is initially lled with a stoichiometrichydrogenairmixtureat ambient pressure and temperature It takes about ve cycles toreach steady cyclic operation Figure 8 shows the xndasht diagram forthe rst cycle of operation obtained by tracing the characteristiclines of the ow eld along the centerline of the tube The timehistories of the ow properties at the head end are also presentedThe detonation wave is directly initiated by a hot driver gas andpropagates downstream at the CndashJ velocity toward the unburnedmixture(region1) It then inducesTaylorexpansionwaves (region2)to satisfy the stationaryconditionat the head end causinga uniformregion (region 3) with constant-ow properties in the upstream

The detonationwave reaches the reactantair interfaceat the tubeexit at t D 0305 ms (point A) which deviates slightly from thefollowing analytical prediction by 06 due to the effect of theexternally imposed ignition source

iquestD D L=u D D 06 m=1956 ms D 0307 ms (9)

The wave then degenerates to a nonreactive shock that is theprimary shock wave proceeding farther downstream into the ex-ternal region followed by a contact surface separating the ambientair and combustion products A sonic region is gradually formednear the tube exit due to the local ow expansion as evidenced bythe clustered characteristic lines in the xndasht diagram Downstreamof the sonic region the ow is expanded to become supersonic and nally leads to the formationof a secondaryshock to match with thesubsonic ow behind the primary shockThis secondaryshockwave

562 WU MA AND YANG

Fig 8 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms 1 = unburned region 2 = Taylor expansion waves 3 = stationary region 4 = non-simple wave regionand 5 = simple wave region

Fig 9 Snapshots of pressure density and their gradients elds at t = 07 ms

moves farther downstreammeeting with expansionwaves originat-ing from the primary shockwaveThese complicated ow structurescan be also observed in Fig 9 which shows the instantaneouspres-sure and density and their gradient elds at t D 07 ms Many salientfeatures are clearly shown including the expansion fans vorticesand rolled-upslip lines that are developedas the shockdiffractsoverthe edge of the tube exit

As the detonation wave catches the reactantair interface and theresultant primary shock wave travels outside the tube a series ofexpansionwaves is generatedwhich propagate upstream resulting

in a nonsimple wave region (region 4) when interactingwith the in-coming Taylor waves A simple wave region (region 5) is recoveredafter passing through the Taylor waves The rst expansion wavereaches the head end at t D 0935 ms (point B) which can be deter-mined by consideringthe interactionbetween the expansionand theTaylor waves and the sound speed in region 3 A similarity solutionhas been derivedby Wintenbergeret al37 to predict this time instantanalytically

t D L=uD C regL=c3 (10)

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

560 WU MA AND YANG

Fig 4 Speci c impulses of ideal PDE Humphrey and Brayton cy-cles as function of static temperature ratio Atilde for stoichiometric H2 airsystem

The freestream velocity and temperature are u0 D 636 ms andT0 D 228 K respectively The nondimensional heat addition q is2247The system performanceincreaseswith increasingstatic tem-peratureratioThe PDE offers the bestperformanceamong the threecycles especiallywhen the static temperatureratio Atilde is smaller than3 This may be because for a given amount of heat addition theMach number of the detonation wave increases with decreasing T1

(or Atilde ) as indicated by Eq (3) The shock-compression effect be-comes more signi cant for a lower T1 leading to a higher increasein the temperature and pressure of the unburned gases before com-bustion The Isp of an ideal PDE reaches 5263 s when T1 D 428 Kthat is Atilde D 1877

III System Performance AnalysisThis section deals with the developmentof a system performance

analysis for airbreathing PDEs as shown schematically in Fig 1The study is based on a modular approach Each module repre-sents a speci c component of the engine and its dynamic behav-ior is formulated using complete conservationequationsThe workinvolves the following three components 1) supersonic inlet dy-namics 2) detonation chamber dynamics and system performanceand 3) effect of nozzle con guration The effects of fuel supply airdistribution and inlet isolator are ignored for simplicity They canhowever be straightforwardlyincluded as submodels in the overallengine performance analysis

A Supersonic Inlet DynamicsThe inlet and its interaction with combustor represent a crucial

aspect in the development of any airbreathing engine includingPDEs The inlet is designed to capture and supply stable air ow at arate demanded by the combustor and to maintain high pressure re-covery and stability margin at various engine operating conditionsThe overallvehicleperformancedependsgreatlyon the energy leveland ow quality of the incoming air A small loss in inlet ef ciencymay translate to a substantial penalty in engine thrust Moreoverany change in the inlet ow structure may modify the downstreamcombustioncharacteristicsand subsequentlylead to undesirablebe-haviors such as ame blowoff and ashback Thus matching inletbehavior to engine requirements is of fundamental importance todesigners50

In addition to its primary function of supplyingair an inlet has adetermining in uence on the dynamics of the entire system throughits intrinsicunsteadinessand interactionswith thecombustioncham-berTypicallypressurewaves areproducedin thecombustioncham-ber and propagate upstream to interact with the inlet ow througha manifold where mixing of air and fuel occurs The resultant owoscillations in the inlet diffuser then either propagate downstreamin the form of acoustic waves or are convected downstream withthe mean ow in the form of vorticity and entropy waves andfurther reinforce the unsteady motions in the combustor A feed-back loop is thus established between the inlet and combustor

Fig 5 Supersonic inlet with mixed compression

Fig 6 Mach number contours with different back pressures understeady-state conditions

The situation is much more complicated in a supersonic PDE dueto the shock-waveboundary-layer interaction and shockacoustic-wave interaction51 [also Oh J Y Ma F H Hsieh S Y andYang V ldquoInteractionsBetween Shock Waves and Acoustic Wavesin a Supersonic Inlet Diffuserrdquo (to be submitted for publication)]The engine dynamics exhibits features that are qualitatively differ-ent from those predicted by treating the inlet and combustor as twoseparate entities

The inlet analysis is based on the axisymmetric Favre-averagedconservationequationsTurbulenceclosure is achievedusing a two-layer model of Rodi calibrated for supersonic ows with shockwaves52 The governingequationsare solved numerically by meansof a density-based nite volume methodology The temporal dis-cretization is obtained using a four-stage RungendashKutta integrationmethod and the spatial discretization employs an upwind total-variation-diminishing scheme developed by Harten53 Speci c de-tails of the numerical algorithm can be found in Ref 54

Figure 5 shows the con guration treated in the present study amixed-compression inlet optimized for a ight altitude of 93 kmand a freestream Mach number of 21 (Ref 50) Figure 6 presentsthe Mach-numbercontoursat two differentbackpressures(pb D 21and 22 atm) which are carefully chosen such that the engine oper-ates at a supercritical condition to provide a suf cient shock stabil-ity margin Under these conditions the two leading conical shocksgenerated by the double-cone centerbody compress the air ow ex-ternallymerge slightlyabove thecowl lip and forma strongobliqueshock which extends into the external-ow region In addition ashockstemming from the cowl inner surface continuesdownstreamhitting and re ecting from both the cowl and centerbodywalls and nally leading to a terminal normal shock The ow undergoes asequence of compression and expansion waves and becomes sub-sonic after passingthroughthe normal shock located in thedivergentsection of the diffuser The inlet recovers a high percentage of thefreestream total pressure by decelerating the air ow through theshock train The pressure recovery coef cients for the two cases are84 and 88 respectively and the Mach numbers immediately infront of the terminal shocks are 142 and 132 respectively

The response of the inlet shock system to downstream distur-bances has also been studied by imposing periodic pressureoscilla-tions at theexit planeA wide rangeof uctuationfrequencyandam-plitude were considered Important phenomena of concern includetemporal and spatial variationsof mass ow rate pressure recoveryand ow distributionas well as shock displacementIn general theacoustic response of the inlet ow increases with increasing ampli-tude of the imposed oscillation but decreases with frequencyAlsoincluded as part of the result is the acoustic impedance function atthe inlet exit a parameter that can be effectively used to character-ize the inletcombustor couplingA more detaileddiscussionon thissubject will be given in subsequent work

WU MA AND YANG 561

B Detonation Chamber DynamicsThe detonationchamberdynamicsis formulatedbasedon thecon-

servation laws for a multicomponentchemically reacting system intwo-dimensionalcoordinatesDiffusive transport is neglected in thecurrent study because of its minor role in determining detonationdynamics and system performance The governing equations andtheir associatedboundary conditionsare solved using a recently de-velopedspacendashtime conservation-elementsolution-elementmethodthat circumvents the de ciencies of existing numerical methodsfor treating detonation waves and shock discontinuities55iexcl59 Theresultant computer code is further parallelized using the messagepassing interface library with domain decomposition to improve itsef ciency

Both simple global and detailed chemical kinetics models areutilized4959 The former involvesonly one progressvariable to char-acterize the chemical reaction rate and assumes constant proper-ties Because of its computational ef ciency and reasonable accu-racy in determining the PDE propulsive performance the model isimplemented in the present work The associated thermochemicalparameters are optimized by comparing the calculated detonationwave properties with those from the NASA chemical equilibriumanalysis48 The relative errors are less than 5 in terms of the det-onation velocity and the CndashJ pressure and temperature

As part of the model validation effort a series of single-pulsecalculations were conducted for a straight tube of 60 cm in lengthinitially lled with a stoichiometric mixture of hydrogen and air atpreconditionedpressure pi and temperature Ti A driver-gas regionspanning 02 mm near the head end with a temperature of 2000 Kand a pressure of 30 atm was employed to initiate the detonationwave directly Four different numerical grids with the sizes of 0201 005 and 0025 mm were used to check the solution accuracyin terms of grid independenceAll of the calculatedpressurepro lescollapsed onto a single curve with the CndashJ propertiesmatching theanalytical values exactly As a result the 02-mm grid was chosenfor the entire study to alleviate the computational burden For asingle-pulse operation the head-end pressure remains at a plateauvalue pp that is p3 in Refs 37 and 46 for certain period soonafter the detonation initiation and then decays gradually to a levellower than the ambient state3746 The impulse can be determinedby integrating temporally the force exerted on the head end fromt D 0 to the instant when the head-endpressure reaches the ambientvalue The contribution to the impulse from the ignition source isestimated to be less than 05 Figure 7 shows the impulse per unitcross-sectionalarea as a function of the plateau pressure pp and thedetonation residence time iquestD (de ned as the tube length L dividedby the detonation wave velocity u D ie iquestD acute L=u D Results canbe correlated well in the following form

I=A D 41pp iexcl pi iquestD (8)

This expression is quite similar to those obtained from the semi-analytical analysis of Wintenberger et al37 and the experimentalwork of Falempin et al14 The constants of proportionality differslightly in the various studies for example 413 in Ref 37 for the

Fig 7 Generalized impulse curve for single-pulse detonation instraight tube with stoichiometric H2air mixture

current conditions suggesting dependenceon the details of experi-mentalprocedureandoperatingconditionsThe generalizedformulaproposed by Kailasanath60 based on his numerical simulations forhydrogenair ethyleneoxygen and propaneoxygen mixtures has alarger constant of proportionality of 465 One factor contributingto this disparity may be differencesin ignition source The width ofthe detonation initiation region is 20 mm in Ref 60 as opposed to02 mm in the present simulationsNonethelessthe precedingpara-metric study demonstrates the capacity and delity of the presentapproach for the PDE performance analysis

The validatedanalysis is then employed to study the performanceof airbreathing PDEs As a speci c example the ight conditioninvolving an altitude of 93 km and a Mach number of 21 is consid-ered The freestream static pressure and temperature are 029 atmand 228K respectivelycorrespondingto a totalpressureof 265atmand a total temperature of 428 K The total pressure and tempera-ture at the combustor entranceobtained from the inlet ow analysisdescribed in the preceding section are 212 atm and 428 K respec-tively The detonation tube is 60 cm long and has a diameter of16 cm which is similar to the dimensions of contemporary ramjetcombustors for air defense applicationsAs a rst approach only asingle straight tube is considered to provide direct insight into thechamber dynamics without complications arising from the nozzleThe interactions among the tubes are also ignored The valve atthe tube entrance is assumed to be either fully open or fully closedThe operationsequenceis thuscontrolledby three time periodsthevalveclose-upperiodiquestclose duringwhich the valve is closedand thetube undergoesdetonationinitiationand propagationand blowdownprocesses the purging period iquestpurge during which a small amountof cold air is injected to prevent preignition of fresh reactants andthe re lling period iquestre ll during which the combustible mixture isdelivered to the chamber The sum of these three periods is equal tothe operationcycleperiodiquestcycle that is iquestcycle D iquestclose C iquestpurge C iquestre llThe purgingperiodshouldbeminimizedto reduceperformancelossA small value of 01 ms is used throughoutall of the calculationsinthe present study

The boundaryconditionsat thehead endof thedetonationtubearespeci ed according to the engine operationDuring the purgingandre lling stages the total pressureand total temperatureare obtainedfrom the inlet analysis The axial velocity is extrapolated from theinterior points and the reactant mass fraction is treated as an inputparameterWhen the valve is closed the head end is simply modeledas a rigid wall

A series of analysesare conductedover a wide range of operationparameters The baseline case has iquestcycle of 3 ms and iquestclose of 24 msThe tube is initially lled with a stoichiometrichydrogenairmixtureat ambient pressure and temperature It takes about ve cycles toreach steady cyclic operation Figure 8 shows the xndasht diagram forthe rst cycle of operation obtained by tracing the characteristiclines of the ow eld along the centerline of the tube The timehistories of the ow properties at the head end are also presentedThe detonation wave is directly initiated by a hot driver gas andpropagates downstream at the CndashJ velocity toward the unburnedmixture(region1) It then inducesTaylorexpansionwaves (region2)to satisfy the stationaryconditionat the head end causinga uniformregion (region 3) with constant-ow properties in the upstream

The detonationwave reaches the reactantair interfaceat the tubeexit at t D 0305 ms (point A) which deviates slightly from thefollowing analytical prediction by 06 due to the effect of theexternally imposed ignition source

iquestD D L=u D D 06 m=1956 ms D 0307 ms (9)

The wave then degenerates to a nonreactive shock that is theprimary shock wave proceeding farther downstream into the ex-ternal region followed by a contact surface separating the ambientair and combustion products A sonic region is gradually formednear the tube exit due to the local ow expansion as evidenced bythe clustered characteristic lines in the xndasht diagram Downstreamof the sonic region the ow is expanded to become supersonic and nally leads to the formationof a secondaryshock to match with thesubsonic ow behind the primary shockThis secondaryshockwave

562 WU MA AND YANG

Fig 8 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms 1 = unburned region 2 = Taylor expansion waves 3 = stationary region 4 = non-simple wave regionand 5 = simple wave region

Fig 9 Snapshots of pressure density and their gradients elds at t = 07 ms

moves farther downstreammeeting with expansionwaves originat-ing from the primary shockwaveThese complicated ow structurescan be also observed in Fig 9 which shows the instantaneouspres-sure and density and their gradient elds at t D 07 ms Many salientfeatures are clearly shown including the expansion fans vorticesand rolled-upslip lines that are developedas the shockdiffractsoverthe edge of the tube exit

As the detonation wave catches the reactantair interface and theresultant primary shock wave travels outside the tube a series ofexpansionwaves is generatedwhich propagate upstream resulting

in a nonsimple wave region (region 4) when interactingwith the in-coming Taylor waves A simple wave region (region 5) is recoveredafter passing through the Taylor waves The rst expansion wavereaches the head end at t D 0935 ms (point B) which can be deter-mined by consideringthe interactionbetween the expansionand theTaylor waves and the sound speed in region 3 A similarity solutionhas been derivedby Wintenbergeret al37 to predict this time instantanalytically

t D L=uD C regL=c3 (10)

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

WU MA AND YANG 561

B Detonation Chamber DynamicsThe detonationchamberdynamicsis formulatedbasedon thecon-

servation laws for a multicomponentchemically reacting system intwo-dimensionalcoordinatesDiffusive transport is neglected in thecurrent study because of its minor role in determining detonationdynamics and system performance The governing equations andtheir associatedboundary conditionsare solved using a recently de-velopedspacendashtime conservation-elementsolution-elementmethodthat circumvents the de ciencies of existing numerical methodsfor treating detonation waves and shock discontinuities55iexcl59 Theresultant computer code is further parallelized using the messagepassing interface library with domain decomposition to improve itsef ciency

Both simple global and detailed chemical kinetics models areutilized4959 The former involvesonly one progressvariable to char-acterize the chemical reaction rate and assumes constant proper-ties Because of its computational ef ciency and reasonable accu-racy in determining the PDE propulsive performance the model isimplemented in the present work The associated thermochemicalparameters are optimized by comparing the calculated detonationwave properties with those from the NASA chemical equilibriumanalysis48 The relative errors are less than 5 in terms of the det-onation velocity and the CndashJ pressure and temperature

As part of the model validation effort a series of single-pulsecalculations were conducted for a straight tube of 60 cm in lengthinitially lled with a stoichiometric mixture of hydrogen and air atpreconditionedpressure pi and temperature Ti A driver-gas regionspanning 02 mm near the head end with a temperature of 2000 Kand a pressure of 30 atm was employed to initiate the detonationwave directly Four different numerical grids with the sizes of 0201 005 and 0025 mm were used to check the solution accuracyin terms of grid independenceAll of the calculatedpressurepro lescollapsed onto a single curve with the CndashJ propertiesmatching theanalytical values exactly As a result the 02-mm grid was chosenfor the entire study to alleviate the computational burden For asingle-pulse operation the head-end pressure remains at a plateauvalue pp that is p3 in Refs 37 and 46 for certain period soonafter the detonation initiation and then decays gradually to a levellower than the ambient state3746 The impulse can be determinedby integrating temporally the force exerted on the head end fromt D 0 to the instant when the head-endpressure reaches the ambientvalue The contribution to the impulse from the ignition source isestimated to be less than 05 Figure 7 shows the impulse per unitcross-sectionalarea as a function of the plateau pressure pp and thedetonation residence time iquestD (de ned as the tube length L dividedby the detonation wave velocity u D ie iquestD acute L=u D Results canbe correlated well in the following form

I=A D 41pp iexcl pi iquestD (8)

This expression is quite similar to those obtained from the semi-analytical analysis of Wintenberger et al37 and the experimentalwork of Falempin et al14 The constants of proportionality differslightly in the various studies for example 413 in Ref 37 for the

Fig 7 Generalized impulse curve for single-pulse detonation instraight tube with stoichiometric H2air mixture

current conditions suggesting dependenceon the details of experi-mentalprocedureandoperatingconditionsThe generalizedformulaproposed by Kailasanath60 based on his numerical simulations forhydrogenair ethyleneoxygen and propaneoxygen mixtures has alarger constant of proportionality of 465 One factor contributingto this disparity may be differencesin ignition source The width ofthe detonation initiation region is 20 mm in Ref 60 as opposed to02 mm in the present simulationsNonethelessthe precedingpara-metric study demonstrates the capacity and delity of the presentapproach for the PDE performance analysis

The validatedanalysis is then employed to study the performanceof airbreathing PDEs As a speci c example the ight conditioninvolving an altitude of 93 km and a Mach number of 21 is consid-ered The freestream static pressure and temperature are 029 atmand 228K respectivelycorrespondingto a totalpressureof 265atmand a total temperature of 428 K The total pressure and tempera-ture at the combustor entranceobtained from the inlet ow analysisdescribed in the preceding section are 212 atm and 428 K respec-tively The detonation tube is 60 cm long and has a diameter of16 cm which is similar to the dimensions of contemporary ramjetcombustors for air defense applicationsAs a rst approach only asingle straight tube is considered to provide direct insight into thechamber dynamics without complications arising from the nozzleThe interactions among the tubes are also ignored The valve atthe tube entrance is assumed to be either fully open or fully closedThe operationsequenceis thuscontrolledby three time periodsthevalveclose-upperiodiquestclose duringwhich the valve is closedand thetube undergoesdetonationinitiationand propagationand blowdownprocesses the purging period iquestpurge during which a small amountof cold air is injected to prevent preignition of fresh reactants andthe re lling period iquestre ll during which the combustible mixture isdelivered to the chamber The sum of these three periods is equal tothe operationcycleperiodiquestcycle that is iquestcycle D iquestclose C iquestpurge C iquestre llThe purgingperiodshouldbeminimizedto reduceperformancelossA small value of 01 ms is used throughoutall of the calculationsinthe present study

The boundaryconditionsat thehead endof thedetonationtubearespeci ed according to the engine operationDuring the purgingandre lling stages the total pressureand total temperatureare obtainedfrom the inlet analysis The axial velocity is extrapolated from theinterior points and the reactant mass fraction is treated as an inputparameterWhen the valve is closed the head end is simply modeledas a rigid wall

A series of analysesare conductedover a wide range of operationparameters The baseline case has iquestcycle of 3 ms and iquestclose of 24 msThe tube is initially lled with a stoichiometrichydrogenairmixtureat ambient pressure and temperature It takes about ve cycles toreach steady cyclic operation Figure 8 shows the xndasht diagram forthe rst cycle of operation obtained by tracing the characteristiclines of the ow eld along the centerline of the tube The timehistories of the ow properties at the head end are also presentedThe detonation wave is directly initiated by a hot driver gas andpropagates downstream at the CndashJ velocity toward the unburnedmixture(region1) It then inducesTaylorexpansionwaves (region2)to satisfy the stationaryconditionat the head end causinga uniformregion (region 3) with constant-ow properties in the upstream

The detonationwave reaches the reactantair interfaceat the tubeexit at t D 0305 ms (point A) which deviates slightly from thefollowing analytical prediction by 06 due to the effect of theexternally imposed ignition source

iquestD D L=u D D 06 m=1956 ms D 0307 ms (9)

The wave then degenerates to a nonreactive shock that is theprimary shock wave proceeding farther downstream into the ex-ternal region followed by a contact surface separating the ambientair and combustion products A sonic region is gradually formednear the tube exit due to the local ow expansion as evidenced bythe clustered characteristic lines in the xndasht diagram Downstreamof the sonic region the ow is expanded to become supersonic and nally leads to the formationof a secondaryshock to match with thesubsonic ow behind the primary shockThis secondaryshockwave

562 WU MA AND YANG

Fig 8 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms 1 = unburned region 2 = Taylor expansion waves 3 = stationary region 4 = non-simple wave regionand 5 = simple wave region

Fig 9 Snapshots of pressure density and their gradients elds at t = 07 ms

moves farther downstreammeeting with expansionwaves originat-ing from the primary shockwaveThese complicated ow structurescan be also observed in Fig 9 which shows the instantaneouspres-sure and density and their gradient elds at t D 07 ms Many salientfeatures are clearly shown including the expansion fans vorticesand rolled-upslip lines that are developedas the shockdiffractsoverthe edge of the tube exit

As the detonation wave catches the reactantair interface and theresultant primary shock wave travels outside the tube a series ofexpansionwaves is generatedwhich propagate upstream resulting

in a nonsimple wave region (region 4) when interactingwith the in-coming Taylor waves A simple wave region (region 5) is recoveredafter passing through the Taylor waves The rst expansion wavereaches the head end at t D 0935 ms (point B) which can be deter-mined by consideringthe interactionbetween the expansionand theTaylor waves and the sound speed in region 3 A similarity solutionhas been derivedby Wintenbergeret al37 to predict this time instantanalytically

t D L=uD C regL=c3 (10)

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

562 WU MA AND YANG

Fig 8 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms 1 = unburned region 2 = Taylor expansion waves 3 = stationary region 4 = non-simple wave regionand 5 = simple wave region

Fig 9 Snapshots of pressure density and their gradients elds at t = 07 ms

moves farther downstreammeeting with expansionwaves originat-ing from the primary shockwaveThese complicated ow structurescan be also observed in Fig 9 which shows the instantaneouspres-sure and density and their gradient elds at t D 07 ms Many salientfeatures are clearly shown including the expansion fans vorticesand rolled-upslip lines that are developedas the shockdiffractsoverthe edge of the tube exit

As the detonation wave catches the reactantair interface and theresultant primary shock wave travels outside the tube a series ofexpansionwaves is generatedwhich propagate upstream resulting

in a nonsimple wave region (region 4) when interactingwith the in-coming Taylor waves A simple wave region (region 5) is recoveredafter passing through the Taylor waves The rst expansion wavereaches the head end at t D 0935 ms (point B) which can be deter-mined by consideringthe interactionbetween the expansionand theTaylor waves and the sound speed in region 3 A similarity solutionhas been derivedby Wintenbergeret al37 to predict this time instantanalytically

t D L=uD C regL=c3 (10)

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

WU MA AND YANG 563

where reg is function of deg and MD and can be calculated as

reg D 1

2

sup31 C 1

M2D

acutecent

(2

microdeg iexcl 1

deg C 1

sup3deg C 3

2C 2

deg iexcl 1

iexcl deg C 12

2cent

M2D

1 C deg M 2D

acuteparaiexcldeg C 1=2deg iexcl 1

iexcl 1

)(11)

Application of Eqs (10) and (11) gives rise to an analytical valueof 0958 ms The slight difference between the numerical and theanalytical solutions may be attributed to the numerical resolutionand dissipation near the tube exit

On the arrival of the rst expansion wave at the head end thepressure begins to decay gradually These expansion waves re ectoff the head end and form anotherseriesof expansionwaves furtherreducing the chamber pressure The downstream-traveling expan-sion waves weaken the secondary shock and eventually cause it tomove upstream

The head-endpressure decays to 023 atm at t D 24 ms at whichpoint the purging stage beginsThe head-end temperature is 1258 Kat this instantBecause of the pressuredifferenceacross the entranceplane a right-running shock wave is established along with a se-ries of central expansion waves and a contact surface between theburned gas and the cold air Another contact surface forms betweenthe fresh reactants and purging air when the re lling stage com-mences 01 ms later The correspondingre lling pressure velocityandMach number are about091 atm 500 ms and 12 respectivelyThe time evolution of the pressure distribution along the centerlineduring the rst cycle of operation is shown in Fig 10

The ow evolution during a steady operation cycle is examinedFigure 11 shows the xndasht diagram and time histories of ow prop-erties at the head end for the fth cycle The main ow featuresremain qualitatively the same as those in the rst cycle Howeverthe secondary shock wave disappears because the ow behind theprimary shock wave is already supersonic In addition the head-end pressure and temperature begin to decay earlier relative to the rst cycle due to the rarefactionwaves produced from the previouscycle Also note that the detonation wave catches the leading freshreactant at x D 512 cm instead of at the tube exit

The impulse of each cycle is calculated by considering the mo-mentum balance over a control volume enclosing the entire engineThe inlet ow loss is properly taken into account as detailed inSec IIIA The cycle-averaged speci c thrust (air based) and spe-ci c impulse (fuel based) are then obtained by dividing the impulseby the air mass and fuel weight for each cycle respectively Forthe baseline case the fuel-based speci c impulse is 2328 s Thismay be compared to a ramjet engine operating at the same ightcondition with perfect nozzle ow expansion which has a speci cimpulse of about 3866 s (Ref 61) A parametric study was carriedout to examine the effect of various operating times on the sys-tem performanceFigure 12 shows the result as a function of iquestcloseThe straight-tube system leads to a speci c impulse far lower thanits theoretical limit of 5263 s based on the thermodynamic cycleanalysis for an ideal PDE Eq (7) which assumes isentropic owprocesses in the inlet and nozzle Although the calculated speci cimpulse can be improvedby optimizingthe operationfrequencyandtiming the net gain appears to be limited with the current designSeveral fundamentalmechanisms responsiblefor such an unaccept-able performance have been identi ed First at high altitudes thestraight-tube design fails to preserve the chamber pressure duringthe re lling stage at a level suf cient to meet the requirements forthe mass loadingdensityof fresh reactantsSecond the low chamberpressure in the re lling stages causes a high-speed reactant streamin the tube and subsequentlyresults in a large performance loss Itis well established that the stagnation pressure drop due to energyaddition is proportional to the square of the Mach number In thepresent case the local Mach number may reach a value of up to12 during the re lling processThe ensuing loss of thermodynamicef ciency becomes exceedingly large compared with conventionalpropulsion systems with subsonic combustion Third the lack of

Fig 10 Time evolution of pressure distribution along centerline dur-ing rst cycle of operation iquestcycle = 3 ms iquestclose = 24 ms and iquestpurge =01 ms

an appropriate ow expansiondevice downstreamof the detonationtube gives rise to an extremely complicated ow structure near thetube exit The internal energy of the exhaust ow can not be effec-tively converted to the kinetic energy for thrust generation furtherdeteriorating the situation

C Effect of Nozzle Con gurationIn light of the limited performance of the straight-tube design

much effort was expended to study the effect of nozzle con gu-ration on the system propulsive performance The nozzle designfor PDEs poses a serious challenge because of the intrinsically un-steady nature of the pulse detonation process Recent studies basedon single-pulsecalculations4142 and experiments122162 indicatethatthe nozzle con guration may signi cantly change the thrust deliv-ered by an engine In addition to its in uence on speci c impulsethroughmodi cationof the gasexpansionprocessthe nozzleaffectsthe chamber ow dynamics and consequentlythe timing of variousphasesof theengineoperationcycle especiallyfor high-altitudeandspace applications

The present work focuses on a choked convergentndashdivergent(CndashD) nozzle because of its effectiveness in preserving the cham-ber pressure during the blowdown and re lling stages In contrastdivergent and plug nozzles do not possess such an advantage espe-cially under high-altitude conditions in spite of their superior per-formance for single-pulse operation at sea level Figure 13 showsschematically the nozzle con guration considered herein measur-ing a length of 20 cm The slope angle is 45 deg for the conver-gent part and 15 deg for the divergent part The ratio of the tube

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

564 WU MA AND YANG

Fig 11 Fifth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation stoichiometric H2 air mixtureiquestcycle = 3 ms iquestclose = 24 ms and iquestpurge = 01 ms

Fig 12 Effect of valve close-up time on speci c impulse iquestcycle = 3 msand iquestpurge = 01 ms straight tube with stoichiometric H2air mixtureh = 93 km and M1 = 21

Fig 13 Con guration of single-tube PDE with CndashD nozzle

cross-sectional area to the nozzle throat area is 178 and the noz-zle expansion ratio is 281 Several calculationswere conducted forthis con guration The baseline case has iquestcycle of 3 ms and iquestclose of21 ms The detonation tube is initially lled with a stoichiometrichydrogenair mixture and the nozzle with air The engine takes vecycles to reach steady operation Figures 14 and 15 show the xndashtdiagrams along the centerline of the tube and the time histories of ow propertiesat the head end for the rst and eighthcycles of oper-ation respectivelyThe ow characteristicsbear close resemblanceto those of the straight-tube case A major difference lies in the re- ection of a shock wave from the convergent section of the nozzle

instead of expansion waves in a straight tube The re ected shockthen propagatesupstream and causes an abrupt increase in pressureat the head end on its arrival as evidencedin the pressurendashtime tracein Fig 14 The nozzle throat remains chokedduring most of the cy-cle thus helping preserve the chamber pressureThe pressure in there lling stage is about 145 atm which is substantiallygreater thanthe straight-tubecase and consequentlyincreases the mass loadingdensity of fresh reactants The relatively lower speed of the re lledmixture also enhances the system thermodynamic ef ciency Thespeci c impulse of 3402 s in the present case is 46 higher thanthe maximum speci c impulse achieved by a straight tube furtherdemonstrating the effectivenessof a chokedCndashD nozzle in improv-ing engine performance

A parametric study is conducted to study the timing effect onsystem performanceby varyingiquestcycle and iquestclose The purge time iquestpurge

is xed at 01 ms Figure 16 shows the effect of iquestclose on the speci cthrust Fsp de ned as the cycle-averaged thrust per unit of air mass ow rate and the fuel-based speci c impulse Isp at four differentcycle frequenciesof 200 250 333 and 400 Hz The correspondingcycleperiodsare 5 4 3 and 25 ms respectivelyWhen the straight-tube design is comparedat the same operatingconditionthe presentsystem with a choked CndashD nozzle can indeed substantiallyimprovethe engine performance by a margin of 45

The speci c thrust increases as iquestclose decreases for all of the fre-quencies considered herein This can be explained as follows Fora given iquestcycle and iquestpurge a smaller iquestclose translates to a shorter blow-down process The resultant higher chamber pressure during there lling stage increases the loading density of fresh reactants Theincreased re lling period also enhances the amount of reactants de-liveredto thechamberCombined these two factorsresult in a highercycle-averagedchamber pressure and consequently a higher spe-ci c thrust Note however that the lower bound of iquestclose is subjectto three practical constraintsThe rst is concerned with inlet over-pressurizationThe head-end pressure must not exceed the stagna-tion pressure of the inlet air to allow for purging and re lling whenthe valve is open The second is related to chamber over lling Thefreshreactantsshouldnot owoutof thenozzleto theexternalregionbefore being burned completely unless afterburning is considered

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

WU MA AND YANG 565

Fig 14 First cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometric H2 airmixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms 1 = uniform unburned region 2 = Taylor expansion waves 3 = uniform region 4 = non-simplewave region and 5 = simple wave region

Fig 15 Eighth cycle xndasht diagram and time histories of ow properties at head end under typical PDE operation with CndashD nozzle stoichiometricH2 air mixture iquestcycle = 3 ms iquestclose = 21 ms and iquestpurge = 01 ms

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

566 WU MA AND YANG

a) Air-based speci c thrust

b) Fuel-based speci c impulse

Fig 16 Effect of valve close-up time at four different operation fre-quencies straight tube with CndashD nozzle for stoichiometric H2air mix-ture h = 93 km M1 = 21

The third constraintalthoughcommonly satis ed in practicalcasesis that iquestclose should be suf ciently long to cover at least the timerequired for detonation initiation and propagation throughout theentire chamber The upper bound of iquestclose (or the lower bound ofiquestre ll is based on the requirement that an appropriate amount offresh reactants be delivered to the chamber to produce thrust

The effect of iquestclose on the fuel-based speci c impulse follows thesame trend as that of the air-based speci c thrust except for a smallrange of iquestclose close to its lower bound The speci c impulse andspeci c thrust satisfy the following relation

Isp DFsp1 C iquestpurge=iquestre ll

f g(12)

As iquestclose decreases the factor 1 C iquestpurge=iquestre ll decreases and mayoverride the increase of Fsp consequently leading to a decrease inIsp as shown in Fig 16b

For a given cycle period iquestclose determines the lling length offresh reactants A larger iquestclose (or smaller iquestre ll leads to a smaller lling length in most cases and consequently decreases the spe-ci c impulse This result however is in contrast to the previousexperimental62 and numerical47 observationsfor single-pulseoper-ations which concluded that the speci c impulse increases as the lling length decreasesOne factor contributing to this discrepancyis that in single-pulsestudies the pressureand temperatureof reac-tants are preconditioned to ambient values whereas in the presentmulticycle study the ow conditions of the re lled mixture dependon the timing of the engineoperationThe use of a chokedCndashD noz-zle also exerted a substantial in uence on the chamber dynamicsThus signi cant differences exist between single-pulse and mul-ticycle operations The conclusions from single-pulse studies maynot be applied to multicycle cases directly

Figure 16 also demonstrates the existence of an optimum fre-quency for achieving a maximum performance for a given PDEcon guration and ight condition At a low cycle frequency morereactants can be recharged into the detonation tube As a conse-quence a higher chamber pressure can be reached and the engine

ef ciency improves However a large re lling time associatedwithlow-frequency operation may cause chamber over lling and thusdegrade the performance These two con icting effects result in anoptimum frequency In the present study the operating frequencyof 250 Hz (iquestcycle D 4 ms) offers the best performance The highestspeci c impulse is 3676 s slightly lower than its ramjet counterpartof 3866 s with optimum nozzle ow expansion

IV ConclusionsA comprehensive analysis has been established to study the sys-

tem performanceof airbreathingPDEs The physical model of con-cern includesinletair-distributionunit detonationtubeand nozzleResults from parametricstudies reveal that for a xed operatingfre-quency a decrease in iquestclose leads to increased engine performancein terms of speci c impulse and thrust The straight-tube designgives rise to unacceptableperformance especially for high-altitudeapplications due to its failure to preserve chamber pressure duringthe re lling stage A choked CndashD nozzle appears to be requiredto deliver performance at a level suf cient to compete with otherairbreathing engines such as ramjets For a given engine con gu-ration and ight condition an optimum cycle frequency and iquestclose

exist for achieving the best performance A thermodynamic cycleanalysiswas also developedto determine the theoretical limit of theengine propulsive ef ciency Results were compared with those ofthe Humphrey and Brayton cycles

For a typical supersonic mission with a ight Mach number of21 and an altitude of 93 km the maximum PDE speci c impulseis 3676 s for a stoichiometrichydrogenair mixture with proper ac-count of the inlet and nozzle performance losses This Isp is lowerthan its ramjet counterpartof 3866 s with perfectnozzle ow expan-sion Furthermorethe intrinsicunsteadinessthrustvectorvariationand other loss mechanisms not considered in the present analysis(such as the energyrequiredfor detonationinitiationand ow lossesassociated with the inlet isolator rotary valve and air distributor)may render the PDE much less attractiveFurther improvement andoptimizationof the system con gurationand operationare required

AcknowledgmentsThis work was supportedby the Departmentof DefenseMultidis-

ciplinary University Research Initiative under the Of ce of NavalResearch Grant N00014-99-1-0744 with Gabriel Roy serving asthe Program Manager

References1Bussing T R A and Pappas G ldquoPulse Detonation Engine Theory

and Conceptsrdquo Developments in High-Speed Vehicle Propulsion SystemsVol 165 Progress in Astronautics and Aeronautics AIAA Reston VA1996 pp 421ndash472

2Hoffman N ldquoReaction Propulsionby Intermittent Detonative Combus-tionrdquo Ministry of Supply Volkenrode Translation 1940

3Nicholls J A Wilkinson H R and Morrison R B ldquoIntermittentDetonation as a Thrust-Producing Mechanismrdquo Jet Propulsion Vol 27No 5 1957 pp 534ndash541

4Krzycki L J ldquoPerformance Characteristics of an Intermittent-Detonation Devicerdquo US Naval Ordnance Test Station NAVWEPS Rept7655 China Lake CA 1962

5Helman D Shreeve R P andEidelman S ldquoDetonationPulseEnginerdquoAIAA Paper 86-1683 June 1986

6Hinkey J B Bussing T R A and Kays L ldquoShock Tube Experimentsfor the Developmentof a Hydrogen-FueledPulseDetonationEnginerdquoAIAAPaper 95-2578 July 1995

7Sterling J Ghorbanian K Humphrey J and Sobota T ldquoEnhancedCombustion Pulsejet Engines for Mach 0 to 3 Applicationsrdquo AIAA Paper96-2687 July 1996

8Litchford R J ldquoDevelopment of a Gas-Fed Pulse Detonation ResearchEnginerdquo AIAA Paper 2001-3814 July 2001

9Shepherd J E Pintgen F Austin J M and Eckett C A ldquoThe Struc-ture of the Detonation Front in Gasesrdquo AIAA Paper 2002-0773 Jan 2002

10Meyer T R Hoke J L Brown M S Gord J R and Schauer F RldquoExperimental Study of De agration Enhancement Techniques in a H2AirPulsed-Detonation Enginerdquo AIAA Paper 2002-3720 July 2002

11Sanders S T Jenkins T P and Hanson R K ldquoDiode Laser Sen-sor System for Multi-Parameter Measurements in Pulse Detonation EngineFlowsrdquo AIAA Paper 2000-3593 July 2000

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002

WU MA AND YANG 567

12Daniau D ZitounRCouguetC andDesbordesD ldquoEffects ofNoz-zles of Different Length and Shape on the PropulsionPerformance of PulsedDetonation Enginesrdquo High Speed De agration and Detonation Fundamen-tals and Control edited by G Roy S Frolov D Netzer and A BorisovELEX-KM Publishers Moscow 2001 pp 251ndash262

13Sinibaldi J O Brophy C M Li C and Kailasanath K ldquoInitiatorDetonation Diffraction Studies in Pulsed Detonation Enginesrdquo AIAA Paper2001-3466 July 2001

14Falempin F Bouchaud D Forrat B Desbordes D and Daniau EldquoPulsed Detonation Engine Possible Application to Low Cost Tactical Mis-sile and to Space Launcherrdquo AIAA Paper 2001-3815 July 2001

15Broda J C Conrad C Pal S Woodward R D and Santoro R JldquoExperimental Results on Air-Breathing Pulse Detonation Studiesrdquo Pro-ceedings of 11th Annual Symposium on Propulsion 1999

16Watts J Conrad C Lee S Y Woodward R D Pal S SantoroR J ldquoFundamental Studies of the DDT Process in Pulse Detonation En-gines Based on SingleMulti-Cycle Operationrdquo Proceedings of 12th AnnualSymposium on Propulsion 2000

17Sinibaldi J O Brophy C M and RobinsonL J P ldquoIgnitionEffectson De agration-to-Detonation Transition Distance in Gaseous MixturesrdquoAIAA Paper 2000-3590 July 2000

18Cooper M Jackson S Austin J Wintenberger E and ShepherdJ E ldquoDirect Experimental Impulse Measurements for Detonations andDe agrationsrdquo Journal of Propulsion and Power Vol 18 No 5 2002pp 1033ndash1041

19Harris P G Farinaccio R and Stowe R A ldquoThe Effect of DDTDistance on Impulse in a Detonation Tuberdquo AIAA Paper 2001-3467 July2001

20Lieberman D H Parkin K L and Shepherd J E ldquoDetonation Ini-tiation by a Hot Turbulent Jet for Use in Pulse Detonation Enginesrdquo AIAAPaper 2002-3909 July 2002

21Stuessy W S and Wilson D R ldquoExperimental Investigation of anAnnual Multi-Cycle Pulsed Detonation Wave Enginerdquo AIAA Paper 96-0346 Jan 1996

22Aarnio M J Hinkey J B and Bussing T R A ldquoMultiple CycleDetonation Experiments during the Development of a Pulse Detonation En-ginerdquo AIAA Paper 96-3263 July 1996

23Hinkey J B Williams J T Henderson S E and Bussing T R AldquoRotary-Valved Multiple-Cycle Pulse Detonation Engine ExperimentalDemonstrationrdquo AIAA Paper 97-2746 1997

24Schauer F Stutrud J and Bradley R ldquoAFRLrsquos In-House ResearchPulse Detonation Enginerdquo Proceedings of 11th Annual Symposium onPropulsion 1999

25Schauer F Stutrud J Bradley R and Katta V ldquoAFRLPRSC PulseDetonation Engine Research Programrdquo Proceedings of 12th Annual Sym-posium on Propulsion 2000

26McManus K FurlongE Leyva I and SandersonS ldquoMEMS-BasedPulse Detonation Engine for Small-Scale Propulsion Applicationsrdquo AIAAPaper 2001-3469 July 2001

27Frankey B Shauer F Bradley R and Hoke J ldquoEvaluation of aHybrid-Piston Pulsed Detonation Enginerdquo AIAA Paper 2002-0474 Jan2002

28Brophy C M Sinibaldi J O and Damphousse P ldquoInitiator Perfor-mance for Liquid-Fueled Pulse Detonation Enginesrdquo AIAA Paper 2002-0472 Jan 2002

29Shimo M Meyer S E Heister S D Weng C S Ji J and GoreJ P ldquoAn Experimental and Computational Study of Pulsed Detonations ina Single Tuberdquo AIAA Paper 2002-3716 July 2002

30Brophy C M Netzer D W and Forster L D ldquoDetonation Studiesof JP-10 with Oxygen and Air for Pulse Detonation Engine DevelopmentrdquoAIAA Paper 98-4003 July 1998

31Brophy C M and Netzer D W ldquoEffects of Ignition Characteristicsand Geometry on the Performance of a JP-10O2 Fueled Pulse DetonationEnginerdquo AIAA Paper 99-2635 June 1999

32Farinaccio R Harris P Stowe R A and Akbar R ldquoMulti-PulseDetonation Experiments with PropanendashOxygenrdquo AIAA Paper 2002-4070July 2002

33Talley D G and Coy E B ldquoConstant Volume Limit of Pulsed Propul-sion for a Constant deg Ideal Gasrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 400ndash406

34Tew D E ldquoIdeal PolytropicRamjet Performancerdquo Proceedingsof 11thAnnual Symposium on Propulsion 1999

35Kent eld J A C ldquoThe Fundamentals of Idealized Airbreathing Pulse-DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 1 2002pp 77ndash83

36Heiser W H and Pratt D T ldquoThermodynamic Cycle Analysis ofPulse DetonationEnginesrdquo Journalof PropulsionandPower Vol 18 No 12002 pp 68ndash76

37WintenbergerE Austin J M CooperM Jackson S and ShepherdJ E ldquoAnalytical Model for the Impulse of a Single-Cycle Pulse DetonationEnginerdquo Journal of Propulsion and Power Vol 19 No 1 2003 pp 22ndash38

38Sterling J Ghorbanian K Humphrey J and Sobota T ldquoNumericalInvestigations of Pulse Detonation Wave Enginesrdquo AIAA Paper 95-2479July 1995

39Mohanraj R and Merkle C L ldquoA Numerical Study of Pulse Detona-tion Engine Performancerdquo AIAA Paper 2000-0315 Jan 2000

40Eidelman S Grossman W and Lottati I ldquoAir-Breathing Pulsed Det-onation Engine Concept A Numerical Studyrdquo AIAA Paper 90-2420 July1990

41Eidelman S and Yang X ldquoAnalysis of the Pulse Detonation EngineEf ciencyrdquo AIAA Paper 98-3877 July 1998

42Cambier J L and Tegner J K ldquoStrategies for Pulsed Detonation En-gine Performance Optimizationrdquo Journal of Propulsionand Power Vol 14No 4 1998 pp 489ndash498

43Li C Kailasanath K and Patnaik G ldquoA Numerical Study of FlowField Evolution in a Pulsed Detonation Enginerdquo AIAA Paper 2000-0314Jan 2000

44Ebrahimi H B Mohanraj R and Merkle CL ldquoMultilevel Analysisof Pulsed Detonation Enginesrdquo Journal of Propulsion and Power Vol 18No 2 2002 pp 225ndash232

45Ebrahimi H B Mohanraj R and Merkle C L ldquoModeling of Multi-Tube Pulse Detonation Engine Operationrdquo AIAA Paper 2001-3813 July2001

46Li C and Kailasanath K ldquoA Numerical Study of Reactive Flows inPulse Detonation Enginesrdquo AIAA Paper 2001-3933 July 2001

47Li C and Kailasanath K ldquoPerformance Analysis of Pulse DetonationEngines with Partial Fuel Fillingrdquo AIAA Paper 2002-0610 Jan 2002

48McBride B J and Gordon S ldquoComputer Program for Calculationof Complex Chemical EquilibriumCompositionsand ApplicationsrdquoNASAReference Publ 1311 June 1996

49Wu Y H ldquoSystem Performance and Thermodynamic Cycle Analysisof Air-Breathing Pulse Detonation Enginesrdquo PhD Dissertation Dept ofMechanical and Nuclear Engineering Pennsylvania State Univ UniversityPark PA May 2002

50Yang V and Cappuccio M ldquoSupersonic Inlet Design for MissilesrdquoDept of Mechanical EngineeringResearch Rept PennsylvaniaState UnivUniversity Park PA 1990

51Yang V and Culick F E C ldquoAnalysis of Unsteady Inviscid DiffuserFlow with a Shock Waverdquo Journal of Propulsion and Power Vol 1 1985pp 222ndash228

52Rodi W ldquoExperience with Two-Layer Models Combining the kndashModel with a One-Equation Model Near the Wallrdquo AIAA Paper 91-0216Jan 1991

53Harten A ldquoHigh Resolution Schemes for Hypersonic ConservationLawsrdquo Journal of Computational Physics Vol 49 No 3 1983 pp 357ndash393

54Oh J Y ldquoNumerical StudyofSteadyandOscillatory FlowStructures inan Axisymmetric SupersonicInletrdquo PhD Dissertation Dept of MechanicalEngineering Pennsylvania State Univ University Park PA Aug 1994

55Chang S C ldquoThe Method of SpacendashTime Conservation Element andSolution ElementmdashA New Approach for Solving the NavierndashStokes andEuler Equationsrdquo Journal of ComputationalPhysics Vol 119 No 2 1995pp 295ndash324

56Wang X Y and Chang S C ldquoA 2D Non-Splitting Unstructured Tri-angular Mesh Euler Solver Based on the SpacendashTime Conservation Ele-ment and Solution Element Methodrdquo Computational Fluid Dynamic Jour-nal Vol 8 No 2 1999 pp 309ndash325

57Chang S C Wang X Y and Chow C Y ldquoThe SpacendashTime Con-servation Element and Solution Element A New High Resolution and Gen-uinelyMultidimensionalParadigmforSolvingConservationLawsrdquo Journalof Computational Physics Vol 156 No 1 1999 pp 89ndash136

58Zhang Z C Yu S T and Chang S C ldquoA SpacendashTime ConservationElement and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler EquationsUsing Quadrilateral and HexahedralMeshesrdquo Journalof ComputationalPhysics Vol 175No 1 2002pp 168ndash199

59WuY H YangV andChangSC ldquoNumerical SimulationofChemi-cally Reacting FlowswithDetailed Kinetics Using the Space-Time MethodrdquoProceedings of 1st International Conference on Computational Fluid Dy-namics Springer-Verlag Berlin 2000 pp 207ndash212

60Kailasanath K ldquoRecent Developments in the Research on Pulse Det-onation Enginesrdquo AIAA Paper 2002-0470 Jan 2002

61Yang V Ma F H and Choi J Y ldquoSystem Performance and ThrustChamber Optimization of Air-Breathing Pulse Detonation Enginesrdquo Pro-ceedings of 15th ONR Propulsion Meeting US Naval Research Of ceWashington DC 2002

62CooperM and Shepherd J E ldquoThe Effect of Nozzles and Extensionson Detonation Tube Performancerdquo AIAA Paper 2002-3628 July 2002