Flame acceleration and DDT of hydrogen-oxygen gaseous mixtures in channels with no-slip walls

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Flame acceleration and DDT of hydrogeneoxygen gaseousmixtures in channels with no-slip walls

M.F. Ivanov a, A.D. Kiverin a, M.A. Liberman b,c,*a Joint Institute for High Temperatures, Russian Academy of Science, Moscow, Russiab Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119991 Moscow, RussiacDepartment of Physics and Astronomy, Uppsala University, Box 516, 751 21 Uppsala, Sweden

a r t i c l e i n f o

Article history:

Received 23 January 2011

Received in revised form

18 March 2011

Accepted 22 March 2011

Available online 23 April 2011

Keywords:

Hydrogen

Flame acceleration

Shock wave

DDT

Detonation

a b s t r a c t

Hydrogeneoxygen flame acceleration and transition from deflagration to detonation (DDT)

in channels with no-slip walls were studied theoretically and using high resolution

simulations of 2D reactive NaviereStokes equations, including the effects of viscosity,

thermal conduction, molecular diffusion, real equation of state and a detailed chemical

reaction mechanism. It is shown that in “wide” channels (D> 1 mm) there are three

distinctive stages of the combustion wave propagation: the initial short stage of expo-

nential acceleration; the second stage of slower flame acceleration; the third stage of the

actual transition to detonation. In a thin channel (D< 1 mm) the flame exponential

acceleration is not bounded till the transition to detonation. While velocity of the steady

detonation waves formed in wider channels (10, 5, 3, 2 mm) is close to the Chap-

maneJouguet velocity, the oscillating detonation waves with velocities slightly below the

CJ velocity are formed in thinner channels (D< 1.0 mm). We analyse applicability of the

gradient mechanism of detonation ignition for a detailed chemical reaction model to be

a mechanism of the deflagration-to-detonation transition. The results of high resolution

simulations are fully consistent with experimental observations of flame acceleration and

DDT in hydrogeneoxygen gaseous mixtures.

Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

reserved.

1. Introduction

Flame propagation and the deflagration-to-detonation tran-

sition (DDT) in channels have been intensively studied both

experimentally and numerically over the past decades. These

studies are inspired by their importance for industrial safety

concerns [1e3] and because their potential application for

micro-scale propulsion and power devices [4,5]. The classical

formulation of the problem in question and an experimental

set-up is an initially laminar flame, which is ignited near the

closed end of the tube and then propagates to an open end.

The flame accelerates, produces weak shocks, may become

turbulent, and eventually DDT occurs, which appears as

a sudden explosion in the vicinity of the flame front [6e14].

From the earlier studies [6,8,12,13] it was thought that the

crucial aspect for DDT is a high intensity of turbulence in the

flow ahead of the flame. A common belief was that the fast

flame acceleration and the DDT are possible only for strongly

turbulent flames. The first explanation of the flame accelera-

tion in tubes with no-slip walls before the DDT goes back to

the work by Shchelkin [13] who argued that due to the wall

friction and the thermal expansion of the burning matter the

flame front becomes curved and therefore accelerates. The

accelerating flame acts like a piston pushing compression

* Corresponding author. Department of Physics and Astronomy, Uppsala University, Box 516, 751 21 Uppsala, Sweden. Tel.: þ46 18329277.E-mail addresses: [email protected], [email protected] (M.A. Liberman).

Avai lab le at www.sc iencedi rect .com

journa l homepage : www.e lsev ie r . com/ loca te /he

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 6 ( 2 0 1 1 ) 7 7 1 4e7 7 2 7

0360-3199/$ e see front matter Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.ijhydene.2011.03.134


waves into the fuel mixture, which may steepen into the

shock waves. The flame acceleration in tubes with no-slip

walls has been widely accepted as an important factor that

influences the DDT [6e16].

DDT is a stochastic process meaning that there is some

uncertainty regarding time and location of the transition.

Exactly how DDT occurs may vary depending on particular

experimental conditions [6e14]. Since the work by Shchelkin

[13] it is known that the presence of obstacles along the

channel walls increases the flame acceleration and shortens

drastically the run-up distance. The experiments demon-

strated that a flame accelerates more rapidly toward the open

end if it passes through an array of turbulence-generating

baffles. This presumably was the reason why the first

attempts to explain DDT were associated with turbulent

flames and were based on assumption that DDT might occur

only in the case of turbulent flames [6,13]. Channels with

rough walls or obstacles are often used to study DDT since in

this case the run-up distance is more or less controlled [17,18].

All the same DDT occurs in channels with relatively smooth

walls [17,19] and in thin capillary tubes [20].

Over the years significant efforts have been devoted to

understand the nature of the flame acceleration and mecha-

nism of the transition from deflagration to detonation.

Although the qualitative picture of the DDT is more or less

clear, however a quantitative theory and the physical mech-

anism of DDT are still poorly understood and require better

theoretical and physical interpretation. Without a doubt the

difficulties are the turbulent burning by itself and the need to

use a detailed chemical model. The conclusions about

possible mechanism of the DDT that have been drawn from

the previous studies were based on a simplified model in

which the reaction proceeds via a single-step exothermic

reaction with Arrhenius kinetics which cannot reproduce the

main properties of the combustion such as the induction time

in chain-branching kinetics and detonation initiation. It is

therefore important to investigate the qualitative and quan-

titative differences of the processes between chain-branching

kinetics and the predictions from one-step models.

The objective of the present work is to examine effect of

the channel width on the flame dynamics and DDT in

hydrogeneoxygenmixtures in channels of different widths. It

is shown that the flame acceleration in channels with no-slip

walls is entirely determined by the features of the flow formed

ahead of the flame. It is shown that the flame dynamics prior

the transition to detonation is considerably different for the

width of the channel less than the critical one in accordance

with the experiments [20]. The results of high resolution

simulations and theoretical analysis are found to be consis-

tent with the experimental studies of the DDT in ethyl-

eneeoxygen [20] and hydrogeneoxygen [21] highly reactive

gaseous mixtures. When DDT occurred, the detonations that

developed always arose from the exponentially growing

pressure pulse, which then triggers the transition to detona-

tion according to themechanism of DDT recently proposed by

Liberman et al. [22,23]. Finally in Section 6 we discuss

a possibility of the DDT to originate due to the Zeldovich

gradient mechanism with the emphasis on comparing the

results using a detailed chemical model with previous studies

that used a one-step chemical model.

2. Formulation of the problem

The high resolution simulations modeled a flame ignited near

the closed end and then propagating to the open end of the

two-dimensional channel. The computations solved the two-

dimensional, time-dependent, reactive NaviereStokes equa-

tions for compressible flow including the effects of viscosity,

thermal conduction, molecular diffusion, the real equation of

state and detailed chemical kinetics for the reactive species

H2, O2, H, O, OH, H2O, H2O2, and HO2 with subsequent chain

branching and energy release in the form

vr

vtþ vðruÞ

vxþ vðrvÞ

vz¼ 0; (1)

vYi

vtþu

vYi

vxþv

vYi

vz¼1r

�v

vx

�rDi

vYi

vx

�þ v

vz

�rDi

vYi

vz

��þ�vYi

vt

�ch

; (2)

r

�vuvt

þ uvuvx

þ vvuvz

�¼ �vP

vxþ vsxx

vxþ vsxz

vz; (3)

r

�vvvt

þ uvvvx

þ vvvvz

�¼ �vP

vzþ vsxz

vxþ vszz

vz; (4)

r

�vEvt

þ uvEvx

þ vvEvz

�¼ �

�vðPuÞvx

þ vðPvÞvz

�þ v

vxðsxxuþ sxzvÞ

þ v

vzðszxuþ szzvÞ þ v

vx

�kðTÞvT

vx

�þ v

vz

�kðTÞvT

vz

�þXk

hk

mk

�v

vx

�rDkðTÞvYk

vx

�þ v

vz

�rDkðTÞvYk

vz

��; ð5Þ

P ¼ RBTn ¼ X

i

RB

miYi

!rT ¼ rT

Xi

RiYi; (6)

e ¼ cvTþXk

hkrk

r¼ cvTþ

Xk

hkYk; (7)

sxx ¼ 2mvuvx

� 23m

�vuvx

þ vvvz

�; (8)

szz ¼ 2mvvvz

� 23m

�vuvx

þ vvvz

�; (9)

szx ¼ sxz ¼ m

�vv

vxþ vu

vz

�: (10)

Here P, r, u, v e are pressure, mass density, x and z compo-

nents of the velocity, Yi¼ ri/r e the mass fractions of the

species, E¼ eþ (u2þ v2)/2 e the total energy density, e e the

inner energy density, RB e is the universal gas constant, mi e

the molar mass of i-species, Ri¼ RB/mi, n e the molar density,

sij e the viscous stress tensor, cv ¼Pi

cviYi e is the constant

volume specific heat, cvie the constant volume specific heat of

i-species, hi e the enthalpy of formation of i-species, k(T ) and

m(T ) are the coefficients of thermal conductivity and viscosity,

Di(T ) e is the diffusion coefficients of i-species, ðvYi=vtÞch e is

the variation of i-species concentration (mass fraction) in

chemical reactions.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 6 ( 2 0 1 1 ) 7 7 1 4e7 7 2 7 7715


The real equations of state for the fresh mixture and

combustion products were taken with the temperature

dependenceof thespecificheats andenthalpies of eachspecies

borrowed from the JANAF tables and interpolated by the fifth-

order polynomials [24]. The transport coefficients were calcu-

lated from the first principles using the gas kinetic theory

[25,26]. The viscosity coefficients for the gaseous mixture are

m ¼ 12

"Xi

aimi þ X

i

ai

mi

!�1#; (11)

where ai ¼ ni=n is the molar fraction, mi ¼ ð5=16Þx� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip bmikT

q=

�pS2

i~Uð2;2Þi

��is the viscosity coefficient of i-species,

~Uð2;2Þ

e is the collision integral which is calculated using

the Lennard-Jones potential [26], bmi is the molecule mass

of the i-th species of the mixture, Si is the effective molecule

size. The thermal conductivity coefficient of the mixture is

k ¼ 12

"Xi

aiki þ X

i

ai

ki

!�1#: (12)

Coefficient of the heat conduction of i-th species ki¼ micpi/Pr

can be expressed via the viscosity mi and the Prandtl number,

which is taken Pr¼ 0.75.

The binary coefficients of diffusion are

Dij ¼ 38

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pkT bmi

bmj=� bmi þ bmj

�qp$r$S2

ij~Uð1;1Þ

T�ij

;

where Sij¼ 0.5(SiþSj), Tij*¼ kT/eij*, e�ij ¼ffiffiffiffiffiffiffiffie�i e

�j

q; e* are the

constants in the expression of the Lennard-Jones potential,

and ~Uð1;1Þij is the collision integral similar to ~U

ð2;2Þ[25,26].

The diffusion coefficient of i-th species is

Di ¼ ð1� YiÞ=Xisj

ai=Dij: (13)

Variation of the concentrations Yi due to chemical reactions

is defined by system of chemical kinetics

dYi

dt¼ FiðY1;Y2;.YN;TÞ; i ¼ 1;2;.N: (14)

The right hand parts of (14) contain the rates of chemical

reactions, which depend on temperature according to the

Arrhenius law in a standard form [25]. The elementary reac-

tions of the Arrhenius type together with pre-exponential

constants and activation energies used in the simulations

are presented in Table 1.

3. Code validation: resolution andconvergence tests

The convergence of the solutions and the proper resolution to

capture details of DDT is of paramount importance, especially

when a detailed chemical mechanism is used. Thorough

convergence and resolution tests were carried out to verify

that the observed phenomena and the mechanism of DDT

remain unchanged with increasing resolution, especially

during late stages of the flame acceleration when the flame

thicknessmay decrease due to pressure rise in the flow ahead.

3.1. The numerical method and the code validation

TheNaviereStokes equations (1)e(10)were integrated using the

second-order numerical scheme based on splitting of the

Eulerian and Lagrangian stages, also known as the coarse

particlemethod (CPM) [27]. High stability of themethod is due to

dividing one time-step calculation into three stages. On the first

stage, the change of hydrodynamic characteristics on the fixed

Eulerian space grid is calculated using the explicit scheme

without regardingofmass,momentumandenergytransfer.The

hydrodynamic variables are transferred through the cell

boundaries on the second stage using the values of hydrody-

namic characteristics from the first stage. The third stage

consists of final calculation of the values of all parameters for

every cell and for thewholesystem. Itwasshown [28] that ahigh

numerical stability of the method is achieved if the hydrody-

namic variables are transferred across the grid boundary with

the velocity, which is an average value of the velocities in

neighboring grids. The overall modified solver is then the

second-orderaccurate,providinghighaccuracyof thesolutions.

The modified CPM and optimal approximation scheme were

thoroughly tested and successfully used for simulation engine

combustion knock occurrence in SI engine [28,29].

The convergence and resolution tests are shown in Figs. 1

and 2. Fig. 1 shows the convergence tests and accuracy of

calculations for different mesh resolutions for the one-

dimensional steady-state laminar flame. The meshes were

taken to resolve the structure of the flame front with 6, 8, 16,

32 and 64 computational cells, corresponding to the compu-

tational cell sizes: of 0.1, 0.05, 0.02, 0.01, 0.005 mm, respec-

tively. Fig. 1 shows that accuracy of the solution is quite

satisfactory already for 8 computational cells per flamewidth.

Even with six cells per the flame width, the flame velocity,

density and temperature differ less than 2% from the

Table 1 e Reactions and reactions rates.

Reactions Af (cm3/mol s) Eaf (kcal/mol) Ab (cm3/mol s) Eab (kcal/mol)

(R1) H2þO2¼ 2OH 2.52� 1012 39.0 1.16� 1013 21.0

(R2) OHþH2¼H2OþH 2.25� 1013 5.24 9.90� 1013 20.3

(R3) HþO2¼OHþO 1.55� 1014 16.7 1.16� 1013 0.705

(R4) H2þO¼OHþH 2.46� 1013 9.84 1.07� 1013 7.90

(R5) 2HþM¼H2þM 3.60� 1015 0.0 1.46� 1016 104.0

(R6) HþO2þM¼HO2þM 3.60� 1015 0.0 3.01� 1015 47.8

(R7) 2HO2¼H2O2þO2 1.0� 1013 0.0 1.30� 1014 40.0

(R8) 2OHþM¼H2O2þM 1.11� 1016 1.92 7.40� 1018 47.0

(R9) HþH2O2¼H2þHO2 1.17� 1014 11.8 1.55� 1014 28.5



converged solution. With resolution of 16 computational cells

per flame width the difference between computed values and

the converged solution is negligible.

Fig. 2 shows the one-dimensional steady-state laminar

flame structure computed using four different numerical

resolutions. Themesheswere taken to resolve structure of the

flame front for temperature and concentration of species

YH2 ; YH; YOH with 6, 8, 16 and 32 computational cells per width

of the flame. The results obtained using more than 8 compu-

tational cells agree very well with the converged solution,

whereas there are some small differences between the

converged solution and the solution using 6 cells to resolve the

flame structure. Similar resolution and convergence tests

were also performed for higher initial pressures to ensure that

the resolution is adequate to capture details of DDT during

late stages.We concluded that a resolution from 8 to 12 cells is

adequate to capture details of the flame acceleration and the

mechanism of DDT.

Table 2 shows the parameters of the shocks calculated

using the RankineeHugoniot adiabatic, the experimental

parameters of hydrogeneoxygen detonation [30] and hydro-

geneoxygen flame and the corresponding parameters

computed using the code. The agreement between the theo-

retical, experimental and computed values is very good. The

slight difference of some theoretical and computed values is

because the theoretical calculations used an ideal equation of

state, while the computations used a real equation of state.

3.2. Chemical reaction scheme: sensitivity analysis

The chemical reactions are the set of ordinary differential

equations (14) for concentrations Yi of eight species H2O, H2, H,

O2, O, OH, HO2, H2O2 per computational cell. Stiff systemof the

differential equations of chemical kinetics was solved using

the Gear method [31]. Third body efficiencies for the recombi-

nation reactions were considered in way similar to paper [32].

Themass diffusivity of themixturewasmodeled using kinetic

theorywithdefault values for the Lennard-Jones characteristic

length and energy parameters for the individual species. The

viscosity and the specific heat of the mixture have been eval-

uated using mass weighted mixing laws. For the individual

species in the mixture these properties have been taken from

JANAF Tables with their temperature dependence approxi-

mated by the fifth-order polynomials.

It was found that the flame structure and the thermody-

namic characteristics of the flame and detonation are only

slightly dependent if any when different detailed chemical

schemes are used in hydrodynamic calculations. Fig. 3 shows

the induction times calculated using different chemical

kinetics schemes for H2eO2 mixture, where curves 1e6 are

calculated for the reaction schemes [33e38] and curve 7 is for

the one-step Arrhenius kinetics, respectively. It is seen that

the various detailed reaction schemes give rather similar

values of the induction times. There is some deviation in the

induction times in the temperature range 850e950 K for the

schemes [35e37] which is not significant. At the same time

there is a significant difference between the induction time

obtained from the single-step Arrhenius model and from the

detailed chemical reaction schemes. The induction time for

a single-step Arrhenius model is several orders of magnitude

0 20 40 60

0.98

1

Lf /�x

ρ b/ ρ

,b0

ρb/ρb0Tb /Tb0Uf /Uf0

Uf/U

f0T b

/ T,

b0

Fig. 1 e Convergence test for the one-dimensional steady-

state laminar flame structure. Comparisons are shown for

density, velocity and temperature normalized to their

ambient values.

T,K

0

500

1000

1500

2000

2500

3000

- 32- 16- 8- 6

Y H2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Y H

-0.05

0

0.05

0.1

0.15

0.2

0.25

x, mm

Y OH

-0.5 0 0.5

0

0.02

0.04

0.06

0.08

0.1

Fig. 2 e Resolution test for the one-dimensional steady-

state laminar flame structure. Comparisons are shown for

temperature and concentration of species YH2 ; YH; YOH

computed with 6, 8, 16 and 32 computational cells per

width of the flame.



shorter than the induction time for the detailed chemical

models in the most important temperature range T< 1000 K.

A one-step Arrhenius model cannot reproduce the chain-

branching kinetics because in order to have a correct induc-

tion time for a one-step chemicalmodel, the activation energy

must be extremely high, but then the thickness of the reaction

zone in the flame becomes extremely thin.

Computations showed that the velocity, thickness and

adiabatic temperature of a laminar flame obtained for

different detailed chemical reaction schemes [33e38] are close

to each other with accuracy better than 10%. The flame

structure and parameters of the flame in Table 2 were repro-

duced with high accuracy using the reactive NaviereStokes

code based on the third order upwind-biased finite volume

method [39], which has been traditionally used for the spatial

discretization of theNaviereStokes equations for DNS andhas

proved to be robust for modeling complex hydrodynamic

flows [40] and combustion problems [41].

4. Effect of the channel width on the flameacceleration

Simulations of the flame propagating in channels with no-slip

walls filled with the stoichiometric H2eO2 mixture at initial

temperature and pressure T0¼ 298 K and P0¼ 1 atm were per-

formedfor thechannelsofwidthsD from10mmto0.5 mmwith

the minimum computational cell size: D¼ 0.02 mm. A laminar

planar flame was initiated near the left closed end of two-

dimensional channel and propagated to the right open end.

Fig. 4 shows the computed velocityetime dependence of

the combustion wave during the flame acceleration and the

transition to detonation in channels of widths: 0.5, 1, 3, and

5 mm. For the channels of width D> 1.0 mm, the veloc-

ityetime dependence plots demonstrate the same feature of

three distinctive stages of the flame acceleration: a short stage

of the exponential increase of the flame velocity; the second

stagewhen the acceleration rate decreases comparedwith the

first stage; the sharp increase of the flame velocity and actual

transition to detonation. On the contrary, scenario of the

flame acceleration is different for thinner channels,D< 1 mm.

The difference in dynamics of the flame acceleration in

a narrow and in awide channels is clearly seen in Figs. 5 and 6,

which show zoomed images of the velocityetime depen-

dences during the initial stages of the flame acceleration in

the channels of width D¼ 2, 3, 5, 10 mm in Fig. 5 and in the

channels of widths D¼ 0.5, 0.8, 1.0 mm plotted in Fig. 6. For

a wider channel, the initial stage which is approximated by

the exponential increase of the velocity shown by the dashed

lines in Fig. 5, turns into the stage of a slower velocity increase,

which can be described by a polynomial function

ULffaþ btn1 þ ctn2 þ/ with the exponents ni< 1. On the

contrary, in a thinner channel, D< 1 mm, the exponential

increase of the flame velocity (the dashed lines in Fig. 6) is not

bounded in time and an initially subsonic deflagration wave

may accelerate exponentially till the actual transition to

detonation.

Table 2 e Shock, detonation and flame parameters forhydrogeneoxygen.

Shock parameters Computed Theory

Pressure, bar 7.06 7.11

Density, kg/m3 1.67 1.60

Gas velocity, m/s 947.45 943.25

Mach number 2.5 2.5

Detonation parameters Computed Experiment

Initial temperature, K 300 298

D, m/s 2722 2713

Pressure ratio, p2/p0 17.3 17.3

T2, K 3463 3278

Mach number 5.05 5.05

Overdriven detonation Computed Experiment

Preheating, K 600 600

Pressure ratio, p2/p0 7.7 7.74

T2, K 3187.0 3187.6

Deflagration Computed Experiment

Flame thickness 0.32 0.26

Adiabatic temperature 3050 3100

Expansion factor 8.2 8.4

Fig. 3 e Induction times for H2eO2 mixture calculated using

different chemical kinetic schemes: curves 1e6 for the

detailed chemical kinetic schemes; curve 7 is for the one-

step Arrhenius kinetics.

t, ms

UfL

,m/s

0 0.5 10

1000

2000

3000

4000

CJ-detonation0.5

1.0 3.0 5.0

Fig. 4 e Computed H2eO2 flame velocityetime

dependences in channels of widths: 0.5, 1, 3, 5 mm.



A propagating flame initiated near the closed end of the

channel controls the flow forming ahead of it, which results in

the flame acceleration compatible with the physical boundary

conditions. It is therefore important to examine the acceler-

ation mechanisms for a propagating flame depending on the

physical conditions such as, thickness of the boundary layer

and the velocity distribution in the flow ahead of the flame.

The feature of the flame acceleration depends on the width of

a channel with no-slip walls and influences the physical

mechanism of the transition to detonation. When the flame is

initiated near the closed end, the expansion of the high

temperature burned gas induces an outward flow of the

unburned mixture with the velocity u¼ (Q� 1)Uf ahead of the

flame front while the flame propagates with the velocity

UfL¼QUf in the laboratory reference of frame [42], where Uf is

the normal velocity of laminar flame, Q¼ ru/rbz 10 is the

density ratio of the unburned ru and burned rb gases,

respectively. Because of the wall friction the flow velocity

vanishes at the channel walls and the flow field ahead of the

flame is not uniform. Every part of the flame front moves with

respect to the unreacted mixture with normal velocity Uf and

simultaneously it is drifted with the local velocity of the flow

ahead of the flame. Thus, the flame shape is defined by the

relative motion of different parts of the flame front. As the

flame front advances into a nonuniform velocity field, the

flame surfacewill be stretched taking the shape of the velocity

profile in the flow ahead, and the flame surface will increase.

The stretched flame consumes fresh fuel over a larger surface

areawhich results in an increase in the rate of heat release per

unit projected flame area. The increase in the rate of heat

release due to the flame stretching gives rise to a higher

volumetric burning rate, and a higher effective burning

velocity based on the average heat release rate per frontal area

of the stretched flame sheet. A higher burning velocity results

in an enhancement of the flow velocity ahead of the flame,

which in turn gives rise to a larger gradient field and enhanced

flame stretching, and so on. In this way a positive feedback

coupling is established between the upstream flow and the

burning velocity as the flame is stretched in the velocity field

ahead of the flame front.

For a wide channel the flow ahead of the flame is nearly

uniform in the bulk, with the flow velocity dropping to zero at

the channel walls within a thin boundary layer of thickness:

dl�D. The flame sheet “repeats” the shape of the upstream

flow velocity profile remaining almost flat in the bulk with the

edges of the flame skirt stretched backward within the

boundary layer. Within the model of a thin flame the increase

of the burning rate (combustion wave velocity) is proportional

to the relative increase of the flame surface (length in 2D case)

which grows linearly in timewith accuracy dl/D� 1 andwhich

in turn gives rise to a larger gradient of the velocity field ahead.

Therefore the combustion wave velocity increases exponen-

tially in time, which can be presented as

UfL ¼ QUf$exp�aUft=D

�; (15)

where a is a numerical factor of the order of unity.

Expression (15) for the combustion wave velocity is similar

to the exponential increase of a finger flame velocity obtained

in [43], but the expression (15) arises due to the specific

structure of the upstream flow with a thin boundary layer

with accuracy dl/D� 1.

The accelerating flame acts as a piston producing

compression waves in the unreacted gas. The time and the

coordinate where the compression wave steepens into a shock

wave are determined by the condition that the velocity u(x,t) in

theRiemannsolution for a compressionwavebecomesamulti-

valued function [44]. From this condition, one can obtain the

distance between the flame (x¼Xf) and the coordinate (x¼Xsh)

where shock is formed (Xsh�Xf)w 5D. Then the thickness of

the boundary layer can be estimated as dlwðXsh � XfÞ=ffiffiffiffiffiffiRe

p,

whereRe¼ (Q� 1)UfD/nxQUfD/n is the Reynolds number in the

upstream flow, and n is the kinematic viscosity coefficient.

For a wide channel the condition dl/D� 1 is satisfied since

Re[ 1. In these cases the Poiseuille flow ahead of the flame is

not established until the actual transition to detonation. The

width of a channel for which the Poiseuille flow is formed

before the DDT occurs can be estimated from the condition

that thickness of the boundary layer dlw5D=ffiffiffiffiffiffiRe

pwill become

of the order of the channel width D, which gives Rew 25. For

numerical estimates it is convenient to express the Reynolds

number in terms of the flame thickness Lf and the normal

velocity of a laminar flame [42] taking into account LfUfx n

[40], so that RewQD/Lf.

t, ms

UfL

,m/s

0 0.2 0.4 0.6 0.8 10

200

400

600

23

510

Fig. 5 e Enlargement of the flame velocityetime

dependence for the channels: D[ 2, 3, 5 and 10 mm.

Dashed lines are approximations by the solution (15).

t, ms

UfL

,m/s

0 0.05 0.1 0.15 0.2 0.25 0.30

200

400

600

800

1000

1200

0.5

0.8

1.0

Fig. 6 e Enlargement of the flame velocityetime

dependence for thin channels D[ 0.5, 0.8 and 1.0 mm.

Dashed lines are exponential approximations.



Taking for H2eO2 flame at P0¼ 1 bar: Lf¼ 0.32 mm,

Ufx 10 m/s, Qx 8, and we find out that the Poiseuille flow is

formed in the channel of width less than Dw 25Lf/Qx 1 mm

long before DDT. The qualitative difference of the flow fields

developing ahead of the flame in a “wide” channel and in

a thin channel is illustrated by Fig. 7, computed for the

channel of width D¼ 5 mm and Fig. 8 computed for the

channel D¼ 0.5 mm. The figures show the velocity contours

and the corresponding longitudinal velocity profiles (right

part) in the cross section AA of the channels.

For a flame propagating in a thin channel the parabolic

velocity profile corresponding to Poiseuille flow can be

established ahead of the flame long before the transition to

detonation occurs. In this case the combustion wave velocity

also increases exponentially similar to Eq. (15) with slightly

different value of a due to a positive feedback coupling

between the flame and the upstream flow. However, contrary

to the wide channels, in a thin channel the exponential

velocity increase is not bounded in time until the transition to

detonation. Fig. 9 shows computed values of a for the veloc-

ityetime dependence (15) for channels of widths from 0.5 mm

to 10 mm. For an ideal case, when small perturbations were

not imposed, the flame shape is symmetrical developing well

familiar tulip shape [45,46]. The corresponding values of a are

shown by the solid line with squares. For more realistic case,

when small perturbations (less then 1% of the flame velocity)

were imposed from the beginning, the shape of the flame

shown in Fig. 10 became asymmetrical, and the corresponding

values of a are shown by the solid line with the triangles. The

linewith circles is for the thin channelsD< 1 mm. It should be

stressed that in a thin capillary channels only a single-mode

(non-tulip) flame develops.

As the flame front is stretched along the walls in a wide

channel, a narrow fold is formed within the boundary layer

between the flame skirt and the wall. As the fold becomes

deeper, the angle at the fold’s tip becomes smaller, and parts

of the flame front near the fold’s tip approaches the wall. A

simple geometric consideration [47] shows that after the time

about t2wðLf=UfÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiD=QLf

pthe edges of the flame skirt near the

walls will be shortened reducing the surface of the flame front

area and thus decreasing the rate of the flame acceleration.

Since this time interval is short, during the first stage the

flame acceleration is nearly constant with accuracy of the first

order terms of series expansion in Lf/D� 1. Therefore during

the next stage, t� t2, the flame velocityetime dependence can

be approximated as

UfLxQUf

1þ b

�t=sf

�n�; where 0 < n < 1: (16)

For a piston moving with the velocityetime dependence (16)

the function u(x,t) in the Riemann solution for a simple trav-

elling wave is multi-valued everywhere for any values

0< n< 1. This means that formally the compression wave

produced by the flame steepens into the shock directly on the

surface of the flame. Specific value of 0< n< 1 in expression

(16) does not matter since we are interested only in the loca-

tion where the compression wave steepens and the shock is

formed.

In contrast to a stationary flame, the flow with the accel-

erating flame is not isobaric. In the latter case pressure is

growing at about the same rate as the flame velocity. From the

time when the compression waves steepen into the shock

close to the flame front, the unreacted mixture of higher

density enters the flame front and produces a narrowpressure

pulse on the scale of the order of the flamewidth. The increase

of pressure enhances reaction rate and the heat release in the

reaction zone creating a positive feedback coupling between

the pressure pulse and the heat released in the reaction. As

a result the peak of the pressure pulse will grow exponentially

increasing the reaction rate. In a thin channel the combustion

wave velocity increases exponentially, so that the compres-

sion waves steepen into the shock at the distance about

several widths of the channel. Because of small width of the

channel, the distance to the shock is comparable to the flame

thickness. Therefore, the pressure pulse in a thin channel

increases exponentially from the beginning of the flame

acceleration. This difference in the dynamics of flame

x, mm410 420 430

5mm

A

z, mm

u, m

/ s

0 1 2 3 4 50

200

400

600

A-A

Fig. 7 e Flow field ahead of the flame front in “wide”

channel, D[ 5 mm at t[ 1 ms (left), and corresponding

velocity profile along the channel in the cross AA (right).

x, mm7 8 9 10 11

A

z, mm

u, m

/s

0 0.1 0.2 0.3 0.40

50

100

150

200

A-A0.5mm

Fig. 8 e Flow field ahead of the flame front in capillary

channel D[ 0.5 mm, at t[ 0.1 ms (left), and corresponding

velocity profile along the channel in the cross AA (right).

D,mm

α

0 2 4 6 8 10

1

1.5

2

2.5

3

3.5

Fig. 9 e Numerical factor a in the expression (15) for the

combustion wave velocity in channels D[ 0.5, 0.8, 1.0, 2, 3,

5, 10 mm. Solid lines with squares are computed for tulip

flames, solid lines with triangles are computed for

asymmetric flames, circles are for single-mode flames in

capillary channels.



acceleration in a wide and in a thin channel is clearly seen in

the experimentally measured velocityetime dependence [20].

In simulations of H2eO2 flames there are three distinctive

stages of the flame acceleration for wide channels, D> 1 mm,

while for D< 1 mm there is only one stage of exponential

acceleration, which ends up with the transition to detonation.

For the experimental studies of ethyleneeoxygen combustion

in capillary tubes [20] the critical diameter of the tube when

the flame dynamics is changed can be estimated about

0.35e0.4 mm, taking into account that for the ethylene/

oxygen flame at atmospheric pressure, Lf¼ 0.075 mm,

Uf¼ 5.5 m/s, Q¼ 14 compared to Lf¼ 0.32 mm, Ufx 10 m/s,

Qx 8 for the flame in hydrogeneoxygen.

5. Formation of the pressure pulse and DDT

The computed overall picture of the flow, the flame develop-

ment and the transition to detonation in the channel of width

D¼ 5 mmare shown in Fig. 10 representing the time sequence

of the density gradient fields. The physical times in millisec-

onds shown for each frame in Fig. 10 are not evenly spaced but

clustered to reveal the most important details of the flame

acceleration and the transition to detonation. As it was said, in

a wider channels, D> 1 mm, if the perturbations were not

imposed the flame is developing symmetrically in a tulip

shape. The shape of the flame depends on the imposed initial

small perturbations, but the overall dynamics of the flame

acceleration is not affected by the initial small perturbations

(less than 1% of the flame velocity). Well known [19,47] sto-

chasticity of the DDT process better corresponds to the model

when small perturbations are imposed on the flame from the

beginning, so that they tend to destroy the tulip shapemaking

the flame front asymmetrical, as it is shown in Fig. 10.

Fig. 10 shows different stages of the flame acceleration and

the transition to detonation. Some frames in Fig. 10 show only

part of the computational domain adjacent to the reaction

front at selected times. The flame propagates with themoving

flow and quickly becomes convoluted and stretched along the

channel walls. The increasing flame surface area results in the

faster energy release, thus increasing the flame speed and

accelerating the flow ahead. The earliest frame at 0.15 ms

shows the perturbed flame in the left end of the computa-

tional domain and the compression waves produced by the

accelerating flame which steepen into the shocks on the right

side. As time passes, after 0.4 ms, the flame acceleration slows

down and compression waves start steepening into the

shocks in the immediate proximity ahead of the flame. During

the next stage, from 0.4 ms till 1.21 ms, the shock waves are

formed close to the flame front. They are coalesced, merged,

amplified and create a layer of compressed and heated

unreacted gas e the preheat zone [47]. During this stage the

flame consumes the unreacted mixture of considerably larger

density. As a result, a pressure pulse forms at the tip of the

flame, it strengthens and grows exponentially in time. From

1.21 ms to 1.211 ms the pressure pulse steepens into a strong

shock which is coupled with the reaction zone forming over-

driven detonation. The last frame in Fig. 10 shows the reto-

nation wave, and the overdriven detonation with

a characteristic cellular structure.

Fig. 11 presents variations of the pressure and temperature

profiles at sequential times from 0.2 ms till 1.2 ms with the

time interval 0.1 ms. The pressure pulse sits on the flame tip

which is fast accelerating. At the beginning the flame accel-

eration is a 2D effect closely related to the 2D shape of the

flame in the presence of a boundary layer. Further fast

Fig. 11 e Temperature (dashed lines) and pressure (solid

lines) profiles corresponding to leading point in the flame

front represent the flame structure and the pressure peak

formation; D[ 5 mm, time instants are from t0[ 0.2 ms,

Dt[ 0.1 ms, tf[ 1.2 ms.

Shock

0 10 20 30 40 50

x, mm30 640 650 660 6706 680

Flame Oblique shock

Detonation

0.065ms

340 350 360 370330 380

Retonation

Flame

0.2ms

0.4ms

1.2065ms

1.21ms

1.21187ms

1.2183125ms

(p ), bar /m0 4E+6 8E+6

2 2Δ

Fig. 10 e Pressure gradient at a sequential times showing

the overall flow development during transition to

detonation in channel D[ 5 mm for P0[ 1 bar. The

computational times are shown on the left of each frame.



acceleration of the flame tip is due to coupling with the

pressure pulse. Eventually, the pressure pulse becomes strong

enough to affect reactions. The pressure pulse and the flame

appear coupled because a positive feedback between the

pressure, the reaction rate and the heat release in the reaction

zone. The pressure at which the coupling begins is around

3 atm and since this time pressure starts to grow exponen-

tially. An ignition arisingwithin the body of the pressure pulse

is synchronic with its growth, which is manifested by the

exponential in time growth of the pressure peak after 0.5 ms

shown by the dashed line in Fig. 12. Figs. 11 and 12 need to be

considered in unison to obtain a clear picture of the pressure

peak evolution, with the time and length-scales of each stage

described on the figures. It should be stressed that tempera-

ture ahead of the flame front does not exceed 600 K till

the moment of the actual transition to detonation, so that the

rates of chemical reactions are negligibly small ahead of the

flame. Thus, a temperature gradient ahead of the flame if any

cannot produce a detonation through the Zeldovich gradient

mechanism involving gradients of reactivity.

From the beginning the amplitude of the pressure pulse is

not enough for steepening effects to be important. Because of

a positive feedback between the pressure rise and the

enhanced reaction, the propagating pressure pulse grows

exponentially and simultaneously drags the reactions with it.

By the time 1.21 ms the amplitude of the pressure peak

becomes large enough, it steepens into a strong shock and

after this moment the transition to detonation occurs in less

than 1 ms. It should be noted that the pressure peak reaches

14 bar at 1.2 ms and it increases above 20 bar at 1.21 ms when

the transition to detonation starts. Such amplitude of the

pressure peak is comparable to a pressure jump across the

shock with the Mach number Mshz 5, which is close to

the strength of the leading shock in a detonation wave.

The last stage of the actual transition to detonation is

shown in Fig. 13, which presents variations of the pressure

temperature and concentration YH of H-radicals profiles at

sequential times, from t¼ 1.2101875 ms till 1.2113125 ms with

the time interval 0.25 ms for the same conditions as in Fig. 10.

The transition to detonation is clearly seen from the

steepening of the temperature profile and the increased

temperature of the products corresponding to a detonation.

One can see that the distribution of H-radicals is different for

a deflagration and after the transition to a detonation. In the

deflagration H-radicals appears within the front of the

combustion wave, while structure of the detonation wave

consists of the well pronounced shock wave with the jump in

temperature and pressure following by the reaction.

In a thin channel the overall picture of the flame velocity

evolution before the DDT differs from that in a wide channel,

but themechanism of DDT is similar to that in a wide channel

except of some details of the detonationwavewhich is formed

after the transition. Fig. 14 shows density gradient evolution at

a sequential times for the hydrogeneoxygen flame develop-

ment and the transition to detonation in channel D¼ 0.5 mm,

P0¼ 1 bar. The compression waves generated by the acceler-

ating flame steepen into the shocks at the distance compa-

rable to the width of the flame and a pocket of compressed

unreactedmixture is formed close ahead of the flame from the

very beginning. As a result, the flame consumes a fresh

mixture of higher density, and the pressure peak starts to

t, ms

p,ba

r

0 0.5 1

10

20

Fig. 12 e Peak pressure history in the channel D[ 5 mm.

Dashed line is exponential approximation on the 2nd stage

of the flame acceleration.

x, mm

T,K

p,ba

r

652 654 656

1000

2000

3000

4000

0

0.02

0.04

0.06

0.08

0.1

0.12

0

20

40

60

80pTYH

H

Fig. 13 e Pressure, temperature and H-radical

concentration distributions in the flame tip during the

actual transition to detonation in the channel D[ 5 mm

from 1.2101875 ms till 1.2113125 ms, Dt[ 0.25 ms.

0.0 4.0E+5

Δρ , kg /m2 2 7

54 55

42 43

50 51 52

t=0.1650ms

t=0.1705ms

t=0.1720ms

Fig. 14 e Density gradient at a sequential times showing

the overall hydrogeneoxygen flame development and the

transition to detonation in channel D[ 0.5 mm,

P0[ 1 atm.



grow exponentially almost from the very beginning. Fig. 15

shows a sequence of the temperature (dashed lines) and

pressure (solid lines) profiles calculated for the flame in the

channel D¼ 0.5 mm from t0¼ 73.75 ms till tf¼ 168.75 ms with

the time interval Dt¼ 5 ms. It shows the exponential increase

of the pressure peak at the flame front until themomentwhen

the transition to detonation starts. The overall picture of the

transition is similar to what takes place in a wide channel, but

there are features associated with specific of a narrow

channel. In a narrow channel there are no shocks, which

runaway and compress the unburnedmixture far ahead of the

flame. Therefore in a narrow channel the ChapmaneJouguet

detonation is formed almost bypassing the stage of an over-

driven detonation and the overdriven detonation is much less

pronounced compared to the case of wide channels. The

characteristic detonation cell size in H2eO2 at the initial

pressure 1 atm is 2.5 mm, which is larger than the channel

width. Therefore in a thin channel a steady detonation wave

cannot be properly formed. This are presumably manifested

by the pressure and density profiles oscillations seen as the

density spikes in Fig. 16, where the density in the detonation

wave are plotted for the channel D¼ 0.5 mm, and in Fig. 17

which show time history of pressure for the same condi-

tions. The average detonation pressure is close to that for the

CJ steady detonation but the average detonation speed is

slightly less than the ChapmaneJouguet velocity and the

retonationwave ismuch less pronounced. It can be because of

the width of the channel is about 5 times less than the deto-

nation cell size and detonation is below the propagation limit.

It is possible also that this effect may be enhanced by the

hydrodynamic resistance in the narrow channels. At present

there is no enough experimental data to make a certain

judgment and the propagation of detonation in thin channels

requires more detailed study. In the present study we are

interested only in the process of the flame acceleration and

the transition from a slow combustion to a detonation regime.

Details of the propagation of a detonation wave in a thin

channel will be considered in a separate work.

6. About possible mechanism of DDT in gas-phase combustion

The question that drove this work was a particular combus-

tion problem: What are the origin and the physical mecha-

nism underlying the deflagration-to-detonation transition in

gaseous combustible mixtures? Uncontrolled DDT has enor-

mous destructive potential and the study of DDT is an

important topic associated with hydrogen and explosion

safety. There were significant prior works made in an attempt

to identify the mechanism of DDT. It was argued that the

accelerating flame creates conditions in nearby unreacted

material that lead to ignition centers in unreacted material

called “hot spots.” If local conditions in unreacted material

allow a spontaneous wave to form then this wave evolves into

a shock that is synchronized with the reaction and became

strong enough to become a detonation through the Zeldovich

gradient mechanism involving gradients of reactivity [48].

Thus the conclusion was that the mechanism of DDT is the

Zeldovich gradient mechanism. It should be emphasized that

all the previous studies of DDT used a simplified reaction

model in which the reaction proceeds via a one-step

exothermic reaction with Arrhenius kinetics. However,

Fig. 15 e Evolution of temperature (dashed lines) and

pressure (solid lines) profiles on the flame tip; D[ 0.5 mm,

time instants are from t0[ 0.07375 ms, Dt[ 0.005 ms,

tf[ 0.16875 ms.

Fig. 16 e Density profiles for the detonation evolution in

channel D[ 0.5 mm; t0[ 175 ms, tf[ 370 ms with Dt[ 5 ms.

t, ms

p,ba

r

0 0.1 0.2 0.3 0.4

10

20

30

40

50

Δt=4.625 sμ

Fig. 17 e Pressure evolution in channel D[ 0.5 mm. Period

of pressure oscillations at the detonation front is 4.625 ms.



a one-step reaction model cannot reproduce correctly the

main properties of the combustion such as the induction time

in chain-branching kinetics and detonation initiation.

The gradient mechanism of detonation initiation proposed

by Zeldovich has been widely studied also using a simplified

chemical model of a single-step exothermic reaction with

Arrhenius kinetics (see [49] for a recent review and numerical

study of the problem). Questions then arise such as: Is the

temperature gradient can develop in unreactedmaterial ahead

of the flame that can give rise to a detonation? Is the tempera-

ture gradient inhot spots canproduce adetonation through the

Zeldovich gradientmechanism? To answer these questionswe

consider the evolution from a linear temperature gradient to

a detonation using high resolution numerical simulations for

combustible materials whose chemistry is governed by

a detailed chemical kinetics model. We employ a model which

has been widely studied representing an initial linear temper-

ature gradient in the fuel. Emphasis is on comparing the results

with that used a one-step chemical model.

The burning chemistry in fuels such as hydrogeneoxygen,

hydrogeneair and the like is governed by kinetics with chain

initiation and branching reactions corresponding to Arrhenius

kinetics with small if any energy release (typically it is slightly

endothermic), and followed by the exothermic chain termi-

nation reactions. Both these stages have well-defined times of

induction zones followed by the stage of exothermic reactions

of chain recombination or termination reactions. The main

difference between the real chain-branching reactions and

a one-step chemical model is that in a single-step Arrhenius

model the induction time is several orders of magnitude

shorter than the real induction time in the important

temperature range, T< 1200 K. In order to have a well-defined

induction zone for one-step chemistry, the activation energy

should be taken very high, but then the reaction zone becomes

exponentially thin. Another difference is that in a single-step

Arrhenius model reaction is exothermic for all temperatures,

while the chain-branching reactions are slightly endothermic

during induction stage. Therefore considerably different

results are obtained using chain-branching chemistry models

than that found for one-step chemistry. For a detonation to be

ignited the temperature gradient must be much shallower to

create a balance between the acoustic and reaction times at

some point in the evolution.

Because of limited space, we shall discuss here mainly

the case of H2eO2, more detailed studies for another fuels

will be published elsewhere. The governing equations are

the one-dimensional version of time-dependent, reactive

NaviereStokes equations (1)e(10). We consider uniform initial

conditionsapart froma linear temperaturegradient. The initial

conditions at t¼ 0, prior to ignition are constant pressure and

zero velocity of the unburned mixture. The temperature at

x¼ 0 is taken maximum and it decreases in the positive

x-direction: T¼ T0(1� ax). The temperature non-uniformity is

large enough for a detonation to ignite or fail inside the

gradient area, so we are not concerned by the conditions

outside it. The boundary condition at x¼ 0 is a reflective

boundary condition so that this boundary is a solid, reflecting

wall. Example of the detonation initiation by the steepest

temperature gradient in H2eO2 mixture at initial pressure

P0¼ 1 atmandT0¼ 1500 K calculatedusing a detailed chemical

kinetics scheme is shown in Fig. 18. The deceleration of the

spontaneous wave and its coupling with the pressure wave

Fig. 18 e Formation of detonation in the temperature

gradient for a detailed chemical model: pressure (solid

lines) and temperature (dashed lines) profiles evolution;

Dt[ 2 ms.

Fig. 19 e Formation of detonation in the temperature

gradient for one-step chemical model: pressure (solid lines)

and temperature (dashed lines) profiles evolution,

Dt[ 1 ms.

T, K

L,cm

500 1000 1500 2000 2500 30000

2

4

6

8

10

12

14

16

18

20

9 reactionsone-step

Fig. 20 e The length of the steepest temperature gradient

for initiating a detonation in H2eO2 at P0[ 1 atm.



takes place along the gradient region where exothermal time

scale becomes higher than the induction one. For a steeper

gradient this coupling fails in the region where the induction

stage is shorter than the exothermal time scale and the

resulting will be a deflagration instead of detonation. The

evolutionof spontaneouswave calculated for the conditions in

Fig. 18 but using a one-step reactionmodel is shown in Fig. 19.

For a one-step reaction there is a heat release at the induction

stagewhichswitchesongasdynamics fromtheverybeginning

along the whole gradient, and if the gradient is sufficiently

shallowthesupersonic spontaneouswave followedby thermal

wave decelerates, couples with the pressure wave and trans-

forms into detonation. In the detailed chemical model if the

temperature gradient is sufficiently shallow so that the reac-

tion initially propagates supersonically, there are no gas

dynamic perturbations at the induction stage since the initi-

ating reactions proceed without heat release and the wave of

exothermal reaction follows the spontaneous wave path with

the delay determined by the time scale of termination reac-

tions. The result is that even when the thermal runaway point

propagates supersonically (it does so through an evolving

induction region) the thermal runaway path differs consider-

ably from the spontaneous wave path. It is seen from Figs. 18

and 19 that the length of the steepest temperature gradient

which can ignite a detonation is considerably longer compared

to the length predicted from a one-step model. The lengths of

the steepest temperature gradients, which can ignite a deto-

nation, depending on the maximum temperature at x¼ 0 for

a detailed chemicalmodel and for a one-stepmodel are shown

inFig. 20. It is seen that for temperature atx¼ 0 less than1200 K

the length of the steepest temperature gradient is at least one

or two orders of magnitude longer than that predicted from

a one-step model. In the case of lower initial pressures, or for

slower reacting mixtures, such as hydrogeneair, this differ-

ence ismuchgreater. For example, forH2eO2 at initial pressure

0.1 atm, this length already becomes several meters.

Compared to highly reactive mixtures (e.g. H2eO2, C2H4eO2)

the real ignition time forH2eair andmethaneeair differsmuch

stronger from that calculated from a one-step model. Accord-

ingly, the steepest temperature gradient which can ignite

a detonation in H2eair and methaneeair mixtures is many

orders of magnitude shallower compared to that predicted

from a one-step model. Thus, we come to the important

conclusion that thegradientof temperaturewhicharises inhot

spots and the like is very unlikely to be the DDT origin or

mechanism of the deflagration-to-detonation transition. For

example, in hydrogeneoxygen at initial pressure 0.1 atm the

temperature gradient which can ignite a detonation would

have to be several meters long, which exceeds the size of the

experimental facility.

7. Conclusions

The major conclusion of this work is that the flame acceler-

ation in tubes with no-slip walls is an important factor in

creating the right conditions for DDT to occur. Effects of the

channel width on reaction propagation and the dynamics of

reaction wave propagating in the channels of different widths

from 0.5 mm to 10 mmhave been systematically investigated.

The critical widths of the channel about 1.0 mm for hydro-

geneoxygenmixture and 0.35 mm for ethyleneeoxygen at the

atmospheric initial pressure have been obtained. It is shown

that dynamics of the hydrogeneoxygen flame before DDT is

different in wide (D> 1.0 mm) and in narrow (D< 1.0 mm)

channels and it is defined by the upstreamflowfield generated

by the accelerating flame.

Insight into how DDT occurs and what is the mechanism of

DDT was obtained by analyzing a series of high resolution

multidimensional numerical simulations with a detailed

chemical reaction model. DDT was simulated from first princi-

ples resolving the scales ranging from the size of the system to

the scales much smaller than the flame thickness. It is shown

that the shock formed close to the flame front is the primary

factor for generating exponentially growing pressure peak

whichstarts-upmechanismofDDT.Thisexponentially growing

pressure pulse then triggers the transition to DDT without the

temperature gradient involving the Zel’dovichmechanism. The

mechanism of DDT appears to be similar in both thin and wide

channels though the flame dynamics is different.

One-dimensional analysis of the detonation initiation by

a temperature gradient using a detailed chemical model

shows that in the DDT events a detonation cannot develop

through the Zeldovich’s gradient mechanism. The minimal

length of the temperature gradient capable to initiate deto-

nation obtained for a detailed chemical reaction model is

several orders of magnitude larger than that predicted from

a one-step Arrhenius model. The widely spread conclusion

drawn from previous studies, which used a single-step

chemical model, was that the detonations always arose

from hot spots in unreacted material formed ahead of accel-

erating flame and that the temperature gradient in these hot

spots is themechanism of the transition to detonation. On the

contrary, the temperature gradient cannot be a mechanism of

the deflagration-to-detonation transition simply because the

actual length of the temperature gradient which can initiate

detonation in real fuels is by orders of magnitude larger than

the size of hot spots and even size of the experimental facility.

The calculations of the present work are 2D and some

features can be expected in a 3D case, for example the tran-

sition could be faster. Another important factor that needs

careful investigation is related to turbulence. It is not very

important in the case of highly reactivemixture, but it is likely

that for slow reactive mixtures (e.g. methaneeair) DDT can be

influenced by turbulence.

Acknowledgments

This workwas supported by the Russian Agency of Innovation

(Award number No 02.740.11.5108) “Study of fundamental

processes of laminar and turbulent combustion”, and sup-

ported by the Russian Foundation of the High education. The

authors are grateful to T. Elperin, S. Frolov, M. Kuznetsov,

V. Lvov, A. Rakhimov, T. Rakhimova, N. Smirnov and

N. Zaretsky for fruitful discussions. The authors wish to thank

Nikolay Popov and Alexander Chukalovskii for insightful

discussions of different chemical schemes.M.L. is grateful to I.

Koshatskii for considerable help and useful discussions.



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Flame acceleration and DDT of hydrogen-oxygen gaseous mixtures in channels with no-slip walls