Numerical Simulation of the Operation Process and Thrust ...frolovs.ru/pdf/2011-19-eng.pdf · reactions in a turbulent flame and the preheat zone. COMBUSTION, EXPLOSION, AND SHOCK

ISSN 1990�7931, Russian Journal of Physical Chemistry B, 2011, Vol. 5, No. 4, pp. 597–609. © Pleiades Publishing, Ltd., 2011.Original Russian Text © V.S. Ivanov, S.M. Frolov, 2011, published in Khimicheskaya Fizika, 2011, Vol. 30, No. 7, pp. 48–61.

597

INTRODUCTION

In 1940, Zel’dovich, having considered the ther�modynamic efficiency of different modes of combus�tion in ramjet engines [1] showed that the highestextent of conversion of the chemical energy of fuelinto useful work is achieved in the detonation combus�tion mode. It is known that, for the burning of a com�bustible mixture of a given composition under thesame initial conditions, the minimum value of theentropy of the combustion products is achieved in thedetonation mode. This means that, for an ideal (isen�tropic) expansion of the detonation products into theatmosphere, the amount of irreversible loss will beminimal compared to other modes of combustion(e.g., at constant pressure or constant volume).

One possible schemes for realizing detonationcombustion in a ramjet engine is to periodically fill thecombustion chamber with a combustible mixture andto initiate detonation, thereby burning the mixture ina running detonation wave [2, 3]. Such a ramjetengine, known as an air�breathing pulse detonationengine (ABPDE) is very promising in its potentialthrust performance. Estimates show [4, 5] that thespecific impulse of an ABPDE operating on hydrogenor a hydrocarbon fuel can reach 5500 and 2500 s,respectively, over a wide range of flight Mach numbers,from 0 to 4–5. Therefore, extensive scientific research

and practical development of such aircraft propulsioninstallations are now under way.

In the published works, the characteristics ofABPDEs have been estimated by grossly simplifying ofthe operation process. In particular, the flow in thecombustion chamber is usually considered one�dimensional or quasi�one�dimensional, the combusti�ble mixture (primarily hydrogen–air) is assumed to bepremixed, and the initiation of detonation is postu�lated to occur instantaneously. In fact, the flow in theABPDE is not one�dimensional, whereas the pro�cesses of fuel–oxidizer mixing and detonation initia�tion take a finite time. In addition, to directly initiatedetonation in mixtures of air with different fuels, it isnecessary to use sources with high energy and power[6], which is unacceptable for aircraft. Therefore,instead of direct initiation of detonation with intensesources, the tendency now is to use weak ignitionsources with subsequent transfer of deflagration todetonation by various means of flame acceleration [7].There are several publications (see, e.g., [8–11]) thatoffer various methods for numerical simulation ofdeflagration�to�detonation transition (DDT). Unfor�tunately, to quantitatively evaluate the ABPDE thrustperformance, these methods are unacceptable, sincethey use models of combustion not taking into accountthe fundamental differences between the chemicalreactions in a turbulent flame and the preheat zone.

COMBUSTION, EXPLOSION, AND SHOCK WAVES

Numerical Simulation of the Operation Process and Thrust Performance of an Air�Breathing Pulse Detonation Engine

in Supersonic Flight ConditionsV. S. Ivanov and S. M. Frolov

Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russiae�mail: [email protected]

Received June 25, 2010

Abstract—Multidimensional simulations of the unsteady gasdynamic flow in the duct of an air�breathingpulse detonation engine (ABPDE) operating on propane gas and the flow around it in supersonic flight atMach numbers M of 3.0 and an altitude of 9.3 and 16 km are performed. It is shown that, at a length anddiameter of the duct of 2.12 m and 83 mm, respectively, an ABPDE with an air intake and a nozzle can oper�ate in a cyclic mode at a repetition frequency of 48 Hz, with a rapid deflagration�to�detonation transition(DDT) occurring at a distance of 5–6 combustion chamber diameters. To determine the thrust performanceof the ABPDE in flight conditions, a series of working cycles were simulated with consideration given to theexternal flow around the engine. Calculations showed that the specific impulse of the ABPDE is approximately1700 s. This value is much higher than the specific impulse typical of ramjet engines operating on conven�tional combustion (1200–1500 s) and substantially lower than the specific impulse obtained for the atmo�spheric conditions at sea level at zero flight velocity (~2500 s).

Keywords: deflagration�to�detonation transition, pulse detonation engine, specific impulse.

DOI: 10.1134/S1990793111040075

598

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B Vol. 5 No. 4 2011

IVANOV, FROLOV

In this work, we for the first time tried to simulatethe thrust performance of prospective ABPDEs withan air intake and a nozzle operating in the DDT modeon hydrocarbon fuels under supersonic flight condi�tions. The basis for multidimensional numerical sim�ulations of the operation process of the ABPDE was asoftware package developed by the authors. The pack�age makes it possible to perform multidimensionalnumerical simulation of turbulent combustion, DDT,and detonation of explosive gas mixtures in channelsof complex configuration. This became possible due tothe development and implementation of a new algo�rithm for explicit tracking of the flame front (ETFF)and a new algorithm for calculating bulk energyrelease based on the particle method [12]. The soft�ware package includes extensive databases of the ther�mophysical properties of substances, laminar flamespeeds at different initial temperatures, pressures, andcompositions of alkane hydrocarbons–air mixtures(from methane to normal tetradecane), and chemical�kinetic parameters for calculating preflame ignition.Unlike the other existing methods for modeling DDT[8–11], the new algorithm takes into account themajor features of the chemical reactions in flames andin preflame regions: the flame reactions occur in thepresence of heat and diffusion fluxes, whereas thereactions in a preflame region are dominated by chaininitiation.

Note that the new algorithm was previously used tocalculate the thrust performance of ABPDEs in atmo�spheric conditions at sea level at zero flight speed [12].It was shown that the specific impulse of the ABPDEoperating on a stoichiometric propane–air mixture inthe DDT mode is ~2500 s.

PHYSICAL STATEMENT OF THE PROBLEM

Figure 1 shows a schematic diagram of an two�cir�cuit ABPDE, 2.12 m in length and 83 mm in outerdiameter, with an air intake, receiver, annular bypasschannel (second circuit), and combustion chamber(first circuit) equipped with a mechanical valve andnozzle. The engine running on propane gas is blownon by a steady supersonic airflow with a Mach numberof M = 3 at a height Z.

The supersonic air intake is designed according tothe scheme proposed in [13]. Adjacent to the air intakeis the receiver, a cylindrical volume intended forsmoothing the wave processes induced by valve open�ings and closings.

The combustion chamber of the ABPDE (first cir�cuit) is a tube consisting of two sections: an expandingconical section with a maximum diameter of d = 83 mmand a cylindrical section of the same diameter. Thetube of the first circuit is recessed into the tube of thesecond circuit so as to leave an annular gap for the out�flow of air from the receiver. The initial section of thecombustion chamber accommodates regularly placedrings with a blockage ratio of 0.3. The distancesbetween ring in the conical and cylindrical parts were50 and 82 mm, respectively. The rightmost obstaclewas positioned 510 mm from the ignition source. Thisobstacle and the nozzle are separated by a smoothcylindrical section of length 1000 mm.

The left end of the combustion chamber isequipped with a mechanical valve. In the positionshown in Fig. 1, the valve is closed, and the entire air�flow passing through the air intake and receiver isdirected to the second circuit. With the valve opened,the entire airflow is directed into the combustionchamber. Due to the specific shape of the inlet sectionof the second circuit, the mechanical valve blocks 73%of the flow when the inlet into the combustion cham�

82

(b)

150 1543 124302

Air intake Receiver Valve Second circuit

Ignition

Nozzle

Fuel supply

83 72 5266 4283

(a)

Fig. 1. (a) Schematic diagram of the simulated two�circuit ABPDE with an air intake and a nozzle and (b) its sizes (in mm).


NUMERICAL SIMULATION OF THE OPERATION PROCESS AND THRUST PERFORMANCE 599

ber is closed, and 48% when it is opened. The right endof the combustion chamber is equipped with a super�sonic nozzle with a critical cross�section diameter of~42 mm.

The cyclic operation process of the ABPDEincludes three stages:

In the first stage, when the valve is opened, thecombustion chamber is filled with fuel–air mixture(FAM). Fuel is fed into a certain section of the com�bustion chamber, located before the first annularobstacle (indicated by the vertical line in Fig. 1a). Toexclude a direct contact of the fresh FAM with the hotcombustion products of the previous cycle, fuel is fedinto the air stream with a certain delay relative to thetime of opening of the valve.

When the combustion chamber is filled with FAM,the valve instantly closes and the second stage of theoperation process begins. The fuel–air mixture in thecombustion chamber is ignited by an annular source inthe recirculation zone behind the first obstacle (indi�cated by the bold points in Fig. 1). The resulting flamepropagates in the turbulent FAM flow with accelera�tion and eventually gives rise to DDT. The detonationwave front propagates downstream and exits throughthe nozzle into the surrounding atmosphere.

The third stage of the operation process is the out�flow of combustion products. This stage lasts until thepressure on the valve from the combustion chamberdrops to a preset value that still ensures a positivethrust. After achieving , the valve instantly opensand the cycle repeats itself.

The problem to be solved in the present work is cal�culate the thrust performance of an ABPDE with anair intake and nozzle in supersonic flight conditionwith account of all the physical and chemical charac�teristics of the oxidation and combustion of propaneand finite times of turbulent flame acceleration andDDT. Note that the mechanical valve in this scheme ofthe ABPDE is used only to simplify the analysis. In thefuture, we plan to extend the analysis to a valvelessABPDE design (see, e.g., [13]).

MATHEMATICAL MODEL

The flow of a viscous compressible gas inside andoutside the ABPDE was described by the Reynolds�averaged two�dimensional unsteady Navier–Stokes,energy conservation, and species continuity equations[14]:

(1)

P*P*

( )

i i ij

j

ij i ji j

DU U UU

Dt t x

P U Ux x

∂ ∂ρ = ρ + ρ

∂ ∂

∂ ∂= − + τ − ρ

∂ ∂' ' ,

(2)

(3)

where is the time; is the coordinate; isthe mean density; is the mean pressure; is the

dynamic viscosity; is the mean velocity; is ther.m.s. fluctuating velocity component; is the viscous

stress tensor; is the mean total

enthalpy ( is the mean static enthalpy); is the ther�mal conductivity; is the mean temperature;

is the mean mass fraction of the lth spe�cies of the mixture ( is the total number of speciessin the mixture); is the molecular diffusion coeffi�cient of species l; is the r.m.s. fluctuation of the massfraction of lth species, and are the mean sourceterms of matter and energy (e.g., chemical reactions).

The turbulent fluxes of matter, momentum, and energyin (1)–(3) were modeled using the standard k–ε turbu�

lence model with Here, is the kinetic energy of turbulence, and ε is its dissipation.

Modeling of the chemical sources, and for tur�bulent combustion and DDT requires taking intoaccount the contributions from both frontal combus�tion (index f) and bulk preflame reactions (index V):

To determine and , we used the ETFF [7, 8].

The contributions from bulk reactions, and weredetermined using the particle method (PM) [12, 15, 16].

jj

ij jj j i

DI I IUDt t x

P TQ Ut x x x

∂ ∂ρ = ρ + ρ∂ ∂

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

l l lj

j

ll l l j

j i

DY Y YU

Dt t x

Yr D Y U

x x

∂ ∂ρ = ρ + ρ∂ ∂

⎛ ⎞∂∂= ρ + ρ − ρ⎜ ⎟∂ ∂⎝ ⎠' ' ,�

t jx j =( 1, 2) ρ

P µ

iU iU '

ijτ

i

i

I H U= + ∑ 21 2

H λ

T( 1, ..., )=lY l N

N

lD

lY '

lr� Q�

( )iU U k i= = =

1 2' ' 2 3 ( 1, 2, 3).k

lr� Q,�

l lf lVr r r= +� � � ,

f VQ Q Q= +� � � .

lfr� fQ�

lVr� VQ� ,

Active particles Inactive particles

Flame front Shock wave front

Fig. 2. Schematic diagram illustrating the particle methodfor simulating deflagration�to�detonation transition.

600


IVANOV, FROLOV

Pressure, Pа

9.1e + 02 6.5e + 04 1.3e + 05 1.9e + 05 2.6e + 05 3.2e + 05 3.9e + 05 4.5e + 05 5.2e + 05 5.8e + 05 6.5e + 06

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 3. Calculated pressure distributions (isobars) in the ABPDE receiver (a) before and (b–h) after valve closure in the secondoperation cycle: (a) 90.0 ms, (b) 90.1 ms, (c) 90.4 ms, (d) 90.7 ms, (e) 91.0 ms, (f) 92.0 ms, (g) 94.0 ms, and (h) 96.0 ms.



С3H8, wt %

0 0.006 0.012 0.018 0.024 0.030 0.036 0.042 0.048 0.054 0.060

Temperature, K

200 430 660 890 1120 1350 1580 1810 2040 2270 2500

Temperature, K

Temperature, K

200 480 760 1040 1320 1600 1880 2160 2440 2720 3000

200 480 760 1040 1320 1600 1880 2160 2440 2720 3000

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4. Calculated (a) propane mass fraction distribution during blow�through of the combustion chamber and (b–f) evolution oftemperature distribution: (b) immediately after ignition, (c) during flame acceleration, (d) immediately before DDT, (e) imme�diately after DDT, and (f) during detonation wave propagation: (a) 82 ms, (b) 90.2 ms, (c) 90.4 ms, (d) 90.56 ms, (e) 90.6 ms,and (f) 90.8 ms.

System (1)–(3) in conjunction with the k–ε turbu�lence model and the coupled ETFF–PM model wereclosed by the caloric and thermal equation of state ofan ideal gas with variable specific heat and supplementedby the initial and boundary conditions. All the thermo�physical parameters of the gas were considered variable.

Let us describe the ETFF–PM model in moredetail. The ETFF algorithm is based on the flameletmodel and the Huygens superposition principle. Theflame front is thought of as a set of infinitely thin ele�mentary areas (elements) separating the fresh mixturefrom the combustion products. In a turbulent flow,

602


IVANOV, FROLOV

each ith element of the flame front moves with localinstantaneous velocity equal to the sum of the tur�bulent burning velocity , and the flowvelocity : Here, is the laminar flamespeed, S is the surface area of the curved flame front,and Smi is the surface area of the ith element of theflame; the length of the velocity vector is defined as

The local turbulent burning velocity can bedetermined using one of the known turbulent flamemodels, for example, the Shchelkin model [17],

(4)

which suggests that depends on the local laminarflame speed and local velocity fluctuation. In theETFF algorithm, the laminar flame speed is takenfrom detailed spreadsheets as a function of the initialtemperature, pressure, and mixture composition [18].Such tables are based on solving the problem of thestructure of a flat laminar flame, using either detailedor semi�empirical kinetic mechanisms, and containinformation on the flammability limits. As regards thecomponents of the local flow velocity, and theyare determined by solving the system of averagedequations of flow using a special interpolation proce�dure.

Thus, the ETFF algorithm makes it possible todetermine the surface area of all the elements of theflame Smi and the corresponding values of the turbu�lent burning velocity at each moment of time for eachchosen volume (computational cell) and, therefore, tocalculate the contribution of the frontal combustion,

and , to the chemical source terms, and

where is the mean mass fraction of species l in thepreheat zone, is the computational cell volume, and

is the heat effect of the combustion reactions; thesummation is performed over all elements of the flamefront in the cell.

One of the advantages of the ETFF algorithm is thepossibility of using it for calculating the initial stage ofpropagation of the flame front (when the flame is lam�inar) and the subsequent stage of flame acceleration,when the flame is curved by turbulent fluctuations.Indeed, the difference between the used models of tur�bulent and laminar burning of a gaseous homogeneousmixture lies only in using the speed ut instead of un. Inaddition, the known formulas for calculating ut give, asa rule, an asymptotic transition from turbulent to lam�inar flame front propagation in the absence of turbu�lence (when 0, ).

u fi

ti n miS S=u u

iu fi ti i= +u u u . nu

i i i= +u U U'.

tu

( )t n nu u U u≈ +

1 22 2'1 ,

tu

U U ',

lfr� fQ� lr� :�Q

lf l mi ti

i

r Y V S u−

= ∑�

1 ,

f lfQ Qr=�

� ,

lYV

Q

U →' t nu u→

In the PM algorithm, used to calculate the contri�butions from bulk preflame reactions, and , tochemical source terms, the instantaneous local statesof a turbulent reacting flow are presented as a set ofinteracting (Lagrangian) particles. Each ith particle

has individual properties: the position in space andthree local instantaneous velocity components

, volume density the static

enthalpy mass fractions of species ,

and the statistical weight used in determining themean values of the variables over the ensemble of par�ticles. For each ith particle, the following system ofequations is solved [12]:

(5)

(6)

(7)

(8)

where is the partial density of the lth species

in the ith particle, is the change of the mass con�centration of the ith species in the course of chemical

reactions, is the mean pressure at the point of loca�

tion of the ith particle, is the rate of energy release

by chemical reactions in the ith particle, is the dif�

fusion flux of the lth species to the ith particle isthe momentum flux to the ith particle due to molecu�

lar viscosity, and is the heat flux to the ith particle.

To determine the flux (exchange) terms ,

and in the PM, the classical model of linear decayto the mean is used [15]:

(9)

(10)

(11)

where С ≈ 2.0 and are coefficients; ,

and are the mean mass fraction of lth species, meanvelocity, and mean enthalpy of the gas at the point oflocation of the ith particle, respectively; is thefrequency of turbulent fluctuations; A(t) =

is the random function describingthe influence of fluctuations of pressure and velocityon the motion of the particle; here, is acontinuous random variable having normal distribu�

lVr� VQ�

ikx

iku k =( 1, 2, 3) iV , i

ρ ,ih , i

ly l N=( 1,..., )iw ,

iikk

dxu

dt= ,

( ),

ρ= ∇ +

i ii ill hl

d VJ J

dti i

i ik

k

du Pdt x

∂ρ = − ∇τ

∂,

ii ii i i i i k

Vk

udh Pq Q Pdt t x

∂∂ρ = −∇ + ρ + −

∂ ∂� ,

i i il lyρ = ρ

ihlJ

iPi

VQ�

∇ilJ

i∇τ

iq∇

ilJ∇ ,

i∇τ

iq∇

i i i i il l lJ C y Y V∇ = − − ρ ω0.5 ( ) ,

i i i i ik kP E u U A t−

ρ ∇ − τ = −ζ − +1( ) ( ) ( ) ( ),

i i i iq C h H∇ = − − ρ ω0.5 ( ) ,

ζ ≈ ω2.075 ilY , i

kUiH

kω = ε

( )C dW t dtε1/2

0( )

C ≈0 2.1, W



tion and satisfying the condition and

( is the Kronecker symbol).

The mean quantities , and are determinedeither by interpolating the corresponding values of

, and (obtained from the solution of averagedequations (1)–(3)) or by averaging over the ensembleof particles using the formulas

where the statistical weight of ith particle is given by

The mean pressure field and the local fre�quency of turbulent fluctuations required for solv�ing the system of equations (5)–(8) supplemented byadditional relations (9)–(11), are determined by solv�ing the averaged equations (1)–(3) and the equationsof the k–ε turbulence model. The most importantadvantage of the PM is the possibility of accuratelydetermining the rates of chemical reactions in a turbu�

lent flow: the source terms and are determinedusing the known mechanisms of chemical reactions

and the instantaneous mass fraction of

and the temperature The instantaneous local rate of change of the mass

concentration of the lth species in the ith particle iscalculated by the formula

where is the molecular mass of the lth species;

and are the stoichiometric coefficients for the lthspecies acting as a reagent or product in the kth reac�tion, respectively; and , and are the preexpo�nential factor, the temperature exponent, and the acti�vation energy for the kth reaction; is the universalgas constant; and are, respectively, the total num�bers of reactions and species in the chemical mecha�nism.

The rate of energy release by chemical reactions inthe ith particle is calculated by the formula

where is the heat effect of the kth chemical reac�tion.

idW t =( ) 0

i j ijdW t dW t dt= δ( ) ( ) ijδ

ilY , i

kU iH

lY ,

kU H

i i i i i i i i il l k k

i i i

Y w y U w u H w h= = =∑ ∑ ∑, , ,

i i i i i

i

w V V= ρ ρ∑ .

kP t x( , )ω,

ihlJ i

VQ�

ily l N=( 1,..., )

iθ .

( )

k

j k

Li i i nhl l l k l k k

k

N i ijk

ijj

J V W A

yEWR

=

ν

=

= ν − ν θ

⎛ ⎞ρ× − ⎜ ⎟

θ ⎝ ⎠

∑

∏,

, ,

1

'

1

'' '( ) ( )

exp ,

lW l kν ,' ,

l kν ,''

kA , kn kE

RL N

j k

k

NL i ii i n jk

V k ki ijk j

yEQ H A

WR

ν

= =

⎛ ⎞ρ⎛ ⎞= θ −⎜ ⎟ ⎜ ⎟⎝ ⎠ρ θ ⎝ ⎠

∑ ∏,'

1 1

1 ( ) exp ,

kH

Knowing and , one can determine the con�tribution of bulk reactions, and , to the chemical

source terms and

METHOD OF SOLUTION

To numerically solve the problem, the coupledETFF–PM model supplemented by spreadsheets oflaminar flame speeds and the kinetic mechanism ofthe preflame oxidation of propane in the particles wereincorporated into the FIRE gasdynamic package.Averaged equations (1)–(3), supplemented by the k–εturbulence model, the equations of state, the coupledETFF–PM model, and the initial and boundary con�ditions were solved numerically by the finite volumemethod, using the successive approximations withpressure correction (SIMPLE�method). The effect ofsolid surfaces on the flow characteristics wereaccounted for through by wall functions.

The combustible gas was propane, which is oftenused as a model hydrocarbon fuel for ABPDEs. Fuelsupply was simulated by introducing a source of massinto all computational cells in the section indicated inFig. 1. The source of mass was selected so that thecombustion chamber would be filled with stoichio�metric propane–air mixture. The combustible mix�ture was ignited at the periphery of the combustionchamber, as shown in Fig. 1. The initial kernel of igni�tion was in the form a quarter�circle with a radius of1 mm instantaneously filled with the combustionproducts of a stoichiometric propane–air mixture at atemperature of 2200 K and a pressure equal to thelocal pressure in the flow.

At the instant of ignition, particles used in the PMfor simulating the preflame processes were randomlyplaced ahead of the flame front. The mean numberdensity of particles was 10 particles per cell. Each indi�vidual particle was characterized by the initial values ofthe variables corresponding to the interpolated meanvalues at the location of the particle, except the veloc�ity, which consisted of the interpolated mean velocityand velocity fluctuation. The latter was determinedusing the interpolated value of the turbulent kineticenergy and the hypothesis of isotropic turbulence witha normal distribution probability density of velocityfluctuations. The PM equations (5)–(8), were solvedusing an explicit scheme and random�number genera�tors. The number of particles in each control volumewas maintained at a given level by means of specialprocedures of cloning and clustering of particles with�out violating the statistical laws.

ihlJ i

VQ�

lVr� VQ�

lr� :�Q

i i i ilV hl

i i

r w J V= ρ∑ ∑ ,�

i iV V

i

Q w Q=∑� � .

604


IVANOV, FROLOV

Table 1 represents a fragment of the spreadsheets of

the laminar flame speed used in the ETFF algo�rithm. The spreadsheets are built for a wide range ofpressures P (from 1 to 120 atm), temperatures T (from293 to 900 K), and fuel�to�oxidizer equivalence ratio

nu

(from the lower to the upper flammability limit). Atthe known values of temperature, pressure, and fuel�to�oxidizer equivalence ratio in the mixture ahead ofthe ith element of the flame front, the local flamespeed was determined by linear interpolationbetween the two nearest values in the spreadsheet. Thespreadsheets are compiled based on solving the prob�lem of the velocity of propagation and structure of aflat laminar flame using a five�step global kineticmechanism, described in detail in [19]. For the sake ofcompleteness, Tables 2–4 list the kinetic mechanism(Table 2) and kinetic parameters of reaction (1) (Table 3)and reactions (2)–(5) (Table 4) for a stoichiometricpropane–air mixture. Note that the kinetic parame�ters of the empirical mechanism (activation energy Eand preexponential factor ) depend on the pressureand the mixture composition. In calculating , thevalues of and for reaction (1) corresponded totemperatures T > 775 K (Table 3).

The kinetic mechanism presented in Table 2 wasalso used in the PM algorithm. For simulating the bulkreactions at T <775 K and T > 775 K, the differentvalue of and were used (Table 3). Thus, the pre�flame two�stage self�ignition with a cool flame andsubsequent hot explosion was simulated [19]. In thePM algorithm, the ith particle was considered experi�encing self�ignition when the temperature rise ratereached 106 K/s.

When simulating the preflame ignition and DDT inthe ABPDE combustion chamber shown in Fig. 1, wetook into account that, at low temperatures, the pre�history of the particles practically has no effect on the

ignition delay. For particles with 350 K, the term

in (8) was ignored. This procedure made it possible tosignificantly reduce the computation time to the onset of

DDT, since the number of particles with 350 K wasrelatively small: such particles were located in the areabetween the flame front and the shock wave ahead ofit. Particles were largely located in the fresh fuel–airmixture ahead of the shock wave and had a tempera�ture close to the initial temperature.

Figure 2 shows how the PM was used in calculatingthe DDT characteristics. Following the above�described procedure, those particles that are located inthe region between the flame front and shock wave arecalled in Fig. 2 active, whereas those ahead of theshock wave are termed passive. When a passive particleoccurs behind the shock wave, it becomes active.When an active particle occurs behind the flame front,it disappears.

The ETFF and PM algorithms were tested by sim�ulating flame acceleration in smooth tubes and tubescluttered with obstacles and the autoignition of ahomogeneous fuel–air mixture at an elevated initialtemperature. The results of simulations of flame accel�eration, described in detail in [20], were compared

Φ

niu

A

nuE1 A1

E1 A1

iθ <

iVQ�

iθ >

Table 1. Laminar flame speed un (cm/s) in a stoichiometricpropane–air mixture (Φ = 1)

Pressure, MPa Temperature, K Flame speed

0.1 300 39

450 78

600 143

750 247

900 451

0.3 300 28

450 55

600 102

750 178

900 306

1.0 300 19

450 35

600 64

750 112

900 191

4.0 300 8.9

450 19

600 36

750 62

900 105

10.0 300 6.0

450 13

600 24

750 41

900 69

Table 2. Global kinetic mechanism of propane oxidation [19]

No. Reaction

1 С3H8 + 3.5O2 → 3CO + 4H2O

2 CO + H2O → CO2 + H2

3 CO2 + H2 → CO + H2O

4 H2 + H2 + O2 → H2O + H2O

5 CO + CO + O2 → CO2 + CO2



with experimental data. The results of calculations ofautoignition were compared with kinetic calculationsbased on a detailed kinetic mechanism of propane oxi�dation and with experimental data on the ignitiondelay time. In all cases, the ETFF and PM algorithmsgave results in satisfactory agreement with experimen�tal data and detailed kinetic calculations.

CALCULATION RESULTS

This section presents the results of simulating theoperation process in the ABPDE in supersonic flightconditions at a Mach number of = 3.0 and altitudesof = 9.3 and 16 km. The parameters of the air at thealtitude of 9.3 km are as follows: pressure, 0.29 atm;temperature, 228 K; speed of sound, 304 m/s.At an altitude of 16 km, the analogous parameters are0.104 atm, 217 K, and 295 m/s, respectively. To deter�mine the thrust performance of the ABPDE, we per�formed end�to�end simulations of several cycles, tak�ing into account the external flow around the engine.When calculating the aerodynamic drag, we allowedfor the pressure drag and the drag of viscous friction onall solid surfaces of the ABPDE.

Simulation of the Flow in the ABPDE

Let us examine the results of simulations of theflight of an ABPDE at an altitude of 9.3 km, as anexample. In this case, due to the compression of the

MZ

aP =

aT = ac =

airflow in the supersonic diffuser, the pressure insidethe receiver reaches ~6 atm, whereas the temperatureand airflow velocity of the air, ~500 K and ~200 m/s,respectively. Figure 3 shows the calculated pressuredistributions (isobars) in the receiver before (Fig. 3a)and after (Fig. 3b–3h) closing the valve in the secondworking cycle of the ABPDE.

Figure 3a shows the distribution of pressure in thereceiver at the first stage of the cycle during blow�through of the combustion chamber with air and itsfilling with FAM (valve is opened). The duration ofthis stage in the second operating cycle was estimatedto be 10 ms.

Figure 3b shows the distribution of pressure in thereceiver 0.1 ms after valve closure, i.e., after the start ofthe second stage of the working cycle of the ABPDE.As can be seen, valve closure (momentary) leads to theformation of a pressure wave that propagates in thereceiver upstream and eventually into the second cir�cuit (Figs. 3b–3d). Approximately 1 ms after valveclosure, the pressure wave comes to the closing com�pression shock in the diffuser (Fig. 3e) and thenpushes it to the inlet of the diffuser (Figs. 3f and 3g). Acomparison of Figs. 3g and 3h suggests that, 2 ms aftervalve closure, the flow in the receiver and second cir�cuit is stabilized. Recall that while the valve is closed,ignition, flame acceleration, DDT, detonation propa�gation (the second stage of the operation process)occur in the combustion chamber, after which thecombustion products outflow through the nozzle into

Table 3. Kinetic parameters of reaction (1)

P, atm

А1, l/(mol s)

E1 = 40 kcal/mol, T ≤ 775 К

E1 = 45 kcal/mol, T > 775 К

1 1.96 ⋅ 1012 1.73 ⋅ 1012

20 1.96 ⋅ 1012 8.77 ⋅ 1011

40 1.96 ⋅ 1012 7.50 ⋅ 1011

80 1.96 ⋅ 1012 6.41 ⋅ 1011

120 1.96 ⋅ 1011 5.85 ⋅ 1011

Note: Reaction (1) is considered to be bimolecular.

Table 4. Kinetic parameters of reactions (2)–(5)

Reaction no.

Forward reaction Reverse reaction

A, mol, l, s n E,

kcal/molA,

mol, l, s n E, kcal/mol

2, 3 1.0 ⋅ 1012/P 0 41.5 3.1 ⋅ 1013/P 0 49.1

4 7.0 ⋅ 1013/P0.5 0 21.0 – – –

5 8.5 ⋅ 1012/P1.5 0 21.0 – – –

Note: Pressure in atm.

606


IVANOV, FROLOV

the surrounding atmosphere (the third stage of thecycle). The calculations were based on the assumptionthat the valve remains closed until the pressure on thevalve from the combustion chamber does not drop to apreset value ( = 0.33 MPa) that yet provides a posi�tive thrust. The combined duration of the second andthird stages of the work cycle was calculated to be11 ms. Thus, the total duration of the second workcycle was 21 ms, which corresponds to an oper�ating frequency of the ABPDE of 48 Hz.

Figure 4 shows the development of the operationprocess in the combustion chamber, in particular,shows the calculated distribution of the mass fractionof fuel in the combustion chamber at 1 ms after thebeginning of its filling (valve is opened). It is seen that,near the air–stoichiometric FAM interface, the com�bustible mixture is diluted with air, with the degree ofdilution near the walls of the combustion chamberbeing higher. The obstacles installed on the chamberwalls increase the degree of dilution of the mixturewith air.

Ignition of the FAM in the second cycle occurredat a time of 90 ms (valve is closed). Figures 4b–4fshows the calculated temperature distribution in thecombustion chamber at different times after ignitionof the mixture. Initially, combustion develops rela�tively slowly (Fig. 4b). Further, the flame front accel�erates rapidly (Fig. 4c), forming ahead of itself a com�pression wave and then a shock wave (Fig. 4d), whichsubsequently gives rise to DDT. Deflagration�to�detona�tion transition occurs at a distance of 400 mmfrom the ignition source, i.e., 4.8d. The deto�nation wave formed initially propagates in the sectionwith obstacles, and then, in the smooth section of thecombustion chamber. Figures 4e and 4f display thecalculated temperature distribution at the initial andlater stages of detonation propagation. Note that thepressure and temperature of the FAM in the combus�tion chamber were ~5 atm and ~500 K. In addition,the FAM was highly turbulized. These factors substan�tially improved the conditions for DDT onset com�pared to normal conditions, which are usually realizedin laboratory experiments on DDT.

As a result of DDT, an overdriven detonation wavefront forms in the combustion chamber, which propa�gates downstream in fresh FAM, gradually weakeningto the Chapman–Jouguet velocity whereas, a ret�onation wave propagates upstream through the com�bustion products. By the time 91 ms, when the detona�tion wave approaches the nozzle, its velocity is

1900 m/s. If we consider that the detonation wavepropagates in a moving FAM (at ~200 m/s) and thatthe combustible mixture near the nozzle is diluted withair, then the resulting velocity agrees well with thecharacteristic value of 1800 m/s for a stoichio�metric propane–air mixture. At the time of occurrenceof DDT, degree of overdrive of the detonation wave

P*

cycleτ =

f =

DDTL ≈

DDTL ≈

CJD ,

D ≈

CJD ≈

reaches a very high value, 1.45. Note that,under these conditions, the diameter of the cylindricalpart of the combustion chamber (d = 83 mm) is muchlarger than the critical detonation diameter dcr (dcr = 30–50 mm for a stoichiometric propane–air mixture undernormal conditions).

Figure 5 shows the calculated temperature distribu�tion at the third stage of the work cycle, the stage of out�flow of combustion products from the combustion cham�ber. Despite the fact that convergent�divergent nozzleused in the ABPDE reduces the thrust due to the reflec�tion of the detonation wave from the convergent portion,the nozzle is necessary to maintain a high pressure andtemperature of FAM and a low flow velocity in the com�bustion chamber, thereby providing favorable conditionfor rapid DDT. Figure 5 shows that, due to expansion, thecombustion products in the combustion chamber cooland that, immediately after the beginning of outflow, aMach disk arises downstream from the nozzle. In flightconditions with M = 3.0, unsteady exhaust jet of theABPDE only two times wider than the nozzle exit.

Calculation of the Thrust Performanceof the ABPDE

Figure 6 shows the calculated time dependence ofthe force acting on the ABPDE in flight for three oper�ation cycles. The force is considered positive if it actsagainst the direction of the incoming flow. It is seenthat the first cycle differs from the two subsequentcycles. This difference is due to the start�up of theABPDE in the calculation procedure: initially, theentire duct of the engine is filled with relatively coldair, which is forced out with hot air and FAM duringblow�through and filling. The first cycle differs fromthe second and third cycles in duration as well: theduration of the first cycle was 30 ms, whereas the dura�tion of the second and third cycles was 21 ms. The sec�ond and third cycles in Fig. 6 are almost identical, i.e.,the operation process sets in after the second cycle.Consequently, the thrust performance of the ABPDEcan be estimated starting from the second cycle.

Integrating the curve presented in Fig. 6, one cancalculate the impulse for each cycle of engine opera�tion. For example, for the second (and third) cycle,the impulse is positive, ~0.043 Ns. Given that the cycleduration is 21 ms, the mean total force actingon the ABPDE in flight is 2.05 N. Note that thisforce is made up of the thrust and the aerody�namic drag of the engine : Since theforce is positive, the ABPDE can move with accel�eration under these conditions.

To determine the thrust produced by theABPDE, it is necessary to know its aerodynamic drag

in flight. This force can be determined by solvingthe same problem as that solved in constructing Fig. 6,but without FAM ignition (without the active stage of

CJD D ≈

cycleτ =

F ≈

( )TF( )RF T RF F F= − .

F

TF

RF



Temperature, K

200 510 820 1130 1440 1750 2060 2370 2680 2990 3300

(a)

(b)

(c)

Fig. 5. Calculated temperature distributions at various moments of time during the outflow of combustion products from theABPDE: (a) 91.2 ms, (b) 91.4 ms, and (c) 92.0 ms.

the cycle). In this case, there are two ways to deter�mine the force (1) to assume that, when the valveis opened, the combustion chamber is filled with thehot products of the previous cycle (this case corre�sponds to one ignition failure) and (2) to assume thatwhen the valve is opened, the combustion chamber isfilled with cold FAM (this corresponds to several suc�cessive ignition failures).

For the second (and third) operation cycle, the cal�culated impulse of the aerodynamic drag was found tobe –0.17246 N s (first method) and –0.19433 N s (secondmethod). Given that 21 ms, we obtained

⎯0.17246/0.021 = –8.21 N and –0.19433/0.021 =–9.25 N, where the indices 1 and 2 correspond to the firstand second method of evaluation of

Thus, with the above two methods for estimating, the thrust created by the ABPDE, yielded, respec�

:RF

cycleτ = RF ≈, 1

RF ≈, 2

RF .

RF

tively, 2.05 + 8.21 = 10.26 N and

2.05 + 9.25 = 11.3 N.

The specific impulse of the ABPDE is determinedfrom the thrust force and the fuel mass con�sumed per cycle [5]:

where is the acceleration of gravity. Given that themass of propane fed into the combustion chamberduring the second (and third) operation cycle wasapproximately 1.28 ⋅ 10–5 kg and 21 ms, weobtain

s (12)

T RF F F= + ≈, 1 , 1

T RF F F= + ≈, 2 , 2

TF fm�

Tsp

f

FI

m g=

�

,

g

cycleτ =

spI−

⋅

= ≈

⋅ ⋅

, 1 510.26 0.021 1720

1.28 10 9.8

608


IVANOV, FROLOV

and

s. (13)

Taking into account the calculation error, we con�cluded that the specific impulse of the ABPDE inflight with M = 3.0 at an altitude of Z = 9.3 km was

= (1800 ± 100) s.Note that, for the first cycle, the total force acting

on the ABPDE was 8.5 N, significantly higher than thecorresponding value for the second and third cycles.This difference is due to the conditions of blow�through and filling of the combustion chamber ofABPDE with the mixture. Before the first cycle, theABPDE was blown through with cold air for 50 ms. Asa result, the pressure in the combustion chamber wasapproximately 2 times higher than in subsequentcycles, a factor that increased the mass of the fuel–airmixture. Thus, in the first cycle, the mass of the fuel inthe combustion chamber was 2.08 ⋅ 10–5 kg. If the dragforce of the ABPDE remains constant, this factorcauses an increase in the force acting on the engine.

Similar calculations were performed for the sameABPDE in flight conditions with M = 3.0 at an alti�tude of Z = 16 km. For comparison, Table 5 lists themain input parameters and the calculation results forZ = 9.3 and 16 km. As can be seen, for the flight at analtitude of 16 km, 1700 s. This value ismuch higher than the specific impulse characteristic of

spI−

⋅

= ≈

⋅ ⋅

, 2 511.3 0.021 1890

1.28 10 9.8

spI

sp spI I≈ ≈, 1 , 2

conventional combustion ramjet (for the selected flightconditions, 1200–1500 s, according to different data).

CONCLUSIONS

Detailed multidimensional calculations of theunsteady gasdynamic flow in the duct of an ABPDErunning on gaseous propane and the flow around it insupersonic flight conditions with a Mach number of3.0 at altitudes of 9.3 and 16 km were performed. TheABPDE consisted of an air intake, receiver, annularbypass channel, chamber equipped with a mechanicalvalve, and nozzle. Unlike all previous computationalstudies, the operation process in the ABPDE wasbased on the ignition of the mixture with a weak sourcefollowed by DDT. Numerical simulation of DDTbecame possible due to the development and imple�mentation of a new ETFF algorithm and a new algo�rithm for calculating the characteristics of bulk energyrelease based on the particle method. It is shown that,for the selected ABPDE configuration with a lengthand diameter of the duct of 2.12 m and 83 mm, respec�tively, it is possible to realize a cyclic operation processwith a frequency of 48 Hz and rapid DDT at a distanceof 5–6 combustion chamber diameters. Deflagration�to�detonation transition occurs at a short distancebecause of a high degree of turbulization of the flow atelevated (compared to normal) pressures and temper�atures.

200

100

0

–100

–200

–300Fo

rce

acti

ng

on

th

e A

BP

DE

, N

1009080706050 110 120Time, ms

Cycle 1 Cycle 2 Cycle 3

Fig. 6. Calculated time dependence of the total force acting on the ABPDE in flight at an altitude of 9.3 km for three work cycles.

Table 5. Basis parameter and calculation results for the ABPDE in flight with a Mach number of 3.0 at altitudes of 9.3 and 16 km

, km , MPa , К , ms , Hz MPa , N , N , N , kg/s , s , s

9.3 0.029 228 21 48 0.331 2.05 8.21 9.25 0.00061 1720 1890

16 0.010 216.7 21 48 0.175 0.77 4.54 4.52 0.00032 1690 1680

Z aP aT cycleτ f P*, F RF ,1 RF ,2 fm� spI ,1 spI ,2



To determine the thrust performance of theABPDE in flight conditions, we simulated a series ofoperation cycles, taking into account the external flowaround the engine. When calculating the aerodynamicdrag, we allowed for the pressure drag and the drag ofviscous friction on all solid surfaces of the ABPDE.Calculations showed that the thrust performance ofthe ABPDE were well reproducible starting from thesecond cycle of the operation process. The mean totalforce acting on the ABPDE in these conditions waspositive. On the one hand, this means that, in a freeflight, this ABPDE will be moving with acceleration.On the other hand, this means that this engine can beinstalled on an aircraft, the additional aerodynamicdrag of which will offset the positive total effect.

Calculations showed that the specific impulse ofthe ABPDE in supersonic flight with a Mach numberof M = 3.0 at altitudes of 9.3 and 16 km is approxi�mately 1700 s. This value is much higher than the spe�cific impulse of an conventional�combustion ramjet(1200–1500 s) and substantially lower than the valueof specific impulse obtained for atmospheric condi�tions at sea level at zero flight speed (~2500 s) [12].

ACKNOWLEDGMENTS

This work was supported by the Federal Target Pro�gram “Research and Pedagogical Cadre for InnovativeRussia” for 2009–2013 (state contract no. P504) andin part by the Russian Foundation for Basic Research(project no. 11�08�01297).

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Numerical Simulation of the Operation Process and Thrust ...frolovs.ru/pdf/2011-19-eng.pdf · reactions in a turbulent flame and the preheat zone. COMBUSTION, EXPLOSION, AND SHOCK