On the Mechanism of the Deflagration-to-Detonation Transition in a Hydrogen–Oxygen Mixture

ISSN 1063�7761, Journal of Experimental and Theoretical Physics, 2010, Vol. 111, No. 4, pp. 684–698. © Pleiades Publishing, Inc., 2010.Original Russian Text © M.A. Liberman, M.F. Ivanov, A.D. Kiverin, M.S. Kuznetsov, T.V. Rakhimova, A.A. Chukalovskii, 2010, published in Zhurnal Éksperimental’noі iTeoreticheskoі Fiziki, 2010, Vol. 138, No. 4, pp. 772–788.

684

1. INTRODUCTION

The wave of a chemical reaction in a gas mixture isone of the most fundamental manifestations of thepropagation of an exothermic reaction and holds acentral position in the study of combustion processes.Two stationary regimes of combustion wave propaga�tion are known [1] to be possible in a gas mixture: slowcombustion (deflagration) and a supersonic detona�tion wave. The deflagration wave propagates throughthe diffusion of heat and (or) free radicals from thereaction zone into the fresh mixture ahead of the front,so that its velocity is much lower than the speed ofsound and the pressure is almost constant. The other(supersonic) combustion wave propagation mecha�nism is associated with shock waves. In the detonationwave, the gas is compressed and heated by a strongleading shock that ignites the gas mixture. It is wellknown [2–5] that when a flame propagates in abounded space, for example, in a shock tube, theflame velocity gradually increases and a sudden, veryrapid formation of detonation occurs at some distancefrom the place of ignition. The deflagration�to�deto�nation transition (DDT) is of fundamental interest asa spontaneous transition between the two stationary

solutions of the complete system of gasdynamic equa�tions. Elucidating the physical processes underlyingDDT is important in understanding the ways of deto�nation prevention in connection with the safety prob�lems of nuclear power plants, for hydrogen storage,and for hydrogen engines [6–8], in initiating detona�tion to design pulsed detonation engines [9], and inastrophysics to explain the explosions of thermonu�clear supernovae [10, 11].

Despite long�term experimental and theoreticalstudies, the physical mechanism of DDT is not yetclear and is considered as one of the most importantunresolved problems in the combustion theory. Thefirst attempts to explain DDT suggested that flow tur�bulence ahead of the front accelerated the flame sothat strong shocks formed ahead of the flame led tomixture self�ignition and detonation wave formation.This view was based on the well�known experimentsby Shchelkin et al. [4, 12, 13], in which strong flowturbulence was produced ahead of the flame with spi�rals that accelerated the flame and reduced consider�ably the detonation formation time. Strong turbulencefacilitates the DDT observation but masks signifi�cantly the physical processes underlying the transition

On the Mechanism of the Deflagration�to�Detonation Transitionin a Hydrogen–Oxygen Mixture

M. A. Libermana,b,*, M. F. Ivanovc, A. D. Kiverinc, M. S. Kuznetsovd,**,T. V. Rakhimovab, and A. A. Chukalovskiib

aDepartment of Physics, Uppsala University, SE�75121, Uppsala, SwedenbInstitute of Nuclear Physics, Moscow State University, Moscow, 119992 Russia

cJoint Institute for High Temperatures, Russian Academy of Sciences, ul. Izhorskaya 13/19, Moscow, 125412 RussiadForschungszentrum Karlsruhe, Karlsrue, 76021 Germany

*e�mail: [email protected]**e�mail: [email protected]

Received March 15, 2010

Abstract—The flame acceleration and the physical mechanism underlying the deflagration�to�detonationtransition (DDT) have been studied experimentally, theoretically, and using a two�dimensional gasdynamicmodel for a hydrogen–oxygen gas mixture by taking into account the chain chemical reaction kinetics foreight components. A flame accelerating in a tube is shown to generate shock waves that are formed directlyat the flame front just before DDT occurred, producing a layer of compressed gas adjacent to the flame front.A mixture with a density higher than that of the initial gas enters the flame front, is heated, and enters intoreaction. As a result, a high�amplitude pressure peak is formed at the flame front. An increase in pressure anddensity at the leading edge of the flame front accelerates the chemical reaction, causing amplification of thecompression wave and an exponentially rapid growth of the pressure peak, which “drags” the flame behind.A high�amplitude compression wave produces a strong shock immediately ahead of the reaction zone, gen�erating a detonation wave. The theory and numerical simulations of the flame acceleration and the new phys�ical mechanism of DDT are in complete agreement with the experimentally observed flame acceleration,shock formation, and DDT in a hydrogen–oxygen gas mixture.

DOI: 10.1134/S1063776110100201

STATISTICAL, NONLINEAR,AND SOFT MATTER PHYSICS

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 111 No. 4 2010

ON THE MECHANISM OF THE DEFLAGRATION�TO�DETONATION TRANSITION 685

from one regime to the other. Experiments show thatDDT in highly reactive mixtures, for example, hydro�gen–oxygen, hydrogen–air, etc., occurs in such a waythat the flow in the tube ahead of the flame becomeslaminar to the point of detonation wave formation. Itshould be noted that the explosion of a gas mixture perse cannot initiate detonation, because the pressureproduced in the explosion is much lower than thatbehind a leading shock with a typical (for detonation)Mach number Msh = 5–6.

Using a one�dimensional numerical model,Zel’dovich et al. [14] showed that detonation could beinitiated by a specified initial nonuniform temperaturedistribution. The idea of the gradient mechanism pro�posed in [14] is based on the concept of a spontaneouscombustion wave [15] proposed by Zel’dovich. If themixture is initially heated nonuniformly with a tem�perature distribution T(x), then ignition begins atpoints where the temperature is higher (and the induc�tion time τind(T) is, accordingly, shorter) and subse�quently propagates in the form of a spontaneous com�bustion wave with velocity Usp(x) = [dτind(T)/dx]–1

along the temperature gradient. The velocity of thespontaneous combustion wave can be arbitrarily high,depending on the temperature gradient. In particular,if the velocity of the spontaneous wave is equal to theChapman–Jouguet velocity, then the formation ofdetonation is possible [15]. Numerical solutions ofone�dimensional gasdynamic problems were used toclassify in detail the regime of a spontaneous combus�tion wave with various specified temperature distribu�tions [16–20]. After [14, 15], Zel’dovich’s gradientmechanism was considered as the most natural mech�anism that explained DDT. It should be noted that theinitiation of detonation by a temperature gradient wasdemonstrated only in one�dimensional numericalsimulations where the initial temperature nonunifor�mity was specified artificially [14–20]. At present,there exist neither experimental data nor multidimen�sional numerical simulations with real chemical kinet�ics showing that the DDT mechanism is actuallyattributable to the formation of a temperature gradientor an induction time gradient. The fundamental ques�tion of how a temperature gradient can be formed andwhich physical processes trigger the gradient mecha�nism if the physical mechanism of DDT is assumed tobe actually related to the formation of a temperaturegradient has also remained unsolved until now.

For the last 20 years, DDT has been studied innumerous works through two� and three�dimensionalnumerical simulations in order to ascertain the DDTmechanism and which processes give rise to a temper�ature gradient (see the review [21]). In their numericalsimulations, the authors used a single�step model of achemical reaction with a low activation energy to sim�ulate DDT in hydrogen–air, acetylene–air, etc. andconcluded that the temperature gradient triggeringZel’dovich’s mechanism and DDT emerge at hotpoints formed ahead of the flame front at the crossing

of the shock waves reflected from the tube walls. Infact, the actual induction times (mixture ignitiontime) for temperatures of 600–800 K at hot spotsobtained in [21] are four or five orders of magnitudelonger than the time of the entire process from theflame ignition to DDT even for the most reactive mix�tures, such as hydrogen–oxygen. Therefore, whetherthe results obtained in [21] are applicable to DDT inhighly reactive mixtures is still an open question.

In this paper, we show that the physical processesleading to DDT can be considerably more complexthan has been though previously. Here, we present ourexperimental results for the combustion wave acceler�ation dynamics and DDT for hydrogen–oxygen mix�tures at various initial pressures. Similar experimentalresults were also obtained for ethylene–oxygen mix�tures [22]. The DDT mechanism is studied by numer�ically solving the two�dimensional gasdynamic equa�tions along with the equations for the kinetics of mainchemical reactions for a hydrogen–oxygen mixture. Itshould be emphasized that the experimentally accessi�ble temporal resolution of high�speed shadow photog�raphy (0.1–0.01 ms) is insufficient to resolve the rapidDDT process and, in this sense, a numerical experi�ment that complements a real physical experimentallows one to see the details and to understand thephysical mechanism of DDT.

2. EXPERIMENTAL STUDIES OF DDT

The DDT experiments in a stoichiometric hydro�gen–oxygen mixture were carried out in shock tubesmade of stainless steel with a rectangular cross sectionof 50 × 50 mm2 and a length from 3.4 to 6.065 m at ini�tial mixture pressures from 0.2 to 0.75 bar. The flameand shock velocities were measured by the method ofhigh�speed shadow (Schlieren) photography with atemporal resolution of 0.1 ms. A spatial resolution of200 μm allows weak compression waves, their crossingand reflection from the tube walls, etc. to be clearlydistinguished. Accordingly, the velocity measurementaccuracy was ±2 m/s. The calibrated pressure sensorsand photodiodes located along the tube axis recordedthe arrival times of the flame and shock fronts at agiven point. The design of the facility and diagnosticequipment, the optical layout, the stroboscopic gener�ator, the high�speed camera, and the method of veloc�ity measurements are described in [22–24].

The mixture was ignited by an electric spark nearthe closed end of the tube and the generated combus�tion wave propagated to the opposite end. The tubelength was chosen to be large enough for the shockwaves reflected from the opposite end to have no timeto return to the flame and, where possible, to have noeffect on the transition process.

The time dependences of the combustion wavevelocity measured in our experiments for various ini�tial pressures and various tube wall roughnesses fromthe time of ignition to the formation of Chapman–

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LIBERMAN et al.

Jouguet detonation are presented in Fig. 1. Threeclearly distinctive stages of combustion wave propaga�tion along the tube were observed in all experiments.At the first stage, the flame moves with an almost con�stant acceleration for a short time (less than 1 ms). Atthe second stage, the combustion wave continues toaccelerate but more slowly than it does during the firststage. The third stage is a rapid, almost abrupt,increase in velocity to the velocity of an overdrivendetonation wave, which then transforms into Chap�man–Jouguet detonation. It should be noted thatFig. 1 and Fig. 2 below show the local velocity of thepoint of the flame front on the tube axis.

Since the overall reaction order for a hydrogen–oxygen mixture is n = 2.73, the reaction rate and the

laminar flame velocity increase with pressure [2]. Thenormal laminar flame velocity increases from 6.8 to8.9 m/s as the pressure changes from 0.2 to 0.75 bar. Atthe same time, the apparent flame velocity, i.e., thevelocity in the laboratory frame of reference increasesfrom 55 to 74 m/s. At the end of the first stage, theapparent flame velocity is 375 and 420 m/s at initialpressures P0 = 0.6 and 0.75 bar, respectively. At the endof the second stage, immediately before the formationof detonation, the apparent flame velocity reaches850–900 m/s. We also see from Fig. 1 that, at the sameinitial pressure, the rougher the tube walls, the smallerthe distance traveled by the flame before DDT. Moredetailed measurements whose results are presented inFig. 2 show that the flame velocity at the first stage isfitted with a high accuracy by an exponential timedependence. Similar dynamics consisting of threeclearly distinguishable flame acceleration stages isobserved for DDT in ethylene–oxygen mixtures withan overall reaction order n = 1.9–2.0. Thus, theobserved flame acceleration dynamics in tubes andDDT are characteristic of at least highly reactive mix�tures.

The decrease in velocity at t = 0.5–0.7 ms seen inFigs. 1 and 2 is observed at low initial mixture pres�sures (below 0.75 atm for H2–O2). To avoid misunder�standings, it should be emphasized that we are dealingwith the decrease in the local velocity of the flamefront measured on the tube axis. The decrease in

v, km/s

4 62t, ms

1

8

123

2

4

1 '2 '3 '

DCJ

0

3

10 12

Fig. 1. Change in the velocity of a combustion wave in ahydrogen–oxygen mixture with time at initial pressures 0.5(curves 1, 1'), 0.6 (curves 2, 2 '), and 0.75 (curves 3, 3 ') barin a tube with wall roughness d ≤ 50 μm and d = 1000 μm(1', 2 ', 3 '). DCJ is the Chapman–Jouguet velocity.

v, m/s

0.4 0.60.2t, ms

100

0.8

12

3200

0

300

1.0 1.41.2

400

Fig. 2. Time dependence of the flame velocity in a hydro�gen–oxygen mixture at the initial stage of flame propaga�tion in the tube at initial pressures P0 = 0.2 (curve 1, v =29exp(3.75t/ms)), 0.6 (curve 2, v = 59exp(3.27t/ms)), and0.75 (curve 3, v = 70exp(2.5t/ms)) bars; the dimensions ofthe preexponential factors are [m/s] everywhere.

BL

BL

Fig. 3. Shadow photographs of the initial flame accelera�tion stage, P0 = 0.75 bar. Starting from the upper frame,the times, positions, and velocities of the flame and shockfronts are, respectively, t = 0.75, 1.3, 1.4, 1.5 ms, Xf = 229,371, 410, 452 mm, Uf = 350, 370, 390, 420 m/s, Xsh = 420,826, 903, 980 mm, MSh = 1.48–1.5. The boundary layer(BL) is everywhere less than 1.5 mm.



velocity on the axis is related to the onset of the forma�tion of a tulip�shaped flame front (as can be seen fromthe shadow photographs presented below in Figs. 3and 4), when the points of the front on the axis lagbehind, while the tulip petals move faster. In contrast,the combustion wave velocity, which is determined bythe flow of burned gas passing through the flame frontsurface and divided by the tube cross section is an inte�gral characteristic that defines the flame dynamics.This velocity grows monotonically, but its growth rateis lower than that at the first acceleration stage. This isconfirmed by the fact that the compression waves pro�duce shock waves not far ahead but immediatelybefore the flame front (see Section 5). The decrease invelocity almost to zero for P0 = 0.5 bar at t = 10 ms(curve 1 in Fig. 1) is related to the interaction of theflame with the counter�propagating shock reflectedfrom the opposite end of the tube.

On the shadow photographs presented in Fig. 3, wesee the shape of the flame front and the leading shockin a hydrogen–oxygen mixture at initial pressure P0 =0.75 bar. On the upper frame, at t = 0.75 ms, the flamefront is seen on the left and the shock is observed onthe right at a distance of about five tube widths. Thewhite band in the lower part of the frame is the bound�

ary layer between the shock and the flame front. Theboundary layer thickness is initially about 1 mm anddoes not exceed 5 mm near the flame front to the pointof detonation formation, which is considerablysmaller than the tube width. On the succeeding framestaken at t = 1.3, 1.4, and 1.5 ms, we see weak compres�sion waves produced by the accelerating flame, theirreflection from the tube walls, and the flow ahead ofthe flame front behind the leading shock. The flowbetween the shock and the flame front is seen toremain laminar and homogeneous in the bulk of thetube cross section, except for the weak compressionwaves and the thin boundary layer.

The evolution of the flame front shape during thetransition from the first acceleration stage to the sec�ond one are shown on the shadow photographs pre�sented in Fig. 4 for initial mixture pressures P0 = 0.75and 0.2 bar. The coordinates and velocities of theflame and the shock (Mach numbers) in Fig. 4 foreach instant of time are given in Tables 1 and 2. A char�acteristic feature that is clearly seen at a lower initialpressure in Fig. 4b is the elongation of the flame frontalong the tube walls, which gives rise to a narrow gapbetween the front surface and the tube wall. Initially,the total surface area of the flame front increases; the

(a)

(b)

Fig. 4. Shadow photographs showing the evolution of the flame front shape and velocity at initial pressure P0 = 0.75 bar, wallroughness d = 50 μm (a) and P0 = 0.2 bar, d = 1000 μm (b). The time, coordinates, and velocity of the flame for each frame aregiven in Tables 1 and 2.

688


LIBERMAN et al.

combustion wave velocity increases accordingly. At theend of the first acceleration stage, whose durationdepends on the initial pressure (0.4 ms for P0 =0.75 bar and 0.9 ms for P0 = 0.2 bar), the depth of thegap between the flame front and the wall decreases, theflame front becomes almost flat, its surface areadecreases, and, accordingly, the growth rate of thevelocity decreases. The Mach numbers for the shocksfar ahead of the flame remain in the range Msh =1.3⎯1.5.

Figure 5 presents shadow photographs of the last(third) stage immediately before the formation of adetonation wave. Since DDT is a stochastic process,

the exact place of detonation nucleation is not knownin advance. Figure 5 shows one of the scenarios for theformation of detonation near the tube wall. Similarphotographs were taken in the cases where detonationnucleated near the tube axis. On the upper frame inFig. 5 at 3.3 ms, we see a compressed layer of gasbetween the flame front and the shock with a Machnumber Msh = 2.3. On the next frame taken 0.1 mslater, DDT has already occurred and the detonationwave is seen. It should be noted that, in contrast to theinitial acceleration stage when the compression wavesproduce shocks at distances of the order of ten channelwidths, at the end of the second acceleration stage theshock waves are formed in the immediate vicinity ofthe flame front. The most important feature of the sec�ond flame acceleration stage is the formation of a layerof compressed and heated gas adjacent to the flamefront by a shock wave. It is seen in the upper Fig. 5 asa lighter strip about 10 mm in width.

The density and temperature inside the com�pressed layer ahead of the flame front at the DDT timedepend on the intensity of the shocks forming thecompressed layer. For the Mach numbers measured inour experiments, Msh = 2–2.3, the density increases bya factor of ρ2/ρ0 = 3–3.5 (ρ2 and ρ0 are the densitiesbehind and ahead of the shock, respectively), while thetemperature inside the layer rises from 550 to 700 K. Itis the formation of the layer of a shock�compressedand �heated mixture adjacent to the flame front that isa key factor determining DDT. As will be shown in ournumerical experiment, the time scale of the processesunderlying DDT is less than or of the order of a micro�

Table 2. Coordinates and velocity of the flame in Fig. 4b

t, ms

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Xf, mm 121 149 179 206 231 252 267 281 294 307 320 330 340 350 358 –

Uf, m s–1 – 280 300 270 245 210 150 140 130 130 120 104 104 93 85 –

Table 1. Coordinates and velocity of the flame and the shock in Fig. 4a (Δ is the boundary layer thickness)

t, ms

0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

Xf, mm 194 229 259 287 316 346 374 401 410

Uf, m s–1 – 350 330 305 290 300 280 275 225

Xsh, mm 194 420 500 581 662 743 823 904 985

Msh 1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.54

Δ, mm – 2.6 2.4 3.0 3.3 3.3 3.0 3.2 3.5

Fig. 5. Formation of a compressed layer of unburned gasadjacent to the flame front and DDT. The initial pressureis P0 = 0.55 bar. The upper photograph is for t = 3.3 ms,Xf = 1143 mm, Uf = 997 m/s, Xsh = 1163 mm, MSh = 2.3.The lower photograph is for t = 3.4 ms, DDT (PZ is thepreheated zone, DW is the detonation wave).

PZ

DDT

DW



second, i.e., two or three orders of magnitude smallerthan the temporal resolution between the frames ofshadow photographs. In this sense, a numerical exper�iment complements a physical experiment, providingan insight into the nature of DDT and the physicalprocesses characteristic of this transition. This featureof the flow dynamics during DDT can be seen even onthe shadow photographs in previous experimentalworks [25–27], although no proper attention wasgiven to this.

3. FORMULATION OF THE PROBLEMOF TWO�DIMENSIONAL NUMERICAL

SIMULATIONS

The dynamics of a flame propagating from theclosed end of a two�dimensional channel and DDTare studied in two�dimensional numerical simula�tions. The boundary conditions on the channel wallsrequire that the velocity become zero inside the vis�cous boundary layer on the walls. In this paper, atten�tion is focused on the physical processes underlyingthe DDT mechanism in a tube with smooth walls. Wenumerically solve the systems of Navier–Stokes equa�tions for a compressible medium by taking intoaccount viscosity, thermal conduction, multi�compo�nent diffusion, and energy release in the chemicalreactions R1–R9 listed in Table 3.

The gasdynamic equations in Eulerian variables are

(1)

(2)

∂ρ∂t�� ∂ ρu( )

∂x�� ∂ ρv( )

∂z��+ + 0,=

∂Ci

∂t�� u

∂Ci

∂x�� v

∂Ci

∂z��+ + 1

ρ�� ∂∂x�� ρDi

∂Ci

∂x��⎝ ⎠

⎛ ⎞=

+ ∂∂z�� ρDi

∂Ci

∂z��⎝ ⎠

⎛ ⎞ ∂Ci

∂t��⎝ ⎠⎛ ⎞

ch

,+

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

Here, P, ρ, u, and v are, respectively, the pressure,density, x and z velocity components; Ci = ρi/ρ are themass fractions of the mixture components involved inthe reaction; E = ε + (u2 + v2)/2 is the total energy

ρ ∂u∂t�� u∂u

∂x�� v

∂u∂z��+ +⎝ ⎠

⎛ ⎞ ∂p∂x��–

∂σxx

∂x��

∂σxx

∂z��,+ +=

ρ ∂v∂t�� u∂v

∂x�� v

∂v∂z��+ +⎝ ⎠

⎛ ⎞ ∂p∂z��–

∂σxz

∂x��

∂σzz

∂z��,+ +=

ρ ∂E∂t�� u∂E

∂x�� v

∂E∂z��+ +⎝ ⎠

⎛ ⎞ ∂ pu( )∂x

�� ∂ pv( )∂z

��+⎝ ⎠⎛ ⎞–=

+ ∂∂x�� σxxu σxzu+( ) ∂

∂z�� σzxu σzzv+( )+

+ ∂∂x�� κ T( )∂T

∂x��⎝ ⎠

⎛ ⎞ ∂∂z�� κ T( )∂T

∂z��⎝ ⎠

⎛ ⎞ hk

k

∑+ +

× ∂∂x�� ρDk T( )

∂Ck

∂x��⎝ ⎠

⎛ ⎞ ∂∂z�� ρDk T( )

∂Ck

∂z��⎝ ⎠

⎛ ⎞+ ,

P RBTnRB

mi

��Ci

i

∑⎝ ⎠⎜ ⎟⎛ ⎞

ρT ρT RiCi,

i

∑= = =

ε cvT

hkρk

ρ��

k

∑+ cvT hkCk,

k

∑+= =

σxx 2μ∂u∂x�� 2

3��μ ∂u

∂x�� ∂v

∂z��+⎝ ⎠

⎛ ⎞ ,–=

σzz 2μ∂v∂z�� 2

3��μ ∂u

∂x�� ∂v

∂z��+⎝ ⎠

⎛ ⎞ ,–=

σzx σxz μ ∂v∂x�� ∂u

∂z��+⎝ ⎠

⎛ ⎞ .= =

Table 3. Reduced scheme of hydrogen–oxygen combustion reactions, preexponential factors, and activation energies forthe forward (f) and backward (b) reaction rates. Here, M denotes any particle to which an excess energy is transferred

Reaction Af, cm3 mol–1 s–1 , kcal mol–1 Ab, cm3 mol–1 s–1 , kcal mol–1

R1 H2 + O2 = 2OH 2.52 × 1012 39.0 1.16 × 1013 21.0

R2 OH + H2 = H2O + H 2.25 × 1013 5.24 9.90 × 1013 20.3

R3 H + O2 = OH + O 1.55 × 1014 16.7 1.16 × 1013 0.705

R4 H2 + O = OH + H 2.46 × 1013 9.84 1.07 × 1013 7.90

R5 2H + M = H2 + M 3.60 × 1015 0.0 1.46 × 1016 104.0

R6 H + O2 + M = HO2 + M 3.60 × 1015 0.0 3.01 × 1015 47.8

R7 2HO2 = H2O2 + O2 1.0 × 1013 0.0 1.30 × 1014 40.0

R8 2OH + M = H2O2 + M 1.11 × 1016 1.92 7.40 × 1017 47.0

R9 H + H2O2 = H2 + HO2 1.17 × 1014 11.8 1.55 × 1014 28.5

EafEab

690


LIBERMAN et al.

density of the mixture; ε is the internal energy density;RB is the universal gas constant; Ri = RB/mi, mi is themolar mass of component i; n is the molar density ofthe mixture; σij is the viscous stress tensor; c

v =

is the specific heat of the mixture at constant

volume; cvi is the specific heat of component i at con�

stant volume; hi is the specific formation enthalpy ofcomponent i; κ(T) and μ(T) are the thermal conduc�tivity and viscosity coefficients of the mixture; Di(T) isthe diffusion coefficient of component i in the mix�ture; and (∂Ci/∂t)ch is the change in the mass fractionof component i through chemical reactions.

The transfer coefficients and the equations of statefor the unburned mixture and combustion productscorresponded to the parameters of real mixtures. Theequation of state for the mixture was derived from thetemperature dependence of the heat capacities of theindividual components, which were determined fromthe JANAF thermophysical tables through interpola�tion by fifth�order polynomials [28]; we calculated thetransfer coefficients based on the kinetic theory ofgases [29, 30]. The viscosity coefficient of the gas mix�ture was defined by the relation

(11)

where αi = ni/n is the molar fraction, μi =

5 /16π is the viscosity coefficient of

component i, is the reduced collision integralcalculated for the Lenard–Jones potential [30], isthe mass of the molecule of component i, and Σi is theeffective size of the molecule. The thermal conduc�tivity coefficient for the mixture is defined in a simi�lar way:

(12)

In this case, the thermal conductivity coefficient κi ofcomponent i is expressed in terms of the correspond�ing viscosity coefficient and the Prandtl number(Pr ≈ 0.71–0.75), which changes only slightly forgases, κi = μicpi/Pr, where cpi is the specific heat ofcomponent i at constant pressure.

The diffusion coefficient of component i in amulti�component medium is defined as

(13)

where

cviCii∑

μ 12�� αiμi

i

∑αi

μi

��

i

∑⎝ ⎠⎜ ⎟⎛ ⎞

1–

+ ,=

πm̂ikT Σi2Ω̃i

2 2,( )

Ω̃i2 2,( )

m̂i

κ 12�� αiκi

αi

κi

��

i

∑⎝ ⎠⎜ ⎟⎛ ⎞

1–

+i

∑ .=

Di 1 Ci–( )αi

Dij

��i j≠

∑⎝ ⎠⎜ ⎟⎛ ⎞

1–

,=

Dij3 2πkTm̂im̂j/ m̂i m̂j+( )

8ρπΣij2 Ω̃

1 1,( )Tij*( )

��=

is the binary diffusion coefficient, Σij = (Σi + Σj)/2,

= kT/ , = , ε* is the constant that

enters into the Lenard–Jones potential, is an

analog of the collision integral [29, 30].The hydrogen–oxygen mixture combustion kinet�

ics was described by the reduced scheme of nine reac�tions [31] presented in Table 3. The chosen reducedscheme is fairly economical in the number of includedequations. The validity of the system of chemicalkinetics equations used is confirmed by good agree�ment with experimental data for the normal velocity,the flame front width, the adiabatic temperature, andthe combustion product composition.

The mass fractions Ci of the mixture componentswere found by solving the system of chemical kineticsequations

(14)

The right�hand sides of Eqs. (14) contain the chemicalreaction rates dependent on temperature according tothe Arrhenius law presented in standard from [29].The rates of the forward, kf(T), and backward, kb(T),reactions related between themselves by the principleof detailed balancing are specified in the form

(15)

The values of Af, b and activation energy in (15)

used in our calculations of the forward and backwardreaction rate constants are given in Table 3.

The flame and detonation wave characteristics cal�culated for a hydrogen–oxygen mixture using thekinetic scheme R1–R9 agree well with experimentalmeasurements [22]. For example, at initial pressureP0 = 0.6 bar, the front width and the normal velocity ofthe laminar flow are Lf = 0.44 mm and Uf = 8.4 m/s,respectively, the adiabatic temperature is Tb = 3012 K,the expansion ratio is θ = ρu/ρb = 8.36 (ρu and ρb arethe densities ahead of and behind the flame), theChapman–Jouguet detonation velocity and tempera�ture are DCJ = 2815 m/s and TCJ = 3590 K, in completeagreement with their measured values [22].

The numerical method used in simulations is theEuler–Lagrange method known as the coarse particlemethod [32]. In this method, all quantities are speci�fied at the centers of the cells of an Eulerian grid. Thetime step is performed in three stages. At the first(Eulerian) stage, the transfer between neighboringcells is assumed to be frozen. The system of gasdy�namic equations is solved without including any con�vective terms using the method of central differencesof the second order of accuracy in space. At the second(Lagrangian) stage, the mass, momentum, and energytransfer through the cell boundaries is computed. Atthe third stage, the contributions to the solution

Tij* εij* εij* εi*εj*

Ω̃i j1 1,( )

Ω̃2 2,( )

dCi

dt�� Fi C1 C2 … CN T, , , ,( ), i 1 2 … N., , ,= =

kf b, T( ) Af b,

Eaf b,

RBT��–⎝ ⎠

⎛ ⎞ .exp=

Eaf b,



obtained at the first and second stages are combined.In [32], it was shown that the computational scheme isstable when the transfer through the cell boundariesoccurs with a rate equal to the half�sum of the rates inneighboring cells. However, in this case, the secondstage and, hence, the entire algorithm has only the firstorder of accuracy in space. The method of construct�ing the Lagrangian stage with the second order ofaccuracy in space proposed in [32] makes the schemeunstable in most cases. In contrast to [32], we used adifferent method of increasing the accuracy of the sec�ond stage. The velocity between the centers of neigh�boring cells was assumed to change linearly. In thiscase, as the transfer velocity of matter between neigh�boring cells, we take the velocity at the point fromwhich the mass reaches the cell boundary in the cho�sen time step. The particle�in�cell method modified inthis way was used to solve various combustion prob�lems and showed good agreement with experimentswhen numerically simulating such effects as the emer�gence of knocks in internal combustion engines [33,

34] and the self�ignition of a hydrogen jet during rapiddepressurization of high�pressure vessels [35]. Thesystem of chemical kinetics equations was solved byGear’s method [36].

4. NUMERICAL SIMULATIONSOF THE FLAME ACCELERATION AND DDT

IN A HYDROGEN–OXYGEN MIXTURE

We numerically simulated the propagation of acombustion wave in a two�dimensional channel with awidth from 5 to 10 mm filled with a stoichiometrichydrogen–oxygen mixture at an initial temperature of300 K and an initial pressure from 0.6 to 1.0 atm. Theminimum number of the computational cells in theflame front width ranged from 6 to 10. At the initialtime, the flame was initiated in the form of a flat layernear the closed end of the channel. Within a short timeafter ignition, a plane flame front is formed whichpropagates from the closed end of the channel to theopen one. The evolution of the flame front in a 5�mm�

640630 650 660 670 680x, mm

1.21187 ms

1.21100 ms

1.21000 ms

1.20650 ms

0.40000 ms

0.20000 ms

0.06500 msSWFlame

100 20 30 40 50

340330 350 360 370 380

Flame

SW

DW

DW

Fig. 6. Results of our numerical simulations showing the evolution of a flame propagating at P0 = 1 bar from the closed end of a5�mm�wide channel, the formation of compression and shock waves, and DDT (the density gradient is displayed). The compu�tational domains in which the combustion wave and the shocks are located are shown. The time is indicated to the right of eachframe.

692


LIBERMAN et al.

wide channel at an initial pressure of 1.0 atm thatdemonstrates the overall flow pattern in the channel,the development of instabilities at the flame front, andthe compression and shock waves emitted by theaccelerating flame ahead is shown in Fig. 6, where thedistributions of the density gradient are presented.

At the first flame acceleration stage, the shocks areformed far ahead of the flame, at a distance of~5 channel widths. The second stage begins afterabout 0.4 ms, when the combustion wave acceleration(see above, for a discussion of the concept of combus�tion wave velocity) slows down and the formation ofweak shocks begins directly at the flame front. At theend of the second flame acceleration stage (at t =1.21 ms), the shocks merging ahead of the flame frontform a layer of compressed unreacted gas adjacent tothe flame. After this moment, DDT lasts less than 1 μs.The instants of time for each frame indicated on theright in Fig. 6 are not evently spaced. Only the compu�tational domains demonstrating the most significantfeatures of the process in which the combustion wave,the shocks, and the compression wave are located areshown. In agreement with the experimental results(see Figs. 1 and 2), the transition time and the pre�det�onation length decrease with increasing initial pres�sure of the gas mixture.

The calculated time dependences of the flamevelocity and DDT in a 5�mm�wide channel for initialmixture pressures P0 = 0.6, 0.8, and 1.0 bar are shownin Fig. 7. During the first flame acceleration stage, att < 0.4–0.5 ms, the flame front elongates along thechannel walls and deforms due to the development ofperturbations. The change in the growth rate of theflame velocity at t = 0.4–0.5 ms is seen from the

change in the slope of the curves Uf(t) in Fig. 7, whichis qualitatively the same as for the curves v(t) in Fig. 1.

The intensity of the shock forming immediatelyahead of the flame front increases to Msh = 2.3–2.4 and a thin (3–4 mm) compressed layer ofunburned gas with a maximum mixture temperaturefrom 500 to 550 K is adjacent to the flame front beforethe onset of DDT at t = 1.21 ms. Since the rates of thecombustion�initiating reactions at such temperaturesare almost zero, the nonuniformity of the temperaturedistribution ahead of the flame front plays no role. Theflow before DDT observed in our numerical experi�ment (see Figs. 6 and 7) corresponds to the formationpattern of the shock and the layer of compressed gasahead of the flame front on the shadow photographs(see Fig. 5).

The calculated temperature, density, and pressuredistributions ahead of the flame front immediatelybefore the onset of DDT for initial mixture pressureP0 = 1.0 atm at t = 1.21 ms and P0 = 0.6 bar at t =3.85 ms are shown in Fig. 8. The flame front may beconsidered as a semi�permeable piston whose temper�ature rises by almost an order of magnitude on theflame front width scale Lf . The unburned mixturecompressed by the shocks in the layer ahead of theflame front penetrates into the front and is heatedalmost to the final temperature behind the front, initi�ating chain combustion reactions. As a result, a narrowhigh�amplitude pressure peak is formed at the leadingedge of the front. Already at the onset of DDT, themaximum pressure at the leading edge of the flamefront is 25 bar for P0 = 1.0 bar and about 20 bar for P0 =0.6 bar. It is worth noting that such a pressure corre�sponds to the jump in pressure behind a shock withalmost the same intensity as that of the leading shockin detonation.

Figure 9 show the pressure profiles at the flamefront from t = 0.2 ms to t = 1.2 ms at steps of 0.1 ms forthe same parameters as those in Fig. 6. During the sec�ond acceleration stage (t ≥ 0.4 ms), the compressionwaves produce shocks at the flame front. Starting fromthis moment, a fresh high�density mixture enters theflame front, which is heated and enters into reaction.As a result, a pressure wave peak is formed at the flamefront. As the density and pressure rise, the normalflame velocity and the amplitude of the compressionwaves produced by the accelerating front increase. Thepeak of the pressure wave with the maximum at theleading edge of the flame front accelerates the chaininitiation reactions, increasing in amplitude through arise in the energy being released in the acceleratingreaction and “drags” the combustion wave behind. Asa result of the positive feedback, the pressure peak atthe flame front grows exponentially with time. Themaximum pressure is reached by the time of detona�tion nucleation, when the narrow high�amplitudepressure peak produces a strong shock immediatelyahead of the flame front. Indeed, as we see from theresults of our computations presented in Fig. 10, the

Uf, km/s

1 2t, ms

1

3

1 2 3

2

4

0

3

4

DCJ

Fig. 7. Time dependences of the velocity of a combustionwave propagating in a 5�mm�wide channel for initial pres�sures P0 = 1.0 (1), 0.8 (2), and 0.6 (3) bar.



pressure maximum at the flame front at 0.5 ms < t <1 ms is fitted with a high accuracy by an exponential,

is the solid curve in Fig. 10 in which the circles indicatethe pressures obtained in our computations. Althoughthe exponent in the above expression for the growth ofthe pressure maximum cannot be found analytically,the exponential pressure growth per se has a simplephysical nature.

The changes in pressure growth rate are seen fromthe change in the slope of the time dependence of the

Pmax P0 1.5464t/ms( )exp=

velocity in Fig. 10 at 0.4 ms. After 1 ms, the compres�sion wave produces a strong shock at 1.21 ms. Sincethe half�width of the pressure wave pulse during thesecond flame acceleration stage is approximatelyequal to the flame front width Lf, the time of the shockformation and DDT can be estimated as the crossingtime of the characteristics emerging from the pointcorresponding to the pressure maximum and from thepoint ahead of the pressure peak. For the DDT time,we obtain

tDDT Lf/ Uf as+( ) 1 μs,≤≈

Fig. 8. Temperature (dot�and�dash curves), density (dashed curves), and pressure (solid curves) profiles before the onset of DDTfor P0 = 1 bar at t = 1.21 ms (a) and P0 = 0.6 bar at t = 3.85 ms (b).

T, K

200x, mm

1000

400

2000

3000

6000

10

6

2

0

p, bar

12

4

8

14

Fig. 9. Pressure (solid lines) and temperature (dashedlines) profiles for the conditions of Fig. 6 (P0 = 1 bar) atsuccessive instants of time from 0.2 to 1.2 ms at steps of0.1 ms.

Fig. 10. Change in maximum pressure in the compressionwave at the flame front with time: the dots represent ournumerical simulations; the solid curve indicates the fitPmax = P0exp(1.5464t/ms).

p, bar

0.5 t, ms

6

1.0

8

30

0

10

20

T, K

640 650x, mm

1000

660

ρ

T

p

2000

630

3000(a)

6700

3

2

1

0

ρ, kg/m3

4

15

10

5

20

0

25p, bar T, K

1570 1580x, mm

1000

1590

ρ

p

2000

1560

3000(b)

16000

2

1

0

ρ, kg/m3

20

10

5

0

p, bar

1610680

T

15

694


LIBERMAN et al.

where as is the speed of sound, in agreement with theresults of our computations presented in Fig. 6, wherethe transition is observed between 1.21 and 1.211 ms.Since the shock emerges at the leading edge of theflame front, it forms a complex associated with thereaction zone that corresponds to the structure of aZel’dovich–von Neumann–During detonation wavefrom the outset.

Figure 11 shows the temperature, pressure, and Hradical mass fraction profiles during DDT for instantsof times from 1.2101875 to 1.2113125 ms at steps of0.125 μs. The formation of a detonation wave is seenfrom the change in temperature gradient and theincrease in combustion product temperature and fromthe relative change in temperature profiles and Hradial distribution “triggering” reactions in H2–O2combustion. In the deflagration regime, the reactionsare distributed over the flame front, beginning approx�imately at 1100 K. In contrast, all reactions in detona�tion begin behind the temperature front, behind theshock.

Thus, the DDT mechanism is the formation of azone of compressed unreacted gas ahead of the flamefront formed during the second stage, when the accel�erating flame produces weak shocks directly at theflame front. As a result, a high�amplitude pressurepeak emerges, which grows exponentially with timeand turns into a strong shock associated with the reac�tion zone. As we see from our computations (seeFig. 11), the temperature gradient does not arise and,hence, is not the DDT mechanism, at least in the caseof highly reactive gas mixtures under the conditionsstudied. It could have been assumed that the gradientmechanism could appear at lower initial mixture pres�sures. Our computations performed for initial mixture

pressure P0 = 0.6 bar showed that the DDT mecha�nism in this case is the same as that at P0 = 1.0 bar.

It should be noted that the initiation of detonationduring the propagation of a narrow high�amplitudepressure pulse was observed in [37], where a one�dimensional numerical gasdynamic model with a sin�gle�step chemical reaction with a high activationenergy was considered. The authors of [37] consideredthe solutions obtained as a surprising effect, notingthat the meaning of the solutions obtained is unclear,because it is not clear how such a narrow and strongpressure wave could be formed and there are no exper�imental data that would explain the formation of sucha narrow pressure peak.

5. FLAME ACCELERATIONAND THE DDT MECHANISM

Experiments and numerical simulations of DDTshow that this transition begins after the formation ofshock waves directly at the flame front, producing alayer of compressed and heated gas adjacent to thefront. Thus, the peculiarity of the acceleration of acombustion wave in tubes with no�slip walls is themain factor determining the physical mechanism ofDDT in highly reactive mixtures that was first pointedout previously in [38–40], where only the gas heatingin the layer was taken into account.

In the thin�front approximation, the combustionwave velocity increases proportionally to the surfacearea (in the two�dimensional problem, the length) ofthe flame front [2]. The surface area of the flame frontincreases through instabilities bending the flame frontand increasing the combustion wave velocity, forexample, the hydrodynamic instability [1, 2]. How�ever, the front stretching along the tube walls due to gasfriction on the walls and a nonuniform velocity distri�

CH, arb. units

652 653x, mm

0.05

654

0.10

651 6550

3000

2000

1000

5000T, K

400060

40

20

80p, bar

656

Fig. 11. Sequences of temperature (dash–dotted curves) and pressure (dashed curves) profiles and distribution of the H radicalmass fraction (solid lines) during DDT at t = 1.2101875 ms < t < 1.2113125 ms at steps of 0.125 µs.



bution over the tube cross section in the flow ahead ofthe flame makes a major contribution to the increasein front surface area and flame acceleration at the ini�tial stage. The flame acceleration due to a nonuniformflow velocity distribution over the tube cross sectionhas already been discussed in early papers [3, 4]. In[41], it was shown that the combustion wave velocityincreases exponentially with time for a parabolicvelocity profile in the flow ahead of the flame. Underthe assumption of a parabolic velocity profile in theflow ahead of the flame, an exponentially unboundedincrease in velocity due to the flame stretching andacceleration was used in [42] to explain the explosionand DDT. However, for highly reactive mixtures, suchas hydrogen–oxygen, acetylene–air, etc., a Poiseuilleflow ahead of the flame is established in a time that isseveral orders of magnitude longer than the entireflame propagation time to DDT. A parabolic velocityprofile in the flow can arise after some period forweakly reactive mixtures. However, it is more likelythat in the latter case, flow turbulence will be the deci�sive factor for DDT.

Consider a simple model of a flame propagatingfrom the closed end of the tube that, nevertheless, pro�vides an insight into the peculiarities of flame acceler�ation in tubes significant for DDT physics. The flamepropagates from the closed end with velocity Ufl = θUfrelative to the tube walls, while the unburned mixtureahead of the front moves with velocity u = (θ – 1)Uf.Here, Uf is the normal flame velocity relative to theimmobile gas, θ = ρu/ρb ≈ 10 is the ratio of the densi�ties ρu and ρb ahead of and behind the flame front,respectively. The flow ahead of the flame is formedbehind the shock wave arising from the compressionwaves emitted by the accelerating flame front. In thiscase, a flow with a thin boundary layer and an almostconstant velocity distribution over the tube cross sec�tion is formed between the flame and the shock. As aresult, the flame velocity is Ufl = θUf almost in theentire tube cross section and decreases in the narrowboundary layer to Uf on the wall, so that the frontedges stretch backward along the tube walls in theboundary layer. Since the increase in flame velocity isproportional to the increase in front surface area,which increases linearly with velocity in the approxi�mation of a small (compared to the tube thickness)boundary layer thickness, the flame velocity growsexponentially with time. The flame velocity relative tothe tube walls in the thin�front approximation can berepresented as

(16)

where α is a numerical factor of the order of unity andD is the channel (tube) width.

Considering the accelerating flame front as a pistonthat generates compression waves, we can find the dis�tance ahead of the flame front at which the compres�sion wave produces a shock. At this distance, thevelocity is a multivalued function in the Riemannian

Ufl θUf αUft/D( ),exp=

solution for a simple compression wave [1]. It is easy tofind through calculations that for a piston moving withvelocity (16), the shock is formed at a finite velocity atan intermediate point of the compression wave:

(17)

where

is the flame front (piston) velocity in the frame of ref�erence comoving with the flow ahead of the front. Thedistance from the flame front to the place of shock for�mation, to within Ueff/as0 � 1, is equal to several tubewidths D:

(18)

Note that the exponential growth of the velocity isrelated not to the parabolic velocity profile in the flowahead of the flame but to the flame front stretching inthe narrow boundary layer. The thickness of the lami�nar boundary layer forming in the flow behind the

shock can be estimated as δ1 ∝ (Xsh – Xf)/ , whereXsh and Xf are the coordinates of the shock and theflame front. Since the distance from the flame to theshock is of the order of several tube diameters, theboundary layer thickness for large Reynolds numbersRe is much smaller than the tube diameter. The for�mation time of a Poiseuille velocity profile in the flowahead of the flame can be estimated as the time whenthe boundary layer thickness becomes of the order ofthe tube diameter. Using a well�known relation [2]between the front width and the velocity of a laminarflame, LfUf ≈ ν, where ν is the kinematic viscosity, theReynolds number in the flow ahead of the flame can berepresented as

The relation LfUf ≈ ν is valid to within an order of mag�nitude. Taking experimental values of

for our estimation, we will obtain Re = 103–104.Hence it follows that the maximum boundary layer

thickness near the flame front is about 5 mm under theconditions of our experiments in Section 2 and about0.5 mm for the conditions of our numerical simula�

u2as0

γ 1+�� 1 γ θ 1–( )

Ueff

as0

��– ,=

Ueff t( ) Uf 1 θ[ αUft/D( )exp 1]–+{ }=

XshD

α γ 1+( )��

as0

Ueff

��=

× γ2 γ 2 2 γ 1+( )2

4�� 1–

2as0

θ γ 1+( )Ueff

��ln–+ +⎩ ⎭⎨ ⎬⎧ ⎫

.

Re

Re Duν

�� θ 1–( )DLf

�� .= =

Lf 0.02 cm, D≈ 5 cm, D/Lf 102,≈=

θ 10, Xsh Xf– 5D,≈ ≈

696


LIBERMAN et al.

tions. The formation time of a Poiseuille flow ahead ofthe flame can be estimated by assuming the flow tobegin in the tube behind the shock. This time is then

which is much longer than the flame propagation timeto DDT, ~1 ms. A Poiseuille flow ahead of the flamecan be formed from the outset in the case of a flame ina capillary with a very small diameter. The criticaldiameter at which a Poiseuille flow emerges, for exam�ple, for combustion in an ethylene–oxygen mixture isapproximately 0.45 mm. The change in the flow pat�tern must be seen from the disappearance of the kinkin the time dependence of the velocity in experiments[43] for a tube diameter smaller than 0.45 mm.

The first stage of exponential flame accelerationlasts a short time interval. Since the characteristiclength scale of thermal conduction is of the order ofthe flame front thickness, we will find that the angle atthe vertex of the fold formed in the boundary layer

between the flame front and the wall at t ≥ τf ≈5τf decreases in such a way that the heat flux from thereaction zone in the transverse direction becomes ofthe order of the heat flux along the tube axis. In thiscase, the surface area of the flame front begins to rap�idly decrease either through mixture heating inside thefold and an increase in the velocity of the part of theflame near the vertex of the angle in the fold or throughquenching of the part of the flame near the fold vertexif the heat losses of the tube wall are high. The decreasein the surface area of the flame front causes a decreasein the rate of increase in the combustion wave velocity.Since, to small terms of the order of Lf/D � 1, it followsfrom (16) that the acceleration during the first stage isconstant, the combustion wave velocity at the second

stage, at t ≥ τf ≈ 5τf, can be represented as

(19)

For a piston moving with velocity (19), the velocity ofa simple compression wave in the Riemannian solu�tion is always a multi�valued function at any values ofthe exponent n < 1. A specific value of n < 1 is unim�portant in this case, because we are interested only inthe place of shock formation relative to the flamefront. Since the solution for the velocity is always amulti�valued function, the compression wave turnsduring the second acceleration stage into a shockdirectly at the piston surface, i.e., in the immediatevicinity of the flame front.

The scenario for the formation of shock waves by aflame accelerating in a tube considered here disregardsthe more complex picture that includes the front insta�bility, the interaction of the flame front with the com�pression waves reflected from the tube walls, etc. Atthe same time, this model shows the overall pattern ofthe processes that lead to the formation of a layer of

tP θ DLf

��⎝ ⎠⎛ ⎞ 2 Lf

Uf

�� 105 Lf

Uf

�� 104 ms,∼ ∼≈

D/θLf

D/θLf

Uf Ueff 1 β t/τf( )n+[ ], 0 < n 1.<≈

compressed and heated gas adjacent to the flame front,which is the decisive process including DDT. The factthat the shocks are formed directly at the flame frontthrough a change in the flame acceleration rate in thetube was first considered previously in [38, 39]. In thiscase, it was assumed that the heating of a fresh gasmixture immediately ahead of the flame front alonecould lead to an acceleration and broadening of theflame front, as a result of which a temperature gradientis formed for DDT according to Zel’dovich’s mecha�nism when the flame temperature profile is restruc�tured.

6. CONCLUSIONS

Experiments and numerical simulations taking intoaccount the H2–O2 mixture combustion chemicalkinetics show that DDT in highly reactive mixtures isinitiated by the formation of a layer of shock�com�pressed gas ahead of the flame front. Our analysis of theDDT mechanism indicates that the gas compression bythe shocks ahead of the flame and the resulting high�amplitude pressure peak at the flame front turn out tobe the most significant processes for the DDT mecha�nism. The exponentially growing (with time) pressurein the compression wave at the leading edge of theflame front accelerates the initiation of combustion,reducing the ignition time and simultaneously amplify�ing the compression wave, so that the latter produces astrong shock associated with the reaction zone.

Detailed numerical simulations show that no tem�perature gradient emerges during the formation ofdetonation and that the gradient mechanism is not theDDT one, at least for the highly reactive mixtures con�sidered here. It should be noted that Zel’dovich’s gra�dient mechanism of detonation initiation was shownto be feasible only in terms of one�dimensionalnumerical models. On the other hand, the layer ofheated gas adjacent to the flame front can be formedwithout shock involvement. Such a configuration canemerge when combustion is initiated in a narrow gap[44] or when the gas trapped inside the gap at the flamefront produced by the front instability is heated [45].A narrow gap at the front can also be formed when aflame interacts with turbulent vortices in the case ofdeflagration during the propagation in a tube withobstacles.

ACKNOWLEDGMENTS

We thank the Russian National Foundation for thepreparation of personnel in the sphere of higher edu�cation and the Federal Agency for Science and Inno�vations for support of this work. The work was per�formed under State contract no. 02.740.11.5108) inthe context of the “Investigation of the FundamentalProcesses of Laminar and Turbulent Combustion andthe Controllability of Combustion Processes UsingExcited Molecules and Radicals of Oxygen Generated



in a Nonequilibrium Plasma” Program. We thankS.M. Frolov, B.E. Meierovich, N.N. Smirnov, andM. Radulescu for helpful discussions. One of us(M.A.L.) is grateful to I.Ya. Koshatskii.

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Translated by V. Astakhov

On the Mechanism of the Deflagration-to-Detonation Transition in a Hydrogen–Oxygen Mixture

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Transcript of On the Mechanism of the Deflagration-to-Detonation Transition in a Hydrogen–Oxygen Mixture