M. Short and J.W. Dodd- Linear Stability of a Detonation Wave with a Model Three-Step Chain-Branching Reaction

8/3/2019 M. Short and J.W. Dodd- Linear Stability of a Detonation Wave with a Model Three-Step Chain-Branching Reaction

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Mathl. Comput. Modelling Vol. 24, No. 8, pp. 115-123, 1996Copyright@1996 Elsevier Science Ltd

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Lin e a r S t a b ili t y o f a De t ona t i on Wa ve w it ha Mode l Th r e e -S t e p C ha in -B ra n c h i n g R e a c t i onM. SHORT+AND J . W. DOLD

School of Mathematics, University of BristolBrist ol BS8 ITW, U.K.

Abstr act-The linear ta bilityof a planar detonation wave with a three-step chain-branching reac-tion is studied by a normal mode approsch. The reaction model consists of a chain-in itiati on step anda chain-branching step governed by Arrhenius kinetics, with a chain-termination step which is inde-pendent of temperature. It mimics the essential reaction dynamics of a real chain-branching chemicalsystem. The linear stabil ity of the steady detonation wave to two-dimensional disturbances is stud-ied with the chain-branching crossover temperature, i.e., the temperature at which chain-branchingand chain-termination rates are equal, ss a bif urcation parameter. This parameter determines theratio of the length of the chain-branching induction zone to the chain-termination zone within thesteady detonation wave. The effect of linear transverse disturbances is considered for two values ofthe chain-branching crossover temperature: in one the planar steady detonation wave is stable toone-dimensional disturbances, while in the other it is unstable to such disturbances.Keywords-Detonation, Stab ili ty, Chain-branching reactions.

P. INTRODUCTIONThe problem of th e one-dimensional linear insta bility of a detonat ion wave in a chemical mixtur ewhich was assumed to react via an idealised one-step Arrhenius reaction was first studied byEr penbeck [1,2]. A Laplace tra nsform technique was used to analyse th e behaviour of sma llam plitude distu rban ces from the plan e stea dy detona tion wave. Later, Lee an d Stewar t [3] useda norma l mode appr oach to addr ess th e linear inst ability problem. Their nu merical shootingtechnique provided a str aightforwar d way of calculating th e sta bility s pectr a.

Experiment al stu dies on one-dimensiona l pulsat ing detonat ion insta bilities su ch as [4] an d onmu ltidimensiona l tr an sverse insta bilities su ch as [5] demonstr at e a distinct s ensitivity to th e typeof th e chemical mixtur e used. In particular, a majority of th e mixtur es u sed in th ese experimentsinvolved th e rea ction of hydr ogen an d oxygen, which occurs t hr ough a chain-bra nchin g rea ctionmechanism . This involves a sequence of cha in-initiat ion, cha in-bra nching, an d cha in-termina tionsta ges. A sma ll am ount of reactan t is convert ed into cha in-car riers, which ma y be either freera dicals or atoms, by means of th e slow chain -initiation reactions. The chain-car riers ar e th enra pidly multiplied th rough chain-bra nching reactions, while th e rise in concentr at ion of cha in-ra dicals is ret ar ded by cha in-terminat ion steps which occur either thr ough a bsorption at , th evessel walls or th rough th ree-body collisions in th e int erior.

In th e following, both th e one- and twedimensiona l linear inst abilities of a plana r detonat ionwave with a model th ree-step chain-bran ching reaction ar e considered. The model consists ofan Arr heniu s type chain-initiat ion step with a large activation energy and an Arr heniu s t ypeM. Short was supported in this work by an E.P.S.R.C. research grant and J. W. Dold was supported by anE.P.S.R.C. Advanced Fellowship.t Present address: Department of Theoretical and Applied Mechanics, University of Ill inois, Urbana, IL 61801,U.S.A.

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11 6 M. SHORTAND J . W. DOLD

chain-branching step with a moderate activation energy. These two stages are assumed to bethermally neutral. Finally, an exothermic temperature-independent chain-termination reactionconverts chain-radicals into products. The reaction model thus retains the essential chemicaldynamics of a real chain-branching reaction, unlike the idealised one-step Arrhenius kinetic modelused previously [l-3]. The linear stability analysis is conducted for varying values of the chain-branching crossover temperature, i.e., the temperature at which the chain-branching and chain-termination rates are equal. This parameter has a major influence on determining the ratio ofthe length of the chain-branching induction zone to the chain-termination zone within the steadyplanar detonation.

2. MODELThe thermodynamic behaviour of the fluid is modelled with the compressible reactive Euler

equations written in nondimensional form as~+pv.u=o,+ -lvp = 0, Dp-g+Pt = ,where the variables p, u = (~1, uz), p, and e are the density, velocity, pressure, and specificinternal energy, respectively. We assume a polytropic equation of state and an ideal thermalequation of state, such that

e=&-Q T=?l P (2)where q represents the local chemical energy and T represents the temperature. The scales for thedensity, pressure, temperature, and velocity are the post-shock density, pressure, temperature,and sound speed, respectively, in a steady detonation wave, described in Section 3. The scalingsfor length and time are given below.

Based on the early reaction model of Gray and Yang [6] in their study of the homogeneousexplosion of a chain-branching reaction, we propose that the essential dynamics of a global chain-branching reaction can be represented by three main stages: chain-initiation, chain-branching,and chain-termination reactions with the rate constants Icr, kB, and kc, respectively. On thisbasis, a model three-step chain-branching reaction can be represented via the following threereaction stages:

I: F + Y (Initiation),B: F + Y -+ 2Y (Chain-branching), kB=eXp($(&-$)) (3)c: Y + P (Chain-termination), kc = 1.

Kapila [7] has used a similar model to study the homogeneous explosion problem in the limitof high activation energies, except now the chain-initiation reaction is included. Based on thereaction dynamics of a number of chain-branching reactions (see [6] and, for example, [8]), thefollowing assumptions are made regarding the model reaction mechanism. The chain-initiationand chain-branching rate constants kr and kB are taken to have Arrhenius temperature dependentform with inverse activation energies cl and eB, respectively. The chain-termination reaction isassumed to be first order, independent of temperature, and is modelled as having a fixed rateconstant. The reaction model (3) is then the simplest realistic mechanism which reproduces thedynamics of a chain-branching reaction. Generalisations such as higher order chain-terminationreactions, like those considered by Gray and Yang [6], could also be considered in order tofurther improve the model. The reference time t, is then scaled such that the chain-terminationrate constant is unity, i.e., kc = 1, with the reference length set to t, times the sound speed


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Detonation Wave Stability 117

behind the detonation shock. The remaining parameters in (3) are T I and T B. These representchain-initiation and chain-branching crossover temperatures, respectively, in which the chain-initiation and chain-branching rates become ss fast as the chain-termination rate. Consumptionequations for fuel and radical then become

DfDt= -rI - rgDYz=rI+TB-Tc, (4where

Tc =y. (5)Finally the chemical energy q is defined [9] as

q=Q-Qf-(Q+D)y, (6)where Q > 0 represents the total chemical energy available in the unreacted mixture and D rep-resents the amount of endothermic energy absorbed by the initiation and chain-branching reac-tions I and B in breaking down the reactant F into the energetic radical Y.

In general, the chain-initiation reaction is energetically inhibited, so that given the steadypost-shock detonation temperature of unity, we assume

T I >l and cI


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118 M. SHORT AND J. W.DOLDThe spatial structure of the detonation is determined by numerical quadrature of the first-orderequations for fuel and radical concentration

(13)Immediately behind the shock, the following conditions apply:

p* = p* = T = 1, u; = M,, u; = 0, f = 1, y* = 0. (14)The detonation overdrive d is defined by

(15)where M& is the Chapman-Jouguet detonation velocity, determined by the velocity being exactlysonic uf = dm at the end of the wave where q* = Q.

Figure 1 shows the steady detonation profiles for (a) fuel f and radical y and (b) temperature T ,respectively, with distance behind the detonation shock for varying values of the chain-branchingcrossover temperature TB = 0.8, 0.85, 0.9, and 0.95 at fixed Q = 3, EI = l/20, &B = l/8, T I = 3,7 = 1.2 and d = 1.2. For the lower values of T B , the steady detonation structure consists of ashort chain-branching induction zone and much longer chain-termination region. As the chain-branching crossover temperature increases, the chain-branching induction zone becomes longer,with a lower peak concentration of chain-radicals being established. For the higher T B c 1, thedetonation structure is dominated by a chain-branching induction zone being much greater inlength than the chain-termination region, with only a very moderate rise in peak concentration ofchain-radicals. For T B > 1, chain-branching is inhibited [9] and the reaction is, for all practicalpurposes, quenched. Only the extremely slow initiation reaction is able to bring about anychanges at all. We will not consider T B 2 1 in this article.

4. LIN EAR STABIL ITY ANALYSISWe now present a normal mode linear stability analysis of the steady detonation structure

(lo)-(13) with the model chain-branching reaction scheme (4)-(5) for a range of values of thechain-branching crossover temperature T B . This proceeds by defining a shock-attached coordi-nate system

z=X+D,St-+(y,t), (16)where 0: is the detonation Mach number relative to the shocked gases and $(y,t) representsthe perturbation to the steady shock location due to a small disturbance. Perturbations to thesteady detonative wave structure are then sought in the form

a = z*(2) + z(z)eateiky, $=e e ,t ik y (17)where

z = (V 211, 212,P, f, YjT , (18)v = l/p is the specific volume, Re(cy) is the disturbance growth rate, Im(cy) the disturbancefrequency, and k the disturbance wavenumber. For brevity, we will omit the superscript * inwhat follows, the quantities without the superscript referring to the underlying steady-wavesolution. Substituting expansions (17) into equations (l)-(6) results in a system of first-orderlinear differential equations for the vector of complex perturbation eigenfunctions z(z), namely

cyz + AZ; + ikBz + Cz - & - ikBb = 0, b=z=, (19)


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Detonation Wave St ability 119

0 . 8

0 . 6G

0 . 4

5 10 15x(a)

5 10 15x

(b)Figur e 1. Stea dy detonat ion profiles showing (a) f (solid lines) an d y (dash ed lines)an d (b) T. The symbols ma rk correspondin g profiles for TB = 0.80 Cl),B = 0.85(O), TB = 0.90 O),nd TB = 0.95 (A).

where

A=0 Ul 0 ; 0 0 112 0 -?I 0 0 0u2 0 0 0 01

0 0 7.4 x 0 0Y (20)0 0 -yp 212 0 00 0 0 0

u200 0 0 0 IL.2I0 Ul 0 0 0 , B=0 -yp 0 111 0 00 0 0 0 211 00 0 0 0 0 Ul II& 215 0 0 0 0 -PZr Ulx 0 0 0 0

cc 0 u2z 0 0 0 0 .- GJ Px 0 YUls 0Pv fz 0 Pp Pf - - i ;

_ - Pv El 2 0 - Pp - Pf - A/ +1 _

(21)


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120 M.SHORTAND J. W. DOLDThe new quantities in C are

and(23)

The pertur bation shock conditions ar e determ ined from th e Rank ine-Hugoniot conditions ss

v= (r+ $iM2&'2 (1+ M2) a

4 = (r+1)M2 7 4 =(D, -Mg)ik,

p = -;;yl;, (24)f = 0, y = 0.

Finally, we apply a rear -boun dar y condition at z = 00 such that th ere should be no acoust icwaves propagating upst rea m from infinity, leading to th e acoust ic ra diation condition

(25)

where Urb, Vb an d cb ar e th e unper tu rbed s-velocity, specific volume, an d sound speed, respec-tively, in th e burnt gases. For a given set of initial therm odynam ic an d chemical p ar am eters cha r-acter ising th e st eady det onat ion st ru ctur e, t he complex eigenvalues (Y an d eigenfunctions Z(Z)ar e determ ined by a nu merical shooting technique based on tha t of Lee an d Stewart [3]. Thisinvolves tak ing an initial guess for (Y, integrat ing equat ions (19) from t he pert ur bed shock condi-tions (24) into the bur nt zone an d itera ting on th e initial guess for a un til th e acoust ic ra diationcondition (25) is sat isfied in th e burn t zone at a point where th e reaction ra te is exponent iallysmall.

5. LINEAR STABILITY RESULTSFigure 2 shows the region of insta bility to one-dimensiona l distur bances of th e stea dy det o-

na tion str uctur e (lo)-(13) an d th e migration of nine un sta ble roots when the cha in-bra nchingcrossover temperatu re varies in the ran ge T B E [0.8,1.0] for Q = 3, er = l/20, Ed = l/8, T I = 3,7 = 1.2, d = 1.2, an d wavenu mber k = 0. Figure 2a shows the behaviour of the growth rateRe(o) against T B while Figure 2b shows th e corr esponding frequen cy Im (o) of th e un sta bleroots. For T B < 0.803, th e steady p lana r detonat ion is foun d t o be stable to one-dimens iona ldisturban ces. In this range, Figure 1 shows tha t th e steady detonation stru ctur e is dominatedby th e temper at ur e-independent cha in-termina tion reaction, with only a short cha in-bra nchinginduction zone. There ar e similar ities here to th e sta bility properties of a detonat ion wave withan ideal&d one-step chemical rea ction m odel ha ving zero activat ion energy as stu died by Er -penbeck [lo]. The genera l conclusion, which is reinforced here, is th at temper at ur e-independentreactions render th e detonat ion more sta ble th an temp erat ur e-dependent chemical kinetics. AtT B = 0.803, a neutr ally st able mode (Y = or is presen t, ha ving Re(crr) = 0. Fr om Figure 2b,it is seen that th e neutr ally st able mode h as a compa ra tively low frequency of oscillation withIm(crr) = 0.219. As T B increases through T B = 0.803, a Hopf bifur cat ion occurs ren dering t hedetonat ion un sta ble to th e low frequen cy distur ban ce. At T B = 0.821, a second un sta ble mode a 2appea rs having a mu ch higher frequency th an th e mode ~1 with Re(as) = 0, Im(crz) = 1.11. AsT B increases fur th er, additiona l un sta ble roots appea r, each having a higher frequency th an th eprevious root. At T B = 0.85, th ere is th e low frequen cy inst ability with Re(oi) = 4.62 x 10d2an d Im(crr ) = 0.12 an d two high frequen cy inst abilities, op with Re(crz) = 4.01 x low2 an dIm (crs ) = 0.77, an d (~3 with Re(as) = 9.97 x 10s3 a nd Im(cYs) = 1.49. Despite th e fact t ha t t he


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Detonation Wave Stability 121

0.80 0.85 0.90 0.95 1.00TB

(a)2.03

1.5 -

3g l.O-

0.5 -

0.0 . . . , . . . . , . . , , . , . . ,0.80 0.85 0.90 0.95

Ts1.00

(b)Figure 2. The variation in (a) the growth rate Re(a) and (b) the frequency Im(a)of one-dimensional instabili ties as TB is varied. Nine roots are plotted. Purely redroots are shown ss dashed l ines.

detonation becomes more unstable as T B is increased, it is the low frequency mode than remainsthe fastest growing mode at any value of T B . At T B = 0.913, the low frequency mode splitsinto two purely real roots, shown by the dashed line in Figure 2a. One branch of the root hasa growth rate which continues to dominate with T B , while the other has a growth rate whichdecays with T B .For the representative parameter values chosen above, the steady planar detonation is unstableto one-dimensional perturbations for su5ciently large chain-branching crossover temperatures,i.e., T B > 0.803. For smaller T B , the model chain-branching reaction scheme produces a detona-tion structure dominated by the chain-termination region. As the chain-termination reaction rateoccurs independently of changes in temperature, there is no strong coupling between gasdynamicperturbations and the chemistry in this region. It is then seen that the detonation is stable toone-dimensional perturbations. As T B increases, the chain-branching induction zone, in whichthere is a stronger coupling between gasdynamic perturbations and chemistry, becomes longer.


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12 2 M. SHOP AND J. W. DOLDThe detonat ion is th en un sta ble initially to a low-frequency one-dimen siona lpertur bation, withaddit iona l high frequency i~tabiliti~ appea ring as TB increasesfurther.

0 1 2 3 4Figure 3. T he variation in &(a) of transverse instabilities as the wavenumber k isvaried for TB = 0.79. Two roots are shown.

0 1 2 3k

Figure 4. The variation in RR(a) of transverse instabil iti- ss the wavenumber k isvaried for ?& = 0.85. Three roots are shown.Althou~ th e sta bility of detona tion waves to on~dimensional distu rba nces is relevan t to t he

one-dimen siona l pulsa ting detonat ion waves seen in th e experim ent s of Alpert an d Toong [4],detonat ion wa ves tr avelling n sh ock tu bes exhibit a complex tra nsversewave stru ctu re [5] an d nota one-dimen siona l pulsa ting type of insta bility. An importa nt quest ion which t hen ar ises is howth e growth ra tes of t ra nsvers e distu rba nces compa re to th ose of one-dimen siona l pert ur bat ions.Figure 3 shows the cha nge in t he growth ra te Re(cu) agains t tr an sverse wavenum ber k for th etwo m odes cyi an d cr2 in a detonat ion with T B = 0.79 an d th e oth er par am eter s fixed as a bove.For T B = 0.79, th e plan ar det ona tion wave is stable to one-dimen siona l pert ur bat ions withR.e(ai) = -2.57 x 10e2 an d Im (cur ) = 0.25 an d Re(a2) = -9.78 x 10m 2an d Im(crs) = 1.67at ic:= 0. As th e wavenum ber k is increased from k = 0, both modes become un sta ble withor becoming un sta ble at k = 0.093 an d 02 becoming u nst able at k = 1.035. Cairn s growthra tes a re IXe(crl) = 0.160 at k = 0.71 an d Refc~) = 0.160 a t k = 2.542. Thu s, th e second


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DetonationWave S tability 123mode (~2 possesses a greater maximum growth rate at a higher wavenumber, k = 2.542. Bothmodes become st able for su fficient ly lar ge waven um bers, i.e., k = 1.56 for al an d k = 3.78 for (~2.

Figure 4 shows the variation of F&(a) against wavenumber k for the three modes ~1, 02,and ~3 with T B = 0.85. Here, the plana r steady detonat ion is un sta ble to each of th ese modesfor k = 0, possessing growth rates as given above. However, as k increases, the growth rates ofeach of th ese modes increases, demonst ra ting th at th e tr an sverse inst abilities ar e more unst ableth an t he one-dimensional insta bilities. The ma ximu m growth ra tes of each of th ese modes ar eR,e(cwl) = 0.120 at k = 0.355, RR(W) = 0.137 at k = 1.13, and Re(a3) = 0.113 at k = 2.06.The ma ximum growth ra tes of th e tr an sverse distur bance ar e significant ly lar ger th an th e corr e-sponding growth ra tes of th e one-dimensional insta bilities. Again t he second m ode (~2 possessesth e highest overall growth ra te and all th ree modes ar e observed to stabilize a t sufficiently highwavenum bers. Thus , in th e former case where TB = 0.79, alth ough th e detonat ion is sta ble toone-dimensional distur ban ces, it is un sta ble to tr an sverse distur bances. In th e latt er cases whereT B = 0.85, the transverse disturbances would be predicted to grow much more rapidly thanth e one-dimensiona l distur bances ultimat ely giving rise to th e complex tr an sverse wave str uctur eobserved in experiments.

6. SUMMARYThe linear sta bility of a plana r stea dy detonat ion wave ha s been stu died where th e chemical

kinetics of th e reactive mater ial are modelled by a th ree-step chain-bran ching reaction. Forsufficiently sma ll cha in-bra nching crossover temp erat ur es, T B, the steady detonation stru ctur e isdominated by th e temp erat ur e-independen t cha in-termina tion zone an d th e detonat ion is foun d tobe sta ble to one-dimensiona l pert ur bat ions. As th e length of th e cha in-bra nching induction zoneis increased by increasing T B, th e detonat ion becomes un sta ble to one-dimensional pertu rba,tionswith t he fast est growing ins ta bility ha ving a low frequen cy of oscillat ion. For th e case TB = 0.79,the steady detonation wave is stable to one-dimensional disturbances, but is found to be unstableto transverse disturbances. For the case T B = 0.85, th e stead y detonation wave is un sta ble toone-dimens iona l distu rban ces, but the maximum growth ra te of tr an sverse instabilities is foun dto dominate th at of th e one-dimensional insta bilities.

REFERENCES1.2.3.4.5.6.7.8.9.

10.

J.J. Erpenbeck,Stabilityof steady-state equil ibrium etonations,Phys. Fluids5, 604-614 (1962).J.J. Erpenbeck,Stabilityof idealized ne-reactiondetonations,Phys. Fluids 7, 684-696 (1964).H.I. Lee and D.S. Stewart, Calculationof linear nstability:One-dimensionalnstabilityof planedetonation,J.F.M. 216, 103-132 (1990).R.L. Alpertand T.Y. Toong, Periodicity in exothermic hypersonic f lows about blunt projectil es, Acta A s&on.17, 538-560 (1972).R.A. Strehlow, Multi-dimensional detonation wave structure, Astro. Acta 15, 345-357 (1970).B.F. Gray and C.H. Yang, On the unif ication of the thermal and chain theories of explosion limits, J. Phys.Chem. 68, 2747-2750 (1965).A.K. Kapila, Homogeneous branched-chain reactions: Initi ation to completion, J. Eng. Maths. 12, 2:!1-235(1978).F.A. Williams, Combustion Theory, Addison-Wesley, Reading, MA, (1985).J.W. Dold and A.K. Kapila, Comparison between shock ini tiation of detonation using thermally-sensi tiveand chain-branching chemical models, Comb. Flame. 85, 185-194 (1991).J.J. Erpenbeck, Stab ili ty of ideal ized on&reaction detonations: Zero activation energy, Phys. Fluids 8,1192-1193 (1964).

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M. Short and J.W. Dodd- Linear Stability of a Detonation Wave with a Model Three-Step Chain-Branching Reaction