Numerical Investigation of Detonation Transition

Mechanisms Using a High-order Discontinuous

Galerkin Method

Yu Lv∗ and Matthias Ihme†

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA

Re-ignition and re-initiation mechanisms of quenched detonation waves passingover a backward facing step are studied numerically using a high-order Discontin-uous Galerkin method in conjunction with a detailed thermochemical model. Astoichiometric Hydrogen/Oxygen mixture with different diluents (Argon and Ni-trogen) is considered, and effects of the mixture-composition and reactivity on theinitiation process are assessed. It is found that the wall reflection of the decoupledshock-wave plays a critical role in controlling the formation and growth of the igni-tion kernels, and the possible transition to detonation. Through parametric studies,it is shown that for more reactive Argon-diluted mixtures, ignition first appearsbehind the regular shock reflection, which is followed by spontaneous re-initiation.By replacing Argon with Nitrogen, the reactivity reduces and re-ignition throughMach-reflection is observed. This ignition mechanism is closely tied to the effectsof adiabatic heating by Mach shock compression and the enhanced mixing alongthe slip-line due to hydrodynamic instabilities. The present numerical investiga-tion shows good agreement with experimental measurement and confirms previouspostulations regarding the role of Mach reflection in the re-ignition process.

I. Introduction

It has been recognized that the pulsed detonation engine (PDE) is a promising propulsionsystem to support high-speed flight. In PDEs chemical energy is transformed into internal energyby a detonation process. From a practical point of view, fast DDT (detonation to deflagrationtransition) has to occur in a short pre-detonator section, before the detonated gas transmits into alarger combustion chamber. Successful transmission is not alway guaranteed. The abrupt geometricexpansion first causes the diffraction of the detonation wave. As the shock gradually weakensduring the expansion, the post-shock state becomes less favorable to ignition. Thus, the flamemay gradually decouple from the shock front, forming a so-called shock-flame complex. Simplyspeaking, detonation diffraction results in a delay or quenching of the detonation wave. In order toachieve successful transmission, re-initiation has to be established over a short distance inside thecombustion chamber.

In earlier studies, the significance of near-wall Mach reflection on re-initiation has been rec-ognized by Teodorczyk et al.1,2 Using high-speed Schlieren photography, they experimentallyobserved that re-ignition of the gas occurs behind the Mach stem. The authors realized that there-ignition might result from the high temperature in the specific region behind the Mach stem orthe turbulent mixing along the slip-line. Similar pattern of successful transmission was also found

∗Research Assistant, AIAA Member†Assistant Professor, AIAA Member

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52nd Aerospace Sciences Meeting

13-17 January 2014, National Harbor, Maryland

AIAA 2014-1505

Copyright © 2014 by Yu Lv, Matthias Ihme. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA SciTech

in the double-layered shock-tube experiments by Liu et al.3,4 In their study, different transmissionpattern were observed by adjusting the reactivity of the secondary mixture that the primary deto-nation transmits into. However, due to limitations of the experimental instrumentation, there wasstill no conclusive answer which mechanism leads to re-ignition.

Recently, Bhattacharjee et al.5 investigated the re-initiation process of the shock-flame complexin the configuration of Mach reflection using a novel experimental system. The strong jetting behindthe Mach shock is recognized as the main contribution to the rapid consumption of fresh mixture andthe subsequent detonation build-up. Apart from the re-initiation through Mach reflection, otherre-initiation processes are also discovered. Obara et al.6 investigated the detonation diffractionand re-initiation behind a slit plate with two inlets, using high-speed Schlieren and soot trackingtechniques. They found that detonation can be directly re-initiated via local explosion by theintersection of diffracted shock fronts, or indirectly via the interaction among diffracted, reflected,and explosion waves. Another interesting finding is from the study of Sorin et al.7 on a diameterstep setup. The soot traces indicated that re-ignition occurs away from the wall and might beinfluenced by the reflected wave, if the tube exit, d, is smaller than a certain ratio of the detonationcell size, klimλ.

Over recent past, several efforts have been made on the numerical investigation of detonationtransmission. Jones et al.8 and Oran et al.9 performed simulations to reproduce the experimentsby Liu et al.3,4 using simplified chemical models. Qualitative agreement has been achieved betweensimulation results and Schlieren images. In the study by Bhattacharjee et al., a simplified chemicalmodel is also employed to simulate the ignition events behind the Mach stem, but higher resolutionis required to investigate the wall jet and its effect on the re-ignition and subsequent transition todetonation. Besides increased numerical dissipation of low-order discretization schemes, anotherdrawback for those numerical studies is the utilization of simplified chemical models. As pointedout by Shepherd,10 the accurate representation of detonation quenching and initiation requiresdetailed chemical kinetic descriptions.

By addressing these issues, the present study utilizes a computational framework based ona high-order Discontinuous Galerkin scheme, to study detonation transmission physics. The lowdissipative character of this scheme provides high resolution and better description for the triggeredvortex behind the Mach stem. The detailed chemical kinetic model is integrated in order to providea more accurate representation of quenching and ignition events. The problem setup is consistentwith the experiment by Ohyagi et al.11 The work aims to shed light on the physical mechanismsof detonation re-initiation processes.

II. Governing Equations

The reactive Euler equations for multi-species combustion can be written in the following form:

∂t(ρYi) + ∂j(ρYiuj) = ωi, (1a)

∂t(ρui) + ∂j(ρuiuj) + ∂ip = 0, (1b)

∂t(ρE) + ∂j [(ρE + p)uj ] = 0, (1c)

where ρ, ui, Yj , E and p refers to the density, velocity of the ith component, mass fraction of speciesj, specific total energy, and pressure. The density is computed from the conservation of elementaldensity,

ρ =

N∑j

(ρYj), (2)

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and the specific total energy is defined as

E =

N∑i=1

Yi

[h0f,i +

∫ T

T0

cp,i(T )dT

]+u2j

2− p

ρ, (3)

in which T is the temperature and h0f,j is the heat of formation of the jth species defined at reference

temperature T0; cp,i is the specific heat capacity at constant pressure of the ith species, which isa function of temperature and evaluated through NASA’s polynomial expressions.12 Equations(1) are supplemented by the ideal gas law, relating pressure to density, temperature and speciesconservation:

p = ρRuTN∑i=1

YiMi

, (4)

where Ru is the universal gas constant, and Mi is the molecular weight of the ith species.

III. Numerical Method

A. Discontinuous Galerkin Discretization

The reactive Euler equations can be written in index notation as:

∂tUj + ∂kFckj − Sj = 0 , (5)

where Uj refers to the jth component of the conservative state vector, and F ckj is the inviscid flux

for the jth component of the state vector in the kth spatial direction; F ckj is a non-linear function

of the full state vector.Equation (5) is locally discretized in a discontinuous Galerkin (DG) spaces. For this, the com-

putational domain Ω is subdivided into a set of elements e and the solution of the reactive Eulerequations is restricted to be represented by polynomials. We then approximate each component ofthe conservative vector by the following space,

Ψh = φh ∈ L2(Ω) | φh|Ωe ∈ PNp(Ωe) , (6)

where PNp(Ωe) is a finite dimensional space on non-overlapping element Ωe, here specified by a setof polynomials with the order not larger than Np. For the sake of clarity, we denote the spatialbasis by φ, or its index form φq, where the subscript q in braces refers to the qth component ofthe basis. Each component of the solution of Eq. (5), U, is then approximated as

Uj(t,x) =

Np∑q=1

Ujqφq(x), (7)

where Ujq refers to the component of the qth basis function. With this derivation, the problem

reduces to solving for Ujq. To do that, Eq. (5) is multiplied by a test function φq, resulting inthe weak form:

∂tUjq

∫Ωe

φqφqdΩe +

∫Ωe

φq∂kFckjdΩe −

∫Ωe

φqSjdΩe = 0 , (8)

where∫

ΩeφqφqdΩe is the mass matrix. The reaction term is local, and convection is treated

using integration by parts,∫Ωe

φq∂kFckjdΩe = −

∫Ωe

∂kφqFckjdΩe +

∫∂Ωe

φ+qF

ckj(U

+j , U

−j )nkdSe (9)

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where nk is the outward pointing normal and ∂Ωe represents the boundary of element Ωe. On ∂Ωe,the notation ()+ and ()− refers to the quantities taken from the interior and exterior of elementΩe, respectively. Elements in each direction are coupled through the flux function F c

kj , and the

convective flux is evaluated using a HLLC Riemann solver.13

B. Algorithmic Development

Although the basic form of the Discontinuous Galerkin scheme has been presented above, sev-eral critical aspects have to be addressed in order to integrate a detailed thermochemcial modelinto the simulation. First, the variable thermodynamic properties, specifically temperature- andcomposition-dependent heat capacity, introduces numerically spurious oscillations in the frameworkwith restrictive energy conservation.14–16 To prevent these oscillations, a relaxation method (theDouble Flux model17) is applied, leading to a framework with quasi-conservation of energy. Second,the detailed chemistry with many species, as a stiff system, requires consideration of the relatedtime scales with disparity of several orders of magnitude. For this, a fractional time-stepping ap-proach is used, in which the convection operator in Eq. 5 is handled explicitly and the reactionsource term is advanced implicitly. Finally, a detonation, essentially a combustion-induced shock,is mathematically a discontinuity, which also challenges the numerical stability in the context of ahigh-order DG scheme. Capturing shocks with DG scheme has been a major research topic and isstill under development. To this end, a WENO-based limiter, recently proposed by Zhong & Shu,18

is used. This limiter is combined with the treatment to guarantee the positivity of density andpressure based on the entropy minimum principle. More details and validation of the developedcomputational platform can be found in the Ref.19,20

IV. Simulation Setup

In this study, we consider the experimental configuration of Ohyagi et al.,11 and a schematicof the experimental setup is shown in Fig. 1. In this facility, a long initiator section is adopted toensure fully-developed cellular structures before the detonation wave expands into the combustionsection. The height of the combustor section is three times the cross-section of the initiator. Toensure consistency with the experimental study, we set the origin of the coordinate system 1.8 cmupstream of the combustor end-wall. The simulations were initialized with a 1D Chapman-Jouguet(CJ) wave near the inlet of the initiator section. Small perturbations were imposed along the shockfront to initiate transition to 2D cellular structures. The detonation wave diffracts after enteringthe combustor section.

Figure 1. Schematic of computational domain.

Low initial pressure and high dilution ratio have been found to be essential in predicting the

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correct detonation size with detailed chemical kinetics. Considering this, in the present study theinitial pressure and temperature are set to 26.7 kPa and 293 K, respectively, and the dilution ra-tio is fixed at 40% by volume. A parametric study is performed by considering different diluentproperties. The fresh mixture is represented as 2H2 + O2 + βAr + (4 − β)N2, where β ∈ [0, 4] isthe diluent parameter. By changing β, the mixture properties and Zeldovich-von Neumann-Doringflame structure are modified, and different re-initiation pattern can be observed. In this investiga-tion, we consider three cases with β = 1.5, 2, 4. This kind of setup results in the same detonationMach number (Mcj = 4.9) but different induction lengths, specifically Lig = 240, 410, 478 µm forβ = 1.5, 2, 4, respectively. The grid resolution is adjusted for each case so that at least 20 degreesof freedom per Lig are guaranteed. The chemistry is described using the mechanism by Burke etal.,21 calibrated for high-pressure applications of hydrogen combustion.

V. Results and Discussions

In all the cases presented here, the quenched detonation configurations are all observed afterthe fully established detonation waves diffract over the backward-facing step. The decoupled shockfront, propagating faster than the flame front, first interacts with the bottom wall, and underdifferent diluent parameters the shock reflection results in a different ignition pattern which mightlead to successful re-initiation or failure of detonation.

A. Case β = 4

As shown in Fig. 2, the detonation diffraction first leads to the formation of a shock-flame complex,in which the shock and flame fronts are completely decoupled. Several “finger-like” flame segmentsare observed behind the shock front. At t = 52 µm, the shock starts to interact with the wall,and subsequently the reflected shock transmits into the burnt mixture. After this reflection occurs,the hot spot is immediately generated around x = 40 mm. This is followed by an extremelyfast transition (∼ 1µs) to an overdriven detonation, visible at t = 59.5 µm in Fig. 2(c). Thenew detonation front grows and rapidly engulfs the inert transverse wave. After the detonationwaves catches up the leading incident shock, a reactive Mach stem is formed due to the mutualinteractions. The induced hydrodynamic instabilities along the slip line is well resolved by thepresent simulation. The forward wall jetting effect behind the Mach stem is clearly observed. Thenew triple point, connecting the reactive Mach stem, reactive transverse wave and inert incidentshock, moves along the curved front of the incident shock, and finally re-initiates the whole front.This re-initiation sequence shows very good agreement with that reported in the experimentalstudy.11

B. Case β = 2

The re-initiation sequence for the case β = 2 is shown in Fig. 3. Compared to the case β = 4,it is noticeable that the number of “finger-like” flame segments is much smaller and the momentwhen the decoupled shock front starts to impact the bottom wall is delayed, due to the reducedreactivity. Furthermore, the formation of hot spots by shock reflection is much more delayed. Att = 68 µs, a new hot spot is generated at the near-wall region around x = 50 mm. Instead of the fastspontaneous transition to detonation observed for the case β = 4, this case reveals substantial flamefront propagation and growth during a considerably long time from t = 68 µs to t = 75 µs beforethe transition occurs. At t = 73 µs, a weak shock is observed ahead of the new flame front, due tothe merging of combustion generated acoustics. It is very likely that the re-initiation mechanismof this case is different from that found in the case β = 4. The same moment also corresponds tothe onset of a Mach stem formation, which produces a small hot spot behind the Mach stem. Its

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(a) (b)

(c) (d)

(e) (f)

Figure 2. Re-initiation sequence of the quenched detonation with β = 4, showing temperature (top)and pressure (bottom). The blue line in the pressure field indicates the flame front defined as theiso-contour of 0.5Y eq

H2O.

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growth can be found at t = 73 ∼ 75.5 µs. After the new detonation front is established, its forwardpropagation is much more pronounced, while the retonation wave that leads to the fast consumptionof upstream fresh gas in the case β = 4 is not generated. The subsequent wave interaction ends upwith the similar triple point configuration as that shown in Fig. 2(e-f). It is worthy emphasizingthat the re-initiation still occurs in the regime of regular shock reflection.

C. Case β = 1.5

Figure 4 shows the re-ignition sequence of the quenched detonation with β = 1.5. This case revealssimilar pattern as those for β = 2, in terms of detonation diffraction, shock reflection and hotspot generation. However, comparatively speaking, the new hot spot close to the bottom wallgrows much slower, which can be examined by comparing Fig. 4(c-e) to Fig. 3(c-e). Therefore, it isconfirmed that as the diluent parameter β decreases, the thermochemical environment created byregular shock reflection becomes less favorable for ignition. Without sufficiently fast reaction andthe induced interaction of acoustic waves, the re-initiation by the regular shock reflection fails in thiscase. Instead, the growth of hot spots behind the Mach stem, coupling with vortices along the slipline and wall jet, are more pronounced after t = 81.5 µs. Mach reflection controlled ignition finallycharacterizes the evolution of the quenched detonation. Although transition to detonation fromthe newly generated hot spots is not observed in this case, the current study confirms the earlierpostulation1 regarding the effects of adiabatic heating by Mach shock and instability-enhancedmixing on re-ignition.

VI. Conclusion

A computational framework based on a high-order Discontinuous Galerkin scheme is appliedto simulate the re-ignition and re-initiation of quenched detonation waves in combination witha detailed thermochemical model. The quenched detonation is generated through the detonationdiffraction over a backward-facing step. A stoichiometric Hydrogen/Oxygen mixture was consideredwith variable diluent composition. A series of simulations has been performed to systematicallyidentify ignition mechanisms that are controlled by the coupling between gas-dynamic processesand chemical kinetics. It is found that shock reflection plays an essential role in forming new hotspots. Mixtures with higher percentage of Argon diluent tend to follow the re-initiation by regularshock reflection. By successively replacing Argon with Nitrogen, the hot spots grow slower. Tocertain extent, if re-initiation is not achieved for regular reflection, the subsequent evolution of thequenched detonation wave is controlled by the re-ignition behind the Mach stem.

Acknowledgments

Financial support through the NSF CAREER program with Award No. CBET-0844587 isgratefully acknowledged.

References

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H2O.

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