Theory of gaseous detonations - Cranstheory over the last ten years. They concern nonlinear dynamics of detonation fronts, namely patterns of pulsating and/or cellular fronts, selection

Theory of gaseous detonationsPaul Clavin Citation: Chaos 14, 825 (2004); doi: 10.1063/1.1784951 View online: http://dx.doi.org/10.1063/1.1784951 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/14/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Indirect detonation initiation using acoustic timescale thermal power deposition Phys. Fluids 25, 091113 (2013); 10.1063/1.4820130 Characterization of complexities in combustion instability in a lean premixed gas-turbine model combustor Chaos 22, 043128 (2012); 10.1063/1.4766589 Modelling spherical explosions with turbulent mixing and post-detonation Phys. Fluids 24, 115101 (2012); 10.1063/1.4761835 Invariant manifolds for dissipative systems J. Math. Phys. 50, 042701 (2009); 10.1063/1.3105924 Semidispersing billiards with an infinite cusp. II Chaos 13, 105 (2003); 10.1063/1.1539802

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Theory of gaseous detonationsPaul Clavina)

IRPHE, Universite´ d’Aix-Marseille I & II and CNRS, 49 rue Joliot Curie, BP 146,13384 Marseille cedex 13, France

~Received 18 February 2004; accepted 2 July 2004; published online 16 September 2004!

The objective of the present paper is to review some developments that have occurred in detonationtheory over the last ten years. They concern nonlinear dynamics of detonation fronts, namelypatterns of pulsating and/or cellular fronts, selection of the cell size, dynamical self-quenching,direct ~blast! or spontaneous initiation, and transition from deflagration to detonation. Thesephenomena are all well documented by experiments since the sixties but remained unexplained untilrecently. In the first part of the paper, the patterns of cellular detonations are described by anasymptotic solution to nonlinear hyperbolic equations~reactive Euler equations! in the form ofunsteady~sometime chaotic! and multidimensional traveling-waves. In the second part, turningpoints of quasi-steady solutions are shown to correspond to critical conditions of fully unsteadyproblems, either for~direct or spontaneous! initiation or for spontaneous failure~self-quenching!.Physical insights are tentatively presented rather than technical aspects. The challenge is to identifythe physical mechanisms with their relevant parameters, and more specifically to explain how thelength-scales involved in detonation dynamics are larger by two order of magnitude~at least! thanthe length-scale involved in the steady planar traveling-wave solution~detonation thickness!.© 2004 American Institute of Physics.@DOI: 10.1063/1.1784951#

Gaseous detonations are supersonic combustion wavespropagating through a reactive mixture or a pure com-pound. They consist of an inert shock followed by a re-acting flow ignited by the post-shock conditions. Detona-tions have been extensively studied since their discovery135 years ago. In ordinary conditions, the shock-reactioncomplex of a planar detonation is unstable to longitudinaland transverse disturbances. Experiments of the late1950s have established that the spatiotemporal structureof unstable detonation fronts differ notably from the dis-sipative structures of other systems out of equilibriumsuch as flames, crystal growth or Rayleigh–Benard con-vection, etc. The problem is here of a nonlinear hyper-bolic nature. Molecular transports are negligible in thereacting gas. The Euler reactive equations have to besolved with boundary conditions given by the shockwave. One striking observation is the formation of Mach-stems with triple point configuration propagating trans-verse to the shock front at nearly the sound speed, yield-ing the so called ‘‘diamond’’ or ‘‘fish scale’’ pattern thatare observed by markings left on soot-coated foils at thewalls. The cell size, which is much larger than the deto-nation thickness, has been empirically correlated to thecritical conditions of detonation initiation „or quenching….Such critical conditions are of great importance in prac-tical applications, specially in nuclear reactor safety. Inastrophysics, the cellular structures of detonations mayhave an impact on the spectra of type I supernovae. Untilrecently, physical insights were elusive and there was noconvincing explanation of these phenomena that are stilltackled in practical applications by empirical correla-

tions. Understanding of the dynamics of overdriven deto-nations has been significantly enhanced by the theoreticalanalyses of the last 10 years, based on asymptotic solu-tions of the nonlinear hyperbolic problem, in the doublelimit of a small difference of the specific heats„Newtonianlimit … and a large propagation Mach number. The natureof the planar oscillatory instability of the shock-reactioncomplex has been identified as resulting from the sensi-tivity of the heat-release kinetic to the temperature. Thisphenomenon has been described by a model integralequation. A purely hydrodynamic „multidimensional… in-stability develops even in the absence of thermal sensitiv-ity by coupling the longitudinal dynamics of the shock-reaction complex with transverse perturbations to theleading shock. A weakly nonlinear analysis describingcusps of the shock front„propagating in the transversedirection… then leads to a description of the ‘‘diamond’’pattern. On the other hand the critical conditions for ini-tiation and quenching have been obtained as turningpoints of the quasi-steady nonlinear solutions of curveddetonation fronts. These theoretical results yield predic-tions that are quantitatively in good agreement with ex-periments and direct numerical simulations. Based onthese results, the relations between all these phenomenamay then be systematically investigated. A review of theseanalyses is presented below, including the new physicalinsights that they have provided.

INTRODUCTION

A brief history of detonations

Detonations were discovered in 1869 in condensed ex-plosives by Abel, an engineer of the Nobel company, who

a!Also professor at Institut Universitaire de France. Electronic mail:[email protected]

CHAOS VOLUME 14, NUMBER 3 SEPTEMBER 2004

8251054-1500/2004/14(3)/825/14/$22.00 © 2004 American Institute of Physics


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measured their propagation velocity in 1873. The discoverywas reported in the journal Nature~issue of October 1873! inthe following terms ‘‘ . . . the detonation of gun-cotton trav-els more rapidly than any other known medium, with theexception of light and electricity.’’1 A short time later, deto-nation waves were observed in gaseous mixtures in 1882 byBerthelot and Vieille2 who also measured the propagationvelocity. The first computation of the propagation velocity ofthe so-called Chapman–Jouguet~CJ! regime, defined as themarginal speed compatible with conservation of mass, mo-mentum and total energy, was first carried out, as early as1890, by a young Russian scientist, Mikhel’son, in his Ph.D.thesis of Moscow University. This theoretical result does notrequire a knowledge neither of the detonation structure norof the chemical kinetic, it involves only computation of thechemical thermodynamic equilibrium in the burned gases. Asearly as August 1900, Vieille3 was the first to report that asupersonic velocity of propagation is possible only if theexothermic reaction proceeds downstream a leading inertshock wave, considered as an hydrodynamic discontinuity~with Rankine–Hugoniot jump conditions!, see Fig. 1. Theexistence of shock waves was predicted forty years before, inthe form of singularities in the solutions of the Euler equa-tions, by the famous mathematician Bernhard Riemann in histheoretical study of nonlinear acoustics~1860!. The first ob-servations of shock waves are reported few time later in theexperiments by Ernst Mach. These examples illustrate twobasic ways for the scientific knowledge to develop: Newphenomena are discovered by experiments and explained bytheoretical analyses afterwards~Michelson’s experiment in1880, and Einstein theory of special relativity in 1906!,and/or theoretical analyses validated by experimental studiesthat are carried out afterward~Einstein’s general relativity1916!. The first computation of the so called Zeldovich–Neumann–Do¨ring ~ZND! detonation structure of a planarwave solution traveling at constant speed was carried out in1940, 40 years after Vieille’s description! This illustrates thetime scale that it may take for understanding and describing

in mathematical terms a phenomenon after its discovery. Un-steady and multidimensional characters of the propagationregimes were observed even earlier, in 1926~spinning deto-nations!, while more systematic experimental investigationsof pulsating and cellular detonations were developed sincethe late 1950s, see the classical monograph of Shchelkin andTroshin.4

State of the art and challenging problems

It is now well established that unsteady and multidimen-sional phenomena with eventual chaotic aspects, are in-volved in the propagation regimes observed in experiments,exhibiting invariably cellular structures, with formation ofMach stems traveling transverse to the front at approxi-mately the sound speed in the burned gas, see Figs. 2 and 3.As for flames, planar detonations propagating at constant ve-locity are unstable in ordinary conditions. Many efforts, in-cluding the pioneering stability analysis of Erpenbeck5 in1964 and the spectacular advances in numerical simulationssince the early 1980s, have been devoted to describe thesephenomena. It is only during the last decade that physicalinsights have been significantly improved.

The main experimental result, that is well establishedsince the 1970s and unexplained until recently, is that the

FIG. 1. Sketch of the inner structure of a planar overdriven wave~piston-supported detonation!.

FIG. 2. ~a! Idealized two-dimensional cellular detonation showing shockwave at various times with associated smoked foil pattern of~b!. ~b!–~d!Transverse structures of gaseous detonations propagating in a rectangularchannel measured by markings left on soot-coated foils at the walls. Thelines are left by the trajectories of Mach stems. The channel width is 38 mm,showing cell sizes that are much larger than the detonation thickness(1021 mm at most!. The detonation propagates from left to right on~a! and~b! and from bottom to top on~c! and ~d!. In experiments, regular patternsare more difficult to observe than chaotic patterns. Pictures reproduced fromthe Strehlow’s book,Combustion Fundamental, McGraw-Hill InternationalEditions ~1988!.

826 Chaos, Vol. 14, No. 3, 2004 Paul Clavin


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order of magnitude of the characteristic length-scales in-volved in the unsteady phenomena is typically larger than thedetonation thickness,l t' l i1 l e , involving the induction-zone lengthl i and the exothermal reaction-zone lengthl e .These length-scales are related to the chemical time scalesthrough the gas particle velocity, see next section. For ex-ample, the cell size of cellular detonations is larger thanl t inordinary conditions, by a factor ranging from 10 for H2–airmixture to 50 for C2H4–air mixture,6 the factor becoming oforder unity only in very particular conditions, nearH2-lean-limit wherel t increases strongly. Other examples arethe critical tube diameter and the critical radius involved indirect initiation, that are larger thanl t by two orders of mag-nitude at least. Simple dimensional analyses cannot explainsuch a discrepancy, nonlinear coupling between gasdynamicsand chemical kinetics are involved in the selection of thelength-scales controlling the dynamics. What are there? Howsensitive are the results to details of the chemical-kinetic? Inother words, what are the relevant parameters and how manyare there?

It was known qualitatively from a long time that, gener-ally speaking, the thermal sensitivity of the reaction rate is anessential ingredient. More detailed quantitative answers havebeen recently provided by analytical studies coupled withwell-conceived numerical simulations of simplified configu-rations. As for all complex systems, a systematic reduction ofthe complexity is a key point for studying combustionwaves. More precisely, understanding is provided by thestudy of simplified models involving a minimum number ofparameters to represent with a good accuracy a maximumnumber of experimental data. Another key point is that, inmany cases, the main qualitative features and accurate quan-titative predictions may be picked up at the leading order ofa perturbation analysis in some limiting values of the param-eters. The most famous example in combustion is the pio-neering asymptotic analysis of Zeldovich and Frank-Kamenestkii~1938! for planar flames in the limit of a largeactivation-energy. This was followed 45 years later by a

fairly complete description of the dynamics of curved and/orwrinkled fronts of laminar flames, carried out by extendingthe method to multidimension cases in which chemical ki-netics, molecular diffusion and hydrodynamics~due to gasexpansion! are coupled.7–9 Advances in detonation theoryhave been obtained during the last decade by following asimilar approach, including pulsating and cellular detona-tions, critical conditions for initiation, dynamical quenchingand transition from deflagration to detonation in confinedchannels. Among the problems that are well documented byexperiments,6,10 but are still immature, at least from a theo-retical point of view, is the transition from deflagration todetonation ~DDT!. Promising numerical and theoreticalanalysis have been carried out recently for DDT in closedtubes, in the same vein as the 30 year old outstanding experi-mental observations on the topic.10 The unconfined casewhich is of great importance in many situations, from safetyof nuclear reactors to supernovae explosions in astrophysics,is still widely open. All these results are review in the presentpaper. The topic of hydraulic resistance and multiplicity ofdetonation regimes recently reviewed by Sivashinsky9 is notconsidered here.

Contents

• Sensitivity to temperature, structure of planar detonationsand detonability limit.

• Pulsating~galloping! detonation. Dynamical quenching.• Cellular detonations. Hydrodynamic instability. Nonlinear

selection of the cell size.• Direct initiation. Dynamical failure.• Spontaneous initiation and quenching. Deflagration to

detonation transition~DDT!.

STRUCTURE OF PLANAR DETONATIONS

Gaseous detonations are supersonic combustion-wavesthat may be either piston-supported or self-sustained. Ac-cording to the Vieille-ZND structure shown in Fig. 1, a pla-nar detonation consists of an inert shock, followed by a re-acting gas flow. The ignition of the reaction downstream theleading shock is produced by the temperature increase due tothe compression by the shock wave. The reacting flow issubsonic relative to the shock, and the local Mach numberM(M<1) increases with the distance from the leading shockdue to the heat-release by the chemical reaction. For piston-supported detonations~overdriven regimes!, the supersonicvelocity of propagation decreases with the subsonic pistonvelocity. Below a minimum velocity of the piston, the wavepropagates in a self-sustained manner at the Chapman–Jouguet~CJ! velocity. The flows becomes sonic at the end ofthe reaction zone of a CJ wave, while it remains subsonic fora piston-supported detonation. For ordinary CJ detonationspropagating in gaseous mixtures, the propagation velocityvaries between 1500 and 3500 m/s, with a propagation MachnumberMo between 4.5 and 9, and the detonation thicknessl t which increases when decreasingMo is typically betweenfew 1022 mm and few 1021 mm.

FIG. 3. Head-on instantaneous view of the chaotic pattern of a cellulardetonation propagating in a tube with a diameter of 30 mm. Picture repro-duced from the book of Shchelkin and Troshin~Ref. 4! ~1964!.

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Sensitivity to temperature

The phenomenon at the root of the Vieille-ZND structureis similar to the one in the ZFK study of flames. It lies on thesensitivity to the temperature of the overall reaction rate,resulting from an activation energyE which is large compareto the thermal energy,E@kBT, a key point to keep a coldmixture far from the composition at chemical equilibrium

1

t r5

1

tcollBe2E/kBT, where E@kBT, ~1!

where 1/t r and 1/tcoll denote the reaction rate and the elasticcollision rate, respectively, andB is a nondimensional pa-rameter of order unity, depending weakly on temperature andpressure. Typical value of the Arrhenius factorBe2E/kBT fortemperature about 1000 K is 1024. Equation~1! means thatmany elastic collisions occur between two consecutive reac-tive collisions. While the strong shock is a few mean freepaths thick, the chemistry associated with the large activationenergy causes the reaction region to be much larger, of mac-roscopic size, typically 1021 mm. The very subsonic natureof flame propagation is also a consequence of Eq.~1!. Thechemical reaction rate being balanced by thermal conduc-tion, dimensional arguments lead to the following order ofmagnitude for the flame velocity,Uo'AD/t r , where, ac-cording to the elementary transport theory in gases, the dif-fusion coefficient is evaluated in terms of the speed of sounda asD'a2tcoll . According to Eq.~1!, a balance of molecu-lar diffusion and chemical reaction leads to a transcenden-tally small Mach number of flame propagation,Mo

[Uo /ao , Mo'Ae2E/kBT!1. On the other hand, the mo-lecular diffusion is negligible in the flow of compressed gasdownstream to the leading shock of an ordinary detonations.The Mach number of the subsonic flow is typically 0.2–0.3,which is too large for the molecular transport to be non-negligible, 1.M@Ae2E/kBT'102321022. Except acrossthe inner structure of the leading~inert! shock wave, molecu-lar transports of mass and energy are negligible, and the flowof compressed gas is governed by the reactive Euler equa-tions. The distributions of heat release rate, species concen-tration, density and temperature are then governed by a bal-ance between convection and chemical kinetics, the velocity

of gas particles spreading the chemical evolution in timealong the normal distance to the front. The growth rate ofentropy production downstream to the leading shock waveresults mainly from the rate of the irreversible reaction.

The complex chemistry involved in combustion of gas-eous mixtures cannot be fully represented by a simple law asin Eq. ~1!. In flame theory, Eq.~1! is used witht r denotingthe characteristic time for the overall exothermal reaction,while in detonations it is more appropriate to consider Eq.~1! as an expression for the time-delay of the induction pe-riod, t i , with T denoting the temperature of the fresh mix-ture at the Neumann state,T5TN , in the compressed gasjust downstream to the inert shock. More precisely, the dis-tribution of heat-release rate characterizing the inner struc-ture of a detonation exhibits an induction period downstreamthe leading shock, where the heat-release is negligible, endedby a run-away where the heat-release rate increases abruptly,followed by a relaxation period where the temperature in-creases, see Fig. 4. However, in ordinary detonations, awayfrom the detonability limit, the chemistry, despite its com-plexity, introduces one single basic time scale. This is due tothe fact that the characteristic time of heat-release,te , elaps-ing from the run-away~at the end of the induction process!to the end of the exothermic reaction, is typically of the sameorder of magnitude as the induction time,t i which variesbetween 1024 and 1026 s. The duration of the run-awaybeing much shorter thant i andte , the transit time for a fluidparticle to cross the entire detonation structure ist t't i

1te . The flow being subsonic, the acoustic waves introducetime scales shorter thant t .

According to Eq.~1!, t i is decreasing~increasing! oforder unity with increasing~decreasing! slightly TN . Theorder of magnitude of the induction-zone lengthl i is givenby the relationl i'uNt i whereuN is the gas velocity relativeto the shock at the Neumann spike. According to Eq.~1!, theinduction length increases of order unity with decreasingslightly the propagation velocity, i.e., with decreasingTN bya small amount, (TN2TNo)/TNo5O(1/b),

l i / l io'e2b(TN2TNo)/TNo, b[E/kBTNo@1, ~2!

where b is a large number~Zeldovich number!, typically

FIG. 4. Distribution of heat release rate computed with a complex chemical-kinetic scheme for a plane detonation propagating at constant speed in a H2-02mixture at ordinary conditions (To5298 K, po51 atm) for a stoichiometric composition and three detonation speeds,D52900– 3100 ms21 ~three differentshock temperaturesTN). ~a! Reduced heat-release ratewo(x,TN) as a function of the distancex from the shock and arbitrarily normalized by the maximumfor D53100 ms21. ~b! Same curves plotted in the forml i

21wo(x/ l i ,TN), showing a simple scaling for taking into account the variations withTN . Figuresreproduced from Clavin and He~Ref. 18! ~1996!.



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between 5 and 10, measuring the thermal sensitivity, andsubscript o denotes a reference state. This description isvalid, except near the detonability limit where the induction-zone length increases even more strongly, see below.

Detonability limit

Beside the effects of confinement~heat and/or momen-tum losses at the wall!, the detonability limit is related to thechemical-kinetic, and more precisely to the existence of awell known transition in the combustion regime defined by across over temperatureT* , about 1000–1200 K for mosthydrocarbons or hydrogen–air mixtures, at which the rate ofchemical heat-release presents a sharp transition. BelowT*the rate becomes smaller, by many orders of magnitude, andmay be neglected in first approximation. The chemical-kinetic developing in detonation waves may be well repre-sented by a three-step chain-branching model, including avery temperature-sensitive chain-initiation reaction, atemperature-sensitive chain-branching reaction and atemperature-independent chain-termination,11 see Fig. 5. Themodel is an extension of a two-step chain-branching model,used extensively in the past,12 the extension being obtainedby adding an initiation step. In this model,T* is defined bythe temperature at which the rate of the chain-terminationreaction equals the chain-branching rate,w51, as for the socalled ‘‘second explosion limit,’’ see classical textbooks. Inthe absence of molecular diffusion, the chain-initiation stepis essential in detonations to initiate the production of radi-cals by the chain-branching reaction. The induction-zonelength l i diverges when approaching the detonability limitslike l r /(12w) where w<1 is the ratio of recombinationreaction rate to the chain-branching reaction rate.13 No deto-nation supported by the three-step model can exist with aNeumann temperature below the cross over temperature,TN,T* .

PULSATING DETONATIONS

The study of one-dimensional nonlinear oscillations ofthe shock-reaction complex, called pulsating detonations, is apreliminary step in studying the cellular structures and otherunsteady phenomena of gaseous detonations. One-dimensional pulsating detonations were first observed duringthe 1960s in experiments and also in direct numericalsimulations.14 Such numerical studies, see Fig. 6, as well asthose of Erpenbeck5 ~1964! for the linear spectrum, includingthe neighborhood of the bifurcation, have been extendedmore recently by different authors.15–17 The physical expla-nation of the oscillatory instability was enlightened in 1996by a nonlinear analytical study of overdriven regimes,18 inthe distinguished limit of a large Mach number of propaga-tion, Mo

2@1 and a Newtonian limit, (g21)!1 (g[Cp /Cv is the ratio of specific heats!, yielding a very smallMach number of the flow at the Neumann state,MN

2 !1,MN

2 '(g21)/21Mo22 . The instability results from a loop

involving acoustic and entropy waves. Such waves carry theflow perturbations emitted from the unsteady inert shockdownstream across the shocked gases, producing modifica-tions to the heat-release rate. The resulting pressure varia-tions are sent upstream, back to the leading shock, by soundwaves. In the above mentioned limit, to leading order, theflow behind the leading shock is quasi-isobaric, and the per-turbation scheme brings in effects of post-shock pressure-wave propagation at first order.18 For strongly overdrivenregimes, the low Mach number approximation is validthroughout the detonation structure, up to the end of the exo-thermal reaction.

Quasi-isobaric mass conservation and integralequation

The approximation of low Mach number leads to a cleardistinction between the effects of compressibility~acoustics!

FIG. 5. Nondimensional distribution of heat release rate computed from the three-step chain-branching model; Initiation:F→R, ki(T)5Aie2Ei /kBT;

Chain-branching:R1F→2R, kbr(T)5Abre2Ebr /kBT; Chain-termination: R→P1Q, kter51, whereki , kbr , andkter are the constants of reaction. The

chain-termination rate is used to define the unit of time.F represents a reactant~fuel or oxidizer!, R a radical andP a product. The cross over temperature isdefined bykbr(T* )51. Hydrogen–air combustion corresponds tokBTNo!Ebr,Ei . Sufficiently away from the detonability limit,TN.T* , one haski(TN)!1!kbr(TN), and the mass-weighted induction-zone length is approximately given in non dimensional form byj i'kbr

21(TN)ln@kbr(TN)/ki(TN)#, which istypically of order unity for the constants of reactions of H2–O2~Refs. 11 and 13!. This shows that the induction delay is of same order as the chain-termination time, and that the parameter characterizing the thermal sensitivity of the induction-zone length in Eq.~2! is b'Ebr /kBTNo . For a fixed value ofj i , the stiffness of the run-away increases by increasingkbr(TN), denoted bykBN on the figures.



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and the quasi-isobaric expansion of the reacting gas. Twosimplifications appear, the pressure effects are negligiblysmall in the balance of internal energy, and the time delaysintroduced in the loop by the sound waves being muchshorter than the one introduced by the entropy wave, theymay be neglected to leading order. According to mass con-servation, the unsteady mass flux across the oscillatory lead-ing shock must be balanced by the space integral from theshock to the end of the exothermal reaction of the substantialtime derivative of the instantaneous density distribution. Atthe leading order of the quasi-isobaric approximation, thislast quantity is the instantaneous distribution of heat-releaserate, written in nondimensional form asw& (j,t), wheret isthe reduced time andj'(rNol io)21*0

xrdx is the reducedmass weighted distance from the shock~related to the transittime, '*0

xdx/u). According to the species and temperatureequations written in the same approximation, this distribu-tion w& (j,t) is simply related to the steady state distributionof planar overdriven waves propagating at a constant veloc-ity ~constant Neumann temperatureTN) written in nondi-mensional form as wo(j,TN), with by definition*0

1`wo(j,TN)dj51. The relation betweenw(j,t) andwo(j,TN) is through the time delay introduced in the instan-taneous Neumann temperatureTN(t) by the propagation ofthe entropy wave from the inert shock to the point underconsideration,w& (j,t)5wo(j,TN(t2j)). The flow velocitybeing imposed in the burned gas (x→1`) by the piston,

and at the Neumann state@x50,u5uN(t)# by the Rankine–Hugoniot condition linkinguN(t) to the shock temperatureTN(t), a space integration of the equation for mass conser-vation in the quasi-isobaric approximation,du/dj'(Q/CpTNo)w& , then yields a nonlinear integral equation forTN(t), written in nondimensional form as18

~TN~t!2TNo!/~g21!

5~Q/Cp!E0

1`

@wo~j8,TN~t2j8!!

2wo~j8,TNo!#dj8, ~3!

whereQ is the heat-release per unit mass. In the left-handside of Eq.~3!, the linear Rankine–Hugoniot relation at theleading shock has been used, for expressinguN(t) in term ofTN(t). A more general equation, limited only by the smallMach number approximation, might have been obtainedwithout using the linear approximation in the left-hand sideof Eq. ~3!. This approximation however is sufficient to pickup the essential~linear and nonlinear! phenomena, because,according to Eq.~2!, small variations ofTN induce strongvariations ofwo(j,TN) in the right hand side of Eq.~3!.

Oscillatory instability

The distinguished limit to be used is exhibited by intro-ducing the thermal sensitivity in the form,wo(j,TN)

FIG. 6. Evolution of the shock pres-sure ~Neumann spike!, normalized bythe initial pressure, obtained by directnumerical simulation of a planepiston-supported detonation for a one-step Arrhenius law and different val-ues of the degree of overdrive,f5(D/DCJ)

2. The time scale is here thehalf reaction time. The shock tempera-tureTN decreases and the pulsating in-stability increases with decreasingf~increasingQ/CpTN). The first picturecorresponds to the instability thresh-old. A route to chaos by period dou-bling is observed in the next four pic-tures. Finally, a large increase of theperiod of slow propagation, represen-tative of a dynamical self-quenchingof strongly unstable waves, is ob-served in the last picture. Quite similarbehavior is observed by the solution ofEq. ~4! when decreasing the parameterb. Picture reproduced from Clavin andHe ~Ref. 18! ~1996!.



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2wo(j,TNo)[Ã(j,Q) where Q[b(TN2TNo)/TNo , withÃ8(j)[]Ã(j,Q)/]QuQ505O(1) andb@1. The large pa-rameterb measures the intensity of the thermal sensitivity,and the attention is limited to small variations of the Neu-mann temperature~Q of order unity!. The spatial distribu-tions Ã~j,Q! and Ã8(j) characterize the unperturbed one-dimensional ~1D! waves, and are fully governed by thechemical kinetics. Equation~3! is then written as

bQ~t!5E0

1`

Ã~j,Q~t2j!!dj, with

b[1

b~g21!Q/CpTNo, ~4!

whereb is a parameter of order unity in the double limitblarge and (g21) small. The oscillatory instability is exhib-ited by the linear approximation of Eq.~4!, and the linearspectrum is obtained as the rootss of Eq. ~5! below, aftersetting Q(t)5est, with s5s1 iv, s denoting the lineargrowth rate~damping rate fors,0) andv the frequency

bQ~t!5E0

1`

Ã8~j!Q~t2j!dj,

⇒b5E0

1`

Ã8~j!e2sjdj. ~5!

For a given distributionÃ8(j), the solutions to Eqs.~4! and~5! exhibit a Poincare´–Andronov~Stuart–Hopf! bifurcationwhenb is decreased, leading to a limit cycle and to a chaoticbehavior whenb is further decreased, as also observed indirect numerical simulations,16–18see Fig. 6. This shows thatthe pulsating instability of overdriven detonations is pro-moted by increasing the heat release and/or the thermal sen-sitivity, and/or by decreasingTNo , which means by ap-proaching the CJ regime. Solutions to Eq.~5! show a discretespectrum of eigenvaluess, with a bifurcation occurring, fora given distributionÃ8(j), at a critical valueb5bc forwhich all the roots have a negative real parts (s,0) except asingle one withs50. The shock-reaction complex is linearlystable forb.bc , and the instability develops whenb,bc .The shape of the distributionÃ8(j) influences also stronglythe stability. Stiffer isÃ8(j), larger is the critical valuebc ,and more unstable is the detonation. The dependence on thechemical-kinetic parameters controlling the distributionÃ~j,Q! has been investigated with the three-step model.13

For fixed values of the parameterb and of the heat-releasetime te , the instability increases with the induction delayt i .13 The finite time of heat-release,teÞ0, is an essentialingredient in the dynamic of pulsating detonations. The limitof the square-wave model, defined by a vanishing ratio of thetime of heat-release to the induction-time,te /t i→0, is sin-gular, Ã8(j)→d8(j2j i) where j i is the nondimensionalmass-weighted induction-zone length~or nondimensional in-duction time!. In this singular limit, the dynamics presents apathological behavior~singularities after a finite time!; thefirst equation in~5! takes the form of a difference-differentialequation of the advanced type,]Q(t)/]t5bQ(t1j i), and

the second equation describes an infinite spectrum of discreteunstable modes having growth rate increasing unboundedlywith increasing frequency.

A general conclusion of these results obtained within theframework of a quasi-isobaric approximation, is that the pul-sating instability is not thermoacoustic, as thought before.

Dynamical quenching

In ordinary conditions when the functionÃ8(j) is suf-ficiently smooth, the oscillatory frequency at the bifurcationis, according to Eqs.~5!, of the same order of magnitude asthe inverse of the transit time for a fluid particle to cross thedetonation structure 1/t t that is of order 1/t i . Near the det-onability limit, the period of oscillations increases witht i ,and the pulsating solutions of Eq.~3! present the so-called‘‘dynamical quenching.’’16,18,19 This phenomenon may bedescribed as follows. During a deceleration period in thenonlinear oscillation,TN(t) decreases sufficiently to ap-proach the cross-over temperatureT* , the induction timet i

increases so much that a very long period of slow propaga-tion occurs. The slightest disturbances may then quench theslow propagation regime by droppingTN(t) below T* . Asomehow similar situation, with a very long slow propaga-tion period is also described by the solutions to Eqs.~3! and~4! far away from the detonability limit, for strongly unstablesituations when the thermal sensitivity and/or the heat re-lease become large.13 More surprisingly, this behavior is alsoobserved with a simple one-step reaction model and a rategoverned by an Arrhenius law18 ~no cross-over temperature!,see Fig. 6. All these phenomena have been also observed indirect numerical simulations using the one-step or the three-step reaction model.16–19

Effects of compressibility

Equations~3! and~4! are limited by the low Mach num-ber approximation, valid throughout the detonation structurefor strongly overdriven regimes in the Newtonian limit. Thebasic instability mechanism is captured by a quasi-isobaricpost-shock approximation and the effects of compressibilityhave been included at the first order of a perturbationanalysis.18 The analysis shows that the sound waves have astabilizing effect through a velocity coupling, indicatingclearly that the oscillatory instability is just the opposite ofan acoustic instability. The effects of compressibility alsoinduce an increase in the delay due to the finite transit timeof the sound wave propagating upstream~back to the leadingshock!.

The corrections introduced by the effects of compress-ibility may become quantitatively non-negligible near the CJcondition, but they do not change the qualitative descriptionas now explained. Ordinary detonations have a subsonic in-duction region with a small Mach number, typically between0.2 and 0.3, so that the subsonic approximation fails only inthe part of the inner structure where the heat-release rate isno longer so strongly sensitive to temperature. Moreover, theinduction delayt i ~transit time of a fluid particle to cross theinduction region! being of the same order of magnitude aste

~transit time of a fluid particle to cross the region of heat-release!, the time delay introduced by the entropy wave is



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thus the longest one. Compressibility having a minor effectupon w& (j,t), oscillatory instabilities are well captured byEqs. ~3!–~5! resulting from the propagation of the entropywave. This is no longer true in the particular case, of a CJwave with a small heat release, so that the flow is near soniceverywhere, which in a sense is the opposite of the quasi-isobaric post-shock conditions used in Eqs.~3!–~5!. The ana-lytical study of the linear stability of this limiting case yieldsan integral equation of the same type as Eqs.~5!, but wherethe time delay is now controlled by the sound waves propa-gating upstream.20 An integral equation has been constructedby interpolation between the results obtained for the twoextreme situations, strongly overdriven regimes and near CJconditions with a small heat-release.20

CELLULAR DETONATIONS

Basic mechanisms

The cellular structures of gaseous detonations have beenextensively investigated since half a century by experimentalstudies that are now well documented.6 Following the pio-neering work of Erpenbeck,5 numerical analysis of the linearspectrum and of the stability limits21 have been carried outduring the last decade, as well as direct numericalsimulations22,23 and nonlinear studies.24,25 Despite these ad-vances, physical insights have remained elusive until re-cently. The multidimensional linear instability mechanismthat leads to cellular detonations has been enlightened foroverdriven regimes by an asymptotic analysis26 in the samelimit as above, (g21)!1 andMN

2 !1. The oscillatory insta-bility of the inner structure of a wrinkled detonation frontthen appears to be related to two phenomena:

• the longitudinal oscillations of the shock-reaction complexwith a frequencyv of the order of the inverse of the transittime, v'1/t t , as in pulsating detonations,

• the underlying neutral oscillatory modes of the leading in-ert shock, governed by a wave equation for disturbancespropagating in the transverse direction with the speed ofsound in the compressed gas,a, yielding frequencies pro-portional to the transverse wave number,k52p/l ~l isthe wavelength!, v'ak, as first described in the Russianliterature27,28 of the 1950s.

Equating the two frequencies leads to the relationl'2pat t , yielding, when using the speed of sound at theNeumann statea'aN , a ratio of the wavelength of unstabledisturbances to the induction-zone length (l i'uNt t) of theorder of the inverse of the Mach number at the Neumannstate,l/ l i'2p/MN , a number ranging from 13 to 20 forordinary detonations.

Hydrodynamic instability

Further physical insights into the asymptotic analysis de-scribing the branches of the dispersion relation,s(k) andv(k), may be summarized as follows:26

• As for the Darrieus–Landau hydrodynamic instability offlames, the isobaric expansion of gas is responsible for theinstability of detonations. The gas compressibility has a

stabilizing influence. A negative feedback is produced bythe sound waves triggered in the burned gases through avelocity coupling. The sound waves have a smaller ampli-tude than the entropy-vorticity wave by a factorMN

2 , andthe damping is small. This damping, however, is essentialin multidimensional geometry to damp out the distur-bances with wavelengths smaller than the detonation thick-ness,l< l . Neglecting sound waves would lead to un-stable disturbances with a positive growth rate going tozero with the wavelength but remaining positive at verysmall wavelength. The order of magnitude of the wave-length of the most amplified disturbances are of the orderof magnitude described just above,l/ l'2p/MN . There isa finite range of unstable wavelengths whose upper boundincreases when approaching the one-dimensional oscilla-tory instability threshold, by increasing either theinduction-zone length or the sensitivity of the chemical-kinetic to temperature.

• The deflection of the streamlines across the wrinkled shockperturbs the heat-release rate distribution and excites lon-gitudinal pulsations, even in the absence of pulsating in-stability of the planar wave, when there is no thermal sen-sitivity of the chemical-kinetic (b50). This pure‘‘hydrodynamic instability’’ of the inner structure of adetonation is due to the quasi-isobaric change of density.Its strength increases with the heat-release and is also re-inforced by the sensitivity of the heat-release rate to tem-perature. When the heat-release is sufficiently small thehydrodynamic instability may be counterbalanced by theeffects of compressibility, leading to the existence of sta-bility limits. The ‘‘hydrodynamic instability’’ is specific tothe multidimensional geometry: A detonation which isstable to one-dimensional disturbances may be unstable tomultidimensional disturbances.

Stability limits and weakly nonlinear analysis

The striking observation in experiments of cellular deto-nations is the formation of Mach stems traveling transverseto the front, see Figs. 2 and 3. The theoretical study of suchcellular structures requires a nonlinear analysis. A systematicapproach of this nonlinear and hyperbolic problem has beencarried out by a weakly nonlinear and asymptotic analysis inthe neighborhood of the instability threshold.29 In order todescribe quasi-singularities, representative of Mach-stemscoexisting with large cells, a wide range of weakly unstablewavelengths is taken into account into the nonlinear analysis,leading to a partial-differential amplitude equation for theshock front.29 Generally speaking, such weakly nonlinearanalyses consist in a systematic reduction of the complexity,by retaining only the leading-order nonlinear terms nearthreshold. Does the same reduction continue to be meaning-ful under real conditions of strongly unstable situations? Noanswer can be given with certainty, but, in general, under-standing of strong instabilities may be enhanced by investi-gating their onset.

A detailed study of the stability limits~instability thresh-old! determined by the competition of quasi-isobaric heat-release and sound waves, is a preliminary step in the weakly



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nonlinear analysis. This was carried out for overdriven deto-nations in the same limit as for pulsating detonations, (g21)!1 and MN

2 !1.30 For a given chemical-kinetic, themultidimensional bifurcation occurs when the ratio of theheat-release to the enthalpy,Q/CpTN , is increased above acritical value which is as small as the two parameters (g21) andMN

2 . The neighborhood of the instability thresholdhas been described in a distinguished limit taking these threesmall parameters to be of same order of magnitude.30 Thereduced heat-releaseQ/RTN whereR[(Cp2CV) is the con-stant of perfect gas, is a bifurcation parameter of order unityin the analysis,

~g21!!1 and Q/CpTN5O~g21!⇒Q/RTN5O~1!,

and its critical value depends on the parameterb and on thetwo distributions characterizing the heat release rate and itsvariation in planar solutions,Ã~j! and Ã8(j), see the pre-cedent section. A particular attention was paid to the desta-bilizing effect of an increase of either the ratiot i /te or thestiffness of the run-away of reaction rate at the end of theinduction period.30

Discussion of the results

Ordinary detonations are strongly unstable with a mod-erate degree of overdrive, and are in principle outside thedomain of validity of the analysis. However, by consideringa large density jump across the leading shock occurring atlarge overdrive, an essential feature in the dynamics of realdetonations is taken into account, namely a large deflectionof the streamlines. A comparison with numerical analyses ofthe stability limits shows that the results of the asymptoticanalysis are accurate at large overdrive, and less accurate butstill meaningful near CJ conditions where the critical valueof the reduced heat-releaseQ/RTN reaches values larger thanunity. In addition to the explanations given earlier, the rel-evance of the results is also explained by the fact thatQ/RTN

is a quantity considered as of order unity in the asymptoticanalysis,30 contrary to stability analyses based on a simpleperturbation method whereQ/RTN is considered as a smallparameter.31

The difference of origin of the leading-order nonlinearterms in the analyses of pulsating and cellular detonations isworth mentioning. In the weakly nonlinear analysis of cellu-lar detonations they come from a pressure effect associatedwith the Reynolds stress of the entropy-vorticity waves,29,30

while in the 1D pulsating detonations they are of a purelychemical-kinetic nature,18 see Eq.~4!.

For cellular detonations, the end result of the analysis isa differential-integral equation for the evolution of the shockfront, quadratic in amplitude of the front wrinkling, and ex-hibiting formation of cusps, representative of Mach-stems,due to a nonlinear term similar to one describing the wavebreaking in the Burgers equation.29 The solution to this equa-tion is a time-dependent pattern quite similar the so-called‘‘diamond’’ or ‘‘fish scale’’ patterns observed inexperiments,4,6 see Figs. 7 and 8. The weakly nonlinear so-lution exhibits also a nonlinear pattern selection in favor ofunstable wavelengths larger than the most unstable one, see

Fig. 7, by a factor about 2 or larger in large boxes, yielding acell size that are 15 to 50 times larger than the induction-zone length, as observed in experiments.6 A nonlinear selec-tion toward the upper bound of unstable wavelengths may beimportant from an applied point of view. In current detona-tion literature the cellular structure length-scale is reported toincrease when approaching the detonability limits. This iseasily explained by the theoretical analysis presented here bynoticing that, as mentioned before, the induction length in-creases. The first consequence is to increase the basic length-scale. The second one is to reinforce the pulsating instabilityand thus to increase the upper bound of unstable wave-lengths. From a fundamental point of view, an interesting~unusual! self-sustained mean streaming motion of the dia-mond pattern of a gaseous detonation, not reported before inthe detonation literature, is exhibited as a by-product of theweakly nonlinear analysis, see Fig. 8. A self-contained re-view paper with all the details of the analysis is available.32

In conclusion the existence of a transonic region at the

FIG. 7. Overdriven detonation with a one-step Arrhenius law. Top: Nondi-mensional linear growth rate versus wave numbern. The largest unstablewavelength (n54) corresponds here to 30 times the detonation thickness.Bottom: Illustration of a nonlinear selection mechanism by the 2D solutionof the differential-integral equation for the dynamic of the cellular front,obtained by the asymptotic analysis~Ref. 29! presented in the text. Theinitial condition is a sinusoidal perturbation of small amplitude with 10wavelengths (n510) plus a much smaller level of noise. After few periodsof oscillations, a well ordered and fully developed nonlinear and regularpattern of 10 pulsating cells is observed during 30 periods of oscillation,followed by a long transient period of chaotic regime in which the numberof cells decrease down to a final stable state of 6 pulsating cells,n56. Thepattern left by the cusps during one period of oscillation has the samediamond shape as observed in experiments, see Figs. 2 and 8. A similarscenario is observed when starting the computation withn515. Picturereproduced from Clavin and Denet~Ref. 29! ~2002!.



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end of the exothermal reaction in the CJ waves, source oftechnical difficulties in theoretical studies, does not seem toplay an essential role in the formation and dynamic of thenonlinear patterns of pulsating or cellular detonations, thathave been well described by analytical studies of overdrivenregimes. On the other hand the transonic region is well-known to be a key point for the long-time selection of theself-propagating CJ regime. Such problems are discussed inthe following sections.

DIRECT INITIATION

A detonation may be initiated by an energy source of asufficiently large intensity. LetEo denote the quantity of en-ergy which is deposited. When the deposition time is smallcompared to the acoustic time scaleta ~for the sound tocross the size of the region of deposition,ta'r o /a), theflow takes the self-similar form of a Taylor–Sedov inert blastwave initiated by an ideal point source. This approximationis accurate at intermediate distances, namely for radius largerthan the size of the region where the energy is initially de-posited, but smaller than a typical radius for which thechemical heat-release is of the same order of magnitude asthe energy initially deposited,r o!r !R(Eo), with in spheri-cal geometryR(Eo)}(Eo /roQ)1/3, Q denoting the heat-release per unit mass. The strength of the blast wave is anincreasing function ofEo . At later times, when radius of theleading shock increases to approach values of order ofR(Eo), the chemical heat-release can no longer be neglected,the blast wave triggers first a strongly overdriven detonation,and different subsequent regimes are identified fromexperiments.6 Below a critical energy,Eo,Ec , the stronglyoverdriven detonation decays rapidly, and the reaction front

finally separates from the leading shock, see Fig. 9. At last, apremixed flame trails behind the inert shock and no detona-tion is initiated in the cold mixture. ForEo.Ec , the over-driven detonation relaxes to an expanding CJ detonation. Theonset of the CJ wave occurs at a radius of orderR(Eo). NoCJ detonation can be observed with a front radius smallerthan a critical radius defined asRc[R(Eo5Ec).

Well documented experiments6,33show that, contrarily toearly predictions,34 Rc is not of the same order of magnitudeas the largest intrinsic length scale in the problem, namelythe total reaction length of the planar CJ wave,l CJo , but islarger by two or three orders of magnitude,Rc / l CJo'102

2103. Motivated by apparent similarities in length scales, anempirical correlation relatingRc to the cell size has beenproposed in the past.6 A more recent theoretical analysis35

has suggested a different physical insight. Before discussingthis point, it is worth recalling the selection mechanism ofthe CJ wave in planar geometry, viewed as a marginal re-gime ~minimum speed of propagation!. In overdriven deto-nations, the flow is subsonic relative to the shock, throughoutthe detonation structure, up to the end of the reaction. Theleading edge of the rarefaction waves~generated by theboundary conditions in the burned gases! propagates up-stream throughout the reacting gas at the speed of sound~relative to the gas!, and catches the leading shock, weaken-ing its intensity and decreasing the propagation velocitydown to the CJ value, while the Mach number of the flowvelocity at the end of the reaction is increasing toward unity.The CJ regime is self-propagating because the flow at theend of the reaction becomes just sonic~relative to the leadingshock! and then screens the inner structure of the detonationfrom further disturbances propagating upstream in theburned gases.

The turning point of quasi-steady curved solutions

The problem of direct initiation is thus fully unsteady,even in the ideal configuration of spherical geometry with apoint source, and even when the formation of cellular struc-tures of the detonation front is disregarded. A quasi-steadystate approximation for curved detonation fronts, however, isworth investigating, essentially because, as we will see now,quasi-steady curved solutions present a turning point corre-sponding to a~static! critical radius below which no marginalwaves~generalized CJ solution! exist.35,36 The analysis maybe summarized as follows. The CJ regime is defined by thecondition of burned gas velocity equal to the sound speed. Inplanar geometry this steady state solution corresponds to amarginal regime of traveling-waves with a velocity of propa-gationDCJo which is the lower bound compatible with con-servation of mass, momentum and total energy~no solutionexist for D,DCJo). When considering a coordinate normalto the front, curvature of the front~or divergence of the flow!introduces source terms in the conservation equations. It wasknown from a long time ago that these terms change thenature of the marginal solution; the transonic point becomesa saddle point in the phase space that appears before the endof the reaction.37 We will denoteDCJ

1 (R) the velocity of thismarginal solution for a curved detonation of radiusR. For agiven R, the flow of curved detonations having a larger

FIG. 8. Details of the 2D detonation front during a final period of oscillationin the computation of Fig. 7. The propagation is bottom to top. Notice thesimilitude with the pattern observed in experiments, see Fig. 2. The averageshape of the front resulting from a self sustained mean-streaming motion inthe diamond pattern~Ref. 32! is shown at bottom. Picture reproduced fromClavin and Denet~Ref. 29! ~2002!.



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propagation velocity,D.DCJ1 , is subsonic, up to the end of

the reaction, as for the overdriven regimes of the planar case.It was shown more recently that the topology of the spectrumof propagation velocities is modified by curvatureeffects.35,36 For a givenR, another marginal solution with asmaller propagation velocity appears,DCJ

2 (R)<DCJ1 (R),

such that quasi-steady curved detonations with radiusR anda subsonic flow up to the end of the reaction do also exist forany propagation velocity smaller thanDCJ

2 , D,DCJ2 . No so-

lution exist for intermediate velocitiesDCJ2 ,D,DCJ

1 .

Critical conditions

The difference (DCJ1 2DCJ

2 ) decreases when decreasingthe radius of the detonation and goes to zero at a radius,different from zero. A turning point of marginal solutionsDCJ

6 (R) thus appears in theD-R phase plane, see Fig. 9. Forradius smaller than the radius of the turning point, there is nomarginal solution at all, and curved solutions do exist for anypropagation velocity with a flow which is subsonic up to theend of the reaction. In this case rarefaction waves in theburned gases may slow down the detonation wave withoutlimit. According to the selection mechanism by the rarefac-tion waves, only the upper branchDCJ

1 of the C-shaped curveof quasi-steady marginal solutions may attract the reactiveblast wave in theD-R plan and the attractor is expected toact from above,D.DCJ

1 . According to this quasi-static pic-ture, a successful ignition is predicted to occur for depositedenergies large enough for the blast wave trajectories in theD-R coordinates to reach the critical radius with a velocitylarger than the one at the turning point. This is why it hasbeen suggested35 to identify the radius of the turning point

with the ~dynamical! critical radius,Rc , observed in the ex-periments on direct initiation of detonations. This leads topredict an order of magnitude of the critical energy in spheri-cal geometry in the form,Ec}roQRc

3 . The unsteady effectsassociated with the intrinsic dynamic leading to pulsatingdetonations in unstable situations, are discussed below.

Let us first compute the radius of the turning point,showing that it is much larger than the induction-zone lengthof the planar CJ wavel iCJo and explain why. This is basicallya consequence of the thermal sensitivity of the reaction-rateand more precisely of the nonlinear variation of theinduction-zone length with temperature, see Eq.~2!. Forcurved detonations represented by the branches of marginalsolutions we have

l i / l iCJo5e2b(TN2TNCJo)/TNCJo with b[E/kBTNCJo

@1,~6!

where subscriptCJo denotes here the planar CJ wave. Equa-tion ~6! suggests that the critical conditions may be expectedto correspond to a Neumann temperature close to the planarCJ wave, (TN2TNCJo

)/TNCJo5O(1/b). Contrary to dynami-

cal properties, static quantities may be correctly evaluated byan analysis based on the square-wave model in which all ofthe heat release occurs instantaneously after a temperature-dependent induction-zone described by Eq.~6!. In such anapproximation, two simplifications occur:

~a! the marginal solutions of curved fronts are still definedby a sonic velocity in the burned gas;

~b! the additional terms~due to spherical geometry! in theequations~for conservation of mass, momentum andtotal energy! relating the conditions in the burned gasto the initial fresh mixture, are simply proportional tol i /R and are easily computed.

The modifications to the propagation velocity and to theNeumann temperature being proportional tol i /R, (TN

2TNCJo)/TNCJo

}2 l i /R, the nonlinear equation for the mar-ginal solutions of curved fronts takes, according to Eq.~6!,the form

~TN2TNCJo!/TNCJo

}2~ l iCJo /R!e2b(TN2TNCJo)/TNCJo.

When all the constants are taken into account, the equationobtained within the framework of the square-wave model inthe limit b@1 is35

Q exp~2Q!524g2

~g221! S bl iCJo

R D , ~7!

where the unknownQ is a quantity of order unity whichrepresents the ‘‘generalized’’ marginal solution for curvedfronts, Q[b(TNCJo

2TN)/TNCJo.0, the positive sign corre-

sponding to a curved detonation propagating outwards.Equation ~7! describes a turning point of weakly curveddetonations,l iCJo /R5O(1/b), corresponding to the follow-ing critical radiusRc and propagation velocityDc :

Rc

l iCJo524e

g2

~g221!b,

DCJo2Dc

DCJo5

1

2b, ~8!

FIG. 9. Numerical results for direct initiation of detonation in sphericalgeometry with a one-step Arrhenius law. The front velocityD is plotted as afunction of the shock radiusRs , for different values of deposited energy,increasing from curve labeled 1 to curve labeled 4. The plane CJ wave isstrongly unstable~pulsating detonation! as shown by curve 3. The marginalquasi-steady curved solutions with the turning point are plotted for compari-son. The upper branch is clearly shown to act as an attractor, and the turningpoint is found to define correctly the order of magnitude of the criticalcondition, even though an unsteady effect is shown to produce a self-quenching, see curve 2. The deposited energy for curve 1~failure! is lessthan 1/2 the one for the curve 2~dynamical failure! and than 1/4 the one ofcurve 3~successful initiation!. Picture reproduced from He and Clavin~Ref.35! ~1994!.



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expressing that a large thermal sensitivity of the reaction rate(b@1) is responsible for a critical radius larger than theinduction length, see Fig. 9. Moreover the numerical coeffi-cient, in principle of order unity in the asymptotic limitb@1, turns out to be also large, typically between 50 and 100,due to a numerical factor coming from 432 je/(g21)where 2/(g21) results from the Rankine–Hugoniot condi-tions andj 53 in spherical geometry and. Thus, except forthe factor 4, the large numerical factor on the right-hand sideof the first Eq. ~8! is not related to the specificity of thesquare-wave model, making the results in Eqs.~8! robust.

The predicted orders of magnitude for the critical radiusand for the critical energy,Ec}roQRc

3 which is, according toEq. ~8!, much larger by many orders of magnitude thanroQliCJo

3 , are in good agreement with direct numerical simu-lations in spherical geometry using a one-step Arrhenius ap-proximation and a self-similar blast wave solution of Taylor–Sedov as initial condition.35 The numerical simulations showclearly that the upper branchDCJ

1 of the C-shaped curve ofmarginal solutions acts effectively as an attractor of theD-Rtrajectories of the leading shock, see Fig. 9. They show alsothat unsteady effects reminiscent of 1D pulsating detonationsmay be involved in the precise definition of the critical en-ergy but without changing the orders of magnitude ofRc ,35,38 except for very large values of the parameterb(g21)Q/CpTN , for which the planar detonation is stronglyunstable, see the section on pulsating detonations. The lessunstable is the detonation the more accurate are the predictedvalues obtained from the turning point. The main unsteadybehavior observed in the direct numerical simulation near thecritical conditions is that the detonation velocity and theNeumann temperature may first decrease below the planar CJvalues to increase after a time delay, in a way similar to theplanar pulsating detonations, see Figs. 6 and 9. In planargeometry, these purely unsteady effects are responsible forthe existence of a critical energy and of a critical distancewhich plays a similar role as the critical radius in sphericalgeometry.38,39The critical distance in planar geometry, how-ever, is smaller at least by an order of magnitude than thecritical radius in spherical geometry, indicating that the non-linear geometrical effects of curvature are dominant here fordetermining the critical conditions of direct initiation.35,38

This is also true for a complex chemistry as shown by nu-merical simulations for hydrogen–oxygen mixtures dilutedin argon.38 The conclusion is less clear in conditions near thedetonability limit of the planar wave, whenTNCJo

becomes

close to the cross-over temperatureT* , (TNCJo2T* )/T*

5O(1/b). Dynamical quenching may now play a non-negligible role as confirmed by the numerical results usingthe three-step chemical-kinetic model.39,40 Sufficiently farfrom the detonability limit, the value ofD at which TN

5T* is well below Dc , and corresponds to a point on thelower branchDCJ

2 . In such conditions, the cross-over tem-perature does not play an essential role, and the critical con-ditions are accurately represented by the turning point, pro-vide that the instability is not too strong.40

In conclusion, except for strongly unstable detonationsclose to the detonability limit, nonlinear quasi-steady curva-

ture effects are the leading order mechanisms controlling thecritical radius and the critical energy for direct initiation ofdetonation in spherical geometry. The critical radius is largecompare to the induction-zone length for a reason differentfrom the cell size, as pointed out by the comparison of thetwo theoretical results. Regarding experiments, definitiveconclusions can hardly be addressed in the absence of theo-retical analyses and/or direct numerical simulations, takingfully into account the multidimensional instability of spheri-cal detonations.

DEFLAGRATION-DETONATION TRANSITION „DDT…

Due to unsteady flows and hydrodynamic instabilities, aflame is invariably turbulent in any large-scale combustionevent. Experiments have shown that a fast turbulent defla-gration may undergo a transition toward a detonation underfavorable conditions.6,10 Such conditions for initiation of adetonation are severe for the following reasons:

~i! the turbulent flame velocity in a tube, which is re-quired to generate a sufficiently strong precursorshock for auto-ignition in the shocked gas,TN>T* ,is high, about 100 ms21;

~ii ! the shock pressure of a CJ detonation is also high,typically twice times larger the maximum pressure ofa constant volume explosion.

Subtle dynamical effects are involved in DDT. One strik-ing observation of the late 1960s and early 1970s by Oppen-heim and his colleagues10 in their outstanding experimentalinvestigations of the onset of a detonation for flames propa-gation in tubes, is the occurrence of localized explosions~called hot spots or explosion centers!, downstream the pre-cursor shock wave, near the tube wall in the boundary layer,and in the vicinity of the flame brush usually. Recently, asimilar phenomenon has been reproduced by a 2D numericalsimulation41 of a laminar flame with a temperature ratio of 5,a laminar flame speed governed by a one-step Arrhenius law,with a propagation Mach number~relative to the fresh mix-ture! of 0.045. The flame propagates in a small close tubewith adiabatic walls and a channel width not larger than 10times the flame thickness. This result shows that turbulentflows and fast turbulent flames are not crucial for under-standing the basic physics of DDT.41 The key mechanism isthe gradient of temperature~gradient of induction delay! thattriggers thermal explosion and transition as discussed below.Sivashinsky and his co-authors9,41 emphasize the role of theadiabatic pre-compression induced by the hydraulic resis-tance exerted by the walls. Another possibility is the tem-perature gradient in the shocked mixture near the tube wallinduced by the viscous dissipation in the boundary layer.

Spontaneous ignition and spontaneous quenching

Auto-ignition by a proper gradient of the induction delayis known to produce strong blast waves and spontaneousinitiations of a detonation through a synchronization of thecompression wave and the chemical heat-release.42,43This isillustrated by the following simplified example in planar ge-ometry. When neglecting both the flow velocity and the gra-



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dient of sound speed for simplicity, the Poisson equation forthe pressure in a reactive medium is written as

]2

]t2 p2a2]2

]x2 p5~g21!roQ

te

]

]tÃ~x,t !. ~9!

A constant gradient of induction time,dt i /dx, produces anunsteady distribution of heat release rate that may be roughlyapproximated by a traveling wave of induction front in theform Ã5V@x2 t(dt i /dx)21#. The feedback upon the rateof chemical heat-release following ignition has been ignoredfor simplicity. For the very specific condition of a speed ofthe ignition front equal to the sound speed, (dt i /dx)21

5a, Eq. ~9! presents a secular solution describing a simplewave for the pressure pulse propagating in the same directionas the induction front with an amplitude growing linearlywith time like (t/te)roQV(x2at), so that the CJ pressure isquickly reached. For an initial condition corresponding to abell shaped profile of temperature,To(x), with its maximumat the origin,Tmax5To(x50), numerical simulations of theconservation equations for a reacting gas in planar geometryshows that, after a time delayt i(Tmax), a constant volumeexplosion proceeds in a thin slab centered at the origin.42–44

Each neighboring slab reacting after a time delay increasingwith decreasing the temperature,t i(T).t i(Tmax), the size ofthe pocket of constant volume explosion increases at a speedwhich is first higher than the speed of sound, (dt i /dx)21

.a, and decreases with the initial temperature as the inverseof the temperature gradient, (1/t i)(dt i /dx) 52b (1/To)3(dTo /dx), see Eq.~2!. This is true until the inductionfront reaches the point where the condition (dt i /dx)215ais satisfied where, according to Eq.~9!, a run-away of thepressure occurs. The pressure reaching very quickly thevalue of the Neumann pressure of the CJ detonation, a CJwave is spontaneously initiated at this point.42,43

The CJ wave, however, propagates subsequently in anonhomogeneous region of fresh mixture with a gradient oftemperature,udTo /dxuÞ0. It was also shown that the sametemperature gradient that has caused the spontaneous initia-tion at high temperature, may quench the detonation at alower temperature.44 This spontaneous quenching mecha-nism is easily described in planar geometry by assuming thatthe induction zone evolves in a quasi-steady state approxi-mation. The variations of the induction-zone length resultsfrom the variations of the Neumann temperature, accordingto Eq.~2!. The quasi-steady approximation is valid when theplanar detonation is stable and when the characteristic timeof evolution is longer than the transit time across the deto-nation structure, namely when the characteristic length-scaleof temperature variation is longer than the induction zonelength, (l i /To) u dTo /dx u!1. The reaction-zone then movesrelatively to the leading shock with a velocity equal to thetime derivative of the induction-zone length. This produces adifference of mass flux across the two zones of density varia-tions, namely the heat-release-zone and the leading shock.Assuming a sonic condition at the end of the reaction, quasi-steady conservation of mass, momentum and energy acrosseach of these zones, leads to a nonlinear equation for theNeumann temperature~or the propagation velocity of theleading shock!, similar to Eq.~7!, but whereR on the right

hand side is replaced by the characteristic length of variationof the temperature, 1/R→ (1/To) u dTo /dx u. The turningpoint here describes the critical condition for spontaneousquenching.44 It occurs for a temperature gradient with a char-acteristic length scale which is much larger than the induc-tion zone-length by a factor of the order ofb measuring thetemperature sensitivity of the induction delay. Such aquenching has been observed in 1D direct numericalsimulations44 under different conditions, including pulsatingdetonations for which the quasi-steady approximation is notvalid.

This shows how subtle and delicate are the conditionsfor a spontaneous initiation of a detonation at high tempera-ture and for a subsequent transmission in the colder medium.This may explain why the experiments on transition fromdetonation to deflagration in tubes often appear as non easilyreproducible, small disturbances or fluctuations of the tem-perature gradients in the shocked gas modifying the condi-tions for both spontaneous ignition and quenching.

CONCLUSION AND PERSPECTIVES

The strongly nonlinear dynamics of detonation frontshave been successfully described by two types of perturba-tion methods:

~1! The oscillatory instabilities and the cellular structuresof detonations with triple-points represented by cusps travel-ing in the transverse direction at the speed of sound havebeen described by a weakly nonlinear analysis of unstableoverdriven regimes in the Newtonian limit. A quasi-isobariccoupling of the reaction-zone with the leading shock hasbeen identified as the basic instability mechanism respon-sible for the unsteady patterns. The effects of compressibilityare found to be stabilizing. The sonic point in the structure ofCJ waves does not appear to be essential in these fully un-steady phenomena which are governed by the time-scale ofthe entropy-vorticity wave, namely the transit time of a gasparticle across the planar detonation structure. The cell sizepredicted by the analysis is larger than the detonation thick-ness by a factor proportional 2p/MN ~between 13 and 20 inordinary conditions! times a number about 2 or larger, result-ing from a nonlinear selection mechanism. This factor fur-ther increases with increasing the induction-zone lengthwhen approaching the chemical-kinetic detonability limit.

~2! On the other hand, the critical conditions for directinitiation in spherical geometry, or for spontaneous initiationand quenching have been obtained as turning points of quasi-steady solutions in the limit of an induction-zone lengthstrongly sensitive to variations of shock velocity (b@1).The critical radius obtained in the limitb@1, see Eq.~8!, islarger than the induction-zone length by a large factor pro-portional to 8jeb/(g21), showing that the thermal sensi-tivity is here an essential mechanism.

Further developments in detonation theory may arise bycoupling these two types of analyses for including the fullyunsteady and multidimensional effects in the description ofcritical conditions. This requires to take into account thetransonic region in the intrinsic dynamics of the self propa-gating CJ waves. Such a program is in process and analytical



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Documents

Theory of gaseous detonations - Cranstheory over the last ten years. They concern nonlinear dynamics of detonation fronts, namely patterns of pulsating and/or cellular fronts, selection