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ISSN 1990�7931, Russian Journal of Physical Chemistry B, 2013, Vol. 7, No. 3, pp. 276–289. © Pleiades Publishing, Ltd., 2013.Original Russian Text © A.E. Zangiev, V.S. Ivanov, S.M. Frolov, 2013, published in Khimicheskaya Fizika, 2013, Vol. 32, No. 5, pp. 62–75.

276

INTRODUCTION

One possible scheme of implementation of detona�tive combustion in ramjet airbreathing engines is toprovide a periodic filling of the combustion chamberwith a gas mixture with subsequent initiation of deto�nation and burning the mixture in the traveling deto�nation wave [1, 2]. Such engines are known as pulsedetonation engines (PDEs). Various estimates [3, 4]show that the theoretical fuel�based specific impulseof a PDE operating on hydrogen or a hydrocarbon fuelcan be very high, 5500 and 2500 s, respectively, over awide range of flight Mach numbers, from 0 to 4–5.Therefore, at present, intense research and designwork concerning power plants for aircraft is under way.

The aims of the present study were (1) to calculatethe thrust characteristics of a PDE with an air intakeand a nozzle under conditions of supersonic flight withconsideration given to the physicochemical character�istics of the oxidation and combustion of hydrocarbonfuel and to the finite times of turbulent flame acceler�ation and deflagration�to�detonation transition(DDT) and (2) to optimize the design of the PDE toenhance its thrust characteristics. The work is a con�tinuation of research started in [2].

STATEMENT OF THE PROBLEM

Figure 1 shows a schematic of an axisymmetricPDE, 2.12 m in length and 83 mm in outer diameter,with an air intake, receiver, annular bypass channel,

and combustion chamber equipped with a mechanicalvalve, nine turbulizing obstacles and a nozzle. Sincethe operation cycle of the PDE was discussed in detailin [2], we give here only a brief description. The oper�ation cycle consists of three stages:

(1) When the valve is opened, the combustionchamber is filled with a fuel–air mixture (FAM). Fuelis fed into the combustion chamber through a distrib�uted “injector”, which is a single transverse layer ofcomputational cells before the first annular obstacle.The supply of fuel into these cells is modeled by volu�metric sources of mass, ensuring the formation of astoichiometric FAM, which is then fed into the com�bustion chamber. To exclude a contact of the freshFAM with the hot detonation products of the previouscycle, fuel is fed into the air flow with a delay withrespect to the time of opening of the valve. In otherwords, a buffer layer of buffer air layer is formedbetween the fresh mixture and the hot products.

(2) When the combustion chamber is filled withFAM to the required level, determined by a fill factorof χ = 0.9, the valve is immediately closed to stop fuelsupply and begin the second stage of the operationcycle. The fill factor was defined as the ratio of themass of the fuel supplied to the combustion chamberof the PDE to the mass of the fuel in the chamberwhen it is completely filled with the stoichiometricmixture, all other things being equal. Thus, the fill fac�tor is 1.0 if, at the time of ignition, the combustionchamber is completely filled with a stoichiometric

Thrust Characteristics of an Airbreathing Pulse Detonation Enginein Supersonic Flight at Various Altitudes

A. E. Zangiev, V. S. Ivanov, and S. M. FrolovSemenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow Russia

e�mail: [email protected] October 24, 2012

Abstract—The main thrust characteristics, such as thrust force, specific impulse, specific fuel consumption,and specific thrust, of a pulse detonation engine (PDE) with an air intake and nozzle in conditions of flightat a Mach number of 3 and various altitudes (from 8 to 28 km above sea level) are for the first time calculatedwith consideration given to the physicochemical characteristics of the oxidation and combustion of hydro�carbon fuel (propane), finite time of turbulent flame acceleration, and deflagration�to�detonation transition(DDT). In addition, a parametric analysis of the influence of the operation mode and design parameters ofthe PDE on its thrust characteristics in flight at a Mach number of 3 and an altitude of 16 km is performed,and the characteristics of engines with direct initiation of detonation and fast deflagration are compared. It isshown that a PDE of this design greatly exceeds an ideal ramjet engine in specific thrust, whereas regardingthe specific impulse and specific fuel consumption, it is not inferior to the ideal ramjet.

Keywords: deflagration�to�detonation transition, pulse detonation engine, ideal ramjet engine, thrust

DOI: 10.1134/S1990793113050126

COMBUSTION, EXPLOSION, AND SHOCK WAVES

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B Vol. 7 No. 3 2013

THRUST CHARACTERISTICS OF AN AIRBREATHING PULSE DETONATION ENGINE 277

FAM, from the place of fuel supply (injector) to thenozzle inlet. The fuel mixture in the combustionchamber is ignited by an external annular source in therecirculation zone behind the first turbulizing obsta�cle. The flame, propagating in the FAM turbulentflow, accelerates and DDT takes place. The detona�tion wave formed because of DDT propagates down�stream and exits through the nozzle into the atmo�sphere.

(3) With the arrival of the detonation wave at thenozzle exit, the outflow of detonation products begins.It lasts as long as the average pressure on the valve fromthe side of the combustion chamber reduces to a cer�tain critical value P* that still provides a positiveinstantaneous total force (effective thrust) acting onthe engine in flight. When P* is achieved, this forcebecomes close to zero, the valve opens, and the cyclerepeats itself. Note that the instantaneous total force isdefined as the integral of the forces of pressure and vis�cous friction on all solid surfaces of the PDE.

MATHEMATICAL MODEL AND CALCULATION METHOD

As in [2], the operation cycle of the PDE wasnumerically simulated in the two�dimensional axi�symmetric approximation. The mathematical modelunderlying the calculations was a set of Reynolds�averaged equations of conservation of mass, momen�tum, and energy for unsteady, compressible, turbulent,reacting flow:

(1)

∂ ∂ρ = ρ + ρ∂ ∂

∂ ∂ ⎡ ⎤= − + τ − ρ⎢ ⎥⎣ ⎦∂ ∂' ' ;

i i ij

j

ij i ji j

DU U UU

Dt t x

P U Ux x

(2)

(3)

where t is the time; xj (j = 1, 2) is the coordinate; ρ isthe mean density; P is the mean pressure; μ is the thedynamic viscosity; Ui is the mean velocity; is thefluctuating velocity component; τij is the viscous stress

tensor; I = H + is the mean total enthalpy

(H is the average static enthalpy); λ is the thermal con�ductivity; T is the mean temperature; Yl (l = 1, …, N)is the average mass fraction of the lth species in themixture (N is total number of species in the mixture);Dl is the molecular diffusion coefficient of species l ofthe mixture; is the fluctuation of the mass fractionof the lth species, and and are the mean sourceterms of mass and energy. The turbulent fluxes ofmass, momentum, and energy in (1)–(3) were mod�eled using the standard k–ε turbulence model [5] (k isthe kinetic energy of turbulence and ε is its dissipationrate).

Modeling the chemical sources and for turbu�lent combustion and DDT requires taking intoaccount the contributions from both frontal combus�tion (index f) and bulk preflame reactions (index Y):

,

∂ ∂ρ = ρ + ρ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂= ρ + + τ + λ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠� ( ) ;

jj

ij jj j i

DI I IUDt t x

P TQ Ut x x x

∂ ∂ρ = ρ + ρ∂ ∂

⎛ ⎞∂∂= ρ + ρ − ρ⎜ ⎟∂ ∂⎝ ⎠�

' ' ,

l l lj

j

ll l l j

j i

DY Y YU

Dt t x

Yr D Y U

x x

'iU

212

iiU∑

'lY

lr� Q�

lr� Q�

= +� � �l lf lYr r r

= +� � � .f YQ Q Q

Air intake Receiver Valve Annular bypass Nozzle

Fuel supply Ignition

66 83 72 52 82 83 42

302 150 1543 124

Fig. 1. Basic scheme of the axisymmetric airbreathing PDE with an air intake and nozzle (dimensions in mm).

278


ZANGIEV et al.

To determine and , we used an algorithm ofexplicit tracking of the flame front (ETFF), whereasthe contributions from the bulk reactions and were determined by the particle method (PM) [2].

System (1)–(3) in conjunction with the k–ε turbu�lence model and the coupled ETFF–PM model wasclosed by the caloric and thermal equations of state ofan ideal gas with variable heat capacity and supple�mented by initial and boundary conditions. All thethermophysical parameters of the gas were consideredvariable. System (1)–(3) was numerically solved by thefinite�volume discretization method with the first�order approximation in space and time [6]. To avoidexcessive mesh refinement near solid surfaces with no�slip conditions, we used the standard wall functionsmethod [7]. All calculations, as in [2], were conductedfor a sector with an opening angle of 5°.

THRUST CHARACTERISTICS OF A BASIC�SCHEME PDE WITH NINE

TURBULIZING OBSTACLES

We first performed two�dimensional axisymmetriccalculations of the unsteady turbulent reacting flow inthe duct of the basic�scheme PDE with nine turbuliz�ing obstacles operating on gaseous propane and theflow around it in supersonic flight at a Mach numberof M = 3.0 and altitudes of Z = 8, 10, 12, 16, 18, and20 km. We examined the operation of the PDE on astoichiometric propane–air mixture, i.e., at a fuelequivalence ratio of φ = 1. To determine the thrustcharacteristics, we calculated the parameters of threeor four cycles (until fully reproducible results wereobtained) with consideration given to the external flowaround the engine. More specifically, the pressureforce (the integral of the absolute pressure over thesurface) and the viscous friction force (the integral ofthe viscous shear stress over the surface) acting on allsolid surfaces of the PDE in both the internal andexternal flows were calculated. The average size of thecells in the standard computational grid was 2 mm.

The calculation results are listed in Table 1, wherethe following notation are used: Z is the altitude, Pa is

lfr� fQ�

�lYr �

YQ

the pressure, Ta is the temperature of the atmosphericair, f is the operation frequency, P* is the critical meanpressure on the valve from the side of the combustionchamber at which the instantaneous effective thrust isstill positive, F is the effective thrust during one cycle,R is the thrust force (the sum of the force F and dragforce), Isp is the fuel�based specific impulse, СP is thethrust coefficient, Rsp is the specific thrust (the thrustreferred to the air mass flow rate), Csp is the specificfuel consumption (per�hour fuel consumption per 1 Nthrust), is the mass flow rate of fuel. To determinethe thrust R produced by the PDE, it is necessary toknow its drag in flight. This force can be determined bysolving the same problem with two modifications: with“one misfire” (method 1) and with “two misfires”(method 2). In the first case, the problem is solveduntil a reproducible mode is achieved (three to fourcycles), and then, in the subsequent cycle, the com�bustion chamber is filled but no ignition is performed.In the second case, the problem is also solved until theonset of a fully reproducible mode (three to fourcycles), after which two cycles are executed withoutignition. The difference between the two methods ofcalculating the drag lies in the fact that, in the firstcase, the filling of the PDE with fresh mixture dis�places the hot combustion products remaining fromthe previous cycle, whereas in the second case, it dis�places the unreacted (cold) FAM from the previouscycle. In both cases, the force acting on the PDE inflight is determined using the last cycle, without igni�tion. Table 1 lists both the values of the parameters R,Isp, CP, Rsp, and Csp, obtained by method 1 (non�paren�thesized) and method 2 (parenthesized). Further in thetext and tables, these parameters are presented in thesame notations. Note that, here, in contrast to [2], theabsolute values of R, F, and are given not for the sectorwith an opening angle of 5°, but for the PDE as a whole.

For completeness, we would like to write formulas forfuel�based specific impulse Isp, specific fuel consumptionCsp, specific thrust Rsp, and the thrust coefficient СP:

,

fm�

fm�

=

�

spf

RIgm

Table 1. Basic parameters and calculation results for the basic�scheme PDE in flight at a Mach number of 3 and various altitudes

Z, km Pa, MPa Ta, K f, Hz P*, MPa F, N R, N Isp, s CPRsp,

kN/(kg/s)Csp,

kg/(N h)

g/s

8 0.036 236.2 49 0.439 255 978 (1078) 1760 (1940) 0.79 (0.87) 1.11 (1.22) 0.21 (0.19) 56.2

10 0.027 223.3 50 0.31 204 688 (757) 1740 (1920) 0.75 (0.82) 1.09 (1.19) 0.21 (0.19) 40.3

12 0.0194 216.7 50 0.243 149 492 (531) 1720 (1860) 0.73 (0.79) 0.88 (0.96) 0.26 (0.24) 35.3

16 0.010 216.7 50 0.175 84 275 (296) 1710 (1830) 0.77 (0.82) 1.05 (1.14) 0.22 (0.20) 16.6

18 0.0075 216.7 54 0.073 8 169 (179) 1670 (1560) 0.64 (0.68) 0.78 (0.83) 0.29 (0.27) 13.7

20 0.0055 216.7 no detonation

,fm�



,

,

Here, g is the acceleration of gravity, is the air massflow rate, ρ is the density of air, V is the flight speed,Smid is the midsection of the engine. Note that for astoichiometric propane�air mixture, + ≈

In all cases shown in Table 1, the fill factor was χ =0.9. It should be noted that the turbulent flow caused asmearing of the boundary between the FAM and thepurge air layer and, therefore, a dilution of FAM withair, especially near the nozzle. This effect of dilutioninfluenced the propagation of the detonation wave inthe final part of the charge, reducing its velocity oreven disrupting the process, so that this part of thecharge burned out in the flame front.

Figure 2 shows a typical time dependence of theinstantaneous total force (effective thrust) acting on allsolid surfaces of the engine in the limiting operation

=

�

sp

3600 fmC

R

=

�

spa

RRm

=ρ

2 .( 2)

Pmid

RCV S

am�

fm� am�16.6 .fm�

cycle. The numbers indicate the periods and keypoints of the process:

(1) The valve is opened (100.5 ms); (2) the chamberis filled with FAM (100.5–108 ms); (3) ignition(108.2 ms); (4) the acceleration of the flame front(108.2–108.64 ms); (5) DDT (~108.64 ms); (6) thearrival of the retonation wave at the valve (108.9 ms);(7) propagation of the detonation wave downstream ofthe combustion chamber, the reflection from the noz�zle (108.66–109.1 ms); (8) upstream propagation ofthe shock wave reflected from the nozzle throat(109.1–110.5 ms); (9) the interaction of a reflectedshock wave with the valve (110.5–110.8 ms); (10) rep�etition the process described above accompanied bythe outflow of products through the nozzle (>111 ms).

In addition to Fig. 2, Fig. 3 shows the calculatedfields of temperature, pressure and propane mass frac�tion at these moments and time intervals (fields of pro�pane mass fraction are shown only to a time of 109.2 mswhen all the fuel has reacted).

The calculations have demonstrated the possibilityof a high�frequency (~50 Hz) cyclic operation of thePDE under conditions of weak�source ignition (~0.1 J)and DDT. The solid curve in Fig. 4 shows the depen�dence of the calculated thrust R for the basic�schemePDE with nine turbulizing obstacles on the flight alti�

–5.7612095 115110105100

–4.32

–2.88

–1.44

0

1.44

2.88

1

2

3

4

5

6

9

10

7

8

Force, kN

Time, ms

Fig. 2. Calculated time dependence of the instantaneous effective thrust; the basic�scheme PDE with nine turbulizing obstaclesfor a supersonic flight at a Mach number of 3 at an altitude of 16 km.

280


ZANGIEV et al.

tude. As can be seen, the PDE thrust decreases withaltitude, remaining positive to about Z = 18 km.

The solid curve in Fig. 5 shows the calculateddependence of the average fuel�based specific impulse

of the basic�scheme PDE with nine turbulizing obsta�cles on the flight altitude. The average fuel�based spe�cific impulse is defined as the arithmetic mean of thetwo values of Isp given in Table 1. It is seen that the

200

Temperature, К

Absolute pressure, Pa

Mass fraction of C3H8[–]

500 800 1100 1400 1700 2000 2300 2600 2900 3200

10000 74000 1.38e+05 2.02e+05 2.66e+05 3.3e+05 3.94e+05 4.58e+05 5.22e+05 5.86e+05 6.5e+05

0 0.006 0.012 0.018 0.024 0.030 0.036 0.042 0.048 0.054 0.060

100.00 ms

104.50 ms

107.99 ms

108.00 ms

108.20 ms

108.30 ms

Fig. 3. Fields of temperature, pressure and mass fraction of propane in one operation cycle of the basic�scheme PDE with nineturbulizing obstacles for a supersonic flight at Mach number of 3 and an altitude of 16 km.



108.40 ms

108.50 ms

108.60 ms

108.62 ms

108.64 ms

108.66 ms

108.68 ms

108.70 ms

Fig. 3. (Contd.).

282


ZANGIEV et al.

108.80 ms

108.90 ms

109.00 ms

109.10 ms

109.20 ms

109.30 ms

109.40 ms

109.50 ms

109.60 ms

Fig. 3. (Contd.).



average fuel�based specific impulse decreases gradu�ally with increasing altitude, from 1850 s at the altitudeof 8 km to 1600 s at 18 km.

At altitudes above 18 km, the effective thrustbecame negative: for the basic�scheme engine with

nine turbulizing obstacles, the calculations pre�dicted the failure of DDT. In what follows, the flightaltitude above which a PDE of certain scheme can�not operate in a periodic mode, with DDT and pos�itive effective thrust, will be referred to as a limiting

109.70 ms

109.80 ms

109.90 ms

110.00 ms

110.10 ms

110.20 ms

110.30 ms

110.40 ms

110.50 ms

110.60 ms

Fig. 3. (Contd.).

284


ZANGIEV et al.

altitude. Thus, for the basic�scheme PDE with nineturbulizing obstacles, the limiting altitude is Z = 18 km.Table 1 and Fig. 5 show that the fuel�based specificimpulse of the PDE at the limiting altitudedecreases dramatically.

THRUST CHARACTERISTICS OF THE BASIC�SCHEME PDE WITH TEN OR ELEVEN

TURBULIZING OBSTACLES

It turned out possible to increase the maximumaltitude of flight of the PDE by simply increasing thenumber of annular obstacles in the combustion cham�ber of the basic�scheme PDE. For example, uponadding one extra turbulizing obstacle, DDT and posi�tive thrust were realized at altitudes of up to 24 km,with a specific impulse of the PDE being even slightlyhigher in comparison with that value at Z = 18 km.

The discovery of this interesting effect (influence ofthe number of obstacles on the operation of the PDE),we performed a series of systematic calculations forPDEs with one or two extra obstacles as compared tothe basic scheme; i.e., the total number of obstacleswas increased to 10 or 11, while retaining uniformspacing between them. The results for this series of cal�culations are presented in Table 2.

For the PDE with one additional obstacle, the lim�iting altitude was 24 km: at this altitude, no DDToccurred in the PDE combustion chamber because ofa low pressure. However, under these conditions, therewas a different mode of operation with positive effec�tive thrust: the mode with detonation initiation by thereflection of the shock wave generated by flame accel�eration from the tapering section of the nozzle. In thiscase, the detonation wave formed propagated

upstream and, passing through the flame front, turnedinto a shock wave. This regime, we called anomalous,is not considered here.

By increasing the number of turbulizing obstaclesin the combustion chamber of the basic�scheme PDEto 11, we were able to provide DDT and a positivethrust at altitudes of 26, and even 28 km, with the latterbeing considered as the limiting for this PDE. Since afully reproducible periodic operation under these con�ditions could be achieved only after the third cycle, allthe thrust characteristics of the PDE in flight condi�tions at altitudes 26 and 28 km were evaluated from theparameters of the fourth cycle.

The dashed lines in Figs. 4 and 5 represent the cal�culated dependences of the thrust force (Fig. 4) andaverage fuel�based specific impulse (Fig. 5) for thebasic�scheme PDE with 10 and 11 turbulizing obsta�cles. It can be seen that, at a given altitude of flight, thecalculated values of the thrust of the PDEs with 9(solid curve in Fig. 4) 10 or 11 (dashed curve in Fig. 4)obstacles are almost identical up to Z ≤ 18 km. Asnoted above, at Z > 18 km, the basic�scheme PDEwith nine obstacles cannot operate in the periodicmode with a positive effective thrust, whereas the PDEwith 10 or 11 obstacles can. As for the average specificimpulse at Z ≤ 16 km, the basic�scheme PDE with nineobstacles is more effective than the same PDE with 10or 11 obstacles.

The calculated values of the average fuel�basedspecific impulse for the basic�scheme PDE (1700–1800 s) at a flight speed of Mach 3 and altitudes of 8 to26 km (hereinafter, the limiting altitude is excludedfrom consideration) turned out to be comparable withthe fuel�based specific impulse of an ideal ramjetengine with conventional combustion, but operating

110.70 ms

110.80 ms

110.90 ms

111.00 ms

Fig. 3. (Contd.).



on a lean mixture with a fuel equivalence ratio of φ =0.7. According to the calculations performed in [8],the fuel�based specific impulse of an ideal ramjetengine at a speed of flight of Mach 3 at altitudes of 11to 29 km is 1770 s.

Tables 1 and 2 show that the calculated specific fuelconsumption Csp for basic�scheme PDE with nine tur�bulizing obstacles ranges within 0.19–0.21 kg/(N h) ataltitudes of up to 16 km and for the basic�schemePDEs with ten or eleven obstacles, within 0.20–0.22 kg/(N h) at altitudes of up to 26 km. Note that, ata flight speed of Mach 3 at an altitude of 11 to 29 km,the value of Csp for an ideal ramjet engine with conven�tional combustion of a lean FAM (φ = 0.7) is approx�imately the same, ~0.21 kg/(N h) [8]. Thus, specificfuel consumption of a basic�scheme PDE operatingon a stoichiometric fuel�air mixture is about on parwith an ideal ramjet operation on a lean FAM.

The values of the specific thrust for a PDE listed inTables 1 and 2 (from 1.01 to 1.19 kN/(kg/s)) at alti�tudes of 8 to 26 km turned out to be ~20–40% higherthan that for an ideal conventional combustion ramjet(~0.85 kN/(kg/s)) at the same speed and altitude [8].

EFFECT OF THE OPERATIONAL AND DESIGN PARAMETERS OF A PDE

ON ITS THRUST CHARACTERISTICS

To determine the influence of the operational anddesign parameters of a PDE on its thrust, we per�formed a series of parametric calculations for a PDEwith nine turbulizing obstacles under supersonic flight

conditions with a Mach number M = 3 at an altitudeof 16 km.

The operational parameters of the PDE that werevaried in the calculations were the fill factor of thecombustion chamber with gas mixture χ and the criti�cal pressure P* at the time of opening of the valve.

The PDE design parameters that were varied in thecalculations were the length of the receiver and thelength of a smooth section of the combustion cham�ber. To study the influence of the scale factor on thethrust of the engine, we also performed calculationsfor a PDE with a proportional change in all its geomet�ric dimensions.

Table 3 lists the results of the parametric calcula�tions. The following additional notations were intro�duced: scheme, the number of the PDE scheme((1) basic scheme with P* = 0.175 MPa, (2) basic schemewith χ = 0.5, (3) basic scheme with P* = 0.184 MPa,(4) scheme with doubled receiver length, (5) schemewith doubled length of the smooth section (withoutturbulizing obstacles) of the combustion chamber,(6) scheme with halved smooth section length, and(7) scheme with 1.5�fold increased sizes of all ele�ments of the PDE).

Table 3 shows that the decrease of the fill factor(scheme 2) from 0.9 to 0.5 (by 44%) increases theoperation frequency from 50 to 81 Hz (by 62%) andlowers the critical pressure of valve opening P* from0.175 to 0.1 MPa (by 43%), at almost constant valuesof the specific impulse, 1710 (1840) s, specific fuel con�sumption, 0.21 (0.17) kg/(N h), and specific thrust,1.10 (1.40) kN/(kg/s) (instead of 1.05 (1.14) kN/(kg/s)for the scheme 1), but decreases the thrust coefficient

0306 282624222018161412108

0.2

0.4

0.6

0.8

1.0

1.2Basic scheme Scheme with additional

Thrust force, kN

Altitude of flight, km

obstacles

Fig. 4. Calculated dependence of the average thrust of thebasic�scheme PDE with nine turbulizing obstacles (solidline) and basic�scheme PDEs with 10 and 11 obstacles(dashed line) on the altitude. The length of vertical barscorresponds to the difference between the calculated val�ues of the thrust for cold and hot blowing of the PDE.

2000

1400306 282624222018161412108

1500

1600

1700

1800

1900

Basic schemeScheme with additionalobstacles

Altitude of flight, km

Mean specific impulse, s

Fig. 5. Calculated dependence of the mean specificimpulse of the basic�scheme PDE with nine turbulizingobstacles (solid line) and basic scheme PDEs with 10 and11 obstacles (dashed line) on the altitude. The length of verti�cal bars corresponds to the difference between the calculatedvalues of the thrust for cold and hot blowing of the PDE.

286


ZANGIEV et al.

of СP from 0.76 (0.82) to 0.51 (0.65) (by 21–33%) andthe thrust force from 275 (296) to 185 (235) N (by 20–33%). Note that a further decrease in the fill factormakes the length of the combustion chamber filledwith FAM shorter than the detonation run�up distance,so that the periodic operation of the PDE with a positiveeffective thrust and DDT becomes impossible.

As the critical pressure P* was increased from 0.175to 0.184 MPa (by 5%) in scheme 3, the operation fre�quency increased from 50 to 60 Hz (20%). However,the specific impulse, in this case, decreased from 1710(1835) s for scheme 1 to 1580 (1800) s for scheme 3,with some increase in the specific fuel consumption,to 0.24 (0.20) kg/(N h), a decrease in the specific thrust,to 0.98 (1.12) kN/(kg/s) (versus 1.05 (1.14) kN/(kg/s)for scheme 1), and a slight change in the thrust coeffi�cient, from 0.76 (0.82) to 0.72 (0.87). The most signif�icant positive change in this case was the increase in

the effective thrust from 84 to 126 N (by 50%) and theincrease in the total thrust from 275 (296) to 287 (333) N(by 4–13%).

The lengthening of the receiver (scheme 4) did notlead to noticeable changes in the characteristics of thePDE, leaving the operation frequency, specific impulse,specific fuel consumption, thrust coefficient, and thrustforce at about the same level as in the basic scheme. Inthis case, only the critical pressure of valve opening P*increased significantly, from 0.175 to 0.233 MPa(by 33%).

The twofold lengthening of the smooth section ofthe combustion chamber (scheme 5) increases thethrust force from 275 (296) to 295 (349) N (by 7–18%)and the thrust coefficient from 0.76 (0.82) to 0.82(0.97) (by 8–18%), decreases the operation frequencyfrom 50 to 30 Hz (40%), increases the critical pressureof valve opening P* from 0.175 to 0.243 MPa (39%),

Table 2. Basic parameters and calculation results for PDEs with additional turbulizing obstacles in flight at a Mach number of 3 and various altitudes

Z, km Pa, MPa Ta, K f, Hz P*, MPa F, N R, N Isp, s CPRsp,

kN/(kg/s)Csp,

kg/(N h)

g/s

8a 0.036 236.2 50 0.406 266 906 (979) 1705 (1845) 0.73 (0.79) 1.06 (1.15) 0.22 (0.20) 54.1

10a 0.027 223.3 50 0.312 228 696 (757) 1715 (1865) 0.75 (0.82) 1.09 (1.18) 0.21 (0.20) 41.1

12a 0.0194 216.7 50 0.232 176 516 (564) 1710 (1870) 0.76 (0.83) 1.06 (1.17) 0.22 (0.20) 30.9

16a 0.010 216.7 50 0.159 121 297 (329) 1680 (1840) 0.82 (0.91) 1.05 (1.14) 0.22 (0.20) 15.1

18a 0.0075 216.7 50 0.091 63 197 (215) 1670 (1820) 0.75 (0.81) 1.01 (1.12) 0.22 (0.21) 12.2

20a 0.0055 216.7 50 0.068 45 143 (156) 1655 (1800) 0.74 (0.81) 1.05 (1.15) 0.22 (0.20) 8.6

22a 0.004 216.7 50 0.048 32 103 (114) 1660 (1810) 0.74 (0.82) 1.03 (1.15) 0.22 (0.20) 6.5

24a,b 0.003 220.6 49 0.032 21 75 (81) 1650 (1790) 0.72 (0.78) 1.03 (1.12) 0.22 (0.21) 4.3

26c 0.002 222.7 49 0.023 14 53 (57) 1630 (1790) 0.71 (0.76) 1.02 (1.10) 0.22 (0.21) 3.4

28c 0.0016 224.7 53 0.015 4 34 (37) 1430 (1585) 0.60 (0.66) 0.89 (0.99) 0.26 (0.23) 2.4

Note: (a) The number of turbulizing obstacles is 10; (b) an anomalous regime is possible; (c) the number of obstacles is 11.

,fm�

Table 3. Results of parametric calculations of the thrust characteristics of the PDE in flight at a Mach number of 3 and an altitude of 16 km

Scheme χ τ, ms f, Hz P*, MPa F, N R, N Isp, s CPRsp,

kN/(kg/s)Csp,

kg/(N h) g/s

1 0.9 20 50 0.175 84 275 (296) 1710 (1835) 0.76 (0.82) 1.05 (1.14) 0.22 (0.20) 16.5

2 0.5 12 81 0.100 27 185 (235) 1710 (1840) 0.51 (0.65) 1.10 (1.40) 0.21 (0.17) 10.8

3 0.9 17 60 0.184 126 287 (333) 1580 (1800) 0.72 (0.87) 0.98 (1.12) 0.24 (0.20) 18.7

4 0.9 20 50 0.233 77 283 (295) 1720 (1785) 0.78 (0.82) 1.09 (1.14) 0.21 (0.20) 16.5

5 0.9 33 30 0.243 74 295 (349) 1375 (1630) 0.82 (0.97) 0.87 (1.03) 0.26 (0.22) 21.6

6 0.9 12.5 80 0.117 73 253 (282) 1690 (1880) 0.70 (0.78) 1.06 (1.19) 0.21 (0.19) 15.1

7 0.9 32 31 0.164 187 649 (690) 1670 (1780) 0.80 (0.85) 1.04 (1.11) 0.22 (0.20) 39.6

,fm�



reduces the specific impulse from 1710 (1835) s to1375 (1630) s (11–20%), increases the specific fuelconsumption from 0.22 (0.20) to 0.26 (0.22) kg/(N h)(10–18%), and lowers the specific thrust from 1.05(1.14) to 0.87 (1.03) kN/(kg/s) (by 10–21%).

The twofold shortening of the smooth section ofthe combustion chamber (Fig. 6) compared to thebasic scheme is accompanied by the increase in theoperation frequency from 50 to 80 Hz (60%) and thedecreases in the critical pressure of valve opening P*from 0.175 to 0.117 MPa (33%), specific impulse from1710 (1835) s to 1690 (1880) s, thrust force from 275(296) N to 253 (282) N (by 5–8%), the thrust coeffi�cient from 0.76 (0.82) to 0.70 (0.78) (5–8%), specific fuelconsumption from 0.22 (0.20) to 0.21 (0.19) kg/(N h),with specific thrust increasing from 1.05 (1.14) to 1.06(1.19) kN/(kg/s).

It was expected that the 1.5�fold proportionalincrease of all sizes of the PDE (Fig. 1) would bringabout a proportional change in the parameters:f = f0/1.5 = 33 Hz, F = F0 ⋅ 1.52 = 190 N, R = R0 ⋅ 1.52 =619–666 N, = ⋅ 1.52 = 37 g/s at a constantthrust coefficient. However, in reality, the change of allsizes of the PDE increased the thrust force by 135%instead of the expected 125% (from 275 (296) to 649(690) N) and the thrust coefficient (from 0.76 (0.82) to0.80 (by 0.85)), but decreased the specific impulse from1710 (1835) s to 1670 (1780) s (2–3%), the critical pres�

fm� � 0fm

sure of valve opening P* from 0.175 to 0.164 MPa (6%),and the specific thrust from 1.05 (1.14) to 1.04(1.11) kN/(kg/s) (1–2%), and the operation fre�quency from 50 to 31 Hz (by 38%), with the specificfuel consumption remaining nearly constant, 0.22(0.20) kg/(N h).

EFFECT OF THE COMBUSTION MODE

Table 4 lists the results of calculations performed todetermine the effect of the combustion mode on thecharacteristics of a PDE in flight at an altitude of16 km and Mach number of 3. For ease of comparisonof the results, the first row of Table 4 replicates the cal�culation results for the basic�scheme PDE with DDT.In addition to this calculation, we performed threecalculations for different organizations of the combus�tion process in the engine (calculations 2–4 in Table 4).In the calculations 2 and 3, only the acceleration of theflame at the obstacles was considered, without regardfor the bulk preignition reactions, i.e., only the contri�bution of frontal combustion to the PDE characteris�tics was taken into account. In calculation 4, a directinitiation of detonation was considered with allowanceonly for the contribution of bulk energy release to thePDE characteristics. Both the modes were realized bycutting off one of the modules in the combustionmodel (PM or ETFF). In this case, the parameters ofcalculations 2 and 4 in Table 4 were fully consistent

(a) (b)

Fig. 6. Fragments of the (a) standard and (b) fine computational grids.

Table 4. Results of calculations of the thrust characteristics of the PDE in flight at a Mach number of 3 and an altitude of 16 km when the PDE operates in the modes of direct initiation and fast deflagration

Combustion mode

Z, km

Pa, MPa

Ta, K f, Hz P*, MPa F, N R, N Isp, s CP

Rsp, kN/(kg/s)

Csp, kg/(N h)

g/s

DDT 16 0.010 216.7 50 0.175 84 275 (296) 1710 (1830) 0.76 (0.82) 1.05 (1.14) 0.22 (0.20) 16.6

Combustiona 16 0.010 216.7 60 0.142 3 193 (225) 1090 (1270) 0.54 (0.62) 0.68 (0.8) 0.34 (0.29) 18.1

Combustionb 16 0.010 216.7 140 0.1 –13 159 (190) 1470 (1750) 0.44 (0.53) 0.94 (1.12) 0.24 (0.21) 10.8

Detonationc 16 0.010 216.7 55 0.11 51 241 (270) 1580 (1770) 0.67 (0.75) 0.91 (0.97) 0.24 (0.21) 15.8

Note: (a) χ = 0.9, (b) χ = 0.4, and (c) direct initiation of detonation.

,fm�

288


ZANGIEV et al.

with the parameters of calculation 1, whereas calcula�tion 3 differed from calculation 1 only by the fill factorof the PDE combustion chamber with gas mixture,χ = 0.4.

Table 4 demonstrates that calculation 2 yields a rel�atively low specific impulse, 1090 (1270) s, high spe�cific fuel consumption, 0.34 (0.29) kg/(N h), and lowspecific thrust, 0.68 (0.80) kN/(kg/s), as compared tocalculation 1. These low characteristics arise primarilydue to the outflow of some unreacted FAM from theengine into the atmosphere together with the shockwave.

To avoid the outflow of unreacted FAM into theatmosphere, the fill factor of the combustion chamberin calculation 3 was decreased from 0.9 to 0.4. Thisensured a complete burnout of the FAM. Table 4shows that the effective thrust in this case is negative;i.e., the thrust developed by the PDE is insufficient toovercome the drag force.

Detonation in calculation 4 was initiated by artifi�cially raising the temperature of one cross�layer ofcomputational cells near the first turbulizing obstacleimmediately after closing the valve. As can be seen forTable 4, a direct initiation of detonation does not gen�erally improve the thrust characteristics of the PDE.The same effect was observed in the calculations car�ried out in [9]. According to [9], it is associated with anincrease in the total loss of momentum during thepropagation of detonation in a channel with turbuliz�ing obstacles.

EFFECT OF THE COMPUTATIONAL GRID ON THE THRUST CHARACTERISTICS

To determine the effect of the computational gridon the thrust characteristics of the basic�scheme PDEwith nine obstacles, we performed calculations forflight at a Mach number of М = 3 and altitude of 16 kmon a fine grid with a twofold reduced average cell size(Fig. 6) and compared to the results obtained at thestandard cell size.

The results on these two different grids are pre�sented in Table 5. It can be seen that the specific thrustcharacteristic of the PDE, Isp, Rsp, Csp, and the thrustcoefficient СP remained virtually unchanged.

CONCLUSIONS

Thus, the main thrust characteristics, such as thethrust force, fuel�based specific impulse, specific fuelconsumption, specific thrust, and thrust coefficient,of a PDE with an air intake and nozzle were calculatedfor flight at a Mach number of 3 and different altitudes(from 8 to 28 km above sea level), with account of thephysicochemical characteristics of the oxidation andcombustion of hydrocarbon fuel (propane) and ofDDT and the finite time of turbulent flame accelera�tion. In addition a parametric analysis of the influenceof the operation�mode and design parameters of thePDE on its thrust characteristics in flight at Mach 3and altitude of 16 km was performed.

It was shown that, under these conditions, the PDEoperates at a high frequency (~50–80 Hz) with weak�source ignition (~0.1 J) and DDT. The specificimpulse and specific fuel consumption for a PDEoperating on a stoichiometric propane–air mixturerange from 1700–1800 s, and 0.19–0.21 kg/(N h),respectively, in flight at altitudes up to 26 km, parame�ters very close to those for an ideal ordinary�combus�tion ramjet operating on a lean FAM, with a fuelequivalence ratio of 0.7. As for the specific thrust of thePDE, its calculated values for these conditions turnedout to be 18–38% higher than those of an ideal con�ventional�combustion ramjet. It was shown that aneffective operation of a PDE at an altitude of morethan 18 km can be ensured by increasing the numberof turbulizing obstacles.

Parametric calculations showed that the thrustdeveloped by the basic�scheme PDE can be enhancedby proportionally increasing all the geometric dimen�sions of the engine. In this case, the specific thrustcharacteristics changes only slightly, but the maximumoperation frequency decreases significantly. It turnedout that a very effective way to increase the operationfrequency of the PDE is to reduce the degree of fillingof the combustion chamber with FAM. Although thisreduces the thrust, the specific impulse remainsalmost unchanged. Special calculations showed that adirect initiation of detonation in the PDE reduces itsthrust characteristics as compared to the same PDEscheme with DDT. This is due to an increase in thetotal loss of momentum during the direct initiationand propagation of detonation in a channel with tur�bulizing obstacles. It was demonstrated that the thrust

Table 5. Results of calculations of the thrust characteristics of the PDE in flight at a Mach number of 3 and an altitude of 16 km by using different computational grids

Grid Z, km Pa, MPa Ta, K f, Hz P*, MPa F, N R, N Isp, s CPRsp,

kN/(kg/s)Csp,

kg/(N h) g/s

Standard 16 0.010 216.7 50 0.175 84 275 (296) 1710 (1830) 0.76 (0.82) 1.05 (1.14) 0.22 (0.20) 16.6

Fine 16 0.010 216.7 50 0.135 70 263 (284) 1710 (1850) 0.73 (0.79) 1.05 (1.15) 0.22 (0.20) 15.9

,fm�



characteristics of an engine operating on fast deflagra�tion are inferior to those of a PDE with DDT.

Thus, our calculations for the first time showedthat, in conditions of supersonic flight at a Mach num�ber of 3 and altitudes of up to 26 km, a PDE with theindicated scheme is much superior to an ideal conven�tional�combustion ramjet in specific thrust, with itsspecific impulse and specific being not poorer thanthose of an ideal ramjet.

ACKNOWLEDGMENTS

We are grateful to V.V. Vlasenko (Central Aerohy�drodynamic Institute) for fruitful discussions.

This work was supported by the Russian Founda�tion for Basic Research, project no. 11�08�01297.

REFERENCES

1. S. M. Frolov, in Pulse Detonation Engines (Torus Press,Moscow, 2006), p. 19 [in Russian].

2. V. S. Ivanov and S. M. Frolov, Russ. J. Phys. Chem. B5, 597 (2011).

3. K. Kailasanath, AIAA J. 38, 1698 (2000).

4. G. D. Roy, S. M. Frolov, A. A. Borisov, and D. W. Netzer,Progress Energy Combust. Sci. 30, 545 (2004).

5. W. P. Jones and B. E. Launder, Int. J. Heat Mass Trans�fer 15, 301 (1972).

6. S. V. Patankar and D. B. Spalding, Int. J. Heat MassTransfer 15, 1787 (1972).

7. B. E. Launder and D. B. Spalding, Comput. MethodsAppl. Mech. Eng. 3, 269 (1974).

8. J. Kurzke, RTO�AVT�VKI Lecture Series 2010�AVT185 (Von Karman Inst. Fluid Dynamics, Belgium,2010), p. 2.1.

9. S. M. Frolov and V. S. Ivanov, in Deflagrative and Deto�native Combustion, Ed. by G. Roy and S. Frolov (Torus,Moscow, 2010), p. 133.

Translated by V. Smirnov

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