numerical experiments on reaction front …...numerical experiments on reaction front propagation in...

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Proceedings of the Combustion Institute xxx (2014) xxx–xxx

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Numerical experiments on reaction front propagationin n-heptane/air mixture with temperature gradient

Peng Dai a, Zheng Chen a,b,⇑, Shiyi Chen a, Yiguang Ju c

a SKLTCS, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing

100871, Chinab Department of Aeronautics and Astronautics, College of Engineering, Peking University, Beijing 100871, China

c Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

Abstract

Usually different autoignition modes can be generated by a hot spot in which ignition occurs earlier thanthat in the surrounding mixture. However, for large hydrocarbon fuels with negative temperaturecoefficient (NTC) behavior, ignition happens earlier at lower temperature than that at higher temperaturewhen the temperature is within the NTC regime. Consequently, a cool spot may also result in differentautoignition modes. In this study, the modes of reaction front propagation caused by temperature gradientin a one dimensional planar configuration are investigated numerically for n-heptane/air mixture at initialtemperature within and below the NTC regime. For the first time, different supersonic autoignition modescaused by a cool spot with positive temperature gradient are identified. It is found that the initial temper-ature gradient has strong impact on autoignition modes. With the increase of the positive temperaturegradient of the cool spot, supersonic autoignitive deflagration, detonation, shock-detonation, andshock-deflagration are sequentially observed. It is found that shock compression of the mixture betweenthe deflagration wave and leading shock wave produces an additional ignition kernel, which determinesthe autoignition modes. Furthermore, the cool spot is compared with the hot spot with temperature belowthe NTC regime. Similar autoignition modes are observed for the hot and cool spots. Different autoignitionmodes in the considered simplified configuration are summarized in terms of the normalized temperaturegradient and acoustic-to-excitation time scale ratio. It is shown that the transition between different autoig-nition modes is not greatly affected by the NTC behavior. Therefore, our 1-D simulation indicates that likehot spot, the cool spot may also generate knock in engines when fuels with NTC behavior is used and thetemperature is within the NTC regime.� 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Autoignition modes; Cool spot; Temperature gradient; Detonation; n-Heptane

http://dx.doi.org/10.1016/j.proci.2014.06.1021540-7489/� 2014 The Combustion Institute. Published by El

⇑ Corresponding author at: State Key Laboratory forTurbulence and Complex Systems (SKLTCS), Depart-ment of Mechanics and Engineering Science, College ofEngineering, Peking University, Beijing 100871, China.

E-mail address: [email protected] (Z. Chen).

Please cite this article in press as: P. Dai et al., Proc.j.proci.2014.06.102

1. Introduction

Recently, advanced engine technologies suchas homogeneous charge compression ignition(HCCI) and low temperature combustion (LTC)

sevier Inc. All rights reserved.

Combust. Inst. (2014), http://dx.doi.org/10.1016/

http://dx.doi.org/10.1016/j.proci.2014.06.102

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2 P. Dai et al. / Proceedings of the Combustion Institute xxx (2014) xxx–xxx

have drawn massive attention due to their excel-lent performance in efficiency improvement andemission reduction [1]. However, in practicalHCCI engines, experiments [2] showed thatinstead of simultaneous thermal explosion of theentire charge, multiple separated spots firstauto-ignite and then are followed by combustionof surrounding charge. These autoignition spotsintroduce certain modes of reaction front propa-gation towards unburned gas due to the spatialdistribution of charge reactivity.

Zel’dovich [3] first analyzed different modes forpropagating combustion waves caused by autoig-nition in the presence of non-uniform reactivity.Later, other researchers extended Zel’dovich’swork using either simplified reaction model [4,5]or detailed chemistry [6–9]. Using 1-D simulationfor syngas/air with detailed chemistry and trans-port, Gu et al. [8] demonstrated that five propaga-tion modes of autoignition front can be initiatedby temperature gradient: thermal explosion,supersonic autoignitive deflagration, developingand developed detonation, subsonic autoignitivedeflagration, and conventional laminar burningdeflagration. They also proposed the limits fordetonation initiation in terms of two non-dimensional parameters, namely the normalizedtemperature gradient (n) and the ratio of acoustictime to excitation time (e) and found a detonationpeninsular in the plot of n versus e. The theory ofdetonation peninsular was further extended in thestudy of noise and knock (or super-knock) in con-ventional engines [10–12], turbo-engines [13], anddeflagration-to-detonation transition (DDT)experiments [11]. It is noted that in the n-e regimemap mentioned above, determination of practicalengine conditions usually requires assumptions oftemperature gradient and hot spot radius, whichmake it quantitatively inaccurate. Nevertheless,it provides important qualitative insights on nec-essary conditions for knock occurrence in practi-cal engines. In addition, Rudloff et al. [12]introduced a new non-dimensional parameter, p,to predict the violence of abnormal combustionin realistic engines along with the location of (n,e) relative to the detonation peninsula.

Previous studies on the autoignition frontpropagation with reactivity inhomogeneity weremainly focused on simple fuels such as H2, CO,and CH4. However, unlike those simple fuels,large hydrocarbons utilized in engines usuallyhave complicated low-temperature chemistry,and therefore manifest complex ignition andburning properties including negative temperaturecoefficient (NTC) behavior [14–16]. Ignition andreaction wave propagation are influenced by thecoupling effects among low- and high-temperaturechemistries, transport, and acoustic waves [16].On the other hand, the non-monotonic depen-dence of ignition delay time on temperature inNTC regime makes ignition first occur at cooler


locations (for example, the position close to theengine wall) instead of hot spots [17]. Zhang et al.[18] studied dimethyl ether auto-ignition in a 1-Dlaminar situation with temperature inhomogenei-ties. They observed ignition from cold spots for amean temperature in NTC regime and assessedthe influence of temperature gradient on moleculardiffusion effects and pressure fluctuations. Yooet al. [19] conducted 1-D and 2-D DNS to examinethe effects of temperature gradient and turbulenceon n-heptane/air ignition for various temperaturesinside and outside of the NTC regime. However,these two studies only considered subsonic modesof reaction front propagation. In practical InternalCombustion Engines, ignition front propagationmight be supersonic when knock or super-knockoccurs. Therefore, a comprehensive study on theautoignition front propagation including both sub-sonic and supersonic modes in temperature inho-mogeneity at low temperatures, especially withinNTC regime, is needed.

The present work aims to extend the theory onautoignition modes proposed by Zel’dovich [3]and Gu et al. [8] by considering autoignition ofn-heptane at temperature within the NTC regimein a one dimensional planar configuration. Unlikeprevious studies considering hot spot, here theautoignition front propagation is initiated by acool spot in which temperature is lower than sur-rounding mixture. Different modes of reactionfront propagation are observed with the increaseof temperature gradient in the cool spot. It isemphasized that unlike hot/cool spot in engines,1-D planar rather than spherical configuration isconsidered here and thereby the present resultsare only of indicative value.

2. Model and specifications

n-Heptane is considered in this study since it isthe main component of Primary Reference Fuel(PRF) for gasoline and it has typical two-stageignition process with low- and high-temperaturechemistries as well as NTC behavior [14,15]. Insimulation, the skeletal mechanism for n-heptaneoxidation developed by Liu et al. [14] is used. Thismechanism has been demonstrated to be able toaccurately predict ignition (including NTC regime)and flame propagation of n-heptane/air mixturesat a broad range of temperatures and pressures[14]. Stoichiometric n-heptane/air initially at40 atm is considered in this study. Figure 1(a)shows the homogeneous ignition delay time, s, asa function of initial temperature, T0. In the NTCregime, 850 < T0 < 960 K, the ignition delay timeincreases with the initial temperature.

To investigate the different modes of reactionwave propagation, we consider the transientignition and wave propagation processes in aone-dimensional, planar, adiabatic, closed




x (cm)T(K)

0 1 2 3880

890

900

910

920 ξ=10

ξ=1

cool spot, x0=0.5 cm

TH=901.6K

TH=916K

T0 (K)

(dT 0/dx)c(K/m)

800 1000 1200 1400

-1000

0

1000640

900

-120

780

τ(ms)

0

1

2

(a)

(b)

Fig. 1. (a) Ignition delay time and (b) critical temper-ature gradient as a function of initial temperature forstoichiometric n-heptane/air mixture at 40 atm. Theinset shows the initial temperature profiles at T0 = 900 Kand n = 1 and 10.

P. Dai et al. / Proceedings of the Combustion Institute xxx (2014) xxx–xxx 3

chamber (adiabatic and reflective boundary con-ditions are adopted for both sides). We limit ourfocus to laminar combustion regimes and therebythe initial flow is static without turbulence. Thissimplified configuration is applied to indicate pos-sible behaviors of real systems. As shown in theinsert in Fig. 1(a), a cool spot with linear temper-ature distribution is located on the left side of thedomain (i.e. 0 6 x 6 x0, where x0 is the size of thecool spot). The temperature outside the cool spot(i.e. x P x0) is uniform everywhere. The length ofthe computation domain is 10 cm (i.e.0 6 x 6 10 cm) and the size of ignition spot, x0,varies from 1 mm to 6.1 mm. For the cool spot,the temperature gradient is positive; and itbecomes negative once a hot spot is considered.

The critical temperature gradient at whichautoignition front propagates at the speed ofsound in absence of heat conduction and mass dif-fusion is [3,8]:

dT 0

dx

� �c

¼ a�1 dsdT 0

� ��1

ð1Þ

where a is local sound speed at which pressurewave propagates. Figure 1(b) shows the criticaltemperature gradient as a function of initial tem-perature. Figure 1(b) shows that two singularpoints exist at the boundaries of NTC regime,and that the critical temperature gradient inNTC regime is positive. Therefore, in NTCregime, locations with lower temperature andproper positive temperature gradient are expectedto ignite first and generate developing detonationsthereafter. We introduce the non-dimensionaltemperature gradient, n:


n ¼ dT 0

dx=

dT 0

dx

� �c

ð2Þ

where (dT0/dx) is the temperature gradient withinthe ignition spot in 1-D simulation. It is noted thatthe critical temperature gradient is based on themean temperature of the ignition spot, T0. Asshown in Fig. 1(b), we consider two typical meantemperatures of ignition spot: one is T0 = 900 Kwithin the NTC regime at which a cool spot withpositive temperature gradient is established; theother is T0 = 780 K outside the NTC regime atwhich a hot spot with negative temperature gradi-ent is established. The inset in Fig. 1(a) shows theinitial temperature profiles at T0 = 900 K andn = 1 and 10.

The 1-D autoignition wave propagation is sim-ulated using the in-house code A-SURF (Adap-tive Simulation of Unsteady Reactive Flow). A-SURF solves the conservation equations of one-dimensional, compressible, multi-component,reactive flow using the finite volume method.The second-order accurate, Strang splitting frac-tional-step procedure is utilized to separate thetime evolution of the stiff reaction term from thatof the convection and diffusion terms. In the firstfractional step, the non-reactive flow is solved.The Runge-Kutta, MUSCL-Hancock (with MIN-BEE flux limiter), and central difference schemes,all of second-order accuracy, are employed forthe calculation of the temporal integration, con-vective flux, and diffusive flux, respectively. Thechemistry is solved in the second fractional stepusing the VODE solver. To maintain adequatenumerical resolution of the compression wave,shock wave, and detonation, a multi-level,dynamically adaptive mesh refinement algorithmhas been developed and used in A-SURF. Thereaction zone, compression wave, shock wave,and detonation wave are always fully covered bythe finest meshes of 2 lm and the corresponding

time step is 4 � 10�10 s. In the Supplementarydocument, numerical convergence is demon-strated by further decreasing time step and gridsize in simulation. A-SURF has been successfullyused in our previous studies on ignition and flamepropagation [20–22]. The details on the governingequations, numerical schemes, and code valida-tion can be found in [23–24]. As shown in the Sup-plementary document, A-SURF can accuratelycapture shock wave propagation. Moreover, thedetonation speed predicted by A-SURF is veryclose to the Chapman-Jouguet value for detona-tion speed which shows that A-SURF can capturedetonation propagation.

3. Results and discussion

The propagation velocity of autoignition frontbased on homogeneous ignition condition is [3,8]:





u0a ¼

dsdx

� ��1

¼ dsdT 0

dT 0

dx

� ��1

¼ a=n ð3Þ

where the superscript 0 denotes that the velocity isdirectly deduced from homogeneous results inwhich heat conduction, mass diffusion, andreacted gas expansion are not taken into consider-ation. The relation between ua

0 and sound speed ais only an approximate indicator of the propaga-tion modes of autoignition front since heatconduction and mass diffusion modify the temper-ature gradient and ignition delay time. For exam-ple, one-dimensional simulation [8] shows thatdeveloping detonation occurs for a range of nlager than unity. In the following we first discussdifferent autoignition front propagation modesat NTC regime (T0 = 900 K). Stoichiometricn-heptane/air mixture initially at 40 atm is consid-ered in this study.

Figure 2 shows the autoignition process atn = 1 (dT/dx = 640 K/m, see Fig. 1b). Asupersonic autoignitive deflagration instead of adetonation wave is observed. Because of heat con-duction and diffusion of radicals [8], the gradientof the ignition delay, ds/dx, decreases duringinduction period, which makes the autoignitionfront propagates at very high speed ofua � 4000 m/s (much greater than the Chapman–Jouguet detonation speed of 1857 m/s). The peakvalues of temperature and pressure are found tobe very close to those of homogeneous ignitionat constant volume (3060 K, 146 atm). Therefore,it is a typical supersonic autoignitive deflagration

P(atm)

50

100

150

1

2

34

5

T(K)

1000

2000

3000

1

2

34

5

x (cm)

Q(J/m

3 s)

0 1 2 3 41012

1013

1014

23

4 5

Fig. 2. Temporal evolution of temperature, pressure,and heat release rate distributions for T0 = 900 K,x0 = 5 mm, and n = 1. The time sequence is 1: 0 ls, 2:646 ls, 3: 647 ls, 4: 648 ls, 5: 649 ls.


controlled by local autoignition (chemical reac-tions) instead of transport.

According to Eq. (3), the propagation velocitydecreases as the temperature gradient within thecool spot increases. When the normalized temper-ature gradient reaches a certain value (e.g. n = 3for T0 = 900 K), the autoignition wave propa-gates at a speed close to the sound speed andthereby chemical reaction couples with the pres-sure wave initiated by local heat release, whichresults in a developing detonation. Figure 3 showsa typical case of detonation development forn = 10. It is observed that the detonation wavearises within the cool spot and then propagatesinto the right until the mixture ahead of it auto-ignites. After the detonation wave is fully devel-oped (at t = 665 ls, denoted by number 2 inFig. 3), it propagates at a constant speed whichis very close to that of C-J detonation speed of1857 m/s. The maximum pressure (�400 atm) ismuch higher than that in homogeneousconstant-volume ignition (146 atm).

With further increase of temperature gradient,the autoignition front speed decreases accordingto Eq. (3) such that the pressure wave initiatedby the original local thermal explosion decoupleswith the autoignition front. When n > 19, the sim-ple C–J detonation shown in Fig. 3 disappears andmore complex shock-reaction interaction occurs.A typical case is shown in Fig. 4 for n = 28. It isobserved that the decoupling of pressure wavewith the autoignition front occurs at t = 670 ls(see the lines denoted by number 1 in Fig. 4).

P(atm)

100

200

300

400

x (cm)

Q(J/m

3 s)

0 1 2 3 410121013101410151016

T(K)

1000

2000

3000

1

2 3 4 5

1

3 45

2

2 3 45

Fig. 3. Temporal evolution of temperature, pressure,and heat release rate distributions for T0 = 900 K,x0 = 5 mm, and n = 10. The time sequence is 1: 643 ls,2: 665 ls, 3: 671 ls, 4: 678 ls, 5: 683 ls.




t (ms)

velocity(m/s)

0.66 0.68 0.7 0.72 0.740

1000

2000

3000

Location(mm)

0

20

40

60

80

100shock wavereaction front

T0=900K ξ=28

IIII II

Fig. 5. Temporal evolution of location and velocity ofthe reaction front and shock wave for the same case as inFig. 4.


The leading pressure wave propagates towards theright side and eventually evolves into a shockwave (the states before and after the wave arefound to satisfy the jump relations for normalshock). Figure 4 indicates that chemical reactionsmainly take place at the deflagration wave behindthe shock wave and there is no heat releasearound the shock. However, due to the shockcompression, the mixture between the deflagrationwave and leading shock wave starts to auto-igniteat t = 705 ls (line #2 in Fig. 4) [25]. This eventu-ally evolves into a detonation wave (line #4 inFig. 4). This phenomenon is similar to the ‘explo-sion in the explosion’ observed by Urtiew et al.[26] in their study of DDT except that turbulenceis present in their experiments.

Figure 5 shows the evolution of the locationand propagation velocity of the reaction frontand shock wave for the same case as in Fig. 4.The reaction front consequently manifests as (I)deflagration, (II) autoignition in front of the defla-gration, and (III) detonation. The jump of thevelocity of reaction front in Fig. 5 is due to fastautoignition of the shock-compressed mixture.At t = 720 ls, the reaction front couples with theshock wave and becomes a detonation whichpropagates to the right at a velocity close to C–Jdetonation value of 1857 m/s until all the rest ofgas auto-ignites at t = 740 ls .

Since the mean temperature in the cool spot,T0, is fixed at 900 K, the lowest temperature atx = 0, TL, and highest temperature at x = x0,TH, (see the insert in Fig. 1a) decreases andincreases, respectively, with the temperature

P(atm)

200

400600

1001

2 3

4 5

x (cm)

Q(J/m

3 s)

0 1 2 3 4 510121013101410151016

12 3 4 5

T(K)

1000

2000

3000

1 2 3 4 5

Fig. 4. Temporal evolution of temperature, pressure,and heat release rate distributions for T0 = 900 K,x0 = 5 mm, and n = 28. The time sequence is 1: 670 ls;2: 705 ls; 3: 712 ls; 4: 720 ls; 5: 722 ls.


gradient. When the temperature gradient is large(i.e. n > 32), the lowest temperature at x = 0 isbelow 850 K which is lower boundary of NTCregime. Consequently, auto-ignition does not hap-pen sequentially from the left side to the right sidewithin the cool spot. Figure 6 shows the results forn = 45. It is observed that autoignition first occursin the middle of the cool spot (see line #1) insteadof at x = 0. This ignition kernel forms two defla-gration and pressure waves propagating towardsthe left and right sides. The left-propagatingpressure wave quickly reaches the wall and getsreflected. Therefore, at t = 675 ls (line #2 inFig. 6), a two-step like temperature profile is

T(K)

1000

2000

3000

12 44 5 6

3

x (cm)

Q(J/m

3 s)

0 1 2 3 410121013101410151016

1 2

4 5

6

3

P(atm)

200

400600

1001 2

4 5

6

Fig. 6. Temporal evolution of temperature, pressure,and heat release rate distributions for T0 = 900 K,x0 = 5 mm, and n = 45. The time sequence is 1: 666 ls;2: 675 ls; 3: 694 ls; 4: 695 ls; 5: 699 ls; 6: 701 ls.




dT/dx (K/m)0 10000 20000 30000 40000

ξ0 10 20 30 40 50 60 70

detonation

shock+

detonation

non-monotonicT in kernel

nomore

ignitionadvance

forcoolspot

detonation

shock+

detonation

supersonicignite

x 0=5mm

x 0=2mm

I II III

II IIIsupersonicignite

I

monotonicT in kernelIII-1 III-2

monotonicT in kernelIII-1

Fig. 7. Autoignition modes at different temperaturegradients and cool spot sizes with T0 = 900 K.

t (ms)

u a(m/s)

0.64 0.66 0.68 0.7 0.72 0.74101102103104105

ξ=1, 5, 10, 14,19, 23, 38

x(mm)

0

50

100

Fig. 8. Temporal evolution of location and velocity ofthe reaction fronts for different n (T0 = 900 K,x0 = 5 mm).


observed ahead of the deflagration wave. Thisparticular pressure wave assemble finally evolvesin to a single shock wave. Like the case inFig. 4, autoignition occurs in the shock-com-pressed mixture between the deflagration waveand shock wave at t = 694 ls (line #3 in Fig. 6).The non-monotonic temperature distribution forthe shock-compressed mixture is the consequenceof two competing factors: although the left part ofthe mixture is compressed by the shock earlier, thestrength of the shock is weaker compared withthe later one which compresses the right part ofthe mixture since the shock wave intensifies itselfduring propagation. Consequently, some pointin the middle ignites first compared with the sur-rounding substance. This autoignition kernelagain forms two deflagration and pressure wavespropagating towards the left and right sides (line#4 in Fig. 6). The deflagration wave movingtoward the left consumes all reactants therein.On the other hand, the deflagration wave propa-gating toward the right evolves into a detonationwave, similar to the case shown in Fig. 4.

With further increase in temperature gradient,the temperature outside the cool spot finallyexceeds the upper boundary of NTC regime andeventually the mixture outside the cool spotauto-ignites first. Therefore, for very large n, thecool spot and the surrounding mixture ignitealmost simultaneously (the temporal evolution oftemperature, pressure, and heat release rate distri-butions is not plotted due to space limit). Thisoccurs around n > 55 for T0 = 900 K andx0 = 5 mm. With the decrease of the cool spotsize, x0, the critical normalized temperature gradi-ent, n, increases (e.g. n > 110 for x0 = 2 mm).

In summary, three autoignition modes areobserved for a cool spot with temperature inNTC regime: I, supersonic autoignitive deflagra-tion (Fig. 2); II, detonation (Fig. 3); and III,shock- detonation (Figs. 4 and 6). Figure 7 showsthe temperature gradient ranges for differentautoignition modes at T0 = 900 K andx0 = 2 mm and 5 mm. It is noted that autoignitionmode III includes two cases which differ fromeach other in terms of the monotonicity of tem-perature profile in cool spot after its autoignition(III-1, monotonic, see Fig. 4; and III-2, non-monotonic, see Fig. 6). For x0 = 5 mm, theautoignition advance of cool spot compared withthe surrounding mixture vanishes when n > 55while the ignition advantage of cool spot withx0 = 2 mm always exists in the temperaturegradient range shown in Fig. 7.

In order to further reveal the influence of initialtemperature gradient within cool spot on theautoignition front propagation, Fig. 8 showstemporal evolution of reaction front location andvelocity for different values of normalized temper-ature gradient, n, with x0 = 5 mm. The dashedhorizontal lines represent C–J detonation velocity


of 1857 m/s. When n is very small, the autoignitionwave propagates supersonically (ua � 4000 m/s forn = 1) and thermal explosion occurs soon after theignition of cool spot. With the increase of n(n = 5–19), a detonation develops and maintainsitself even outside of the cool spot until thermalexplosion occurs. A further increase of n, however,leads to decouple between the leading shock waveand the reaction zone. Autoignition kernel appearsin the mixture between the deflagration wave andleading shock (Figs. 4 and 6). The appearance ofthis kernel is manifested in Fig. 8 by the suddenjump of reaction front velocity for n = 23 andthe jump of both reaction front location and veloc-ity for n = 38. The deflagration wave formed bythe additional ignition kernel soon evolves into adetonation and propagates rightward until ther-mal explosion of the system occurs.

Although cases with x0 = 5 mm shown in Fig. 8clearly show the propagating process of both det-onation and shock-detonation, the fourth modeof shock-deflagration does not exist in this config-




ε

ξ

0 5 10 15 200

20

40

60

80

100

I: supersonic autoignitivedeflagration

II: detonationIII: shock-detonationIV: shock-deflagration

1mm

1mm2mm

5mm x0=6.1mm

2mm 4.1mm x0=5mm

Circle:T0=900KDelta: T0=780K

0.5mm3mm

3mm0.5mm

I II

III

IV

Fig. 9. Autoignition modes initiated by a cool spot(T0 = 900 K) and a hot spot (T0 = 780 K) in terms of nand e.


uration due to the limitation of upper boundary ofthe NTC regime. For cases with smaller cool spot(x0 = 2 mm) and thus smaller temperature differ-ence, all the four autoignition modes includingsupersonic autoignitive deflagration, detonation,shock-detonation and shock-deflagration areobserved. It is noted that the last mode is quitesimilar to the shock-detonation, except that itsvelocity never reaches C–J value (i.e. supersonicdeflagration) before thermal explosion. In addi-tion, its maximum pressure (�200 atm) is muchlower than that of detonation case.

Unlike the cases studied by Gu et al. [8] andBradley et al. [10], the decouple of pressure waveand reaction zone does not necessarily lead tothe failure of detonation development here. Thisis because the spherical geometry was consideredtherein while the planar configuration is consid-ered in the present study. In Fig. 8, the subsonicdeflagration waves at large n before additionalignition kernels appear (velocity and locationjump) are basically the same as the ‘subsonic def-lagration’ in Ref. [8]. The autoignition caused bywave-compression was also observed by Weberet al. [7] in H2/O2 mixture. They emphasized thatthis situation happens only when the mixture tem-perature outside the hot spot is high enough. Con-sidering that the mixture is above 1000 K beforethe shock wave arrives (see Figs. 4 and 6),shock-compression in this work plays a dominantrole in changing autoignition modes. Similar con-clusion was also drawn in a recent review paper[25] on the interactions between flame propaga-tion and pressure waves. In addition, autoignitionfront of deflagration wave caused by shock wavewas observed by Urtiew et al. [26] in their earlyexperimental study on DDT. However, theso-called ‘explosion in the explosion’ in theirexperiment was the result of turbulence, walleffects, gas dynamics (transverse shock wave)and combustion chemistry, which was intrinsicallymulti-dimensional. Bradley [11] analyzed thisexperimental phenomenon in his ‘detonation pen-insular’ framework using one-dimensional theoryconsidering turbulent flame and the goodagreement was achieved. Therefore, the shock-detonation and shock-deflagration modesobserved in this work could be instructive for rel-evant experimental and multi-dimensional simula-tion observations despite of its one-dimensionaland laminar nature.

Besides the cool spot within NTC regime, wealso investigate flame propagation caused by hotspot outside NTC regime. A low temperature,T0 = 780 K, is chosen as the mean temperaturewithin the hot spot, which corresponds to a criti-cal temperature gradient of �120 K (see Fig. 1).The negative value indicates that a hot spot withhigh temperature rather than a cool spot can gen-erate propagating autoignition wave. It is foundthat despite of the different dependence of ignition


delay time on temperature within and outside theNTC regime, the hot and cool spots show similarautoignition modes with the increase of the nor-malized temperature gradient, n. Figure 9 summa-rizes the autoignition modes in terms of n and e.Following Gu et al. [8], e is the ratio of acoustictime (x0/a) to excitation time se (defined as the timeinterval between 5% and maximum heat releaserate), which assesses the rapidity of reactionenergy release. It is observed that the boundariesbetween two adjacent modes for the cool spot(T0 = 900 K) are close to those for the hot spot(T0 = 780 K). This implies that the NTC behaviordoes not significantly change the autoignitionmodes as long as n and e are fixed. Moreover,Fig. 9 shows that n separating autoignition modesII and III slightly decreases with e, implying that alarger e (or a larger x0 for fix cool/hot spot meantemperature T0) promotes the decouple of pressurewave from reaction zone. With the increase of e,the range for n in autoignition mode III expandsgreatly, indicating that the compression effect ofshock wave is enhanced. This is due to more rapidreaction energy release at higher e which enhancesthe strength of pressure wave.

4. Conclusions

Different autoignition modes caused bytemperature gradient are investigated numericallyin a one dimensional planar configuration forn-heptane/air mixture at initial temperaturewithin and below the NTC regime. In previousstudies, hot spot with negative temperaturegradient was used to generate different autoigni-tion modes. Here autoignition modes caused bya cool spot with positive temperature gradient(the initial temperature is within NTC regime)are studied and different supersonic autoignitionmodes are identified. The initial temperature gra-





dient is shown to have a great impact on autoigni-tion modes. With the increase of the temperaturegradient of the cool spot, four autoignition modesare identified: supersonic autoignition deflagra-tion, detonation, shock-detonation, and shock-deflagration. It is found that in the present config-uration, the shock compression of the mixturebetween the deflagration wave and leading shockwave produces an additional ignition kernel,which evolves into either a detonation wave or asupersonic deflagration wave. Therefore, shock-compression could be an important factor deter-mining the autoignition modes. Furthermore, thehot spot with temperature below the NTC regimeis also studied and similar autoignition modes areobserved for the hot and cool spots. Differentautoignition modes in one dimensional planarconfiguration are summarized in terms of twonormalized variables n (normalized temperaturegradient) and e (the ratio between acoustic timeand excitation time). It is shown that the NTCbehavior does not greatly change the autoignitionmodes as long as n and e are fixed. Therefore, sim-ulations in the present academic configurationsuggest that like hot spot, the cool spot may alsogenerate knock in engines when fuels with NTCbehavior is used and the temperature is withinthe NTC regime.

Acknowledgement

ZC would like to thank the support fromNational Natural Science Foundation of China(Nos. 51322602 and 51136005) and State KeyLaboratory of Engines at Tianjin University(No. K2014-01). YJ would like to thank the ArmyResearch Grant W911NF-12-1-0167 for multi-scale modeling. Helpful discussions with ProfessorYipeng Shi and Mr. Bin Bai at Peking Universitywere appreciated.

Appendix A. Supplementary data

Supplementary data associated with this articlecan be found, in the online version, at http://dx.doi.org/10.1016/j.proci.2014.06.102.


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