Understanding explosions â€“ From catastrophic …...tron stars, or black holes. The transition...

Understanding explosions – Fromcatastrophic accidents to creation of the universe

Elaine S. Oran

Department of Aerospace Engineering, University of Maryland, College Park, MD 20742, USA

Available online 22 October 2014

Abstract

This paper focuses on two extremes of explosions: the general class of Type Ia supernova (SNIa), whichare surprisingly uniform thermonuclear explosions of white dwarf stars, and a specific gasoline vapor-cloudexplosion that occurred at the Buncefield fuel depot in 2005. In both cases, recurring questions are whetheran initial spark or small, local ignition could result in a detonation, and if so, how could this happen? Thebroader question is: What is the origin of the deflagration-to-detonation transition (DDT) in confined,partially confined, and unconfined systems? The importance of DDT to SNIa is based on the use of theseobjects as cosmological “standard candles” that are used for measuring distances and curvature in theuniverse. The importance of DDT to Buncefield is related to design and operational safety of industrialplants and fuel storage facilities. Combinations of observations, specific laboratory experiments, andselected numerical simulations have given us information and some understanding of the DDT processand its likelihood. Numerical simulation both of large- and small-scale phenomena in these reactive flowswere important ingredients in the studies. The invention and discovery of numerical algorithms, including(but not limited to) monotone methods, implicit large-eddy simulation, and adaptive mesh refinement,enabled these simulations certainly as much as the increase in computer speed and memory. Unresolvedissues that arose in these studies include the nonequilibrium, non-Kolmogorov properties of the turbulenceand turbulent fluctuations in these flows, how these prepare the system for transitions, and how torepresent the chemical reactions and energy release in the high temperatures and pressures that are nearand might signal a transition.Published by Elsevier Inc. on behalf of The Combustion Institute.

Keywords: Chemical explosions; Type Ia supernova; Deflagration-to-detonation transition; Numerical simulations ofhigh-speed compressible flow

Preface

This is not a review paper. Instead, it is aselective compilation of information and resultsfrom studies of the extremes of combustion thatrange from cosmology to micropropulsion. Thispaper presents a story of progress in two specific

areas of combustion – accidental explosionsand astrophysics – and attempts to show howknowledge from one field informs and advancesanother. The paper also slips in several issuesfrom a long list of problems that confront thefuture use of numerical simulation of reactingflows.

http://dx.doi.org/10.1016/j.proci.2014.08.0191540-7489/Published by Elsevier Inc. on behalf of The Combustion Institute.

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The form of this paper was partially inspired byfrustrating e!orts to create a recent edited volumefor the Philosophical Transactions of the RoyalSociety, Special Issue on The Physics, Chemistryand Dynamics of Explosions [1]. The contents ofthat volume of collected papers could only skimthe surface by presenting a relatively small slicethrough a universe of information on chemicaland astrophysical explosions. It described someof what we have done to understand these incred-ible, magnificent, malicious, and creative events.

Several reviewers commented that the paperwas “rather qualitative,” or to paraphrase, “toomany words and not enough equations.” Onreflection, I think that might be true,but it is mis-leading. The format is the result of an e!ort tobring the reader up to speed in many areas ofresearch as quickly as possible. In fact, an enor-mous amount of experimental, theoretical, andcomputational work, enough for a long reviewarticle or several books, is too often summarizedin a sentence or two. Unless that was done, Isaw no way to make some of the points neededto proceed with the arguments. In an e!ort to sat-isfy the reviewers, I added a small, more quantita-tive section describing one of the most importantconcepts, the ignition of gradients of reactivity.

I gratefully acknowledge the family, friends,and colleagues with whom I have worked closelyover the years to build up a base of knowledgeand capability. If you do not read anything else,please read the acknowledgments and savor everyname there.

1. Introduction

At every scale, explosions are defining featuresof our universe, as they “destroy order and createnew states and directions” [2]. In combustion, anexplosion can signal a transition from a deflagra-tion to a detonation or the formation of a strongshock front by a turbulent flame. Explosions are afundamental part of astrophysics [3,4]. For exam-ple, an explosion signals the transformation of astar from a white dwarf to a supernova, whichenables the creation of the heavier elements, neu-tron stars, or black holes. The transition in theearly universe from quarks to baryons has beenstudied as a detonation wave (see [5,4] has morereferences). We might argue that the Big Bangitself is an explosion that marks the beginning ofour universe. Besides their creative aspects, explo-sions have a downside. Large-scale accidentalindustrial explosions are rare events with disas-trous consequences for human life and property.Recent occurrences, such as Buncefield [6] Jaipur[7], Sago Mine [8], or Toulouse [9], made interna-tional headlines. Many other smaller-scale, lesspublicized events have also cost dearly. Sometimeswe can decipher the chain of events that led to the

explosion. Sometimes we can identify a way tochange the background in some way to decreasethe likelihood of a recurrence. Sometimes, anycontrol is beyond us, and we can only observewith awe or shock.

Whereas the common use of the term explosioncan be ambiguous, we can give it a more technicaldefinition [2]. An explosion . . .

“ . . . refers to any type of scenario in which energyis injected into a system faster than it can besmoothly [acoustically] equilibrated through thesystem, that is, energy is deposited faster than adynamical time scale. For chemical or nuclearexplosions, this time scale, l=cs, is based on thecharacteristic size l of the system and the acousticvelocity cs. For magnetohydrodynamic explo-sions, cs is replaced by the square root of thesum of c2

s and the magnetoacoustic time scalec2

ma, where cma is proportional to the speed ofAlfven waves, the square of which, in turn, isproportional to the magnetic pressure. The resultof this rapid injection of energy is a local pressureincrease. If the system is unconfined, or if theconfinement is weak and can be broken, strongpressure waves (shock waves) develop and spreadoutward, traveling considerable distances before theyare dissipated. As this happens, over-pressurizedmaterial begins to expand, and heated materialcools. This very general description coversscenarios that range from the Big Bang, to ther-monuclear explosions in stars, to magnetohydro-dynamic explosions on the sun, to most of thechemical explosions on earth.”

This description of an explosion resulted fromdiscussions that came about as we tried to find adefinition inclusive enough to cover the broadrange of situations we were considering. In theprocess, we argued whether or not the Big Bang,which creates space-time as the universe expands,was really an explosion that fits into any normaldefinition of the word. This paper, however,focuses on chemical explosions, such as those thatoccur in fuel depots or coal mines, and by analogyon thermonuclear explosions, such as those thatoccur in supernovae.

1.1. Steady-state physics – review and terminology

Common ideas of chemical explosions identifytwo distinct, steady-flow combustion regimes,subsonic flames and supersonic detonations.Many types of transitional states occur as a sys-tem undergoes transitions between these steadystates. Figure 1, an example of the “p ! vdiagram,” encapsulates much of our currentunderstanding and is a starting point in studiesof combustion. An initial, unexploded mixture is

2 E.S. Oran / Proceedings of the Combustion Institute 35 (2015) 1–35

characterized by a pressure, p0, and specific vol-ume, v0. The final state results in a change in thesequantities due to energy release in the system.Then if we assume steady, planar, one-dimen-sional flow, the principles of conservation of mass,momentum, and energy can be used to find possi-ble final states after heat is released in an explo-sion. The locus of the final states, the Hugoniotcurve, has two disconnected parts, an upperbranch and a lower branch. For exothermic pro-cesses, these two parts are separated. When thefinal state is on the lower branch of the Hugoniotcurve, the process is called a deflagration, andwhen it lies along the upper branch, it is called adetonation. These are the two very di!erent, dis-tinct types of one-dimensional, steady reactionwaves that can propagate through homogeneousexothermic, reactive material. Detonations arecharacterized by large pressure increases andsmall volume decreases, and deflagrations by largevolume increases and small pressure decreases.Propagation velocities of detonations, which aresupersonic, are thus faster than propagationvelocities of deflagrations, which are subsonic.

There are several points that are important forthe material presented in the remainder of thispaper. Tangents of the Rayleigh line and theHugoniot curve represent the minimum velocityfor detonations and a maximum velocity for

deflagrations. These are the Chapman–Jouguetpoints, marked as SCJ and DCJ on the figure. Thesteady-state detonation velocity DCJ is commonlymeasured and easily computed for idealized deto-nations. The CJ deflagration velocity SCJ is a moreobscure quantity that is generally given less time orconsideration. As shown in the figure, there can betwo types of detonations and two types of defla-gration, as the Rayleigh line can intersect each partof the Hugoniots twice. Both types of deflagrationstravel slower than SCJ , and both types of detona-tions travel faster than DCJ . The final fluid veloci-ties behind the combustion fronts are supersonicfor strong detonations and weak deflagrations.The final fluid velocities behind the combustionfronts are subsonic for weak detonations andstrong deflagrations. We generally consider strongdetonations and weak deflagrations as the steadystate for chemical reaction waves in usual combus-tion problems.

The origins of the concepts summarized inFig. 1 were developed over a hundred and thirtyyears ago with observations and experiments ofBerthelot and Vieille [10] and of Mallard and LeChatelier [11], followed by theoretical explana-tions of Chapman [12] and Jouguet [13] . Notableadvances bring us up to about 70 years ago withthe work of Zel’dovich [14], von Neumann [15]and Doring [16]. All of the theoretical work hasunderpinnings in the even earlier work of Rankine[17] and Hugoniot [18], who are associated withthe detonation part of the diagram that showslarge changes in pressure. Then the addition ofRayleigh lines complete the steady-state picture.An excellent review of the development ofshock-wave theory and its relation to the p ! vdiagram is given by Salas [19], and applicationsto combustion are discussed in combustion text-books (see, e.g., [20]).

1.2. Focus on transients and nonequilibrium

But Fig. 1 is really a steady-state picture, andexplosions are truly dynamic events that encom-pass processes that are neither the classical defla-grations nor the classical detonations describedby approximations of steady-state flows in homo-geneous, completely open space. Many of thesteady-state rules are violated in some of the mostinteresting and relatively long-lived intermediatecomplex states that often precede transitionsamong the steady states. Notable among theseare the shock-flame complexes that drive super-sonic fronts at approximately half of DCJ , andthe very high pressures, actually many timesgreater than those of DCJ , that can occur in tran-sient processes that precede the completion of atransition to detonation [21–23]. Some of these willbe discussed in the material that follows. Thereview article, Origins of the Deflagration-to-Deto-nation Transition in Gas-Phase Combustion [23],

Fig. 1. Schematic of the p ! v (pressure–volume) dia-gram for exothermic processes. Initial state of the systemdenoted by point p0 ! v0. Application of conservationconditions show that there are two lines types of finalstates possible, the upper branch for a detonation (largechanges in p give small changes in v), and the lowerbranch for a deflagration (large changes in v corre-sponding to small changes in p). The Rayleigh lines showfinal possible states. two for deflagrations and one fordetonations. The locations where Rayleigh lines aretangent to the Hugoniot curves, SCJ and DCJ . Thequantity DCJ is the usual steady state value of adetonation speed, the Chapman–Jouguet velocity. Thequantity SCJ is the Chapman–Jouguet flame speed, theimportance of which is discussed in the text.

E.S. Oran / Proceedings of the Combustion Institute 35 (2015) 1–35 3

has background material useful to the discussionsin this paper.

The dynamic timescale, sdynamic, the parameterthat determines whether an explosion occurs, isdefined for discussions of both the chemical andthermonuclear explosions that follow in this paperas sdynamic " l=cs, where l is the characteristic sizeof the system and cs is the sound speed. In thesecases, the dynamic time scale is the same as theacoustic time scale. Rapid energy injection causesan increase in the local pressure, and this canresult in shock waves that propagate through thesystem. For this reason, considerations of chemi-cal explosions must take into account the creationand development of shock waves and their inter-actions in exothermic systems.

Other important features of the development ofexplosions are the growth of fluid instabilities andtheir interactions with shocks. Throughout thecourse of this paper, we will be referring toKelvin–Helmholtz, Rayleigh–Taylor and Richtm-yer–Meshkov instabilities. The Kelvin–Helmholtz(KH) instability occurs when there is a velocityshear in the flow or a velocity di!erence betweenadjacent fluids. The Rayleigh Taylor (RT) instabil-ity occurs at the interface between two fluids withdi!erent densities, and occurs when the lighterfluid is pushing the heavier fluid. The Richtmyer–Meshkov (RM) instability occurs when if twofluids with di!erent densities are accelerated, asoccurs when a shock interacts with a a flame.Examples of these will be indicated in the materialpresented below. In more realistic systems, how-ever, these fluid instabilities occur simultaneously,or one sets up the conditions in which the other canoccur. All of the instabilities are extremely impor-tant in generating vorticity, which enhances thelevel of turbulence in the flow, all of which contrib-utes to creating an explosion. Another instabilitythat is sometimes important for laminar flames,usually in the early time of flame propagation, isthe Landau–Darrieus (LD) or hydrodynamicinstability. In the initial stages of laminar flamepropagation, especially in an open environment,the instability can cause wrinkling and growth ofthe surface area. It is finally stabilized when cuspsare formed. In closed environments, as we shall seebelow, it is often superseded by RT or RMinstabilities.

There has been an amazing growth in the capa-bilities of computers and available computermemory, and there have also been significantadvances in software and algorithms for comput-ing. The result is that it is now quite usual toattempt to use some form of numerical solutionof the governing equations to investigate explo-sions. These simulations are used either to provideinsights into what might have or actually didoccur, or even for designing devices or entire facil-ities for producing, storing, and distributing fuels.There is a very wide range of computational

models based on specific flow structures, othersbased on statistics and very general global systemproperties, some asking for very specific informa-tion, and others for only large-scale features. Aclassification of these models has been given in[24], and specific applications to mine simulations,have been discussed in [25]. The next section ofthis paper is a brief description of the history ofthe underpinning numerical algorithms that haveallowed us to compute shock waves, shockinteractions, and therefore, explosions anddetonations.

Type Ia supernovae are the thermonuclearexplosions of white dwarf stars. These stars arecomposed primarily of fully stripped 16C and12O ions, which comprise the fuel that burnsthrough a series of nuclear reactions to produceheavier elements on the periodic table. The fullystripped ions are immersed in a sea of electrons.A driving question in studies of supernovae hasbeen: How does a Type Ia supernova explode? Asthis question is discussed, a new way to create athermonuclear detonation is presented.1 For thisparticular flame structure, passing a shockthrough successive flame fronts could intensifythe shock, and how this can lead to a detonation.Astrophysical Combustion, presented at the 25thInternational Symposium on Combustion [4], isa review and assessment of the state of our under-standing as of 2005.

The other type of explosion discussed in somedetail is more specific, the large vapor-cloudexplosion that occurred in the fuel storage depotat Buncefield, in the UK, in December 2005.The explosion occurred after a liquid-fuel tankoverflowed and enough vapor was distributed bythe wind over a large area around the depot.Investigators were able to collect extensive datashortly after the explosion. For this large-scaleaccidental explosion, major technical and securityquestions focused around: How could so muchdamage have been done?

Discussions of the issues raised by SNIa andBuncefield naturally lead to discussion of whetherthere could have been a detonation formed atsome stage of the explosion process. To addressthis, a number of physical mechanisms for thetransition of a deflagration to a detonation aredescribed and evaluated in these contexts. All ofthese mechanisms require, result from, or causeshocks and shock-flame interactions.

The last section of this paper describes severalcurrent concerns about reactive flows, all of whichpoint to future research problems. The first twoare related to fundamentals of fluid dynamicsand raise questions about properties of the typesof intensely turbulent reactive flows that occurin explosions. The third concern, which is about

1 Not that new (2005), but just unpublished.


chemical reactions and chemical reaction mecha-nisms, questions basic concepts in chemicalreaction pathways and chemical equilibrium inhighly turbulent flows when shock waves arepresent.

2. Numerical necessities for computing explosions

The basis of much of our analysis of explosionsis the set of Navier–Stokes equations (NSE),which can be written as

@q@t#r $ qv " 0 %1&

@qv

@t#r $ qvv " !r $ P %2&

@E@t#r $ %Ev& " !r $ %v $ P& !r $ q %3&

The primary variables in the equations are thedensity q, momentum qv, total energy E, pressuretensor P, and heat flux vector q. Written this way,these equations can be thought of as a set of cou-pled continuity (or conservation) equations forq; qv, and E. The terms on the right describe thephysics that couples these primary variables andact to close the set of equations. In addition tothese source terms, each equation can have other,rather complex sets of terms representing addi-tional physical processes. The evolution of a flowis determined by initial and boundary conditionson the variables, as well as sources and sinks thatmay evolve naturally in the flow or through achanging physical environment.

The equations have been written to emphasizetheir structure as continuity equations, and havethe general form:

@u@t#r $ uv " S; %4&

where u is a conserved variable, and S may be afunction of u, position, and time. Starting withthe compressible Navier–Stokes equations in aconservation form is an important first step inunderstanding the origin of the development ofmodern numerical methods for their solution.For problems involving chemical explosions, theconservation equations for conserved primaryvariables are often supplemented to reflect thepresence of additional physical processes, species,or interactions.

Some of the most important terms are forconstituent species of the gas that can react chem-ically, either exothermically or endothermically,and so change both the energy distribution andthe composition of the gas. When constituentspecies are considered individually, an additionalset of coupled partial di!erential equations areoften added to describe the flow of these species.In addition, a set of ordinary di!erential equations

is used to define the source term S, and these canadd source terms in the energy and momentumequations. When there are additional conservedchemical species in the flow, we can writeconservation equations for these, and S then refersto production and loss terms.

For example, the crudest form of a chemicalreaction, in which a fuel goes to a product, canbe modeled by a very simple additional equation:

dYdt" AqYe!Q=RT " _x %5&

where Y is the mass fraction of fuel, A is the pre-exponential factor, Q is the activation energy, and_x is the rate of transformation of fuel to product.Adding this term to the equations usually meansthat an additional term is required as a sourceterm in the energy equations, specifically !qq _x,where q reflects the chemical energy released intothe flow. In addition to solving this equation, itis necessary to add another conservation equa-tion, @qY =@t # . . . :. to the system that convectsY with the fluid.

Often we consider a series of sequential reac-tion steps taking the system from fuel and oxidizerto a series of intermediate states and finally toproducts. Then for each species considered, weadd N s additional continuity equations of theform @qY i=@t # . . . so that the convection of eachspecies Y i in a reaction mechanism is taken intoaccount. Additional equations to describe thechemical interactions are required, and these areusually ordinary di!erential equations of the formdY i=dt " F %Y 1; . . . Y Ns&. Exactly how to treat thecomplications and explicit uncertainties associ-ated with this procedure has been discussed nowfor many years in text books (e.g., [24]), researcharticles, and lectures. These complications rangefrom simply not knowing which reacting speciesand chemical reactions to include, to how to eval-uate the important chemical-fluid couplingparameters (such as specific heats of reacting spe-cies), for multispecies flow on a computationalgrid.

The Navier–Stokes equations, as writtenabove, are deterministic in the sense that an exactsolution of the set of equations for a given set ofinitial and boundary conditions, with a specifiedset of sources and sinks of variables, shouldalways show the same results. The solution ofthese equations can change when supplementaryconditions change, and, under some circum-stances, the solution can be extremely sensitiveto very small changes. As a result of varying thesesensitive conditions, stochastic properties canemerge. This is the case for turbulent flow, whichis discussed further below.

A sensitivity to small changes can be exacer-bated when there is localized energy release intothe flow arising, for example, from chemicalreactions among the constituents or some kind


of localized energy injection of mass, a laser, or anelectron beam. Localized energy release is mostoften modeled by proposing a set of exothermicchemical reactions, and the energy release thendepends on the background conditions. The com-plexity of these models varies greatly, dependingon the problem and level of the solution required,although from the point of view of the fluid equa-tions. Sometimes, additional reactive or tracerspecies are added that react within and are con-vected with the flow.

A fundamental question asked of any model is:How well does it represent reality? The more lim-ited but important question is: What does the solu-tion of this model tell us about reality? Aspects ofthese questions are touched on later in this paper.Here, however, we will address the two basic ques-tions of numerical simulation: How do we solve thegoverning equations? How well are we solving them?In order to address these questions at least par-tially, we will summarize several of the most rele-vant aspects of computational fluid dynamics bygiving a brief history of computing for compress-ible fluid dynamics, and through this, emphasizingthe most important parts that have enabled com-putational fluid dynamics and given us the currentstate-of-the-art.

2.1. A critical concept – flux limiting

The importance of having the ability to usenumerical methods to solve explosion problemswas realized in the mid-1900s, when the complex-ities of actual explosions of chemical and nuclearfuels became important issues in the developmentof nuclear weapons and weapons e!ects. Theseearly e!orts produced the first finite-di!erencemethods for solving the NSE. The methods weredesigned to be solved by early computers, bothof the human variety (banks of all women whowere good in mathematics) and early machinevariety (computers such as ENIAC).2

The specific approach for converting the NSEinto a form suitable for computers was to approx-imate the coupled partial di!erential equations asa set of coupled algebraic equations, ones thatcould be solved by a computer. This numericalapproach, originally called “finite di!erencing,”takes a sample time and space and tessellates itinto discrete, continuous volume elements, thenexpresses the derivatives of the variable in the“obvious” form. For example, the change in umight be expressed in a straightforward way as%ut#Dt

x ! utx&=Dt and %ut

x#Dx ! utx&=Dx. Many

sophistications can be added to this representa-tion of the derivatives in Eqs. (1)–(3).

In fact, all of the these early methods hadintrinsic di"culties with achieving su"cient accu-racy or even converging to a solution. Here con-vergence has the usual numerical meaning: adecrease in computational time step and grid sizeproduces solutions that approach limiting (andhopefully correct) values. There were underlyingissues with numerical stability in early methods,too. These instabilities became particularly evi-dent when the most obvious finite-di!erencingmethods were used to compute the properties ofshock waves and detonations (see, e.g., [24], chap-ter 4). Additional serious problems arose becauseof both extreme numerical di!usion, caused by theway in which the spatial derivatives wererepresented, and a lack of conservation of the sup-posedly conserved variables, a sure sign that thesolution is going wrong. Without some specificrecognition of the source of the errors and someway to control them, the solutions will also“explode” on the computer3 because the methodshad inherent numerical instabilities. The instabili-ties are catastrophic to the solution when themethods are used to solve for the evolution offlows containing discontinuities, such as shocksor contact surfaces. In such cases, they produceunphysical oscillations that can quickly degener-ate into nonsense.

A discontinuity is, in reality, a gradient or tran-sition region in which the physics is not resolvedon the scale of the computation. For example, ashock is a narrow transition region in which theproperties change very quickly and equilibriumassumptions in any quantities, including the exis-tence of an equation of state might not apply.There are molecular properties at a material inter-face that we do not resolve. The art and science ofcomputational physics then was to devise a way ofcomputing the interface propagation, using givenphysical laws (here the NSE) that will propagatea discontinuous front in a realistic way, withouthaving to resolve essentially “unresolvable”details. An additional problem inherent indi!erence methods themselves was “numerical di!u-sion.” This basic consequence of finite-di!erencingspatial derivatives spreads out the solution in timeover a larger area. It again is a result of errors thatoccur when representing the gradients in the equa-tions in an algebraic form. If the entire solution isconsidered from the point of view of a Fourierdecomposition, we see that as a solution isadvanced in time, there are both di!usion and dis-

2 For example, Von Neumann is associated with thedevelopment of early methods [26] and his wife was anearly computer.

3 As an example for the student reader, I recommendstarting with a single continuity equation, as in Eq. (1),and a simple central di!erencing algorithm for thevariable, and solving the standard test problem of amoving square wave. This is a lesson itself in bothconservation and stability. (See [24], chapter 4, for somedetails on how to do this.)


persion errors in the various Fourier modes thataccumulate and interact [24]. As this in itself is atopic of much research and discussion, and some-thing of a diversion here, we refer the reader to anytextbook on numerical methods for partial di!er-ential equations (see, e.g. [24] for more referencesand explanations).

The first attempt to remedy the instability prob-lem used the standard di!erencing methods, butadded artificial damping terms (artificialdi!usion). This did not always work as desiredand required additional parameters to controlthe added di!usion and dispersion at a shock front([26]; see also discussions in [27,28]). Often theworst e!ects of the instability were eliminated,but there were still numerical oscillations in thesolutions. Even with added controls, thesemethods often became unstable when they werecoupled to the type of exothermic chemicalreactions that occur near the leading shock in adetonation wave.

The major algorithmic breakthrough in ourability to handle these numerical issues for com-puting explosions occurred in the early 1970s withthe invention of flux-limiting methods [29,30].These methods directly addressed the di"cultiesof propagating a discontinuity by imposing physi-cal constraints on the solutions to maintain threeimportant properties of continuity equations: con-servation, monotonicity, and causality. Viewingthe NSE as a set of continuity equations, andimposing the three conditions as constraints, is acritical part of flux-limiting concepts. Here mono-tonicity means that the numerical algorithm usedto solve a continuity equation itself will not imposeany new, unphysical maxima or minima on thesolution. In fact, monotonicity was e!ectedthrough flux limiting. Application of monotone(or monotonicity-preserving) or flux-limitingmethods to astrophysical problems [31] in additionto weapons research further aided their develop-ment. Over the next ten years, many communitiesadopted, expanded, and improved this approach.Now all modern shock-capturing methods forsolving the NSE with shocks either directly or indi-rectly use flux correction, especially near fluid-dynamic discontinuities.

After the introduction of the concept of fluxlimiting, the entire field of numerical analysis ofthe continuity equations for fluid dynamics blos-somed and went in a number of di!erent direc-tions. The more mathematical approaches led tothe development of a string of methods called thetotal-variation diminishing (TVD) methods [32].The more physically based approaches, such asthe partial parabolic method (PPM) [33], led toincluding the e!ects of the flow characteristicsdirectly into the algorithms and considering howto improve the representation of the physical spacebetween computational cells. These and manymore methods, each addressing one of the myriad

issues in solving coupled continuity equations, arediscussed in many books. Even today, after fortyyears of trying, there are e!orts to find the “per-fect” method by twisting here and there in combi-nations of fluxes, limiters, and incorporatingsolution characteristics. Insights into the historyof modern CFD methods have been summarizedin the whimsical drawing created by Bram vanLeer, shown in Fig. 2.

With advances in computer capabilities (suchas speed, memory, and architecture) combinedwith new numerical algorithms, the ability to per-form meaningful numerical simulations advancedeven faster than Moore’s Law. When the newmethods for solving the NSE were combined withnew methods for representing chemical reactionsand energy release and new computational capa-bilities, applications opened up in almost everyarea of science, including atmospheric and solarphysics, combustion, and astrophysics, and almostevery branch of engineering. Nonetheless, thevalidity and accuracy of numerical solutions ofthe NSE for representing a real physical scenario– issues often related to the required level ofcomplexity – are limited by speed and memory ofavailable computers and by knowledge of theappropriate input data.

2.2. A critical discovery – MILES or ILES

The next major step in computation was therecognition that there are other advantages tousing monotone methods in addition to betterways to capture shock waves and other disconti-nuities or gradients in the flow. Relatively early(1970s), we found that when we were solving aflow with turbulence, it was simply not necessary(and of course not possible) to resolve flows tothe viscous scale to obtain useful and “accurateenough” numerical solutions. This discovery waslater formalized and called MILES (monotone-integrated large-eddy simulation) or ILES (impli-cit large-eddy simulation) ([34,35]). These used amonotone method to solve the underlying fluidequations and focused on the physical scales thatwe believed covered phenomena of interest. Thisapproach was used by necessity in the weaponsand astrophysics communities, where importantproblems involved shock waves and supersonicflows spanning ten orders of magnitude in spacescales. It was, however, not considered in otherfields that in which the numerical methods werebased on nonmonotone methods, of which theearly spectral algorithms are good example. Thosealgorithms were forced to add special models(“turbulence models”) to eliminate spuriousenergy accumulation at the smallest scales of thecomputations, and so ensure physically correctdissipation at those scales. This, in fact, was theorigin of turbulence modeling and filteringprocesses for computed variables.


The explanation of why and how ILES worksis based on an interesting property of monotonemethods: At the computational grid scale, mono-tone methods are constrained by conservation,monotonicity, and causality – imposed by the dif-ferencing and flux-limiting – such that they areforced to dissipate energy in a way analogous tothat of the natural viscous dissipation. It is there-fore not necessary to include additional models(“turbulence models”) to avoid the types ofunphysical pile-up of energy at high frequencies,a problem that plagued the spectral and non-monotone methods. Thus if we are solving theNSE for a flow and the features of interest of thisflow are occurring far enough from the viscousdissipation scale, we usually do not have toresolve the flow down to the smallest scales. It issu"ciently accurate to resolve some portion ofthe inertial range4 The flow is then resolved fromthe largest scale down to a grid scale, at which theenergy is dissipated by a numerically imposed vis-cosity. The viscous dissipation scale is e!ectivelyreplaced by a grid scale. The e!ects on the solu-tion of using such an artificial cuto! must be con-sidered or even, if possible, estimated, often on acase-by-case basis. This feature is illustrated inFig. 3.

The use of ILES methods means that theextreme turbulence in an explosion can, in princi-ple, be computed accurately enough and without

resorting to ad hoc models with empirical con-stants [35]. This concept, and the growth of ourunderstanding of when and how it applies, havebeen tested repeatedly over for many years andmany types of problems. Now we believe we havea reasonably good understanding of how, why,and when it works [36]. It is now widely acceptedthat if we resolve enough scales in a turbulentflow, and we are not in the range of viscous

Fig. 2. Bram van Leer’s history of CFD Part II. Figures on pedestals on left are Richard Courant, Kurt Friedrichs, andHans Lewy. Figures in front are Robert MacCormack, Philip Roe, John von Neumann (framed), Stanley Osher, andAmiram Harten. Figures on beach chairs are Peter Lax and Sergei Godunov. Figures climbing the pyramid are JayBoris, Vladimir Kolgan (perhaps mythical), and Bram van Leer. Antony Jameson is flying. (Figure curtesy of Bram vonLeer. Comments in this caption are the author’s.)

4 Defined here simply as the range of length scales forwhich viscous dissipation is not important. In Kolmogo-rov theory, turbulence statistics are scale invariant andfollow a scaling law in the inertial range.

Fig. 3. Energy spectra computed using ILES algorithmsand how they approach a dissipation scale determinedby the grid spacing. Di!erent e!ective flux limitersproduce computations that approach the grid scaleslightly di!erently, as shown by the green lines whichcome close to each other but deviate slightly near thegrid scale. (For interpretation of the references to colorin this figure legend, the reader is referred to the webversion of this article.)


dissipation or where new physics cuts in to domi-nate the system behavior, we do not have toresolve all length scales down to the viscous dissi-pation scale to obtain a useful solution.

2.3. The last piece – dynamically adaptive gridding

The final numerical advance that changed ourability to compute explosions is dynamically adap-tive gridding. Explosion problems are character-ized by a wide range of physical scales (oftenover ten orders of magnitude) and a concentrationof important small-scale features, such as flames,shocks, and detonation waves, in relatively smallareas of the computational domain. Sincesmall-scale features require small computationalcells to resolve, a uniformly spaced mesh through-out the entire domain usually means that there arelarge volumes of space with many smallcomputational cells where nothing of much inter-est happens.5 This can mean a huge waste of com-putational resources. To increase the e"ciency ofcomputations, it makes sense to use a fine mesharound important flow features and a coarse mesheverywhere else. Because of the dynamic nature ofexplosions, the adaptive grid needs to be continu-ally adjusted to resolve evolving flow features.

In the 1970s and 1980s, when the need fordynamically adaptive gridding became evidentfor complicated reacting flows, we developedand tested many approaches. These methods ran-ged from fully Lagrangian methods implementedon dynamically changing tetrahedral or triangulargrids, to a string of di!erent finite-element-basedmethods, to mixed Eulerian–Lagrangian methods.(A review of some of these earlier methods andtheir underlying principles is given in [24].) Amajor issue with these approaches for computingexplosions is that they could not be easily andaccurately enough combined with the types ofhigh-order monotone algorithms without addingsignificant and sometimes unacceptable di!usionand other inaccuracies. Early solutions used slid-ing Cartesian grids in one or two dimensions,and these worked well for certain types of prob-lems, such as computations of detonation cells.These approaches, however, were not a generalanswer to the problem in which there were manyregions of the computational domain that neededgrid refinement and others that could tolerate gridcoarsening. Significant advances finally came fromdeveloping algorithms for local refinement in mul-tidimensional cartesian grids [37,38]. Figure 4 isan example of dynamical refinement for the prob-lem of a flow over a obstacles and how regions of

increased structure may be resolved by increasingthe resolution at in and around these regions.

The current state-of-the-art is to use dynami-cally changing local refinement, at whatever levelrequired or specified. This approach is calledadaptive mesh refinement (AMR). To date, anumber of methods have been streamlined wellenough to be made available to all interestedusers. Some of these methods for Cartesian gridsinclude the FFT [39] (limited in use, not com-monly available), Paramesh [40] (available, easyto use, unfortunately not currently supported),and BoxLib [41] (available, currently supported).The ease of using these varies from nearly impos-sible and not very general to relatively straightfor-ward for a sophisticated user. The selection ofwhich to use depends on the particular problemand the size and architecture of the computer tobe used.

3. Type Ia (thermonuclear) supernovae

White dwarf (WD) stars are a stage in the evo-lution of a red giant star when the central hydro-gen and helium have burned to produce carbonand oxygen. A WD may appear to remain in thesame stage for millions or billions of years, butthen it may suddenly erupt into a very powerfulexplosion, the Type Ia supernovae (SNIa). Thiscan occur if the mass of the WD somehow reachesthe Chandrasekhar limit, which is '1.4 times themass of the Sun. Then, in about two seconds, aWD explodes and releases '1051 ergs of energyin an event that can be as bright as an entire gal-axy. The current picture of what happens in thisexplosion is that the mass of the star has increasedjust enough, and the density and temperaturesomewhere in the star have become just highenough to cause thermonuclear ignition. The igni-tion sets o! a chain of reactions that convert theinitial carbon and oxygen to heavier elements upthrough the iron group. The heavy elements arepropelled by the explosion into interstellar space.

What is both fortunate and amazing is that, toa very good approximation by astrophysical stan-dards, all SNIa have similar spectra and lightcurves. (In fact, it is spectra and the total energyrelease that define the explosion as a SNIa). TheSNIa explosion is over in seconds, but the subse-quent e!ects are visible much longer, as it takesabout fifteen days for the luminosity to peak. Evenhundreds of years later, remains of SNIa’s areobservable as beautiful nebulae. Because of thespectral uniformity, no matter where or when theyoccurred in the universe, they have become impor-tant standards for cosmological measurements.

The detailed structure of the state of a WDbefore ignition, that is, in the state of the “progen-itor” WD star, are major questions. Speculationson possible initial conditions leading to SNIa

5 These small scales cannot be cut o! from thecalculation if, for example, new physical interactionsor processes become at those scales, and the results thenare extremely expensive computations.


cover a wide range of ever more complicatedphysical states, ranging from static (unlikely) todynamic (likely), from a single WD star becomingunstable enough to explode, to events invokingmultiple astrophysical bodies interacting or collid-ing. The most general properties of current work-ing initial models for the progenitor are given inTable 1 (taken from [4]).

The progenitor is about the size of the earthand composed of a mixture of fully stripped 12Cand 16O ions, in about equal proportions,immersed in a degenerate electron gas, a sea ofelectrons detached from nuclei. The electrons

therefore occupy all possible quantum states upto the Fermi level, so that the pressure is essentiallyindependent of temperature. The equation-of-stateis P " E=%c! 1&, where c " 4=3–5=3, dependingon the density, which varies from '109 g/cm3 atthe center of the WD to '106 g/cm3 at the surface.Gravity varies from zero in the center of the star to'106 g (where 1 g is earth gravity). The ambienttemperature of the star is uniform and low,'105 K, except at the point of carbon ignitionwhere it is '108 K.

Some progenitor models do not assume thereare any significant ongoing large-scale motions

Fig. 4. Dynamically adaptive grids used for adaptive mesh refinement are designed to add or remove computational cellsand reinterpret the solution onto finer or coarser meshes.This figure is taken from a calculation of an unstable flamefront propagating in a channel containing a stoichiometric mixture of hydrogen and air and a series of obstacles on thebottom wall.

Table 1Type Ia Supernovae: Initial conditions and combustion properties (taken from [4]).

Property Value Comments

Initial conditions of white dwarf progenitorLifetime of the star 108–1010 yearsRadius of the star R0 " 2( 108 cm About the size of earthDensity 109 g/cm3 at center Decreases from center to 106 g/cm3

Gravity 0–106 g varies from center to near surfaceTemperature 105 K (108 K at ignition)Material 12C–16O ions Degenerate gas, fully stripped

Properties of the explosionExplosion time 2 sLeaves behind nothing All material ejected into spaceLuminosity Same as a galaxy Peaks in 10–15 days, decays in 12–18 months

Properties of flames and detonations in SNIaLaminar flame-front thickness dl " 10!4–102 cmLaminar flame speed Sl " 107–104 cmDetonation front thickness 10!1–103 cmCJ detonation velocity DCJ " 1–1:4( 109 cm=sReaction temperature 109–1010 KEquation of state (degenerate matter) P " %c! 1&E c " 1:3–1:6Major energy-releasing reaction 12C + 12C

Produces He, a particles, Ne, MgReleases 50% of nuclear energyReaction proceeds to produce Mg, Si, . . . up to Ni


of the fluid, though this is not necessarily correct.Now there are models that assume there is a smol-dering state and include large-scale motions. Weknow, however, that there is some level ofturbulence, part of which could be generated bylarge-scale motions or star rotations. We also sus-pect that there might be substantial MHD e!ects,but this is, as yet, an unchartered territory.Reviews of the state of our knowledge and howit has evolved have been given by [3,42–47]. Anearlier topical review appeared in the Proceedingsof the Combustion Institute [4].

The important information for the current dis-cussion is that SNIa involve ignition of nuclearflames and detonations in flame regimes that areprobably analogous to weak ignition, a transitionfrom laminar to turbulent flames, a probable tran-sition from deflagrations to detonations, and thenthe eventual quenching of combustion when eitherthe fuel runs out or the burning front reaches verylow fuel densities. This entire explosion processinvolves many types of fluid instabilities in ahighly compressible, exothermic flow. Below weshow that even though SNIa explosions occurunder extreme conditions compared to anythingwe see on earth, we have been able to apply prin-ciples of combustion theory and practical knowl-edge of the behavior of flames and detonationsto explain significant aspects of the explosion. Inturn, what we have learned through studies ofSNIa has provided rather intriguing insights intocombustion processes on earth.

3.1. Ignition and explosion

Here is a possible scenario for ignition: Whenthe mass of the WD star is about at the Chandra-sekhar limit, a small increase in the mass of a WDresults in a substantial contraction of the star.6

This compresses material near the center, whichin turn increases the temperature and densityand initiates further exothermic thermonuclearreactions. The initial temperature increase in theWD does not a!ect the pressure (which is domi-nated by that of the degenerate electron gas),and so there is no substantial expansion to slownuclear reactions and prevent thermonuclear run-away. The heating is slowed by escaping neutrinos(neutrino loss), convection, and (primarilyelectron) thermal conduction. Eventually, thetemperature in the WD center increases to thelevel where the thermal pressure and the degener-ate-electron-gas pressure become comparable, andthen the material begins to expand. By thistime, the expansion cannot quench the fast

thermonuclear burning. This well-establishedthermonuclear runaway mechanism in degeneratematter [48] is now a key component of SNIa the-ory [47,49–53].

A network of thermonuclear reactions startswith 12C and 16O nuclei and ends in 56Ni andother iron-group elements. Elements of intermedi-ate mass, such as Ne, Mg, Si, S, Ca, are also cre-ated. The peak luminosity occurs about fifteendays after the initial explosion because of the slowradioactive decay 56Ni! 56Co! 56Fe coupledwith the increasing transparency of the expandingmatter. The luminosity fades as Co decays (half-life 77 days). Observational data include spectraand their evolution, light curves, some informa-tion about the environment of the progenitorWD, that is, its host galaxy. Spectra give us veloc-ities of ejected material, composition, and densi-ties of the species in the ejected matter. The riseand fall of light curves put constraints on theamount of 56Ni produced by the explosion, thevalues of density, and the expansion rate. The gen-eral environment of the WD, that is the nature ofthe galaxy and the location of the SNIa in the gal-axy, helps determine an approximate age of theWDs, and therefore provides some evidence aboutthe C/O ratio of the progenitor. To this mix ofinformation, and from knowledge of nuclear reac-tions and nucleosynthesis, we know that there arethree main burning stages, related to the carbon,oxygen, and silicon, respectively. The rolesand scales of each of these di!er by orders ofmagnitude.

3.2. How does a Type Ia supernova explode?

In the past ten years, there have been extensivee!orts to answer this question using numericalsimulations. The physical model for an SNIaexplosion assumes a compressible gas, anexothermic reaction network, thermal conduction(primarily due to electron gas), essentially nomolecular di!usion, a degenerate-electron-gasequation of state, and strong, spatially varyinggravitational forces. The numerical models mustcover a very large range of physical scales and sohave naturally evolved to combine various mono-tone methods, ILES, and AMR, and have beenenabled by advances in computer technology.

We have long known (since 1970s) that a deto-nation alone does not reproduce observations.One-dimensional detonation models, assumingthe detonation starts in the center of a WD wherethe density is highest, do not reproduce theobserved spectra [54,48]. Since burning behind asupersonic detonation front does not allow anyexpansion of the material ahead of the front, reac-tions would only occur at high density, resulting intoo much iron and not enough intermediate-masselements. Multidimensional detonations, (evenallowing for the presence of detonation cells) also

6 This is the limit at which the self-gravity of the starexceeds the internal pressure generated by the body, inthis case, the internal pressure is that of the degenerateelectron gas.


do not give the right composition [55,56]. One-dimensional models of thermonuclear flames, the“deflagration models,” were one of the earliestconcepts tested [50,53,57,58], but they do notresult in enough energy release or the correct spec-tra. They do, however, allow the outer unburnedmaterial to expand ahead of the deflagration, sothat it burns at a lower density and can create someintermediate mass elements. Patches for thismodel, which assume that the flame was turbulentand proposed a turbulent burning speed for flamepropagation, could give reasonable results, but notnecessarily the right late-time velocities or shocksobserved in the interstellar medium. Two-dimen-sional simulations of deflagrations do, however,show the importance of the gravity-induced Ray-leigh–Taylor instability in creating turbulence.They also showed how large Rayleigh–Taylorlobes form early in the evolution of the flame,and how they can mix core and outlying material.All of these results introduced more parametersand uncertainties related to input, dimensionality,and resolution.

In the last ten years, there has been a substantialinvestment in developing three-dimensionalnumerical models to simulate deflagrations inWD explosions [59–65], and these have been car-ried out on the world’s largest and fastest comput-ers. Nonetheless, they are still plagued by the verylarge range of physical scales required for the fluiddynamics combined with the complexity of thereaction mechanism. Figure 5 shows that the rangeof space scales needed to perform a direct numeri-cal simulation of this problem covers twelve ordersof magnitude [4]. The largest scale needed, is thesize of the star. Then, based on ILES concepts ofnumerical simulation of turbulence, the smallestimportant scale to resolve is assumed to be thethickness of a laminar thermonuclear flame. Anynuclear reaction mechanism extending from car-bon and oxygen through to iron is extremely large(see, e.g., [66]), and solving the full mechanismsfor the reaction pathways is an orders of magnitudesti!er problem than any chemical mechanism wedeal with for terrestrial combustion.

Progress has been made, however, because mul-tidimensional simulations have relied on reducedthermonuclear burning mechanisms that canrelease the approximately correct amount ofenergy for a flame or detonation on approximatelythe correct time scales. This is combined with analgorithm to “track” the flame, that is to modelthe position of the flame using a turbulent flamemodel appropriate for a WD explosion dominatedby Rayleigh–Taylor instabilities [59,67]. Still, noneof the deflagration models quite work, the spectrathey produce are not quite right, and the adjust-able parameters are disconcerting. Furthermore,observed velocities of ejected material indicatedthat shocks are generated and that there could bea detonation.

The key feature shown in Fig. 6 is the highlyturbulent flame that allows folding and convolu-tions of the flame surface, and so there is signifi-cant interpenetration of burned and unburnedmaterials. By the end of the explosion, 80! 90%of the material near the center of the WD hasnot completely burned out. This means that thefinal ejected material contains unburned carbon,oxygen, and intermediate-mass elements at all dis-tances from the center. This material would pro-duce spectral signatures in a range of expansionvelocities, including velocities close to zero. Obser-vations, however, show oxygen only at high veloc-ities, as would be produced by the acceleration ofexpanding outer layers. For intermediate-mass ele-ments, the lowest observed velocities are lower, butstill substantial enough to rule out the presence ofthese elements near the WD center.

An example of calculations in which a detona-tion was artificially ignited at various locations inthe WD are shown in and Fig. 7. These computa-tions gave strong indications that DDT wasrequired, as these models could produce the rightssorts of spectra and ejecta. But how does thistransition occur in an unconfined environment?Models that propose DDT in the SNIa arereferred to as “delayed-detonation-models,” aterm introduced to the astrophysics communityby Khokhlov [68] that emphasizes the importanceof DDT to SNIa.

Fig. 5. Length scales in the Type Ia supernova problemspan twelve orders of magnitude, from the size of the awhite dwarf star to the thickness of a laminar flame nearthe center. (Figure courtesy of Vadim Gamezo, takenfrom [4].)


The sequence of events in the delayed-detona-tion model could then be something like this:The WD is ignited as a deflagration somewherenear the center of the star as the temperatureand pressure increase there. As the flame movesaway from the center, it is subject first to Lan-dau–Darrieus and then, and primarily, due tothe RT instability (discussed in the Introduction),which is strong due to strong gravitational forces.The figure shows the characteristic expandingmushroom-shaped structures typical of the RTinstability. There are, of course, other instabilitiespresent. In some cases, we might expect the e!ectsof thermo-nuclear instabilities that would beappropriate for high-Lewis-number flames, andwe always expect secondary e!ects due to KHinstabilities as shear flows are induced aroundthe RT structures. In summary, the evolving flowis highly turbulent, and there are many instabili-ties occurring simultaneously. This is no idealizedflow where we can separate and study their occur-rence and e!ects one at a time.

Eventually, there is a transition to a detona-tion. Thermonuclear reactions occurring during

the explosion provide energy for the expansion.Most of the energy released during the explosionis transformed into kinetic and thermal energiesof expanding material. When and if the sum ofkinetic (Ek) and thermal (Et) energies exceeds thepotential energy of gravitation Eg, the starbecomes “unbound”. This means that the expand-ing material is no longer bound by gravity, contin-ues to expand, and propels the products ofcombustion into interstellar space.

3.3. Unconfined deflagration-to-detonation transi-tion – how could it happen?

To a combustion and explosion scientist, thedeus ex machina for explaining SNIa is the con-cept of DDT, which could solve major questionsof how and when ejecta of form. A stumblingblock is that we all know full well how di"cultit would be to produce a detonation withoutobstacles or confinement. In about 2005, when itwas conceded that DDT would produce reason-able spectra and pure deflagration models wouldnot, a famous statement was made by Wolfgang

Fig. 6. Development of thermonuclear deflagration in carbon-oxygen white dwarf [4]. The gray surface shows theturbulent thermonuclear flame. The color scale shows the radial velocity of unburnt material scaled by 1000 km/s.Distances are scaled by the computational domain size xmax " 5:35( 108 cm. The last frame shows the star after theflame reached the boundary of the computational domain [4,59].


Hildebrandt, “We know there has to be a detona-tion, but where’s the shock [coming from] ??”

Significant advances in understanding combus-tion in SNIa and the origins of DDT in terrestrialcombustion had been made in previous yearsthrough combined e!orts in simulation, theory,and experiments. Numerical studies of DDTbegan in the early 1990s and are being carriedthrough to today. Perhaps the key idea that canbe stated here is that, in fact, DDT is initiated ina relatively small region of space that has been“prepared” in some way by an increase in pressureor temperature. This preparation could be animposed gradient in reactivity, or it could be somesort of localized explosion (energy input) that cre-ates a shock on a small scale, that in turn creates agradient in reactivity. The major point, however, isthat there has to be a localized input of energy on avery small scale. By “very small,” this means onthe scale of a local flame thickness, or possibly less.

Figure 8 shows how successive hot spot ignitions ina flame can eventually lead to DDT. The informa-tion accumulated up until '2006 is summarized ina review article for the 50th anniversary of Com-bustion and Flame [23] (see Fig. 9).

As of about 2010, there had been extensive 3Dnumerical simulations of WD deflagrations, usingsome of the largest computers in the world, alllooking for information about DDT in SNIa.These included programs at the University of Chi-cago, Lawrence Berkeley Laboratories, StonyBrook, and in the Max Planck Institute for Astro-physics. None of these simulations spontaneouslyevolved into a detonation, only events that wereeither purposely created to look at the end prod-ucts or the result of triggering by a numerical arti-fact. Figure 8 is take from a simulation by Jacksonet al. [69], one of the most advanced of thesecomputations, which shows the complexity thatcan arise in a three-dimensional turbulent flow

Fig. 7. Development of a turbulent thermonuclear flame (colored surface) and a detonation (gray surface) in a carbon-oxygen white dwarf. Numbers show time in seconds after ignition. Flames at 0.30, 0.61, and 0.90 s are plotted at thesame scale. Further flame growth is shown by colors that change with distance from the flame surface to the WD center.xmax " 5:35( 108 cm [4,60].


when many orders of magnitude of spatial scalesare resolved.

The reason that these large-scale simulationsdo not predict DDT is that DDT is initiated onscales smaller than these simulations are able toresolve. They cannot span the entire range ofscales from the size of the star to the laminar flamethickness, the range shown in Fig. 5, and DDTmay be initiated on even smaller scales! A possiblemechanism for unconfined DDT was proposed in2005 (Gamezo, private communication), and it isimportant to understand what this mechanismshows. The thermonuclear reactions of C and O,coupled to models for thermal conduction and acorrect equation of state, produce a complex lam-inar thermonuclear flame structure that varies sig-nificantly with density (from '109 g/cm3 (centerof star) to '106 g/cm3 (close to the surface ofthe star). There are distinct multiple stages to thismechanism, in which C burns to produce addi-tional O, then the O burns to produce other ele-ments, etc. This is illustrated in Fig. 10 for adensity of 2 ( '109 g/cm3, the density at whichwe expect a detonation to occur, based on whenwe would find a reasonable composition in theejecta.

At '2 ( 109 g/cm3, there are distinct reactionregimes. Figure 10 shows that the C–C reactionsoccur in a first few millimeters of the flame, releas-ing about 50% of the energy, and this is followedby a somewhat slower O-burning region that, atthis density, extends for centimeters. The key con-cept proposed is that a shock wave, or even anacoustic wave, passing through this flame struc-ture compresses the O-burning region, increasesthe rate of energy release, and amplifies the wave.Then the same shock or acoustic wave repeatedlypasses through such flame structures and repeat-edly gains strength. The acoustic wave couldamplify to become a shock, and weak shock couldbecome a stronger stock, and eventually a detona-tion might emerge. Figure 11 is a result of preli-minary tests of this concept. Here a shock movesthrough a background sea of O bubbles and, forthis set of conditions, emerges as a detonation.

Now consider a turbulent flame with this lam-inar flame structure, and a weak shock that passesthrough flame repeatedly and from many angles.An emerging detonation seems like a possibleand perhaps even inevitable outcome. Questionsarise then as to where this shock can comefrom and whether an acoustic wave, as it moves

Fig. 8. DDT inside a small funnel of unburned material surrounded by a flame [70]. As shown by density, energy-releaserate, and pressure at selected times. Frames 0.5 cm ( 0.1 cm extracted from a much larger computation. Top of domainis a symmetry plane, bottom, left and right extend into a much large computational domain. Flow is generally movingfrom right to left. In each frame, unreacted material is on the top and burned material on the bottom. Physical time isshown in ls on the left side. S: shock fronts; F: flame; D: detonation; HS: hot spot [23].


through enough O-burning regions, could becomea strong enough shock to form a detonation. Soagain, the question comes down to how to createan initial shock.

Therefore, by the '2005, we realized that theanswer to “unconfined” DDT in SNIa was thatsomething is happening in the small scales, not

the large scales of the explosion and combustionprocesses. These are the scales that thegrand-challenge computations were not resolving.The importance of this for both astrophysical andlarge-scale vapor-cloud explosions on earth isconfronted below.

Fig. 9. Frame from a movie of the deflagration phase ofan SNIa showing the ratio of the turbulent velocity atthe scale of the laminar flame width to the laminar flamespeed (spanning 10!2 to 10#2). The surface shown is ofthe flame, in which blue indicates regions where thetimescale for combustion is faster than the timescale forturbulence evaluated at the flame width. Red indicatesregions where the timescale for turbulence is faster thancombustion. This simulation was performed usingresources of the ALCF at Argonne National Laboratoryprovided by an INCITE allocation supported by theDOE [69]. Fig. 10. One-dimensional structure of a laminar ther-

monuclear flame propagating in a 50% mixture of C andO at 3 ( 107 g/cm3 (V.N. Gamezo, privatecommunication).

Fig. 11. Shock-flame interactions in thermonuclear reacting material. A Mach 1.5 shock propagates through a coldermaterial containing warmer thermonuclear O bubbles. Energy released by the compressed, reacting O can accelerate theshock, which can emerge as a detonation.


4. The Buncefield explosion

Now consider an accidental vapor cloud explo-sion that occurred on earth, one that released a“mere” 1020–1021 ergs of energy in about two sec-onds, resulted in about three billion dollars inproperty damage, caused untold personal damageand inconvenience, physically injured about 40people (amazingly, not that seriously), and, mirac-ulously, took no human lives. It also provideddata for study, speculation, and debate aboutlarge-scale vapor-cloud explosions. The explosionoccurred on December 11, 2005 at 6:01 am.

Buncefield was a fuel storage depot in Hert-fordshire, north of London. The British PipelineAgency, a consortium of BP Oil UK and ShellUK, pumped gasoline, diesel, and aero fuels downa pipelines from Stanlow (near Chester) Englandto Buncefield. From there, the fuel was distributedto other locations, including Heathrow and Gat-wick Airports and various locations the south eastof England.

Approximately 200 Mt (megatons, where “t” isone metric ton) of fuel can be stored at Buncefieldat three sites. At the time of the explosion, it held'82 Mt of gasoline, composed of '50% low-vola-tility hydrocarbons, and a mixture of C4 to C6

hydrocarbons. It is estimated that, in total,approximately 60 metric tons burned by the timethe flames extinguished, and therefore the totalenergy released by the end of all of the burningwas '1025 ergs. Figure 12 is a montage of photo-graphs of the depot, the explosion as seen fromsurrounding locations, and some pictures of thedamage caused [6].

Shortly after the explosion, the UK govern-ment initiated a Major Incident Investigation thatwas established by the Health and SafetyExecutive (HSE). Investigators from the HSEwere able to examine the result by personally vis-iting the area, noting the extensive damage, andreplaying the closed circuit TV recordings takenfrom cameras located around the depot. Thiswas unusual in that the investigators and forensicsteam were able to visit the site soon after theexplosion, and, most important be able to seethe result before there was significant externalinterference. A comprehensive and detailed sum-mary of the investigation was published in 2010[6] describing the event, the studies and proce-dures for the investigation, and preliminary stud-ies, and suggesting long-term follow up studies toanswer the most serious of the remaining issues.

4.1. A reconstruction of the events

The investigation found that a faulty switch ona storage tank was responsible for fuel overflow-ing from a storage tank. Because there was so lit-tle wind, the vapor could spread relatively evenlyand unimpeded by any winds that could otherwise

dilute and disperse it. It was estimated that '300metric tons of fuel had leaked from a tank over aperiod of '40 min. Cameras indicated that atabout that time, a 400-m pancake-shaped vaporcloud formed over the area.

After this vapor cloud had formed, a sparkfrom an electrical circuit in the pump houseignited the cloud. The resulting turbulent flamedeveloped quickly and ran through the depot.Two large explosions were heard as separateblasts at least as far as 10 miles away. (The secondexplosion measured 2.2 on the Richter scale.) Itwas not clear where or how the second explosionoriginated, though there is some evidence that it isassociated with a large fire ball. The ambiguitywith respect to the fire ball arose because the cam-eras were directed downwards. Pictures takenfrom a distance showed a large fire ball.

The result was devastation. There was evidenceof very strong pressures and forces that would beneeded to destroy buildings and parked vehicles.Trees and hedges were destroyed. A large smokecloud was released and interfered with the atmo-sphere for many kilometers. The explosions lastedseveral seconds, burning lasted almost a week, thedepot was destroyed, and smoke plumes reachedas high as 2.5 km. After the forensics team inves-tigated the site and examined the damage at closerange, the question arose: “Can the severity of thisexplosion be explained based on current knowl-edge?” This question was addressed in an in-depthstudy of the cause of the level of pressures thatcould cause specific damage, and estimate fromthis of the local conditions required to producethat damage, and an investigation into possiblecontrolling mechanisms in the explosion. Againwe heard the question: Was there DDT?

In October 2009, there were two explosionssimilar to Buncefield. One was due to an overflow-ing tank in San Juan and the other was due to afaulty valve in a gasoline tank in Jaipur [7]. TheSan Bruno explosion occurred due to faulty instal-lation of a NG pipeline in 2010 in San Francisco.

4.2. E!ects of obstructions and confinement

Obstructions and confinement in highly exo-thermic reactive flows can produce shock wavesin the system, and these shock waves can enhanceturbulence and create the background in whichDDT can occur. There have been many studiesof this in the literature, originally theoretical andexperimental (see review by Ciccarelli and Dorofeev[70]) and, within the last ten years, computational.

In order to illustrate the range of e!ects of con-finement, consider an example taken from a seriesof computations done by Gamezo, Ogawa, et al.(e.g., [71–74]) in which the geometry, fuel compo-sition, and method of ignition were systematicallyvaried. Figure 13 shows the channel geometry forone of the simplest cases in which a premixed


stoichiometric hydrogen-air gas at atmosphericconditions is confined in a channel with obstacles.(Note that the system evolution described here isvery similar for methane and other hydrocarbongases.) The channel is closed at the left end andopen at the right end. It is partially obstructedby rectangular obstacles designated as O1, O2,. . .On (or equivalently as O(1), O(2), . . .O(n)), hereevenly spaced along the entire channel length L.(Here the obstacle height d=4 corresponds to theblockage ratio 0.5.) A flame is ignited by placinga circular region of hot, burned material nearthe closed end of the channel.

Each frame in Fig. 14 shows the part of thecomputation in the region surrounding theleading shock and reaction fronts. Initially,

the laminar flame propagates at a velocity veryclose to the laminar flame velocity to unburnedmaterial. Hot reaction products expand and push

Fig. 12. Collection of photographs showing Buncefield before and after the explosion [6].

Fig. 13. Computational setup. Obstacles O(1), O(2),. . .(sometimes designated O1, O2, . . .) are evenly spacedalong the whole channel length. Walls and obstaclesurfaces are adiabatic no-slip reacting boundaries. Sizesare in centimeters. Initial flame radius is 0.5 cm. (Takenfrom [71].)


unreacted material towards the open end of thechannel. The flame front propagates in and withthe moving flow and becomes convoluted as theflow interacts with obstacles and develops struc-ture. The increasing surface area of the flameresults in faster energy release, and so the flowand flame speed increase. (Such a smooth ignitionprocess, as seen here, is often bypassed when theflame ignition mechanism is more energetic orirregular. In some cases, shocks can be reformedquickly at this stage.) The evolution of this systemfrom a time soon after ignition is controlled by aseries of fluid instabilities generally caused by theinteracting shocks and flame fronts. At a particu-lar time and location, one particular interactionmight appear to be the dominant mechanism forflame acceleration and energy release. In fact,many types of interactions occur simultaneouslythrough the domain. At some point it is onlypossible to point them out for consideration. Thiswill become more obvious in the description thatfollows.

Figure 14 shows that as the flame passes obsta-cles, it becomes wrinkled, more energy is released,and the flow accelerates. This initial acceleration isprimarily due to the RT instability, which ariseshere because the hot, less dense burned gas isaccelerating through the cold, heavy gas. Theresult of subsequent KH instabilities is alsoevident in the flow, although at some point, allof these interacting processes become di"cult toseparate.

The flow becomes sonic by 1.4 ms, just past O5,and shocks begin to appear ahead of the flame pastO7 at 1.85 ms. The shocks reflect from obstaclesand side walls and interact with the flame to triggerfluid instabilities. Once shocks have formed RMinstabilities become important as shocks interactwith flames. KH instabilities develop at the flamesurface when a jet of hot burned material passesthrough a narrow part of the channel and a shearlayer forms downstream of the obstacle. RT, RM,and KH instabilities, and flame-vortex interactionsin obstacle wakes are the mechanisms responsible

Fig. 14. Accelerating flame (left column), DDT, and quasi-detonation (two right columns) in 2D half-channel withobstacles computed for d=2 "2 cm, L "64 cm, S " 4 cm. Times in milliseconds are shown in frame corners. (Taken from[71].)


for increases in flame surface area, energy-releaserate, and, eventually, shock strength. The elevatedtemperature behind shocks also contributes to theincreased energy-release rate both because thelaminar flame speed increases and because shockspassing through the reaction zone release addi-tional energy. The average flame velocity graduallyincreases to 800 m/s by 2.1 ms, which is about 0.8of sound speed in the burned material, or 0.4 DCJ .

As the shock and the flame accelerate, the lead-ing edge of the flame remains '1 cm behind theleading shock, which di!racts at every obstacleand reflects from the bottom wall after each dif-fraction. The reflection type changes from regularto strong as the reflection point approaches thenext obstacle, and the resulting Mach stembecomes stronger after each di!raction. The tem-perature increases as each shock-heated regionforms when a Mach stem collides and reflects froman obstacle. At 2.1 ms, the reflection of the Machstem from O12 creates a region with temperaturesabove 830 K. Two hot spots in this region igniteto produce two small flame kernels. Then a deto-nation appears near one of the kernels and propa-gates through the unreacted material.

Detailed numerical studies of detonation initia-tion in hot spots [75,76] showed that spontaneousreaction waves propagating through temperaturegradients create both new flames and detonations.A detonation appears when the gradient profileallows the source of chemical energy to travel forsome time with the same speed as the shock wavegenerated by the energy release, so the shock canbe amplified to a detonation strength [77,78]. Ifthis does not happen, the hot-spot explosion pro-duces a decoupled shock and flame. The shockwave propagating into surrounding hot materialcan trigger additional hot-spot explosions thatcan eventually initiate a detonation, as we observehere.

Because of the way it is formed from the hotspot ignition, the newly formed detonation is over-driven (that is, traveling much faster than DCJ ). Itpropagates through the gap between the flame andthe obstacle, and so moves into the shock-com-pressed material ahead of the flame. As the detona-tion passes around the obstacle, the lower part ofthe front decouples to form a separated shockand a flame. The upper part of the front remainsessentially undisturbed and develops the trans-verse shock-wave structure that is characteristicof the type of detonation-cell structure found ina propagating gaseous detonations. As the detona-tion collides with the upper boundary and this cre-ates a strong reflected shock that triggers adetonation in both the shock-compressed layerbetween the leading shock and the decoupledflame, and the uncompressed material. The strongdetonation wave in the uncompressed materialdevelops detonation cells, collides with O13 at2.125 ms, and di!racts. As the di!raction weakens

the detonation wave, detonation cells grow andform an irregular two-level structure. The di!rac-tion on O14 completely decouples the shock andflame by 2.164 ms, and e!ectively kills the detona-tion. A new detonation is ignited in the shock-compressed material by the collision of the Machstem with O15 at 2.179 ms, but this detonation isunable to propagate through the very narrowgap between the obstacle and the flame. The lead-ing shock and the flame remain decoupled until theMach stem hits O16 and triggers a new detonationat 2.217 ms that spreads past the obstacle.

As the system evolves past what is shown in thefigure, the ignition-quenching sequence repeatscyclically. This is caused a quasi-detonation, a sit-uation in which “the shock and the flame are cou-pled at some times and locations where thereaction is triggered directly by shock compres-sion. At other times and locations, the shockand the flame decouple as the detonation di!ractsover the obstacles. Quasi-detonations areobserved when the width of unobstructed part ofthe channel is d s larger than a few of the charac-teristic detonation cell sizes L, where L is relatedto the spacing of the transverse detonations thatform at the detonation front). [In these cases,]The propagation velocity increases with d=L andreaches DCJ when the detonation propagationbecomes independent of di!raction e!ects,” asdefined in [71].

The final state of the reaction wave, whether itbecomes a detonation, quasi-detonation thatignites and dies repeatedly due to di!raction, ora complex of a leading shock and a turbulent flamefollowing behind, depends on the height of thechannel (the blockage ratio), the spacing of theobstacles, and the composition of the flow. Forexample, increasing height of the channel makesthe DDT transition more likely. This is becausethere is more chance that the detonation candevelop to the point where it can sustain itself afterthe interruption of its flow by the obstacle. Increas-ing the blockage ratio might lead to a shock-flamecomplex, but not to a detonation. Changing thefuel or fuel stoichiometry changes the location ofDDT. Similar scenarios to the one described herewere reported for channels with diameters up to'1 m.

4.3. How do we explain the damage at Buncefield?

Local and time-dependent variations in geom-etry and fuel composition throughout a regionsuch as the Buncefield depot are a much morelikely situation than a perfectly uniform fuel andobstacle distribution used for the computationshown in Fig. 14. Nonetheless, calculations suchas those shown in Fig. 14 teach us about the typesof reaction waves that can exist in explosionscenarios. Thus in a real explosion, such as theone that actually occurred at Buncefield, many


possible regimes of shock-flame complexes likelyexisted both in time and space through the fewseconds of the explosion itself.

As explained by Puttock [79] there wereobstructions at Buncefield in the form of pipe-work, buildings, and other obstacles, but the den-sity of these was relatively low. It was not believedthat these could generate high enough pressures toproduce the damage observed in the wreckage.The investigators then focused on adjacent roads,which were densely lined with trees and under-growth that had not been cleared for many years.Taking into account the placement of the foliageand evidence in particular parts of the depot ofthe passage of strong shock waves, it wasproposed that these trees might have had a majore!ect intensifying the conditions during theexplosion. Nonetheless, the question remained asto how to explain the level of damage, given ourcurrent understanding of explosions. Could therehave been detonation at some point, even in sucha partially confined space as a fuel depot? If DDToccurred, it would not have to lead to a fulldetonation, but could quickly evolve to a quasi-detonation or some form of shock-flame complex.

5. Hot spots, spontaneous waves, and transition todetonation

In the simulations shown that producedFig. 14, detonations arose behind shock reflec-tions. The conditions behind these reflected shocksled to hot spots (or ignition centers) in heated,compressed, unburned material. In other simula-tions, we saw hot spots arising in compressed,heated unburned materials in boundary layers,wakes intense vortices behind Mach stems, etc.(Many of these are described in [23]). The physicsof this process is interesting and worth describinghere, both because it was so controversial in thepast and because it is important in facilitatingtransitions between combustion states (on the dif-ferent portion of the Hugoniot on Fig. 1).

To explain these hot spots, we consider a typ-ical, isolated hot spot — that is, a gradient of reac-tivity – and show how this gradient undergoes atransition to a spontaneous wave (a phase wavein autoignition), which then can evolve into a det-onation or decay into a flame and a shock. Thisdiscussion follows from a large body of work onthe behavior of gradients of reactivity. It is basedon generalizations of a concept described by Zel-dovich et al. [80], which includes those presentedlater by Lee et al. [81]]. The specific analysis pre-sented here is given in much more detail in [23]and especially in [82].

We start by specifying the gradient as a distri-bution of the temperature and density (constantpressure) as shown in Fig. 15. Inside the hot spot,the temperature T " T %r&, where r is the position.

Such a nonuniform distribution could be createdin a highly compressible flow with pressure fluctu-ations, shocks, shears, turbulence, etc. In any suchcomplex, dynamic flow, we can expect gradientswith almost any spatial shape. In some cases, thegradient can be altered by interactions with weakshocks and pressure waves even as ignition occurs.Therefore the most general description of the tem-perature distribution could be something likeT " T %r; t&.

For simplicity, suppose that the chemicalinduction time, sc, be computed from the Frank-Kamenetskii approximation [83]. In addition, weuse the simplest one-step chemical model, of thetype used in the calculations shown above inFig. 14 and shown above in Eq. (5). Then thechemical induction time is

sc "CvT

AqqY

! "RTQ

! "exp

QRT

! "; %6&

where Cv " R=M%c! 1& is the specific heat at con-stant volume [82].

Now consider the case where the temperaturegradient is perturbed from the outside very slowlycompared to the reaction time, so that T " T %r&leads to the spatial distribution of induction timesc " sc%r&. The hot-spot explosion begins at thepoint of minimum sc, and then spreads at varyinglocal spontaneous velocity,

Dsp " !rsc

jrscj2%7&

in the direction of the gradient of the inductiontime. This is true when Dsp is the fastest speed ofthe propagation of the reaction. For deflagrationand detonation modes of burning shown inFig. 1, the propagation speed is limited fromabove by DCJ . The speed of the spontaneous wave,however, can only be limited from below by thelaminar flame speed, or, if detonations are pres-ent, by DCJ . For example, Dsp is infinite for a uni-form distribution of sc, the situation that describesinstantaneous burning at constant volume. (Amore detailed discussion of Eq. (7) is given in[84], starting on page 294).

Focus now on the case for which Dsp is finitebut greater than DCJ and assume that we have spe-cific temperature distribution,

T " max 1!

#######################x2 # %y=a&2

q

L

0

@

1

AT max; T min

0

@

1

A:

%8&

This is two-dimensional and elliptical, and hereL isthe characteristic scale of the temperature gradient.For a " 1, the distribution becomes cylindricallysymmetric. The temperature decreases outwardsuntil it reaches T min, and then stays constant. FromEqs. (7) and (8), we have the temperature gradient


0.1X (cm)

0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5

0.1

0.3

0.4

0.5

0.6

0.2

Temperature Density

K

Y (

cm

)

g/cc

X (cm)

2.776 ms 3.156 3.216

3.395 3.792 4.193

DSP

= DCJ

0.6 1.1 1.6 x 10-3Density(g/cm3)

(a)

(b)

Fig. 15. Initial conditions and evolution of a hot spot [6]. (a) Initial temperature and density distributions in an initiallyisobaric hot spot described by Eq. (12) with T max " 800 K, T min " 600 K, L " 1 cm, and a " 3=2. The contour Dsp " DCJ

shows the boundary of the spontaneous region. (b) Density distribution at di!erent times during the explosion of hotspot described by (a). Contour Dsp " DCJ shows the boundary of the spontaneous region.


rT " T max#######################x2 # %y=a&2

qL

x;ya2

$ %; %9&

and the absolute value of the spontaneous velocity

Dsp "@sc

@T

! "!1 1

jrT j

" @sc

@T

! "!1 T max

L

! " #########################x2 # %y=a2&2

x2 # %y=a&2

s

: %10&

The temperature distribution described by Eq. (8)for T max " 800 K, T min " 600 K, L " 1 cm, anda " 3=2 is based on the predetonation conditionfound in one of our earliest simulations. Figure15b uses the density distribution at selected timesduring the explosion to illustrate the evolution ofthe system. The superimposed contourDsp " DCJ is elongated in the y-direction. Insidethis contour, a spontaneous wave propagatingwith a speed Dsp > DCJ is possible. We call thisthe “spontaneous region.” The reaction in thisregion spreads from the point of minimum sc withthe speed predefined by Eq. (10).

At 2.776 ms from the beginning of the simula-tion, the spontaneous wave is inside the spontane-ous region. There is no shock wave present, andthe distribution of density is continuous. Theshape of the spontaneous wave is elongatedtowards the y-direction, and the overpressure ishigher in the x-direction where Dsp is lower. At3.156 ms, the spontaneous wave has almostreached the boundary of the spontaneous regionin the x-direction. In both frames, the shape ofthe spontaneous wave is elliptical (aspect ratio’ 1 : 3&.

By 3.216 ms, the spontaneous wave has partlyexited the spontaneous region. A weak shockwave has been formed and is present within thepart of the reaction wave located outside the spon-taneous region. There is no shock present in thepart of the wave that is inside. The next twoframes, 3.395 ms and 3.792 ms, show the furtherevolution of the reaction wave with the shockwave developing outside the spontaneous regionand growing in strength continuously. The bound-ary separating the parts of the wave with andwithout the shock follows the contour of constantDsp " DCJ . In the last frame, the wave exited thespontaneous region completely and developed ashock discontinuity in all locations.

Figure 16 shows the density and pressure pro-files at di!erent times along three lines, all with theorigin at x " y " 0, and then going along thex-axis, the y-axis, and the 45! diagonal direction.Along the x-direction, (also see Fig. 15a) profile2 in Fig. 16 shows the smooth structure of thespontaneous wave without a shock. Material isbeing continuously compressed, and it reacts andreaches maximum compression when the fuel iscompletely reacted. Then the material begins to

expand and decelerate. The shock emergesbetween the times corresponding to profiles 3and 4. Profile 4 shows the structure of the wavethat is intermediate between the spontaneouswave and a detonation. The material is first com-pressed continuously, and only then passesthrough the shock. Profile 6 shows the developeddetonation wave.

The same process takes place in the diagonaland y-directions, but with di!erent timing. The lat-est profile, 6, corresponds to the time in betweenthe fourth and fifth frames in Fig. 15b. At thistime, the wave in the y-direction (Fig. 16c) hasnot exited yet from the spontaneous region andhas not yet developed a shock discontinuity. Thewave in the diagonal direction shows that a shockhas formed.

The spontaneous reaction waves that weobserved are supersonic reaction waves in whichburning material is compressed and acceleratedin the direction of the wave propagation. Thespontaneous waves that lead to detonations ini-tially propagate with the velocity Dsp > DCJ .When Dsp approaches DCJ , a shock front developsinside the reaction zone. After that, the internalstructure of a spontaneous wave transformssmoothly to that of a detonation through the ser-ies of intermediate regimes with increasingstrength of the shock which eventually movesahead of the reaction zone.

Many past analyses (see, e.g., [78], [[79], [81,81–87], and many others referenced, e.g. in [23]) showthat there are also critical conditions for the gradi-ent of sc and the size of the hot spot. If the distri-bution of sc does not meet these conditions, thedetonation does not develop, and the spontaneouswave degenerates into a shock and a residualflame. We routinely observe these failed DDT phe-nomena in the simulations. Our multidimensionalsimulations often show transitions that start andthen fail. The spontaneous wave velocity com-puted for the hot spot on the right side is below0:5DCJ everywhere. In [75], we described a hot spotthat produced a shock and a flame behind itbecause the spontaneous wave was too weak tobecome a detonation. The shock it created, how-ever, a!ected the second hot spot, which met thecritical conditions for successful transition todetonation.

6. Tying it together

How does a supernova explode? Can the amountof damage caused by the Buncefield explosions beexplained, given our current knowledge of combus-tion and vapor-cloud explosions? Both of thesequestions have an underpinning issue related tothe strongest form of combustion, a detonation.For a SNIa, the question is how can DDT occurwithout any sort of physical confinement? For


Thomas Epalle

Thomas Epalle

Thomas Epalle

Thomas Epalle

the Buncefield explosion, the question is similar,how does DDT occur with only limited confine-ment? For the SNIa, the occurrence of DDT isthe simplest (and probably the only reasonable)way to fit the data. At Buncefield, there wereobstacles in the flow and partial confinement fromthe ground. Some form of detonation wave, even acomplex transient wave such as a shock-flame com-plex or a quasi-detonation, existed for even a limitedperiod of time, would help explain observations.

Now we discuss and relate two mechanisms ofspontaneous DDT, that is, DDT that can arisenaturally in any general turbulent system thatcontains a flame and in which there are no physi-cal obstructions, such as walls or obstacles. Thefirst is the reaction-enhanced acoustic wave orshock shown in Fig. 11, which is a mechanismseemingly peculiar to a multistage flame in whichthe two stages are well separated. Here the firststage is fast and prepares the system for the slowersecond stage. For the conditions shown in the fig-ure, the first stage, the C burning, is essentiallycomplete and prepares the background for the Oburning, which has a much larger flame thicknessbut is just as energetic. The key to DDT is the wayin which the shock wave is enhanced as it movesthrough the reacting O. For this mechanism towork, it is necessary to have a long enoughenergy-release stage in the flame and manyrepeated shock-flame interactions. Although theexample given is for thermonuclear C–O burningin a SNIa, a similar situation could be createdand studied in a laboratory by finding an analogfuel.

The second mechanism is more general andcan apply to any system containing a flame andstrong enough turbulence. Let us assume thatthe turbulence is subsonic, compressible fluid tur-bulence, there are no significant magnetic fieldspresent, and the medium is essentially homoge-neously mixed. Then we can ask the questions:How does the turbulence a!ect the flame? How doesthe flame a!ect the turbulence? Or more specifi-cally, can a reasonable level of turbulence gener-ate a shock strong enough to cause a detonation?

These were the questions we began to exploreabout five years ago, and these studies resulted ina body of work on flame-turbulence interactions[88–92]. This work was based on three-dimen-sional numerical simulations that were enabledby the availability of large, fast, computers, high-order monotone methods, and ILES. The primaryvariable explored was the turbulence intensity,although there was some test of the e!ects ofchanging the type of chemical reaction. The simu-lations most relevant to the questions posed aboutSNIa considered hydrogen-like flames propagat-ing in a turbulent background media that mim-icked “free space” to the greatest extent possible[92]. (That is, homogeneous, isotropic backgroundturbulence was generated on a doubly periodiccomputational grid, with inflow-outflow boundaryconditions, and containing fluid that resembled astoichiometric mixture of hydrogen and air atatmospheric temperature and pressure.) The tur-bulence was of the form that would be created bydriving the system on a much larger scale thanthe small system simulated (say, the scale of the

Fig. 16. Density profiles (left) and pressure profiles (right) along three di!erent lines passing through the center of thehot spot at various times during the hot spot explosion [6]. (a) along the X-axis; (b) along the diagonal, and (c) along theY-axis. Times are: (1) 0 s; (2) 2.776 ls; (3) 3.016 ls; (4) 3.213 ls; (5) 3.395 ls; (6) 3.587 ls since the beginning of thesimulation.


star), but which had decayed through an inertialrange to some scale that is smaller than a flamethickness. Then we ask: What happens to a flameas it propagates in this background?

The answer to this question is shown in thesequence of diagrams in Fig. 17 [93,92]. As thesystem evolves, the turbulence intensifies as aresult of interaction with the flame. The averageflamelet in the flame brush becomes more andmore distorted from its laminar one-dimensionalstate. As the flame and turbulence self-intensified,energy release increased faster and faster through-out the domain. At some point, it is released sofast that it can no longer be acoustically equili-brated. At this point, a shock forms. This intensi-fication of local energy release can occur when theflame speed reaches the velocity SCJ , the Chap-man–Jouguet flame speed marked on Fig. 1. Thisspeed can be related to the acoustic speed, cs, inthe burned and unburned gas by

S ' cs

a" SCJ ; where a "

qf

qp; %11&

were subscripts f and p refer to fuel and product,respectively.

From the time at which an initial shock is gen-erated somewhere in the flame brush, there are anumber of ways in which a supersonic combustionwave can emerge. These range from direct initia-tion to formation of hot spots, or, for the thermo-nuclear flame, the passage of a shock throughenough of a turbulent flame so that the shock isrepeatedly intensified. As shown in [92], this abil-ity of intensely turbulent flame systems to undergoa transition to detonation has significant conse-quences for standard combustion diagrams thatare used to di!erentiate the di!erent regimes ofturbulent flames.

Now we can ask how this relates to theBuncefield explosion. In Buncefield, there wereseveral strong indications that there was a tran-sition to detonation at some point in the event.These include debeaded tires in the vicinity, andthis was followed by separate tests that showedthe amount of pressure or force it took todebead tires [94]. Another indication of DDTwere structures observed on vehicles that looklike the result of a detonation propagatingthrough a medium with a gradient of fuel ([95], [96]). And another indication is the directionin which objects were pointed and hurled, andthat could only be reproduced by very strongpressure gradients that would only arise fromDDT or a detonation.

In addition to these indicators, there isevidence from video cameras indicating that theturbulent flame accelerated as it passed along treesand bushes lining roads adjacent to the depot. InBuncefield in December, hard, deciduous treeswould be essentially bare. Then one scenarios

proposed is that the branches behaved as obsta-cles in the flow, or as grates that intensified theturbulence on small scales. Obstacles, as we haveseen from the description of Fig. 14, can generateturbulence in the flow that accelerates the flameand produces shocks waves. Another situationto consider is that the trees acted as a grate thatintensified the turbulence on small scales, andthe flame propagated into this and furtherincreased the intensity of the turbulence. Then,according to the theory and simulations shownabove in Fig. 17, when turbulence is intenseenough, especially on the scales that encompassthe local flamelet thickness, it is possible to formshocks, and these are precursors to DDT.

The scenarios in which trees and branchesaccelerate turbulent flames was tested by ShellResearch at Spadeadam, a large test facility innorthern England, and the results were describedat conferences [98,99] and in the popular press[97]. In these experiments, piles of sticks wereimmersed in an environment of propane gas,and then ignited. When the sticks were made ofsoft pine, there was no detonation. When theywere made of hardwood, more like the trees inBuncefield, there was a distinct DDT.

Evidence similar to what was seen in Bunce-field has also been seen in other vapor-couldexplosions. Such indicators, including anoma-lous flame acceleration to high speeds approach-ing sonic (i.e., greater than '600 m/s) anddirections in which objects were thrown, appearin, for example, the events at Port Hudson andJaipur ([7,100]). If DDT occurred at some loca-tion or locations at Buncefield, the questionthen arises as to how long this detonation per-sisted, or whether it decayed quickly to aquasi-detonation or to some sort of strongshock-flame complex, or whether it died as thefuel concentration decreased and the shock andflame separated. If there were a detonation,the question would be whether it persisted longenough to have the type of sustained impact ondepot needed to explain the damage seen. All ofthis depends, of course, on details of the fuelcomposition as function of time during theexplosion.

7. Issues and diversions

Several additional issues related to explosionswill be mentioned now. These concern propertiesof the fluid, the chemistry model, and the energyrelease. They relate to finding models and repre-sentations that would allow us to interpret obser-vations and measurements and compute thebehavior of explosions. The discussion is in noway complete, rather it is selective and reflectscurrent concerns.


7.1. Nonequilibrium, Non-Kolmogorov turbulence

The flows that occur leading up to and duringexplosions are turbulent. The calculations shownin Fig. 14 is a good example. This is not the kindof equilibrium, isotropic, Kolmogorov turbulencedescribed in texts, but one that may begin ratherbenignly as even a laminar flow, but then evolvesinto one driven by shocks and shock interactions.

The explosion turbulence is not just driven onsome large scale, but it is driven on all scales ofthe flow simultaneously. The stage preceding tran-sition to detonation is probably best described asa nonequilibrium broadband turbulence.

In many numerical simulations, such as thatshown in Fig. 14, we observed that general trendswith regard to when and where DDT occursagree reasonably well with results of shock-tube

Fig. 17. Structure of the turbulent flame and the corresponding pressure distribution during DDT in a stoichiometricmethane-air mixture. (a) (e): isovolume of the fuel mass fraction. (f) (j): volume rendering of pressure normalized by theinitial pressure in the domain (note a di!erent colormap range in each panel). Horizontal axis scale gives the distancefrom the right boundary of the domain in cm. The time from the start of the simulation is indicated in each panel in unitsof the large-scale turbulent eddy turnover time. Note that panels (b)–(f), (c)–(g), (d)–(h), (e)–(i) correspond to the sametime instants. (Based on material first presented in [88].)


experiments (summarized in [23]). There was,however, an additional nagging concern becausewe did not expect the results to be in such goodquantitative agreement for simulations that wereonly two-dimensional, and not three-dimensional.Limited tests on three-dimensional computationsshowed that there was general agreement betweentwo-dimensional and three-dimensional simula-tions for a significant number of geometries. Oneexplanation for this could be the unusual type ofturbulence generated in this system by repeatedshock-flame interactions. Here we examine thisin slightly more detail.

The primary origin of the turbulence in the ini-tially laminar flame was not the relatively slow,natural flame instabilities usually studied for lam-inar flames, but Richtmyer–Meshkov instabilitiesinduced by repeated shock-flame interactions,other shock interactions and instabilities that gen-erate vorticity at many scales simultaneously. Asan example, consider just the RM e!ects of ashock-flame interaction, which is the most obvi-ous in Fig. 14. The first interaction has a relativelymild e!ect on the increase in the burning rate[101], but it sets the stage for the e!ects of subse-quent interactions. After this, the already convo-luted flame is shocked repeatedly. Because ofthis frequent, continual shocking, vorticity is gen-erated on a range of scales simultaneously, fromthe scale of the system to the smallest scale ofthe flame thickness, in both the two-dimensionaland three-dimensional calculations. This is inde-pendent of and can be much more e!ective in pop-ulating di!erent scales than the energy cascade.Another source of turbulence on small scales isthe Kelvin–Helmholtz instability, but it alsoappears to be less important than the RM instabil-ities for the flows under consideration.

The behavior of the RM instability in two andthree dimensions has been studied by manyauthors in a number of di!erent scientific commu-nities (see, e.g., [102]). The results most relevant tothese arguments here are the theoretical analysesand numerical simulations of the RM instabilityfor the Euler problem by Li and Zhang [103].These authors compared the two- and three-dimensional growth rates as a function of timefor reflected shocks and rarefactions. The resultsrelated to both the linear and nonlinear phase ofthe interaction. First, as long as the initial ampli-tude and wavelength of the perturbation are thesame, the growth rate of the instability in twoand three dimensions are essentially the same inthe linear regime. Then, in the nonlinear regime,the three-dimensional growth rates are about20–25% larger and faster, respectively, than for2D. This is consistent with the type of growth insurface area and energy release computed for thetwo- and three-dimensional flames [102].

Thus the fact that computations in two dimen-sions show agreement in trends and behavior with

those in three dimensions and with experimentsappears to be consistent with the RM instabilitybeing the major mechanism for generating the tur-bulent flame. This may be because, as arguedabove, at least the qualitative behavior of a systemdominated by RM should be similar in two andthree dimensions. Flow dominated by repeatedshock-flame interactions occurring on all scalescannot be expected to have the isotropic, homoge-neous, equilibrium spectrum characteristic of thetype of turbulence that has been the topic of intensestudy in the past thirty years. In fact, this kind offlow perhaps cannot even be technically called tur-bulence. We refer to it as nonequilibrium turbulence,or non-Kolmogorov turbulence. The properties ofthis nonequilibrium, essentially broadband turbu-lence and how it decays to the more standard equi-librium state is a topic that requires considerablymore work and is ripe for direct numerical simula-tion and theoretical analyses.

7.2. Intermittency – an issue with turbulent flows

“Intermittency” is an important concept influid turbulence. It is almost as important as vor-ticity, and probably of more practical importanceto us than backscatter, which relates to questionsof how perturbations on small scales a!ects thelarge scales. In common language, a phenomenonis intermittent if it occurs at irregular intervals,that is, if it is not continuous or steady. It has atechnical definition in fluid theory that is basedon ways of describing the fluctuations of fluidquantities and the e!ects of these fluctuations onthe flow (see, for example, definitions given in[91]). Properties of intermittency can be observedby either structural (morphological) and statisticalapproaches.

A structural or morphological approachdescribes the dynamics of vortices, or flamelets,or other physical structures in the flow, and howthese interact and change in time and space.Images of the flow can help to identify structures,such as vortices, cusps, or “hot spots” in the flow.Flow structures show how an event is happeningand might give some hint of mechanisms andhow to change the dynamics. Consider Fig. 18,which shows the computed vorticity in two sys-tems of compressible turbulent flow (based onanalysis presented in [104,91] and used in [105]).These frames, taken from two computer-gener-ated movies made from three-dimensional vortic-ity contours, show that large gradients andvelocity di!erences appear and evolve intermit-tently and spontaneously in the flow at seeminglyrandom locations. As these systems evolve in time,the vortical structures appear, dissipate, anddisappear. In fact, most of the dissipation in thesystem is concentrated around these tubes anddo not spread uniformly throughout space. Thisexpresses the intermittency of turbulence.


A statistical approach deals with averaged andcorrelated properties of the flow, often in terms ofprobability density functions or pdfs of quantitiesdescribing the flows. Thus we consider scalarssuch as temperature or density, vectors such asvorticity, and even higher tensor moments. Apdf tells us the relative likelihood for a flow vari-able to take on a given value. They show us thatfor flows with strong intermittency, i.e., strong

fluctuations, there is a higher probability of eventsoccurring than would be predicted if, for example,vorticity were distributed uniformly in space andtime. In turbulence terminology, there is a higherprobability for event that are in the tails of thepdf’s.

There are studies of intermittency in cold flowand through a premixed flame brush in relativelyintense, but subsonic compressible turbulence

Fig. 18. Computed vorticity in two three-dimensional simulations showing localized structure of vorticity concentratedin tubes. Left: from a 3D simulation of homogeneous isotropic compressible turbulence in a triply periodic domain [81].Right: from a 3D simulation of turbulence interacting with a flame brush [90].

Fig. 19. Conditional pdfs of (a) enstrophy X, (b) energy dissipation rate !, scalar dissipation rate v, and the ratio of theturbulent flame width to the laminar flame width dt=dl conditioned on Y [91]. The insets show pdfs of log variables, anddashed black lines show Gaussian distributions.


[91]. Fig. 19, taken from this paper, shows boththe pdf and the deviation from uniformity (Gaus-sianity) for this problem. The larger figures showconditioned pdfs of turbulence quantities such asthe enstrophy, energy dissipation rate, scalar dissi-pation rate, and the ratio of the turbulent flamewidth to the laminar flame width conditioned onthe fraction of fuel Y. The insets show pdfs oflog variables, which can be compared to theGaussian distributions (dashed black lines). Aconclusion is that as the Reynolds numberincreases, fluctuations increase and intermittencybecomes stronger: there is more of a chance of alarge fluctuation, or rare event are simply less rare.The introduction of energy release into the flowstrengthens intermittency, that is, increases thestrength of the fluctuations, something that weall know from our experience.

An important point for explosions is thatlarge fluctuations in fluid properties can lead tothe formation of hot spots, and these in turn,can potentially develop into a detonation. If ahot spot fails to detonate, it can still produce ashock that can increase the chance that the nexthot spot will detonate, as is illustrated in Fig. 8.We have seen above that turbulence can bedirectly responsible for a initiation of a detona-tion [92].

7.3. Including chemical processes

Numerical solutions of reacting flow problemsproceed by mathematically separating the solutionof the fluid dynamics from the solution of thechemical kinetics and energy release, and thencombining the results of each. This process is car-ried out for each fluid element for a sequence ofshort periods of time, called a global time step.The solutions will be stable, accurate enough,and even converge in time as long as these timesteps are small enough. This type of time-stepsplitting, in which the chemical reactions andenergy release are computed separately from thefluid dynamics (or other physical processes, suchas various types of di!usion e!ects) and then cou-pled, has been extremely successful, although ithas almost obvious limitations and inaccuraciesassociated with any application. Nonetheless,time-step splitting has been a vast improvementover the global implicit methods originally used,and it has enabled the computation of a broadclass of problems. (See [24] for an in-depthdiscussion.)

Early in this paper, we summarized some of theimportant advances in computational fluiddynamics that enabled solutions of explosionproblems. There are separate sets of numericalproblems with solving the chemical parts of theproblem numerically, and there is a substantial lit-erature on solving sets of coupled ordinary di!er-ential equations. Currently, there are very

adequate methods of integrating ODEs bothseparately and as part of the reacting-flowproblem. One observation here is that part ofthe problem, the “sti!ness” of these equations isa result of unphysical separation of processes inthe numerical algorithm.7 Perhaps in the futurethese will be addressed from the point of view ofthe physics and not the mathematical construct.

The overriding problem for including chemicalreactions and energy release is finding an adequatemodel or representation of the process, one that isappropriate to include in a fluid dynamics calcula-tion. Two issues that arise from this concern willbe addressed briefly here. The first is the seeminglyexponential growth in the number of chemicalreaction rates required to represent the evolutionfrom fuel and oxidizer to product of reactionand the e!ect of this on combustion modelinge!orts. The second issue is concerned with findinga way to represent this process when normalassumptions break down and we do not, andmay never even know all of the steps in the pro-cess. There is a third issue that arises as a resultof the first two, that of finding the right level ofchemical model consistent with a large-scale fluidcomputation.

The first issue is a bit of a mystery wrappedin a conundrum. Our concern is e"ciently burningCn fuels, where n is a relatively high number. Ourapproach has been to try to model combustioncorrectly for the lower-n fuels, such as those basedon methane, ethane, or propane, and use thesemechanisms as ingredient for models of go evenbeyond n " 10. But here is the issue: A typicalhigh-temperature combustion reaction mecha-nism for methane and ethane is represented byabout 200 elementary chemical reactions stepsand 30 intermediate species, propane by anadditional 100 reactions and 15 additional species,and normal butane by 200 more additional reac-tions and 25 additional species. Higher-hydrocarbonfuels sometime involve hundreds of thousands ofreaction rates and over 100 intermediate species.

Motivations for developing this approachinclude the needs to enhance combustion e"-ciency even a few percent and to control pollutionformation. We hope that if we can understand andtrack the pathways of reaction, that is, get into theprocess deeply enough, we might find ways to dis-rupt or adjust it appropriately. This approach hashad success, if the resources that have gone intothe approach and the myriad positive resultsreported are indicators. A question asked now iswhether this approach is a sensible solution to

7 This actually might originate in the basic formula-tions of the chemical rate constants. For example, somereaction rates imply that we need chemical time stepsthat could become faster than the speed of light. Ofcourse this cannot be physically correct.


the problem, as the uncertainties grow with theaddition of more and more “fit” variables untilthe soup of constants could be made to fit any-thing. Are we approaching the e"ciency and pol-lution problems in the right way? Are there otherways? Can we re-express in some way that wouldlead to more useable models?

Besides making coupled chemistry and fluidcomputations prohibitively expensive, there areother reasons to question the validity of manyof the complex, detailed reaction mechanismsfor simulating explosion conditions. The bestway to explain this is to summarize recent workof Taylor et al. [106,107] and their attempts to

use computations of the cellular detonationstructure of a gas to compare reduced and fullchemical models against each other and measure-ments. The numerical ingredients required tocompute detonation cells are a monotonemethod for solving the compressible flow equa-tions, such as any of the algorithms describedin Section 2.1 chemical reaction mechanism. Inmost cases tested, viscous and di!usion e!ectswere second order at best. The usual objectiveis to reproduce the cellular structure and size towith a factor of two using a two-dimensionalunsteady monotone fluid algorithm and somemodel of energy release.

Fig. 20. Reactive flow under extreme conditions in the reaction zone of a propagating multidimensional detonation.Computations are taken from [106,107]. (a) Temperature map for on instant showing multiple transverse waves, and ablow up of a small portion of the computation. (b) and (c) Profiles of temperature and pressure at two locationssurrounding the detonation front.


The initial results by Taylor et al. were con-founding. First, they were able to reproduce theearlier results [108] (which agreed with subsequentresults of a number of others) to show that adetailed hydrogen-air chemical model can be used,in conjunction with a good fluid algorithm, tocompute a structure and size for detonation cellsthat is very close to those experimentally observedat low pressures (e.g., '0.1 atm). As the initialbackground pressure was increased, however, thecomputations were not able to reproduce experi-mental results either quantitatively or qualitatively.For hydrogen-air initially at atmospheric condi-tions, and for quite a series of supposedly moreaccurate reaction mechanisms, they found cellsizes ranging from factors of 5 to 10 di!erent fromthe observations, and with a complicated “multi-level hierarchal structure” reminiscent of what isseen in marginal detonations. For comparison,there are the experimental results of Bull et al.[109] which were recently reproduced in Poitier(Yves Sarrazin, private communication), all ofwhich showed fairly regular cells of easily deter-mined size.

A closer look at the conditions behind the lead-ing shocks and Mach stems of a computed deto-nation with cellular structure is illustrated inFig. 20. This calculation requires the chemicalreaction mechanism to extrapolate and computeenergy release at both extremely high pressures(>40 atm) and temperatures (>2000 K), whichare parameter ranges in which the mechanismswere never intended to perform. An analysis ofthe relevant scales in the computations showedthe computed chemical autoignition times in theseregimes were on the same order as vibrationalrelaxation times, so that there was definitely some-thing wrong with the chemical mechanism if itwere to be used for computing high-pressure,high-temperature conditions, of just the sortoccurring in explosions. In fact, the chemical reac-tions are not in equilibrium during an explosion,and there currently is no simple way of accuratelyrepresenting what is happening. Perhaps in theinitial and final phases, the processes can be repre-sented as thermal equilibrium of vibrational androtational states, but not at, approaching, or evensoon after an explosion.

Figure 21 is a scatter plot of the temperaturesand pressures of unburned material in a simula-tion of hydrogen-air detonations at standardatmospheric conditions [107] compared to a data-base of high-pressure, high-temperature ignitiondelay measurements in hydrogen mixtures [110].The absence of any significant overlap betweenthe conditions in the simulations (shown in red)and experimental data from shock tubes (black)and rapid compression machines (green) highlighttwo points about chemical kinetics models anddetonations. First, there is a dearth of experimen-tal data that can be used that is relevant to

detonations. Second, simulating detonations withcurrent chemical models requires extrapolationfar outside the range of validity of these models.This problem not only with temperatures andpressure, but also with mixture composition.Due to experimental limitations, much of the datain Fig. 21 was obtained in highly dilute mixtureswhich are far from characteristic of realistic deto-nations in conditions, yet it is assumed it can beextrapolated to these conditions.

How can we compute and predict the chemicallyreacting flows in explosions? Any attempts toinclude all of the excited vibrational and rotationalstates of the molecules would turn even the rela-tively simple hydrogen-oxygen mechanism into ahorror of just as many unknowns and uncertaintiesthat there are in current equilibrium hydrocarbonmechanisms. Another approach could be to modifythe time and shape of an energy release function in achemical reaction mechanism to reflect the creationof nonequilibrium vibrational or rotational statesby passing shocks. Another approach would be toisolate the culprit molecules and more specificexcited states that are causing the disagreement,and adjust the chemical mechanism to reflect these.All of this is speculation and leaves room for clever,further research.

8. Concluding remarks

How does a Type Ia supernova explode? Howcan we account for the damage done by the Bunce-field explosion? And there is third, more specificbut related question that was not directlyaddressed here: Given a large enough volume of aflammable mixture of natural gas and air, such asmay exist in a coal mine or vapor-cloud explosion,can a weak spark ignition develop into a detonation?Questions such as these drove investigations thatreally asked: What is the physical origin of DDT?These questions were addressed with observa-tions, measurements, and numerical simulationsand analysis that allowed us to reach our currentstate of (as yet incomplete) understanding.

The viewpoint we were forced to adopt is thatbackground turbulence in unreacted or partiallyreacted materials sets up conditions in which atransition to detonation may occur. The originof the actual transition, however, arises from asequence of interactions on scales that are small,and in some cases orders of magnitude smallerthan a laminar flame width of the fuel in the back-ground material. This conclusion holds for theorigins of DDT in confined, partially confined,and unconfined scenarios.

There are multiple mechanisms that can pro-duce detonations in a reactive gas system. Theseinclude but are probably not limited to: hot spotscreated through a gradient in reactivity, laminaror turbulent flame acceleration to the point at


Thomas Epalle

which the flame or flame brush reach the funda-mentally unstable Chapman–Jouguet flame veloc-ity, and direct ignition by shock acceleration in areactive material. In situations involving highlyenergetic and turbulent gases, when there aremany shocks and intense turbulence and thetemperatures and pressures are high, it is hardto separate exactly what chain of interactionsactually cause the transition. Often, however, wecan separate and illustrate these mechanisms insimulations, as we have done in the materialpresented above.

Given the plethora of mechanisms than canoccur and how they can a!ect one another, wehave reached the point where we are seriously ask-ing: How can a supernova NOT explode? and Arevapor cloud explosions inevitable, or can we learnto stop or mitigate them? Although not discussedin this paper, we could have also asked similarquestions about the confined explosions in coalmines and other industrial facilities.

The contributions to the study of explosionsmade by numerical simulations have been enabledby enormous advances in computational capabili-ties, both in terms of algorithms and computertime and memory available. In this paper, we triedto present a brief history of three algorithmicinventions or discoveries that proved to be justas important than the availability of bigger andfaster computers and their peripheral equipment.And sometimes, because they are enablers, theyare even more important. The three algorithmsor inventions discussed were flux limiting andthe development of monotone methods, implicit

large-eddy simulation, and flexible and fast meth-ods for adaptive mesh refinement. These were allcritical elements in all of the turbulent reactingflow computations described for supernova explo-sions and deflagration-to-detonation transition inchannels.

The study of vapor-cloud explosions, Type Iasupernova, coal mines, pipelines, flying manholecovers, and other types of explosions havebrought up several questions about the nature offluid turbulence, turbulent combustion, and chem-ical reaction pathways. The turbulence in thetypes of high-speed flows we have studied is non-equilibrium and non-Kolmogorov. They containmany shocks and fluid instabilities that drive tur-bulence at many scales simultaneously. What isthe structure of this e!ectively broad-band turbu-lence, and how does it di!er from standard, clas-sical, Kolmogorov turbulence?

Turbulence appears as fluctuations. What hap-pens if these fluctuations become large enough todominate the background? What does intermit-tency really mean for us? What can we reliablycompute about a turbulent (stochastic) flow? Thisbecomes important when there are safety issuesinvolved. How do we deal with high-temperaturehigh-pressure chemical reactions and energyrelease? What is the correct level of coupling thechemistry to the fluid dynamics? All of these ques-tions are waiting for us to address in the future.

Acknowledgments

I an grateful to Thierry Poinsot and Suk HoChung for the invitation to present the HottelLecture and to prepare this paper. I gratefullyacknowledge Jay Boris, J. Craig Wheeler, FormanWilliams, Derek Bradley, Martin Sichel, VadimGamezo for many years of close collaboration,(generally) good advice, and (sometimes) painfulcriticism. For many years of friendship and col-laboration, I would like to thank David Mott,Geraint Thomas, Takanobu Ogawa, David Kess-ler, and R. Karl Zipf. For their collaborations anddeep insights into problems presented in thispaper, I thank Alexei Poludnenko, Peter Ham-lington, Melissa Green, Aaron Jackson, BrianTaylor, and Ryan Houim. Bram van Leer kindlyallowed me to use his famous graphic of the his-tory of CFD, which after some manipulation,now appears as Fig. 2 in this paper. Brian Taylorand Carsten Olm provided the composite Fig. 21.Paul Ronney and J. Craig Wheeler providedalmost infinite useful comments and suggestionsfor this paper. I thank Mike Johnson for his helpinterpreting the Buncefield explosion. I thank K.Kailasanath, Fernando Grinstein, and AlexeiKhokhlov whose collaborations were instrumen-tal in starting much of this work. Finally, I thankDaniel Oran and Elizabeth Boris, for their help

Fig. 21. Scatter plots of temperature and pressure incomputational grid cells containing only unburnedmaterial, for computations such as that shown inFig. 18 [106]. Overlap of experimental data [107]. Blackrectangle represents the ambient state at 300 K and1 atm. Dashed line indicates post-incident-shock statesas shock speed is varied; circle indicates post-incident-shock state at DCJ . Solid lines show post-transverse-shock states for initial conditions indicated by the opensquares.


Thomas Epalle

and support in providing the wonderful environ-ment that allowed me to write this paper. Specialthanks always to Johann Bark for his insightfulcomments and unfailing companionship.

The research presented in the paper was pri-marily done in the Laboratory for ComputationalPhysics and Fluid Dynamics (LCP) at the NavalResearch Laboratory (NRL), where many yearsof continuous support was provided NRLthrough the O"ce of Naval Research. Parts ofthe work presented was were supported byAFOSR grant F1ATA09114G005 and a priorNASA grant NRA-02-OSS-01-ATP) through theAstrophysical Theory Program. Many of the com-putations presented here were performed on facil-ities provided by LCP or the DoD HighPerformance Computing Modernization Pro-gram. Preparation of this paper was supportedby the University of Maryland through MintaMartin Endowment Funds in the Department ofAerospace Engineering, and through the GlennL. Martin Institute Chaired Professorship at theA. James Clark School of Engineering.

References

[1] E.S. Oran, F.A. Williams, eds., Philos. Trans. Roy.Soc. A (2012) 370.

[2] E.S. Oran, F.A. Williams, Philos. Trans. Roy. Soc.A 370 (2012) 534–543.

[3] J.C. Wheeler, Philos. Trans. Roy. Soc. A 370 (2012)774–799.

[4] E.S. Oran, Proc. Combust. Inst. 30 (2005) 1823–1840.

[5] M. Kamionkowski, K. Freese, Phys. Rev. Lett. 69(1992) 2743–2746.

[6] Buncefield Major Incident Investigation Board(2008). The Buncefield incident 11 December2005. ISBN 978 0 7176 6270 8, <http://www.buncefieldinvestigation.gov.uk>.

[7] D.M. Johnson, Loss Prevent. Bull. 229 (2013) 11–18.

[8] R.A. Gates, R.L. Phillips, J.E. Urosek, et al., FatalUnderground Coal Mine Explosion January 2,2006, Sago Mine, West Virginia. US Departmentof Labor, Mine Safety and Health Administration(MSHA), ID No. 46-08791 (2006).

[9] P. Barbier, Urban Growth Analysis Within a HighTechnological Risk Area: Case of AZF FactoryExplosion in Toulouse (France), Ecole Nationaledes Sciences Geographiques (2003).

[10] M. Berthelot, P. Vieille, Comptes Rendus Acad.Sci. Paris 93 (1881) 18–22.

[11] E. Mallard, H.L. LeChatelier, Comptes RendusAcad. Sci. Paris 93 (1881) 145–148.

[12] D.L. Chapman, Philos. Mag. 47 (1899) 90–104.[13] E. Jouguet, On the propagation of chemical reac-

tions in gases, J. Mathematique Pures et Appliquees6 (1905) 347–425.

[14] Y.B. Zel’dovich, J. Exp. Theor. Phys. (USSR) 10(1940) 542–568.

[15] J. von Neumann, Theory of Detonation Waves,Prog. Rept. No. 238, O"ce of Scientific Research

and Development (OSRD), Rept. No. 549 (April1942).

[16] W. Doring, An. Physik 43 (1943) 421–436.[17] W.J.M. Rankine, Philos. Trans. Roy. Soc. Lond.

160 (1870) 277–288.[18] H. Hugoniot, J. Ecole Polytechnique 57 (1887), See

also: H. Hugoniot, J. Ecole Polytechnique 58(1889) 1–125..

[19] Salas, M.D., Proceedings of the 17th ShockInteraction Symposium, Rome Italy, September,2006. <http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20060047586.pdf>.

[20] F.A. Williams, Combustion Theory: The Funda-mental Theory of Chemically Reacting Flow Sys-tems, Benjamin/Cummings, CA, 1985.

[21] E.S. Oran, V.N. Gamezo, Proc. Sixth InternationalSeminar on Fire and Explosion Hazards, in: D.Bradley, G. Makviladze, V. Molkov (Eds.),Research Publishing, Singapore, 2011, pp. 79–92.

[22] T. Ogawa, V.N. Gamezo, E.S. Oran, Comput.Fluids 85 (2013) 63–70.

[23] E.S. Oran, V.N. Gamezo, Combust. Flame 48(2007) 4–47.

[24] E.S. Oran, J.P. Boris, Numerical Simulation ofReactive Flow, Cambridge, 2001.

[25] E.S. Oran, V.N. Gamezo, D.A. Kessler, Deflagra-tions, Detonations, and the Deflagration-to-Deto-nation Transition in Methane-Air Mixtures, NRLMemorandum Report 6404–10-9300, NavalResearch Laboratory, Washington, DC, 20375,September 2010.

[26] J. von Neumann, R.D. Richtmyer, J. Appl. Phys.21 (1950) 232–257.

[27] D.E. Potter, Computational Physics, Wiley, NewYork, 1973.

[28] D. Drikakis, W. Rider, High-Resolution Methodsfor Incompressible and Low-Speed Flows, Springer,New York, 2010.

[29] J.P. Boris, D.L. Book, J. Comput. Phys. 11 (1973)38–69.

[30] B. van Leer, J. Comput. Phys. 14 (1974) 361–370.[31] C.R. Devore, J. Comput. Phys. 92 (1991) 142–160.[32] A. Harten, J. Comput. Phys. 49 (1983) 357–393.[33] P. Colella, P.R. Woodward, J. Comput. Phys. 54

(1984) 174–201.[34] J.P. Boris, On large eddy simulations using subgrid

turbulence models, in: J.L. Lumley (Ed.), WitherTurbulence? Turbulence at the Crossroads, LectureNotes in Physics, vol. 257, Springer-Verlag, Berlin,1989, pp. 344–353.

[35] J.P. Boris, F.F. Grinstein, E.S. Oran, R.L. Kolbe,J. Fluid Dyn. Res. 10 (1992) 199–228.

[36] F.F. Grinstein, L.G. Margolin, W.J. Rider,Implicit Large Eddy Simulation, Cambridge,New York, 2007.

[37] M.J. Berger, J. Oliger, J. Comput. Phys. 53 (1984)484–512.

[38] M.J. Berger, J. Oliger, J. Comput. Phys. 53 (1984)484–512.

[39] A.M. Khokhlov, J. Comput. Phys. 143 (1998) 519–543.

[40] P. MacNeice, K. Olson, C. Mobarry, R. deFainchtein, C. Packer, Comput. Phys. Comm. 126(3) (2000) 333–354.

[41] J.B. Bell, Center for Computational Sciences andEngineering, Lawrence Berkeley Laboratory,(2014), <https://ccse.lbl.gov/index.html>.


http://refhub.elsevier.com/S1540-7489(14)00403-9/h0010










http://www.buncefieldinvestigation.gov.uk

http://www.buncefieldinvestigation.gov.uk

























http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20060047586.pdf

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20060047586.pdf




































































http://https://ccse.lbl.gov/index.html

[42] D.A. Howell, Nature Commun. 2 (2011). http://dx.doi.org/10.1038/ncomms1344 (2011) id. 350.

[43] B. Wang, Z. Han, Astron. Rev. 56 (2012) 122–141.[44] W. Hillebrandt, M. Kromer, F.K. Rope, A.J.

Ruiter, Front. Phys. 8 (2) (2013) 116–143.[45] D. Maoz, F. Mannucci, Publ. Astron. Soc. Aust.

29 (2012) 447–465.[46] W. Hillebrandt, J.D. Niemeyer, Ann. Rev. Astron.

Astrophys. 38 (2000) 191–230.[47] J.C. Wheeler, R.P. Harkness, Rep. Prog. Phys. 53

(1990) 1467–1557, 18..[48] C.J. Hansen, J.C. Wheeler, Astrophys. Space Sci. 3

(1969) 464–474.[49] D. Branch, Annu. Rev. Astron. Astrophys. 36

(1998) 17–55.[50] S.E. Woosley, T.A. Weaver, Annu. Rev. Astron.

Astrophys. 24 (1986) 205–253.[51] K. Nomoto, K. Iwamoto, N. Kishimoto, Science

276 (1997) 1378–1382.[52] W. Hillebrandt, J.C. Niemeyer, Annu. Rev. Astron.

Astrophys. 38 (2000) 19–230.[53] J.C. Wheeler, R.P. Harkness, A.M. Khokhlov,

P.A. Hoflich, Phys. Rep. 256 (1995) 211–235.[54] W.D. Arnett, Astrophys. Space Sci. 5 (1969) 180–

212, [AC-44].[55] J.R. Boisseau, J.C. Wheeler, E.S. Oran, A.M.

Khokhlov, Astrophys. J. 471 (1996) L99–L102.[56] V. Gamezo, J.C. Wheeler, A.M. Khokhlov, E.S.

Oran, Astrophys. J. 512 (1999) 827–842, [55].[57] K. Nomoto, D. Sugimoto, S. Neo, Astrophys.

Space Sci. 39 (1976) L37–L42.[58] K. Nomoto, R.K. Thielemann, K. Yokoi, Astro-

phys. J. 286 (1984) 644–658.[59] V.N. Gamezo, A.M. Khokhlov, E.S. Oran, A.Y.

Chtchelkanova, R.O. Rosenberg, Science 299(2003) 77–81.

[60] V.N. Gamezo, A.M. Khokhlov, E.S. Oran, Phys.Rev. Lett. 92 (2004) 21102.

[61] M. Reinecke, W. Hillebrandt, J.C. Niemeyer,Astron. Astrophys. 386 (2002) 936–943.

[62] M. Reinecke, W. Hillebrandt, J.C. Niemeyer,Astron. Astrophys. 391 (2002) 1167–1172.

[63] A. Nonaka, A.J. Aspden, M. Zingale, A.S. Alm-gren, J.B. Bell, S.E. Woosley, Ap. J. 745 (2012) 73(22 pp) doi:10.1088/0004-637X/745/1/73.

[64] G.C. Jordan IV, C. Graziani, R.T. Fisher, et al.,Ap. J. 759 (2012) 53 (15 pp) doi:10.1088/0004-637X/759/1/53.

[65] I.R. S_eitenzahl, F. Ciaraldi-Schoolmann, F.K.Rpke, et al., Mon. Notices Roy. Astron. Soc. 429(2012) 1156–1172.

[66] W.R. Hix, F.-K. Thielemann, Astrophys. J. 460(1996) 869–894.

[67] A.M. Khokhlov, E.S. Oran, J.C. Wheeler, Com-bust. Flame 105 (1996) 28–34.

[68] A.M. Khokhlov, Astron. Astrophys. 245 (1991) 14–128.

[69] A. Jackson, see <http://www.astro.sunysb.edu/ajackson/flash.html>.

[70] G. Ciccarelli, S. Dorofeev, Prog. Energy Combust.Sci. 34 (2008) 499–550.

[71] V.N. Gamezo, T. Ogawa, E.S. Oran, Proc. Com-bust. Inst. 31 (2007) 2463–2471.

[72] V.N. Gamezo, T. Ogawa, E.S. Oran, Combust.Flame 155 (2008).

[73] T. Ogawa, V.N. Gamezo, E.S. Oran, Proceedingsof the Seventh International Symposium on

Hazards, Prevention, and Mitigation of IndustrialExplosions (17th ISHPMIE), Volume II, pp. 179–188, 2008.

[74] T. Ogawa, V.N. Gamezo, E.S. Oran, J. LossPrevent. Process Ind. 26 (2013) 355–362.

[75] A.M. Khokhlov, E.S. Oran, Combust. Flame 119(1999) 400–416.

[76] E.S. Oran, A.M. Khokhlov, Phil. Trans. Roy. Soc.Lond. A 357 (1999) 3539–3551.

[77] Ya.B. Zeldovich, V.B. Librovich, G.M. Mak-hviladze, G.I. Sivashinsky, Astro. Acta 15 (1970)313–321.

[78] J.H.S. Lee, I.O. Moen, Prog. Energy Combust. Sci.6 (1978) 359–389.

[79] D.M. Johnson, J.S. Puttock, S.A. Richardson, S.Betteridge, in: D. Bradley, G. Makhviladze V.Molkov (Eds.), Fire and Explosion Hazards: SixthInternational Seminar, Research Publishing, Sin-gapore, DATE79, pp. 40–51.

[80] Ya.B. Zeldovich, V.B. Librovich, G.M. Mak-hviladze, G.I. Sivashinsky, Astro. Acta 15 (1970)313–321.

[81] J.H.S. Lee, R. Knustautas, N. Yoshikawa, Astro.Acta 5 (1978) 971–982.

[82] A.M. Khokhlov, E.S. Oran, Adaptive MeshNumerical Simulation of Deflagration-to-Detona-tion Transition: The Dynamics of Hot Spots,AIAA Fluid Dynamics Meeting, Paper AIAA-99-3439, 1999, AIAA, Reston, VA.

[83] D.A. Frank-Kamenetskii, Di!usion and HeatTransfer in Chemical Kinetics, Nauka, Moscow,Soviet Union, 1967.

[84] Ya.B. Zeldovich, G.I. Barenblatt, V.B. Librovich,G.M. Makhviladze, The Mathematical Theory ofCombustion and Explosions, Consultants Bureau,New York, 1985.

[85] S.B. Dorofeev, V.P. Sidorov, A.E. Dvoinishnikov,W. Breitung, Combust. Flame 104 (1996) 95–110.

[86] X.J. Gu, D.R. Emerson, D. Bradley, Combust.Flame 133 (2003) 63–74.

[87] A.M. Khokhlov, E.S. Oran, J.C. Wheeler, Com-bust. Flame 108 (1997) 503–517, 38–50.

[88] A.Y. Poludnenko, E.S. Oran, Combust. Flame 157(2010) 995–1011.

[89] A.Y. Poludnenko, E.S. Oran, Combust. Flame 158(2011) 301–326.

[90] P.E. Hamlington, A.Y. Poludnenko, E.S. Oran,Phys. Fluids 23 (2011) 125111-1–125111-18.

[91] P.E. Hamlington, A.Y. Poludnenko, E.S. Oran,Phys. Fluids 24 (2012) 075111, I.

[92] A.Y. Poludnenko, T.A. Gardiner, E.S. Oran,Phys. Rev. Lett. 107 (2011) 054501.

[93] A.Y. Poludnenko, E.S. Oran, HPC Insights, Fall2012, DOD Supercomputing Resource Centers,pp. 2–5.

[94] M.F. Haider, R.J. Rogers, J.E.S. Venert, Proceed-ings of the Sixth International Seminar on Fire andExplosion Hazards, in: D. Bradley, G. Mak-hviladze, V. Molkov (Eds.), Research Publishing,2010, pp. 66–75.

[95] R.J. Rogers, J.E.S. Venert, Proceedings of theSixth International Seminar on Fire and ExplosionHazards, in: D. Bradley, G. Makhviladze, V.Molkov (Eds.), Research Publishing, 2010, pp.52–65.

[96] D.A. Kessler, V.N. Gamezo, E.S. Oran, Philos.Trans. Roy. Soc. A 370 (2012) 567–596.


http://dx.doi.org/10.1038/ncomms1344

http://dx.doi.org/10.1038/ncomms1344






































































































http://www.astro.sunysb.edu/ajackson/flash.html

http://www.astro.sunysb.edu/ajackson/flash.html







































































































[97] W. Gray, New Scientist 31 (2012) 44–47.[98] D.M. Johnson, G.B. Tomlin, D.G. Walker, Proceed-

ings of the Tenth International Symposium on Haz-ards, Prevention, and Mitigation of IndustrialExplosions (X ISHPMIE) (Bergen, 10–14 June 2014),pp. 19–44, 2014, ISBN: ISBN 978-82-999683-0-0.

[99] A. Pekalski, J. Puttock, S. Chynoweth, Proceed-ings of the Tenth International Symposium onHazards, Prevention, and Mitigation of IndustrialExplosions (X ISHPMIE) (Bergen, 10–14 June2014), pp. 847–857, 2014, ISBN: ISBN 978-82-999683-0-0.

[100] R.J. Harris, M.J. Wickens, Understanding VapourCloud Explosions – An Experimental Study,Institution of Gas Engineers 55th Autumn Meet-ing, Communication 1408, 1989, U.K.

[101] A.M. Khokhlov, E.S. Oran, A. Yu. Chtchelkanova,J.C. Wheeler, Combust. Flame 117 (1999) 99–116.

[102] M. Brouillette, Ann. Rev. Fluid Mech. 34 (2002)445–468.

[103] X.L. Li, Q. Zhang, Phys. Fluids 9 (1997) 3069–3077.

[104] P.E. Hamlington, E.S. Oran, Signatures of Tur-bulence in Atmospheric Laser Propagation, Pro-

ceedings of the Active and Passive SignaturesConference, SPIE Defense, Security, and SensingMeeting, SPIE, 2010.

[105] E.S. Oran, Numerical Simulations of IndustrialExplosions: Discussions of the Impossible and theImprobable, Proceedings of the Tenth Interna-tional Symposium on Hazards, Prevention, andMitigation of Industrial Explosions (X ISHPMIE)(Bergen, 10–14 June 2014), pp. 3–18, 2014 ISBN:ISBN 978-82-999683-0-0.

[106] B.D. Taylor, D.A. Kessler, V.N. Gamezo, E.S.Oran, Proc. Combust. Inst. 34 (2013) 2009–2016.

[107] B.D. Taylor, D.A. Kessler, V.N. Gamezo, E.S.Oran, The Influence of Chemical Kinetics on theStructure of Hydrogen-Air Detonations, AIAAPaper AIAA-2012-0979, American Institute ofAeronautics and Astronautics, Reston, VA, 2012.

[108] E.S. Oran, J.W. Weber Jr., E.I. Stefaniw, M.H.Lefebvre, J.D. Anderson Jr., Combust. Flame 113(1998) 147–163.

[109] D.C. Bull, J.E. Elsworth, P.J. Shu!, Combust.Flame 45 (1982) 7–22.

[110] C. Olm, I.Gy. Zsely, R. Palvolgyi, et al., Combust.Flame (2014), submitted.




























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