Available online at www.sciencedirect.comProceedings

ScienceDirect

Proceedings of the Combustion Institute 35 (2015) 1963–1972

www.elsevier.com/locate/proci

of the

CombustionInstitute

Computational analysis of re-ignition andre-initiation mechanisms of quenched detonation

waves behind a backward facing step

Yu Lv ⇑, Matthias Ihme

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-3024, United States

Available online 5 August 2014

Abstract

Re-ignition and re-initiation mechanisms of quenched detonation waves passing over a backward facingstep are investigated using a high-resolution discontinuous Galerkin method, in which the reactionchemistry is described using a detailed chemical model. A stoichiometric Hydrogen/Oxygen mixture withdifferent diluents (Argon and Nitrogen) is considered to assess effects of the mixture composition and reac-tivity on the initiation process. It is found that the mixture reactivity and the spatio-temporal evolution ofthe diffracted detonation wave are critical in controlling the development of the ignition kernels, and thepossible transition to detonation. Due to curvature effects and the increasing shock angle, the incidentshock wave transitions from a regular reflection to a Mach reflection. Through parametric studies, it isshown that for more reactive Argon-diluted mixtures, ignition first appears behind the regular shock reflec-tion, which is followed by spontaneous re-initiation through the SWACER (Shock Wave Amplification byCoherent Energy Release) mechanism. By replacing Argon with Nitrogen, the reactivity reduces and re-ignition through Mach reflection is observed. The subsequent re-initiation is primarily controlled byflame-acoustic and wave-wave interactions. Hot-spot re-initiation behind the Mach stem is not founddue to the short ignition window that exists over the limited range of shock angles. A theoretical modelis developed to confirm these findings and to quantify the coupling among the incident shock angle, thethermodynamic state of the shock-compressed mixture, and the ignition delay. Effects of viscous-diffusivetransport are also assessed, and it is shown that the viscous dissipation at the wall promotes ignition andtransition to re-initiation.� 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Re-ignition; Re-initiation; Detonation; Diffraction

1. Introduction

A quenched (or decaying) detonation waveappears when a detonation front diffracts over

http://dx.doi.org/10.1016/j.proci.2014.07.0411540-7489/� 2014 The Combustion Institute. Published by El

⇑ Corresponding author.E-mail addresses: ylv@stanford.edu (Y. Lv), mih-

me@stanford.edu (M. Ihme).

an abruptly expanding geometry, resulting in aso-called decoupled shock-flame complex.According to Matsui and Lee [1], the detonationcell size, k, has to be large enough (typicallyk > dc/13, with dc being the diameter of the orifice)in order to trigger quenching. The detached shockfront, propagating faster than the detached flame,will subsequently interact with the geometric

sevier Inc. All rights reserved.

1964 Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972

boundaries, resulting in complex shock/wallreflections. Under certain conditions, the reflec-tion is able to re-establish a detonation frontdownstream of the expansion section. This re-ini-tiation process is of particular interest for severalpractical applications, including, among others,quenching in detonation arrestors, deflagration-to-detonation transition (DDT), and the ignitionin multi-dimensional detonation systems. Overrecent years, considerable progress has been madeon the qualitative analysis of re-initiation mecha-nisms, and different processes have beenproposed. With relevance to the present work,these mechanisms can be categorized as:

1. Autoignition and re-initiation by regular shockreflection: Autoignition of a quenched detona-tion wave is induced by regular reflection of theflame-detached shock wave. Lee [2] pointedout that this mode exhibits similarities to thetriple-point collision in multi-dimensional det-onation configurations. Fast autoignition willre-initiate the detonation with a strongly over-driven explosion. The onset and dynamics ofthis re-ignition mode have been indirectlydeduced from experimental soot-foil records[3,4], and observed from numerical simulationsusing one-step chemical mechanisms [5,6].

2. Autoignition and re-initiation by Mach reflec-tion: When the incident shock angle exceeds acritical value, Mach reflection can occur. Thisresults in the formation of a high-temperatureregion behind the Mach stem, which promotesignition and re-initiation. This mechanism wasdeduced from experimental investigations ofdetonation waves passing over a backward-fac-ing step [7]. However, even in subsequentexperiments [3,4,8], the distinction to ignitionby regular reflection could not be isolateddue to the limited spatio-temporal range ofmeasurement techniques. Ignition by Machheating was also emphasized in a computa-tional study on DDT [9] in channels with peri-odically positioned obstacles. The simulationsshowed that by varying the spacing betweenobstacles to promote Mach reflection, the heat-ing of the fresh mixture by the collision of astrong Mach stem with the downstream obsta-cle leads to the onset of detonation. However,by excluding these confinement effects and geo-metric interactions, it remains unclear whetherand how the transition to detonation isfacilitated.

3. Re-initiation by hydrodynamic instabilities: Thismechanism, first postulated by Teodorczyket al. [7], emphasizes the effect of enhancedmixing between reaction products andunburned gases by Kelvin–Helmholtz or Rich-tmyer–Meshkov instabilities. This is similar tothe ignition mechanism proposed by Radule-scu et al. [10] for irregular detonation waves.

Recently, a more detailed study has been con-ducted by Bhattacharjee et al. [11,12], whostudied the re-ignition process of a quencheddetonation behind a circular obstacle. At cer-tain conditions, they found that the occurrenceof a rapidly burning mixture is strongly cou-pled to the presence of a wall jet, whichenhances the mixing and growth of the flamefront behind the Mach stem. As postulated,this might lead to the onset of transverse deto-nation. However, the transition to detonationfrom hot-spot ignition appears non-determinis-tic under those operating conditions since there-initiation could not be systematicallyreproduced.

The objective of this work is to characterize themechanisms for re-ignition and re-initiation ofquenched detonation waves. To this end, a canon-ical configuration of a backward facing step isconsidered, and a range of operating conditionsand mixtures are studied to isolate relevant transi-tion pathways. Compared to previous calculations[5,8,9,11], the present simulations are performedusing a detailed chemical mechanism. The mathe-matical model, computational setup, and modelvalidation are presented in Section 2. The resultssection, Section 3, is structured in four parts: Wefirst examine the re-ignition process of quencheddenotation waves. Different mixture reactivitiesand diluents are hereby considered to investigatethe role of the diffracted shock reflection on there-ignition process. These investigations are thencomplemented by theoretical analyses to quantifyeffects of the incident shock topology, the state ofthe shock-reflected mixture, and the chemicalreactivity on the ignition delay. This analysis isfollowed by investigating the transition from igni-tion to detonation, and effects of the diluent gason the detonation initiation process are character-ized. Finally, results from viscous simulations arediscussed to provide insights on the role of vis-cous-dissipative effects in promoting ignition andin accelerating re-initiations.

2. Numerical method and validation

Two-dimensional reactive Euler equationswith detailed chemistry are solved using the dis-continuous Galerkin (DG) discretization [13].The present DG-method employs linear and qua-dratic basis functions, resulting in, respectively,second and third-order accuracy in smoothregions. Shocks and contact discontinuities arerepresented using a WENO-based DG-limiter.To assess viscous-diffusive effects, additional sim-ulations are performed by accounting forthermo-viscous-diffusive transport using a penaltymethod. A Strang-splitting scheme is used fortime-advancement, and the high-pressure Hydro-

3000

3500

Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972 1965

gen/Oxygen mechanism due to Burke et al. [14],consisting of 13 species and 27 elementary reac-tions, is used to describe the reaction chemistry.Thermal properties of the mixture are consideredusing a temperature-dependent polynomial repre-sentation [15].

In this study, we consider the experimentalconfiguration of Ohyagi et al. [3], and the sche-matic is shown in Fig. 1. In this facility, a long ini-tiator section is adopted to ensure fully developedcellular structures before the detonation waveexpands into the combustion section. The heightof the combustor section is three times the crosssection of the initiator. To ensure consistency withthe experimental study, we set the origin of thecoordinate system 1.8 cm upstream of the com-bustor end-wall. All simulations were initializedwith a 1D Chapman–Jouguet (CJ) wave near theinlet of the initiator section. Small perturbationswere imposed along the shock front to initiatetransition to 2D cellular structures. The detona-tion wave diffracts after entering the combustorsection.

In this context it is noted that the correct pre-diction of the cell size k is critical since the physicsof the detonation diffraction is sensitive to theratio k/h (where h is the height of the initiator;for the present configuration h = 1 cm), accordingto the recent studies by Pintgen and Shepherd [19].Therefore, the operating conditions for the simu-lation have to be carefully selected. Specifically,simulations with detailed reaction chemistry[20,21] have shown that for conditions corre-sponding to high initial pressure (p0 > 1 atm)and reduced dilution ratio (<50 %) discrepanciesbetween predicted and measured cell size areobserved. To overcome this issue, lower pressureconditions and higher dilution ratios are consid-ered. To confirm the model accuracy, predictedshock-cell sizes are compared with measurements.The initial pressure and temperature are set to26.7 kPa and 293 K, respectively. A parametricstudy is performed by considering different diluentproperties. The fresh mixture is represented as2H2 + O2 + bAr + (4�b) N2, where b 2 [0,4] isthe diluent parameter. By changing b, the mixtureproperties and Zeldovich–von Neumann–Doring(ZND) detonation structure are modified (seeFig. 2), and different re-initiation patterns can beobserved. In this investigation, we consider fourcases with b = {0, 1.5, 2, 4}. Flame properties

Fig. 1. Schematic of the computational domain.

for these mixture conditions are obtained usingCANTERACANTERA [22] and the detonation toolbox [23].These results are summarized in Table 1. Simula-tions are performed on meshes with quadratic ele-ments, and the resolution is adjusted for each caseto have at least 20 degrees of freedom per induc-tion length. This resolution is consistent to thatused in Ref. [13], where it was found that the det-onation cell width is insensitive to further gridrefinement. For b = 0–4, we observed 4–8 com-plete cells in the initiator section, ensuring thatthe detonation waves are fully developed beforeentering the combustor section.

Numerical soot-foil results in the initiator sec-tor for different values of b are presented in Fig. 3.The numerical predications for the cell sizes arecompared with experimental data [16,17] and the-oretical predictions [18]. These comparisons arepresented in Table 1. For all cases, good agree-ment is observed; however, it is noted that forthe case with b = 0 the size of the detonation cellbecomes comparable to the channel height, whichmight result in cell locking.

To further validate the computational model,simulations of the backward facing step of Ohyagiet al. [3] are performed. Comparisons of numericalSchlieren results with measurements for b = 4 arepresented in Fig. 4, showing three flow-fields atthree different time-instances. Since time was notreported in Ref. [3], the frames in Fig. 4(b) havebeen selected to closely match the numericalSchlieren results. Simulation results in the top ofFig. 4(a), show the instance at which the shockenvelope first impacts the bottom wall and thelocation of the leading shock is at x = 50 mm. Asimilar structure is observed in the top panel ofFig. 4(b). The onset of the Mach stem and theleading shock location (middle panel ofFig. 4(a)) at x = 65 mm and x = 70 mm, respec-tively, is comparable with those shown in the mid-dle of Fig. 4(b). Similar comparisons can beperformed to show that the frame in the bottomof Fig. 4(a) exhibits qualitative similarities to thatin the bottom of Fig. 4(b). This comparisonindicates that the computational method inconjunction with the detailed chemistry capturesrelevant physical processes.

0 0.5 1 1.5 2 2.5 3 3.5 4

1500

2000

2500

β=4β=2β=1.5β=0

Fig. 2. Effects of diluent parameter b on ZND detona-tion structure.

Table 1Properties of diluted H2/O2 CJ-wave (Lig is the inductionlength; kC, kM and ksim refer to the cell sizes obtainedfrom theoretical calculations [18], measurements, andpresent DG-simulations, respectively). Nomenclature:Ea – activation energy; TvN – temperature at vonNeumann state; T0 – temperature of fresh gas; cvN –heat capacity ratio at von Neumann state; MCJ – Machnumber of Chapman–Jouguet wave.

b = 4 b = 2 b = 1.5 b = 0

Ea/RTvN 5.1 6.0 5.9 6.4TvN/T0 6.7 5.8 5.6 5.0cvN 1.44 1.37 1.35 1.32MCJ 4.9 4.9 4.9 4.8Lig [lm] 240 410 478 786kC [mm] 4.3 8.2 8.1 10.6kM [mm] �6 [16] – – �14 [17]ksim [mm] �3.7 �7.8 �7.8 �9.7

1966 Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972

3. Results

In the following, we present simulation resultsto analyze the re-ignition and re-initiation dynam-ics with different dilution parameter b. Section 3.1

(a) β = 4. (b) β =

Fig. 3. Numerical soot foil inside initiator sector for different

(a) Simulations.

Fig. 4. Comparison of Schlieren results for b = 4 between (a) siand (b) experiments [3] at comparable instances. Numerical Sc

examines the ignition behavior. This investigationis complemented by theoretical analyses in Section3.2 to delineate the coupling between incidentshock angle and ignition delay. This is followedby analyzing relevant re-initiation pathways inSection 3.3. Possible mechanisms for initiationfailure are discussed, and the influence of vis-cous-diffusive transport processes is examined inSection 3.4.

3.1. Re-initiation and formation of ignition kernel

Figure 5 compares the formation of hot spotsand re-ignition sequences for three different valuesof b. Animations for these ignition sequences areprovided as supplementary material. The struc-tures of the shock-flame complex, resulting fromthe quenched detonation, are comparable to thatreported by Gallier et al. [24]. For each case, ahighly curved shock front is formed after the dif-fraction, propagating along the combustor sectionat Mach numbers between 2.7 and 2.9. As thefront expands radially, the initial interaction withthe wall leads to a regular reflection (RR). As the

2. (c) β = 0.

dilution parameters: (a) b = 4, (b) b = 2, and (c) b = 0.

(b) Experiments.

mulations at times t = 52, 66, 80 ls (from top to bottom)hlieren results are computed as exp{�j$qj/j$qjmax}.

Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972 1967

wave propagates along the wall the shock angle acontinuously increases. This incident shock angle,which is relevant for the subsequent theoreticalanalysis, is indicated in all panels. When the inci-dent shock exceeds an angle of approximatelya = 40�, the regular reflection transitions to aMach reflection (MR), which is accompanied bythe formation of a Mach stem, in the case thatre-initiation by RR has not occurred.

For the case with pure Argon dilution (b = 4),an ignition kernel (F1) appears near the bottomwall just slightly after RR occurs. At this condi-tion, the incident shock angle is small (a = 30�),so that no Mach stem is observed before the tran-sition to detonation occurs. For the case withb = 2, the ignition kernel, F1, appears consider-ably later and is correlated with a larger shockangle. The initial formation of the Mach stemcan be observed at t = 66.3 ls, and its subsequentgrowth leads to the inception of a new hot spot F3behind the Mach-stem front. The thermodynamicenvironment created by MR and the mixing ofreactants at the slip line and the connected walljet contribute both to this hot spot formation.Compared to previous computational studies onthe Mach-stem formation [5], the present simula-tions in conjunction with the complex chemistryrepresentation and the high resolution providedetailed descriptions of the ignition onset, con-firming recent studies on the role of wall-jetentrainment [11] and earlier postulations on the

t = 54.6 μs t = 66.3 μ

t = 56.6 μs t = 70.3 μ

t = 57.4 μs t = 72.3 μ

(a) β = 4. (b) β = 2

Fig. 5. Comparison of the ignition sequence for different diluand (c) b = 1.5 (Animations for those ignition sequences are pflame; TP – triple point; W – shock wave. (For interpretation oversion of this article.)

Mach-stem-induced ignition process [7]. It is alsointeresting to note that the development of theignition kernel F1 triggers a precursor shock waveW1, which is a result of the merging of combus-tion-generated acoustic waves. This intensewave–wave interaction is most pronounced att = 70.3–72.3 ls, and leads to the formation ofan isolated hot spot F4. The inception of thishot spot is critical for the subsequent transitionto detonation.

By further shifting the dilution ratio towardsnitrogen (with b = 1.5), the occurrence of a shockwave W1 cannot be observed, and an initial igni-tion kernel F3 is seen only at considerably largershock angles. At t = 81.3 ls, the formations of adouble Mach-reflection pattern and two triplepoints become apparent. The triple points eventu-ally merge as the shock wave propagates down-stream. Compared to the previously discussedignition behavior, it is noted that F3 grows veryslowly. By further reducing the dilution parameterb, we were not able to observe new hot spots afterthe shock is reflected from the bottom wall. Withthis, we can conclude that RR-controlled ignitionbecomes weaker as b decreases, so that the igni-tion process becomes increasingly dominated byMR. This shift in the ignition behavior is a resultof the coupling between the thermodynamic stateof the shock-compressed gas mixture and theinduction chemistry. To substantiate this observa-tion further, a theoretical model is presented in

s t = 71.3 μs

s t = 81.3 μs

s t = 97.3 μs

. (c) β = 1.5.

tion parameters before re-initiation: (a) b = 4, (b) b = 2,rovided as supplementary material). Nomenclature: F –f the color in this figure, the reader is referred to the web

3

4

5

6MRRR

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

RR MR

10 20 30 40 50 6050

60

70

80β=4β=2β=1.5β=0

MRRR

Fig. 6. Results from quasi-stationary analysis, showing(top) temperature and (middle) ignition delay in state 3as a function of shock angle at Mach number 2.8; thestate definition is clarified in the supplementary material.Textured areas indicate range of computationallyobserved shock angles before the transition or the endof the simulation: blue, b = 4; black, b = 2; purple,b = 1.5. The shock angle as a function of time, evaluatedfrom the simulations, is shown in the bottom. (Forinterpretation of the color in this figure, the reader isreferred to the web version of this article.)

1968 Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972

the next section, which explains the couplingbetween shock reflection, thermodynamic state,and the ignition delay in the re-ignition region.

3.2. Quasi-steady analysis of shock reflection andignition behavior

To obtain a better understanding about theignition environment that is created by the gasdynamics, a quasi-steady analysis of the shock-compressed state is conducted. The stationarycoordinate system is anchored at the reflectionpoint (or triple point for MR) and propagates atthe speed of Msc0 sin(a), in which a is the shockangle, c0 is the speed of sound of the pre-shockstate, and Ms is the shock Mach number. By uti-lizing inviscid two-shock theory for RR andthree-shock theory for MR [25], the thermal statesbehind the reflected shock or the Mach stem (state3) can be computed analytically. Details on thetheoretical model are provided as supplementarymaterial.

Simulation results for the temperature ratiobetween the state 3 (post-shock) and the state 0(pre-shock, see supplementary material for nota-tion and schematic) as a function of the incidentshock angle a are shown in the top panel ofFig. 6 for different b values. It can be seen thatwith increasing value of b (corresponding to ashift from diatomic Nitrogen to monatomicArgon), the temperature behind the reflectedshock or the Mach stem increases. However, withincreasing shock angle, the temperature decreasesin the RR-regime. This is followed by an abruptincrease at the transition from RR to MR. Inthe MR-region, the temperature shows a strongerdependence on the shock angle, and rapidlydecreases with increasing a. This analysis eluci-dates competing effects between the dilutionparameter and the shock angle on the thermody-namic state in region 3. To further quantify theinfluence of the thermodynamic state on the igni-tion, we compute the ignition delay time based ona constant-volume reactor with initial conditionscorresponding to the state in region 3. Resultsare presented in the middle panel of Fig. 6, show-ing the ignition delay time as a function of shockangle a for different dilution parameters. Consis-tent with the temperature results, the ignitiondelay decreases with increasing value of b. Forthe case of pure Argon dilution (b = 4), the igni-tion delay is sufficiently short, compared to theresidence time derived from the bottom panel ofFig. 6. This corresponds to the early re-ignitionand subsequent re-initiation in the RR-regime,and this is consistent with the simulation resultsshown in Fig. 5(a). As the dilution value isreduced to b = 2, the ignition delay is still belowthe residence time (around 8 ls before RR-MRtransition), promoting the formation and the sub-sequent growth of ignition kernels. However, with

further reduction of b, the ignition time increasesdramatically in the RR-regime. This prohibits thegrowth of local ignition kernels, which was alsoobserved by comparing Fig. 5(c) and (b). At theseNitrogen-rich dilution conditions (b! 0), thesimulation results in Fig. 6 show that Mach-stemignition is more favorable because of the localminimum at low incident shock angles aroundthe RR-MR transition. Following this transition,as shown in the bottom of Fig. 6, the shock anglesreach values around 40–50�. At those conditions,the ignition delay becomes comparable to the res-idence time of the shock-compressed mixture.However, with increasing a the ignition environ-ment behind the Mach stem becomes less favor-able with increasing shock angle. This is evidentby the exponential growth of the ignition-delaytime, which prevents further growth and transi-tion of hot spots. These theoretical results arequalitatively consistent with the detailed simula-tions, reported in Fig. 5. However, it is noted thatthe mechanisms for re-initiation that inherently

Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972 1969

rely on unsteady and multi-dimensional effectscannot be fully explained with this analysis.

3.3. Re-initiation and transition to detonation

Figure 7 shows the spontaneous transition todetonation for the case b = 4 (pure Argon dilu-ent). At t = 56.9 ls, a weak pressure wave is gen-erated by the ignition kernel in the near-wallregion. Quantitative results along the radial dis-tance (indicated by the arrow and originating atx = 36 mm), show that the peak heat-release rate(red line in inlay) is detached from the shockfront. As the initial precursor wave propagatesaway from the wall, the reactants are consumed,resulting in the formation of an induction gradi-ent. Within less than approximately 1 ls, a deto-nation front is established, which is shown in theinlay of Fig. 7(b). This detonation onset is wellexplained by the SWACER mechanism [2], andis the same as the “strong” ignition mode thatwas studied by Oran et al. [26]. Following theonset, the detonation front engulfs the inert trans-verse wave and part of the front advances into theburnt products (see Fig. 7(c)). At this instance,the presence of a forward-running detonation(D1) and a backward-running retonation (D2)are observed. With further evolution of the D1front, the original transverse wave is completelyengulfed and replaced by the transverse detona-tion (see Fig. 7(d)). In addition, due to the inter-action between D1 and the incident shock, a new

(a) t = 56.9 μs.

(c) t = 58.6 μs.

Fig. 7. Re-initiation sequence and transition to detonation fowave; F – flame; IS – incident shock; D – detonation wave; MSflame location where Y H2O ¼ 0:5Y eq

H2O. (For interpretation of tversion of this article.)

detonation front D3 is formed along with a newtriple point that connects the reactive Mach stem,inert incident shock, and transverse detonationfront. At this moment, D1 and D3 propagate atspeeds of 1.15UCJ and UCJ, respectively. As thetriple point moves downstream, the inert incidentshock re-initiates.

A different re-initiation process is observed forthe case with b = 2 (equal volumetric diluent mix-ture of Ar and N2, and snapshots of the transientre-initiation sequence from hot-spot ignition areshown in Fig. 8. Instead of a spontaneous transi-tion to detonation (as observed for the case withb = 4), this transition process is primarily con-trolled by wave-wave interaction and shock-flamecoupling. From this initiation sequence, the inter-action of acoustic waves ahead of the flame frontis evident, resulting in a precursor shock waveW1. The flame holes and wrinkles indicate thepresence of secondary mixing processes by Kel-vin–Helmholtz and Richtmyer–Meshkov instabil-ities at the flame front. These instabilities increasethe flame area and enhance the flame propaga-tion. At t = 73.5 ls (Fig. 8(c)), both waves mergetransversely, leading to the detonation onset.Within 1 ls, the detonation wave is fully estab-lished (Fig. 8(d)), which eventually results in sim-ilar detonation patterns as shown in Fig. 7(d).

Results for the cases with b 6 1.5 are not fur-ther discussed since the transition to detonationcould not be observed for the present geometricsetup and operating conditions.

(b) t = 57.8 μs.

(d) t = 67.0 μs.

r the case with b = 4. Nomenclature: TW – transverse– Mach stem; W – shock wave. Blue isocontour denotes

he color in this figure, the reader is referred to the web

(a) t = 70.8 μs. (b) t = 72.3 μs. (c) t = 73.5 μs. (d) t = 74.5 μs.

Fig. 8. Re-initiation sequence and transition to detonation for the case b = 2; nomenclature follows that of Fig. 7. (Forinterpretation of the color in this figure, the reader is referred to the web version of this article.)

(a) t = 65 μs (b) t = 66.5 μs (c) t = 68 μs (d) t = 69 μs

Fig. 9. Comparison of re-initiation process between viscous (top) and inviscid (bottom) calculation for the case b = 1.5;nomenclature follows that of Fig. 7. (For interpretation of the color in this figure, the reader is referred to the webversion of this article.)

1970 Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972

3.4. Effects of viscous-diffusive transport onre-ignition

Additional simulations are performed to quan-tify effects of viscous-diffusive transport. In thesesimulations, the transport properties are calcu-lated using CANTERACANTERA [22]. With these simulations,it was confirmed that the viscous-diffusive trans-port has a marginal influence on the structure ofthe fully established detonation and the diffractionpattern over the separation corner. However, apronounced sensitivity of the ignition and the re-initiation process to these transport propertieswas observed for mixtures with b = {1.5,2}, corre-sponding to the RR-MR transition regime. This isillustrated in Fig. 9, showing comparisonsbetween viscous (top) and inviscid (bottom) simu-lations for the case of b = 1.5. Results from theviscous computation indicate an earlier formationof multiple ignition spots in close vicinity to thewall. These ignition kernels, which are notobserved for the corresponding inviscid case, rap-idly expand, eventually leading to the onset of det-onation. The viscous dissipation promotes theformation of ignition kernels and the potential

transition to re-initiation, although the transitionmechanism to detonation is similar to thatobserved in Fig. 8. It is also noteworthy tomention that the expanding flame front amplifiesthe growth rate of the Mach stem. By furtherreducing the dilution parameter (shift towardsNitrogen as diluent), the influence of viscouseffects diminishes. This suggests that the energyaddition by viscous heating cannot overcome theextended ignition delay. In this context, it is notedthat the viscous boundary layer was not fullyresolved, so that the present simulations most-likely underestimate the viscous dissipation.Therefore, it is anticipated that these effectsbecome more pronounced by further refining thenear-wall region.

4. Conclusions

The ignition and re-initiation of quenched det-onation waves has been investigated using numer-ical simulations under consideration of detailedreaction chemistry. A stoichiometric Hydrogen/Oxygen mixture was considered with variable

Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972 1971

diluent compositions. A series of simulations wereperformed to systematically identify ignitionmechanisms that are controlled by the couplingbetween gas-dynamic processes and chemicalkinetics. These investigations were complementedby a theoretical analysis to explain the dependenceof the re-ignition process on the shock angle andthe dilution parameter. It was shown that thismodel qualitatively captures the detailed simula-tion results. From this investigation, the followingconclusions can be drawn:

� The re-ignition process is controlled by thecoupling between mixture reactivity and thestate of the shock-compressed mixture that isdetermined by the angle of the incident shockwave. With increasing reactivity (using Argonas diluent), the regular reflection is able to gen-erate a favorable thermodynamic state toenable ignition and subsequent re-initiation.By successively replacing Argon with Nitrogenas diluent, the reduced temperature behind theshock and the increased activation energy sig-nificantly prolong the ignition time, prohibit-ing the growth of ignition kernels.� Transition to detonation by Mach reflection

was not observed for the operating range thatwas considered in the present investigation.The reason for this is the rapid temperaturedecrease and the extended ignition delaybehind the Mach stem with increasing shockangle.� Re-initiation and spontaneous detonation

transition via the SWACER-mechanism wasobserved for the Argon-diluted mixture thathas the highest reactivity. In addition, it wasfound that flame-acoustic interaction andwave-wave amplification can assist the re-initi-ation processes under the condition that theignition delay is comparable to the residencetime.� Heating by viscous dissipation was shown to

have significant effects on the ignition andre-initiation processes for certain mixturecompositions. Through direct comparisonswith inviscid calculations, it was shown thatviscous heating promotes the growth of hotshots and amplifies the wave-wave interaction.

Acknowledgments

Financial support through the NSF CAREERprogram with Award No. CBET-0844587 is grate-fully acknowledged. The authors would like tothank Tetsuro Obara for sharing the experimentalimages in Fig. 4(b), and helpful discussions withMatei Radulescu are gratefully acknowledged.

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.2014.07.041.

References

[1] H. Matsui, J.H. Lee, On the measure of the relativedetonation hazards of gaseous fuel-oxygen and airmixtures, Symp. Int. Combust. 17 (1) (1979) 1269–1280.

[2] J.H.S. Lee, The Detonation Phenomenon, Cam-bridge University Press, Cambridge, 2008.

[3] S. Ohyagi, T. Obara, S. Hoshi, P. Cai, T. Yoshih-ashi, Diffraction and re-initiation of detonationsbehind a backward-facing step, Shock Waves 12(2002) 221–226.

[4] T. Obara, J. Sentanuhady, Y. Tsukada, S. Ohyagi,Re-initiation process of detonation wave behind a slit-plate, Shock Waves 18 (2008) 117–127.

[5] D.A. Jones, M. Sichel, E.S. Oran, Reignition ofdetonations by reflected shocks, Shock Waves 5(1995) 47–57.

[6] C.J. Wang, S.L. Xu, Re-initiation phenomenon ofgaseous detonation induced by shock reflection,Shock Waves 16 (2007) 247–256.

[7] A. Teodorczyk, J.H.S. Lee, R. Knystautas, Propa-gation mechanism of quasi-detonations, Symp. Int.Combust. 22 (1) (1989) 1723–1731.

[8] R. Sorin, R. Zitoun, B. Khasainov, D. Desbords,Detonation diffraction through different geometries,Shock Waves 19 (2009) 11–23.

[9] V.N. Gamezo, T. Ogawa, E.S. Oran, Flame accel-eration and DDT in channels with obstacles: effect ofobstacle spacing, Combust. Flame 155 (1-2) (2008)302–315.

[10] M.I. Radulescu, G.J. Sharpe, J.H.S. Lee, C.B.Kiyanda, A.J. Higgins, R.K. Hanson, The ignitionmechanism in irregular structure gaseous detonations,Proc. Combust. Inst. 30 (1) (2008) 1859–1867.

[11] R.R. Bhattacharjee, S.S.M. Lau-Chapdelaine, G.Maines, L. Maley, M.I. Radulescu, Detonation re-initiation mechanism following the Mach reflection ofa quenched detonation, Proc. Combust. Inst. 34 (1)(2013) 1893–1901.

[12] R.R. Bhattacharjee, Experimental Investigation ofDetonation Re-initiation Mechanisms Following aMach Reflection of a Quenched Detonation. Mas-ter’s thesis, University of Ottawa, 2013.

[13] Y. Lv, M. Ihme, Discontinuous Galerkin method formulti-component chemically reacting flows and com-bustion, J. Comput. Phys. 270 (2014) 105–137.

[14] M.P. Burke, M. Chaos, Y. Ju, F.L. Dryer, S.J.Klippenstein, Comprehensive H2/O2 kinetic modelfor high-pressure combustion, Int. J. Chem. Kinet. 44(7) (2012) 444–474.

[15] B.J. McBride, M.J. Zehe, and S. Gordon. NASAGlenn coefficients for calculating thermodynamicproperties of individual species. NASA/TP-2002-211556, NASA Glenn Research Center, Cleveland,OH, 2002.

[16] T.J. Anderson, E.K. Dabora, Measurements ofnormal detonation wave structure using Rayleigh

1972 Y. Lv, M. Ihme / Proceedings of the Combustion Institute 35 (2015) 1963–1972

imaging, Symp. Int. Combust. 24 (1) (1992) 1853–1860.

[17] M.J. Kaneshige. Gaseous Detonation Initiation andStabilization by Hypervelocity Projectiles. PhDthesis, California Institute of Technology, 1999.

[18] A.I. Gavrikova, A.A. Efimenkoa, S.B. Dorofeev, Amodel for detonation cell size prediction from chem-ical kinetics, Combust. Flame 120 (1-2) (2000) 19–33.

[19] F. Pintgen, J.E. Shepherd, Detonation diffraction ingases, Combust. Flame 156 (3) (2009) 665–677.

[20] N. Tsuboi, S. Katoh, K. Hayashi, Three-dimen-sional numerical simulation for hydrogen/air detona-tion: Rectangular and diagonal structures, Proc.Combust. Inst. 29 (2002) 2783–2788.

[21] B.D. Taylor, D.A. Kessler, V.N. Gamezo, E.S.Oran, Numerical simulations of hydrogen detonationswith detailed chemical kinetics, Proc. Combust. Inst.34 (2013) 2009–2016.

[22] D. Goodwin. Cantera: An object-oriented softwaretoolkit for chemical kinetics, thermodynamics, and

transport processes. Available at http://code.goo-gle.com/p/cantera, 2009.

[23] S. Browne, J. Ziegler, and J.E. Shepherd. Numericalsolution methods for shock and detonation jumpconditions. Technical Report FM2006-006, Gradu-ate Aeronautical Laboratories, California Instituteof Technology, 2008.

[24] Gallier S., Mevel R., Davidenko D., Pintgen F., andShepherd J.E. Numerical study of detonation wavediffraction in hydrogen based mixtures. Proceedingsof the Twenty-fourth International Colloquium onthe Dynamics of Explosion and Reactive Systems,Taipei, Taiwan, July 28-August 2, 2013.

[25] G. Ben-Dor, Shock Wave Reflection Phenomena,Springer, 2007.

[26] E.S. Oran, T.R. Young, J.P. Boris, A. Cohen,Weak and strong ignition. I. Numerical simulationsof shock tube experiments, Combust. Flame 48 (1982)135–148.