Mechanisms of Detonation Formation due to a Temperature Gradient

Transcription

1 Mechanisms of Detonation Formation due to a emerature Gradient A. K. Kaila, D. W. Schwendeman Rensselaer Polytechnic Institute, roy, New York J. J. Quirk Los Alamos National Laboratory, Los Alamos, New Meico and. Hawa, University of Minnesota, Minneaolis, Minnesota Dedicated to John F. Clarke. he reactive gasdynamics community is the richer for him. Abstract Emergence of a detonation in a homogeneous, eothermically reacting medium can be deemed to occur in two hases. he first hase rocesses the medium so as to create conditions rie for the onset of detonation. he actual events leading u to reconditioning may vary from one eeriment to the net, but tyically, at the end of this stage the medium is hot and in a state of nonuniformity. he second hase consists of the actual formation of the detonation wave via chemico-gasdynamic interactions. his aer considers an idealized medium with simle, rate-sensitive kinetics for which the reconditioned state is modelled as one with an initially rescribed linear gradient of temerature. Accurate and well-resolved numerical comutations are carrried out to determine the mode of detonation formation as a function of the size of the initial gradient. For shallow gradients the result is a decelerating suersonic reaction wave, a weak detonation, whose trajectory is dictated by the initial temerature rofile, with only weak intervention from hydrodynamics. If the domain is long enough, or the gradient less shallow, the wave slows down to the Chaman-Jouguet seed and undergoes a swift transition to the ZND structure. For shar gradients gasdynamic nonlinearity lays a much stronger role. Now the ath to detonation is through an accelerating ulse that runs ahead of the reaction wave and rearranges the induction-time distribution there to one that bears little resemblance to that corresonding to the initial temerature gradient. he ulse amlifies and steeens, transforming itself into a comle consisting of a lead shock, an induction zone, and a following fast deflagration. As the ulse advances, its three constituent entities attain rogressively higher levels of mutual coherence, to emerge as a ZND detonation. For initial gradients that are intermediate in size, asects of both the etreme scenarios aear in the ath to detonation. his study is guided by, and its results comared with, eisting asymtotic results on detonation evolution.

2 Introduction here are a number of ractical ways in which a homogeneous combustible, initially in an essentially inert state, can be made to detonate. Eamles include (i) subjecting the material to a strong (incident or reflected) shock, (ii) injecting a hot turbulent jet of combustion roducts into the unreacted material, (iii) generating a flame in a rough-walled or obstacle-filled ie where it becomes turbulent, undergoes distortion, accelerates and transits into a detonation, (iv) inducing hotochemical initiation by irradiating the medium with a burst of energy, and (v) holding the material at a sufficiently elevated temerature for a sufficiently long time (cook-off). Lee [] has classified the modes of initiation into two categories, self initiation and direct initiation. Self-initiation is a slow mode, where the igniting stimulus is weakand the energy resonsible for driving the medium to detonation comes from the chemical energy of the medium itself. Direct initiation is a fast mode and is rovoked by a sufficently strong igniter. In either case, it is generally agreed that rior to the onset of detonation the reactive medium has been reconditioned into a hot, satially nonuniform state. his nonuniformity leads to a gradient in induction times, and detonation is the outcome of the chemico-gasdynamic rocesses associated with this gradient. he urose of this workis to resent a comutational study of the variety of ways in which chemical heat release and comressibility of the nonuniform, reconditioned medium act in concert to roduce a detonation, and to comare the numerical results with available asymtotics. he descrition is based on a simle mathematical model, that of an ideal fluid undergoing a one-ste, eothermic, Arrhenius reaction in a lanar configuration. he nonuniformity is modelled by a linear temerature gradient imosed at the initial instant. he activation energy in the Arrhenius law is assumed large, in view of the observed strong temerature-sensitivity of the reaction rate. he medium is confined between two arallel walls, and the searation between the walls is such that over the time scale of interest, signals generated at the hot wall have not yet arrived at the cold wall so that effectively, the configuration is semi-infinite. he temerature gradient results in the initial reaction rate diminishing with distance away from the hot wall, at a rate determined by the size of the gradient. Our interest is in situations where a detonation is born within a distance from the hot wall comarable to an acoustic length based on a characteristic chemical time, the induction time at the hot wall. he temerature-gradient model has received considerable attention in the literature, and two concetual frameworks have been advanced to relate the size of the gradient to the evolutionary outcome. he first is due to Zeldovich [], who found it eedient to suress gasdynamics altogether, thereby introducing the concet of sontaneous combustion. In this situation the arcel at osition, oblivious of its neighbor, ignites at its rescribed induction time t I (), so that the initial gradient of induction times, t I (), translates into a notional wave seed, /t I (), which Zeldovich termed the seed of sontaneous roagation, U S. He then identified four regimes of roagation by comaring U S with the Chaman-Jouguet detonation seed D CJ, the normal flame seed U f and the sound seed c. He argued that I. for U S >D CJ the reaction wave is a weakdetonation with ressure increase across it falling between the Chaman-Jouguet and the constant-volume-elosion limits, II. for c<u S D CJ there aears a shockwave and a transient rocess culminates in a CJ detonation, III. for U f <U S c<d CJ the reaction wave roagates at the sontaneous wave seed with little ressure change across it, and IV. for U S <U f the rocesses of diffusion come into lay to move information ahead of the sontaneous wave, resulting in a flame. Zeldovich s ideas, while etremely instructive, are limited in accuracy. As already indicated, nonuniformities awaken gasdynamics which, in turn, can modify the initial gradient significantly, thereby altering the evolutionary scenario. he second concet, termed SWACER (ShockWave Amlification due to Coherent Energy Release) is due to Lee and colleagues [], [], []. It does take gasdynamics into elicit consideration and argues that if the initial gradient is to rovoke a detonation, the shock generated by the elosion of the hottest arcel must amlify as it roagates down the gradient. his requires, in turn, a certain synchrony between the

3 rogress of the shockand the sequential release of chemical energy by successive arcels of the medium, a situation reminiscent of Rayleigh s criterion for the amlification of a single ressure ulse. It is clear that while SWACER envisages a shock accelerating to form a detonation, Zeldovich s criterion also admits the ossibility of a detonation emerging from the deceleration of a reactive wave. It makes sense, therefore, to conduct a careful study in which a sequence of rescribed initial gradients is considered, and the corresonding modes of evolution analyzed, to see whether one or the other of the two mechanisms dominates, and if so, how the degree of dominance shifts as the size of the gradient is varied. Asects of the roblem have been eamined, both comutationally and analytically, in a number of studies over several decades. One of the earliest numerical investigations, due to Zeldovich et al [], had a low resolution and a considerable amount of noise. Nevertheless the authors demonstrated successfully that for very shallow gradients the ressure rise across the domain was nearly uniform, for larger gradients a fast reactive wave decelerated to form a detonation, and for gradients that were larger still the reaction wave and the shockfailed to coule together. Gelfand et al [] rovided more detail, concentrating on moderate gradients and establishing in articular that a shockless reaction wave decelerated to a sub-cj seed before develoing a shockand then accelerated to form a quasisteady detonation comle. Makhviladze and Rogatykh [7] sought to quantify the influence of gasdynamics on the urely chemical (sontaneous) evolution by eamining the amount by which ressure in a chemico-gasdynamic calculation would eceed that in a urely reactive calculation. From the numerical comutations a critical value of this measure as a function of the imosed gradient was identified, searating gradients that led to a sontaneous wave from those that would roduce a detonation. Khokhlov et al [] considered a semi-infinite domain in a uniform state, ecet for an embedded kernel in which both temerature and reactant concentration varied linearly, thereby roducing an induction-time gradient across it. A shockless reaction wave, decelerating away from the oint of least induction time, emerged. he goal of the investigation was to find two critical values of the size of the kernel: one for which the reaction wave develoed a shock-reaction comle, and the other for which the shock-reaction comle survived and assed the detonation into the cold region outside the kernel. A similar configuration was eamined by Clavin and He [9] who concluded that once the detonation had initiated within the gradient ocket, it would roagate into the uniform, cold region beyond only if the temerature of this region eceeded a critical temerature determined by the gradient. In a later study these authors [] develoed critical conditions for the initiation of a H O miture subjected to satially nonuiniform hoto-chemical irratiation. Among analytical studies, those of an asymtotic nature based on the limit of large activation energy (Kaila and Dold [], Dold, Kaila and Short [], Kaila and Dold [] and Short []) have led to artial but significant advances. In this limit, small initial nonuniformities suffice to create significant initial gradients in induction time. he early hase of evolution can then be eamined by means of a smalldisturbance treatment of the equations of reactive gasdynamics. he onset of elosion (i.e., substantial and raid variation) is signalled by the develoment of finite-time singularities in the solution of the reduced equations. For weakgradients in induction time the singularity locus is suersonic. hen, further asymtotic analysis on the full set of governing equations is feasible and reveals that the ath to detonation is through the deceleration of a high-seed reactive wave, in agreement of the first of the Zeldovich criteria. For strong gradients in induction time the singularity locus is subsonic. In this case a fuller asymtotic develoment is unavailable but artial results indicate that rovided the initial gradient is not too large, the outcome is a SWACER-like scenario. When the asymtotic analysis succeeds, it does so because it is able to divide the evolutionary event into a sequence of distinct stages with the following convenient roerties: when a stage is unsteady, it is amenable to a small-disturbance analysis, and when a stage involves large variations in roerties, these occur quasisteadily. In either case, a more-or-less analytical descrition of the stage is available, and the ieces can be fitted together to solve the uzzle. he descrition is available only at leading order, and therefore increasingly accurate for larger activation energies. When the activation energies are moderate the fit is less than erfect, and the scenario more comle than the analysis would suggest. A well-resolved comutational study analyzed in light of the asymtotic results is therefore in order, and is the object of this aer. Results of this study will also cover the situations for which an asymtotic treatment is unavailable, and in addition, rovide an accurate quantitative estimate (rather than an asymtotic order estimate) of the size of the gradient for which a articular mode of evolution holds.

4 here are other idealized setus that have been devised to study the assage to detonation, and these corresond to boundary-driven (rather than the resent, initial-condition-driven) configurations. Here the initial state is satially uniform, cold, and stagnant, and disturbance is introduced into the system across the boundary of a semi-infinite domain either through mechanical means (by imulsively moving the boundary into the domain as a iston) or thermal means (by eosing the stationary boundary to sufficiently raid heating). he ensuing events have been comutationally eamined and analyzed in equisite detail in a series of ioneering aers by Clarke and colleagues [], [], [7]. Also noteworthy is the investigation by Sileem, Kassoy and Hayashi [] who eamined the consequences of raid energy deosition in a boundary layer at the wall. In another related work, Short [9] considered the evolution of a eriodic initial disturbance in any one of the state variables. Asymtotic results for a large activation energy and a small, slowly-varying initial disturbance were sulemented by comutational results, which showed in articular the eistence of unsteady domains not redicted by the asymtotics. In the concluding section of the resent workwe shall comare our results with those in these earlier studies and show that the temerature-gradient model catures all the essential asects of the evolutionary scenarios described therein. his aer begins by introducing the governing equations, their nondimensionalization, and identification of the relevant arameters. he Zeldovich sontaneous-wave concet is resented net. his is followed by a brief account of the very considerable amount of information that the asymtotic analysis rovides. he numerical strategy is resented net, followed by a dislay and discussion of the numerical results for a broad range of initial gradients. Governing Equations he state of the medium is secified by the ressure, density ρ (or secific volume v =/ρ ), seed u and temerature. Progress of reaction is measured by the variable, which increases in value from to as the reaction roceeds from initiation to comletion. A dimensional quantity is signified by attaching a rime to the corresonding symbol, while absence of the rime indicates that the quantity is dimensionless. he reference values for the rimitive state variables are,,ρ,u = ρ, () where u is the isothermal sound seed at the reference state. Pressure, temerature and density are related via the gas law = ρ R W, where R is the universal gas constant and W the molecular weight (deemed identical for the reactant and roduct secies). he secific heats at constant volume and at constant ressure, C v and C resectively, obey the relation R W = C C v. For the simle one-ste Arrhenius kinetics adoted here, an aroriate unit of time is t = R E a B C v Q e(e a/r ), () where B is the re-eonential frequency factor, E a the activation energy and Q the secific heat of reaction. he reference length is taken to be the acoustic length L = u t. ()

5 Since diffusive transort lays a negligible role in the situation under study, the governing equations are simly the equations of reactive gasdynamics, written below in the dimensionless, conservation form, with ρ t +(ρu) =, () (ρu) t +( + ρu ) =, () (ρe) t +[( + ρe)u] =, () (ρ) t +(ρu) = ρr, (7) = ρ, () E = (γ )ρ + u q, (9) [ ( ɛ R = ( ) e )]. () (γ )q ɛ he variable E aearing above is the sum of the secific internal and kinetic energies, measured in units of u. Also making their aearance are the dimensioness heat release q and the dimensionless inverse activation energy ɛ, defined as q = Q u, ɛ = R E a. () he roblem is eamined in a semi-infinite domain,. (In comutations the domain is finite, with the right boundary far enough away so as not to interfere with the solution in the window of interest.) he left boundary is always taken to be a rigid wall, so that the aroriate boundary condition there is u(,t)=. () he initial conditions are quiescent and uniform, ecet for a linear gradient in temerature (and hence density), (, ) =, () (, ) =, () u(, ) =, () (, ) = α, () with α measuring the size of the gradient. It is useful to kee in mind that ɛ is a small arameter, reflecting the strong temerature-sensitivity of the reaction rate. It follows immediately from equation () that temerature changes of order ɛ roduce relative changes in reaction rate R of order unity. hat in turn suggests choosing α in equation () to be of order ɛ, so that in the initial temerature rofile, reaction rate varies by multiles of order unity over lengths of order unity. herefore we take α = ɛa, where now a measures the gradient on the ɛ scale. hen the initial temerature rofile reads (, )= ɛa. (7) he sontaneous wave When hydrodynamics is suressed, evolution occurs only in time and aears as a arameter through the initial condition. he governing equations reduce to u =, ρ = ɛa, = ( ɛa), =, ɛa (γ )q

6 while (, t) satisfies the initial-value roblem t = ɛ (γ )q [ ɛa +(γ )q ] e [ ɛ ( )], (, ) = ɛa.. rogress temerature reaction rate t Figure : he sontaneous elosion at =. his is Semenov s constant-volume thermal elosion roblem, for which the eact solution at each is monotonic, rising from = ɛa to = ɛa +(γ )q, as shown in figure for = and for the secific set of hysical and kinetic data listed in able. he reaction rate eaks at the induction time t i (; a), and is weakeverywhere ecet in a narrow interval containing t i. (Since only the relative size of the reaction rate is imortant, here and elsewhere R is normalized by a multilicative constant before lotting, with the constant chosen urely for grahical convenience.) As a result the temerature rise is small and gradual in the induction hase t<t i, etremely raid and substantial near t i, and ends with a leisurely relaation through a small, final increment. Figure dislays the variation of induction time with for a number of gradient values a. Each curve in the figure can be thought of as the locus of a sontaneous reaction wave for the articular choice of a. he corresonding wave seeds are dislayed in figure, and their starting values at the hot wall given in able. q = 7, γ =., ɛ =., e =., e =., c = 7., c =., D c =., n =.7, n =.7. able : Proerties of samle elosive.

7 .... t.... seed Figure : Sontaneous-wave loci and seeds for various a. a U S at = able : Sontaneous wave seed at = for various a. he asymtotic aroach We now give a very brief account of the substantial amount of information that has been obtained by ursuing the limit ɛ. his aroach divides the evolutionary event into several distinct hases that can be analyzed in sequence. he first hase, induction, involves small deartures from the initial state. For the sontaneous wave it has the small-disturbance eansion + ɛφ, with φ(, t) satisfying φ t = e φ,φ(, ) = a, and yielding the solution ɛ ln{t i (; a) t}, t i (; a) e a. In this aroimation, the temerature rise through the induction stage is of order ɛ, and the induction eriod is aroimated by the time-to-blowu of the small-disturbance solution. It is also a simle matter to see that the raid evolution following induction occurs on an eonentially small time scale, with the new time variable σ given by t i t = e σ/ɛ. hus the two major features of the evolution, i.e., the induction time and the tendency for a raid ost-induction rise, are both catured well by the asymtotics. Furthermore, there is the additional, and more imortant, imlication that an O(ɛ) small-disturbance theory could also be a natural vehicle to elore the early hase of the reactive-gasdynamic roblem; it would allow small gasdynamic disturbances to interact with weakreaction effects and otentially alter the local reaction rates by order unity. With gasdynamics switched on, we eect all state variables to undergo O(ɛ) changes during the induction stage, leading to the eansion +ɛφ, +ɛp, v +ɛv, u ɛu, ɛ Z, φ = P + V. () (γ )q 7

8 At leading order the governing equations yield a reduced set which can be maniulated into the characteristic form ( Pt ± γp ) ± γ ( Ut + γu ) = e φ = γφ t (γ )P t = Z t, (9) where the characteristics are simly the article aths d/dt = and the sonic lines d/dt ± = ± γ. he factor γ aears because the isothermal, rather than the adiabatic, sound seed was chosen as the reference seed. As anticiated, these equations model the interaction between linear acoustics and weak(but nonlinear) chemistry, and collectively, may be thought of as the satially nonhomogeneous generalization of the induction stage of Semenov s elosion. Note that the reactant-consumtion equation is secondary. First derived by Clarke, these equations, their occurrence in various contets, and their solution, have been discussed etensively [], [], [], []. Although the roblem is only numerically tractable in general, it is useful to obtain an imlicit reresentation of its solution for additional analysis. his is done by integrating along the characteristics to get and φ = γ [ P ln e a γ γ t { } ] γ e P (, t ) dt, () γ P = I + I + I, γu = I + I + I, () where I = t e φ(+ γt γt,t ) dt, () I = t (t / γ)h(t / γ) (t / γ)h(t / γ) e φ( γt+ γt,t ) dt, () I = e φ( + γt γt,t ) dt. () It is now ossible to eamine this solution as a function of the gradient arameter a. We consider three cases.. Moderate gradient, a = O() For a fied and order unity, the induction solution has been eamined in considerable detail ([], [], []). Blowu occurs first at the hot wall =, at time t I that decreases monotonically with a but is always bracketed by the constant-volume blowu time of unity and the constant-ressure blowu time of γ. A local analysis of the blowu structure reveals, at =, the limiting behavior φ ln(t I t)+o(), P ln(t I t)+o() as t t I, imlying that the density erturbation V = φ P remains bounded as ressure and temerature erturbations eerience a singularity. hus the blowu is locally at constant volume. For t>t I, the singularity moves away from the hot wall into the colder fluid, along a ath t = ˆt() which can be determined numerically. he singularity emerges from the boundary with infinite seed, but undergoes deceleration as it moves further into the medium. Behind the singularity, i.e., for t>ˆt(), the induction equations are of course no longer valid, but the full equations show, much in the same way as for the Semenov elosion discussed earlier, that chemical reaction goes to comletion in a thin zone with eonential raidity, and ressure, temerature, velocity and density all rise. hus the singularity locus describes to an ecellent aroimation the ath of a reaction wave into the medium. he wave, because of its very thin structure, is quasisteady, and can therefore be reresented as a succession of weak-detonation Rayleigh lines, with increasingly shallow sloes, on a ressure-volume lane, figure.

9 = C = v v Figure : Decelerating, quasisteady, weakdetonations. here must, of course, be an eansion behind the wave to accommodate the velocity boundary condition at the stationary wall. However, as long as the reaction wave remains suersonic relative to the state behind it, it remains oblivious to the following flow and its ath is determined entirely by the reconditioning in the induction region ahead. Signals from the backbegin to influence the ath of the wave only after it has slowed down sufficiently for a sonic oint to aear in the reaction zone. his haens when the sloe of the Rayleigh line has descended to the CJ-value, corresonding to the lower bound on the seed of a quasisteady wave. Unsteadiness must now intrude, and it does so at the rear of the reaction zone where a weakshock aears. With little change in wave seed, the shockstrengthens and swiftly moves to the head of the wave to imart it the conventional structure of a ZND detonation. hus the entire evolutionary rocess, from the initial gradient to the aearance of the full-fledged detonation, is amenable to asymtotic analysis. Later we shall resent a detailed numerical comutation of this case, for finite activation energy, and comare the numerical results with the analytical redictions.. Shallow gradient, a As a is lowered, the singularity locus tends to flatten, i.e., blowu occurs in quicksuccession across a broad region. Uon using a rather than as the satial variable, an asymtotic analysis of the induction roblem in the limit a by Short [9] finds the solution φ a ln ( te a), P ln ( te a), yielding t = e a as a first aroimation to the ath of the reaction wave. he situation is now akin to that for the a = O() case discussed above, ecet that the wave must now travel a long distance, of order (/a) ln(/a), before it slows down sufficiently for transition to detonation. A transition will not materialize if the domain is only order unity in length. We shall resent a comutational study of this situation as well.. Large gradient, a For large a the singularity locus suffers a raid deceleration uon emergence from the hot wall. his and other features of the induction hase can be understood by an eamination of the induction solution in the limit a [], which divides the domain into an outer region and an O(/a) thickboundary layer at the wall. Anticiating (subject to verification) that the ressure disturbance will now be o() throughout, equation () simlifies to yield ) φ a ln ( te a, γ 9

10 an aroimation that is uniformly valid to order unity. When substituted into equations (-), the above eression allows the integrals I, I and I to be evaluated asymtotically, whereuon equations () yield P and U. he results are collected below, for t>/ γ. Analogous results may be obtained for t</ γ. Within the boundary layer, a = ξ = O(), ( γ P at ln t ), γ U at [ ln ( t ) ln γ ( te ξ γ )], while outside, = O(), P γ a(t / γ) ln U a(t / γ) ln [ ( t γ [ ( t γ )], γ )]. γ As anticiated, these results confirm that the ressure disturbance is indeed of order smaller than unity, and is in fact, O(/a). he same is true of the disturbance in U. One is now in a osition to return to () and obtain φ to order /a, and the results are as follows. Within the boundary layer, ) φ ξ ln ( te ξ + O(/a) γ ( γ a γt ln t ), γ and without, [ γ φ a a γ(t / γ) ln ( t )]. γ γ hus φ enjoys a larger, O() increase in the layer, thanks to the layer being the sole site of chemical activity at this stage. Pressure rises in a satially uniform manner within the layer, which therefore acts as an eanding kernel that drives an acoustic flow in the region outside. At the constant-ressure induction time t = γ, two singularities aear. he stronger one, at the O() level in φ, emerges from the wall and travels along the slow ath t = γe ξ, reresenting a thermal wave determined entirely by the initial temerature gradient to leading order. he weaker one, in all the layer variables at O(/a), develos instantly throught the layer and then travels acoustically in the outer region along the ray t = γ + / γ. It is this singularity that limits the domain of validity of the induction solution to t<γ. As t γ, the order-unity temerature erturbation within the boundary layer undergoes blowu locally at ξ =, while the ressure disturbance blows u throughout the layer, albeit at order /a. here is no evidence against this relative ordering ersisting even when the induction stage eires and the full nonlinear equations are brought into lay, leading one to conclude that the order-unity increase in the temerature, corresonding to full reactant deletion at the wall, brings about only a smaller, O(/a) rise in ressure in the boundary layer. his result has two imlications. First, the acoustic waves roduced by the increasing ressure, being of small amlitude, continue to eit the boundary layer along the linearized characteristic ath t = γ+/ γ. Second, the O(/a) thermal disturbance associated with these waves must overcome the O(ɛa) temerature deficit (caused by the initial gradient) in the outer region in order to induce significant chemical activity there. Such a couling between gasdynamics and chemistry, a rerequisite for the creation of a detonation wave, therefore leads to the uer bound O <O(/ ɛ) uon the size of the gradient arameter a. Once the acoustic ulse reaches the outer region, its further evolution is outside the scoe of the small disturbance theory outlined above, and one must turn to numerics for further eloration.

11 Numerical aroach We have seen that the asymtotic analysis gives a rather comlete but aroimate descrition of the evolutionary rocess for values of a that are either of order unity or very small, whereas only a artial icture restricted to early-time behavior is available for large values of a. In order to obtain a full and accurate descrition of the behavior, highly resolved numerical calculations are needed. For this urose, we have adoted a numerical aroach based on a conservative, finite-volume discretization of the governing equations together with a scheme of adative mesh refinement (AMR) ressure..... Level Level Level Level Level Level rogress Figure : Profiles of and and the diagram for three different refinement levels, at t =.99 for a =. he numerical solutions used in our study are obtained on a set of grids of various levels of refinement. On each grid, the equations are discretized using a second-order accurate, Godunov-tye, shock-caturing scheme with numerical flues comuted using a Roe Riemann solver. he choice of time ste for each grid is based on a CFL stability constraint. he source term is handled numerically using an order two-three Runge-Kutta scheme which selects its own, ossibly sub-cfl, time ste based on an estimate of the local truncation error for the integration of the source term for each oint on the grid. In regions of high satial or temoral solution activity, the grid is refined recursively in order to accurately resolve fine-scale behavior. Raid satial behavior is detected by monitoring the absolute value of the second difference in the density at each oint on the grid, while fast temoral behavior is indicated by the estimate of the truncation error in the integration of the source term based on the CFL time ste. A tolerance on the truncation error is set sufficiently low so that at most or sub-cfl time stes are taken for the integration of the source term for each oint on the grid. If the truncation error is too large, then the grid is refined thus reducing the CFL time ste and the corresonding truncation error for oints on the refined grid. Numerical eeriments have been carried out in order to determine the number of grid levels needed to achieve well-resolved solutions. For eamle, figure shows rofiles of and at a articular value of t and for a =. (his case is discussed in detail later.) For this calculation there are cells on the base grid for between and. Solutions are shown for three different refinements, corresonding to maimum refinement level l ma =, and. We use a refinement ratio of, so that the finest level of refinement is equivalent to aroimately, cells on a uniform grid. Figure shows the reaction-zone structure on the versus lane at a fied time for increasing values of l ma. hese figures show that while the lowest level of refinement suffices to locate the region of raid ressure variation reasonably well, both in the hysical domain and in the domain of reaction rogress, higher resolutions are needed to resolve the rofiles, and to obtain accurate eakvalues. hese resolutions are tyical for the calculations resented in the aer.

12 Numerical solutions for rogressively increasing gradient We now resent comutational results for three reresentative values of a, covering the cases a, a = O() and a. he asymtotic analysis does not, of course, rovide recise quantitative estimates, and the actual reresentative values of a for each case has required some eerimentation. We shall find that the comutational results are in full agreement with the asymtotic redictions for the first two cases. For the third, where the asymtotic descrition is incomlete, numerics agrees with the asymtotics but goes further to comlete the icture. After covering the major cases, we resent results for several additional values of a to reveal further variations. In all cases, the asymtotic analysis has guided the resentation and interretation of the voluminous amount of data generated by the comutations. We shall frequently contrast the aths (and roerties) of the numerically-comuted wave (also referredto as the reactive-hydrodynamic wave or RH wave) and Zeldovich s sontaneous wave (abbreviated as the S wave). While the ath of the latter is always identified by the locus of ositions of the eakin reaction rate, the ath of the former is identified, as aroriate, by one of the following: locus of ositions of eak reaction rate, eakressure, or shock. he seed of the S wave has already been denoted by U S ; that of the RH wave will be denoted by D. In all cases, the window for viewing the comutational results is <<. here are two ecetions for which the change in the structure of the wave was so slow that the longer window << was deemed more informative. For most cases the total time of travel across the viewing window is divided into several subintervals, for which the results are resented in sequence.. Shallow gradient, a his situation is tyified by the selection a =.. In general terms, both the sontaneous and the reactivehydrodynamic scenarios consist of an induction delay, followed by a raid swee of the domain by a wave of reaction originating at the hot boundary. Wave trajectories are shown in figure and the corresonding seeds in figure. wo different trajectories for the RH wave are shown, one corresonding to the eak in reaction rate and the other to the eakin ressure. Although the two aear to be very nearly coincident on the scale of the grah, a magnified view, figure, reveals that the former leads the latter. Either can be used to comute the wave seed; we have used the eak-ressure trajectory for the grah of D in figure. We note that the S wave leaves the wall at a high seed of 9.7 units, and although it slows down, its seed remains above the CJ-detonation value D CJ =. before it eits the domain at =. According to Zeldovich s criterion, therefore, the corresonding RH wave must turn out to be a weakdetonation, and the numerical results show this to be true. he RH wave emerges later but travels faster than the S wave, overtaking it at. and maintaining a small lead thereafter. he longer delay associated with the RH wave, and its faster initial seed, are the results of weak chemico-acoustic adjustments eerienced by the domain during the early hase of evolution. he situation is illustrated by the rofiles shown in figure 7(a,b). At t =. about % of the reactant has been consumed at the wall. emerature is rising everywhere, slightly faster near the wall than away from it. he differential rate of heating creates a broad zone of eansion, indicated by the ositive gradient of u revailing at this instant in the interval <<., figure 7. he eansion, in turn, revents the local temeratures from rising as much, or as raidly, as they would in the S wave for which eansion is suressed; hence the reduced rate of reaction and the longer induction delay. By t =. the region of eansion (whose edge is marked by the eak in u ) has shrunkto a thin boundary layer, <<.. he region outside is now in a state of negative velocity gradient, and the comression that now sets in begins to moderate the eansion that had develoed at earlier times. By t =.9 the layer of eansion has narrowed even further, figure 7. his effect is also aarent in figure 7, where the lackof coincidence between the corresonding airs of and rofiles (reflecting the increase in secific volume) is restricted to increasingly thinner near-wall regions. he ushot is that ignition delay is longest at the wall and diminishes away from it, turning even mildly negative in regions beyond.7, thereby causing a crossover between the aths of the RH and S waves. As the wave rogresses, its suersonic character revents any disturbances from behind to intrude into the

13 t. Wave aths for a =. Seeds: a =. RH: ma reaction-rate ath RH: ma ressure ath S wave S wave RH wave.. seed Figure : Wave aths and wave seeds for a =....7 RH: ma reaction-rate ath RH: ma ressure ath.. t Figure : Magnified view of wave aths at early times for a =.. region ahead, while its high seed does not give that region enough time to evolve further to any significant degree. he wave ath, therefore, is determined entirely by the state of the domain revailing at the time the wave was born. As this state is a rogressively weaker erturbation of the initial state away from the wall, the wave ath and the wave seed are increasingly well-aroimated by those of the S-wave, the further away from the wall the wave has travelled. he structure of the RH wave can be gleaned from figures and 9, where the comuted information is dislayed in a variety of ways. Figures (a,b) show rofiles of,,, u, v and reaction rate against at successive times in the interval.9 t.77. he overwhelming imression is that of a wave in which eaks in, and reaction rate travel in near-erfect synchrony as the wave swees across the domain. he structure is reminiscent of the sontaneous elosion of figure ; a broad induction zone followed by a thin elosive zone in which temerature and ressure both rise. However, the ultimate levels of and are no longer identical, figure, the difference reflecting the degree of comression through the wave. Although increasing with time, the etent of comression, as reflected by the decrease in v, figure, remains small. On the other hand, substantial article velocities are generated, with u behind the wave taking on a value as high as. (a considerable fraction of the initial sound seed,.) at the final time ste t =.77, figure. Behind the wavehead there is a long tail of eansion, clearly visible in the, v and u rofiles of figure. his eansion undergoes reflection at the rigid wall and turns into a forward moving, and strengthened, rarefaction wave.

14 velocity.... O Figure 7: Profiles of u and of, and at early times for a =.. he circular marker in identifies the eakin u at t = r r v u Figure : Profiles of, and reaction rate and of u, v and at later times for a =.. Additional insight into the character of the wave is obtained by referring to figures 9(a, b). he lots in figure 9 show a urely comressive reaction zone, with ressure increasing monotonically to its eakat =, the small increase from one rofile to the net being the result of wave deceleration. he v -lane in figure 9 shows that ecet for the slow ram in front of the wave, the main art of the wave structure at each instant of time falls on a straight (Rayleigh) line, whose sloe diminishes (in magnitude) from one time ste to the net as the wave advances. he Rayleigh-line structures attest to the quasisteadiness of the wave. here is a striking resemblance to the scenario redicted by the asymtotic analysis and dislayed in figure, ecet that each Rayleigh line in figure 9 corresonds to a slightly different initial state created by the induction rocess. he earliest line dislayed in figure 9 is nearly vertical, with a value of v slightly higher than, suggesting again that the wave is born as a near-constant-volume elosion (a wave of infinite seed) but in a slightly eanded near-wall atmoshere. he increasingly flatter sloes down the sequence indicate a rogressive deceleration of the wave (and the associated small increase in comression referred to above) as it advances. he eansion behind the wave is entirely devoid of reaction.

15 .... ressure. time ressure. time rogress secific vol Figure 9: and v diagrams for a =... Wave aths for a =.. Seeds: a =... RH: ma reaction-rate ath RH: ma ressure ath S wave. S wave RH wave u+c at ressure ma.. t. seed Figure : Wave aths and wave seeds for a =... Moderate gradient, a = O() Some eerimentation led to the choice a =. as reresentative of values of a of order unity, as regards matching closely with the asymtotic redictions. While some asects of the behavior for this case remain similar to those for a, others are altered substantially. A sense of the overall evolution can again be acquired from the lots of wave trajectories and seeds, dislayed in figure (a,b). he S wave now leaves the wall at U S =. <D CJ =.. his corresonds to Zeldovich s second regime and therefore, one eects the aearance of a shock. Of secial interest is the manner in which the shock is born, a matter not addressed in any detail by the S-wave concet. For the RH wave the eak reaction rate continues to lead the eakressure in osition, figure, but the two trajectories continue to remain very close on the scale of the lot in figure. As such, the eak-ressure locus continues to be the basis for the lot of D in figure. In contrast to a =. the RH wave now aears after a longer delay, starts out more slowly, overtakes the S wave at a shorter distance (. ) from the wall, and most significantly, decelerates to asymtote D CJ rather than U S. hus RH roagation, at least in its later stages, is determined not by the gradient revailing at the instant of wave birth, but by a different mechanism altogether. It is convenient to address the details of evolution by grouing the comutational results into a sequence of hases labelled birth, re-transition, transition and ost-transition.

16 . ressure ma reaction-rate ma..9. t Figure : Magnified view of wave aths at early times for a =.... Phase I:birth he very-early-time near-wall eansion continues to lay a rominent role during the eriod rior to the birth of the RH wave. he gestation eriod is longer because the eansion is now stronger, as shown by the higher eakvelocities in figure, and by the larger near-wall differences in the corresonding airs of, rofiles in figure. We note that as comared to figure 7 these lots are drawn on a magnified scale, to show more clearly that the boundary layer is now thinner u Figure : Very-early-time rofiles of u and of, and for a =.... Phase II:re-transition In addition to the grahs of U S and D, figure also dislayes the grah of u + c, the seed of the forward sonic disturbance associated with the locus of ositions of maimum ressure. Secifically, u + c is comuted at the osition of maimum ressure if the maimum is smooth, and immediately behind it if the maimum occurs at the shock. he figure shows that U S <u+ c, i.e., the S wave is subsonic relative to the following flow, throughout the domain. On the other hand, D (u + c) changes sign from being ositive to negative at., indicating that the RH wave undergoes a transition from being suersonic to subsonic relative to the following flow. Results for the re-transition stage are gathered in figures and. It is at this stage that the behavior most resembles that for a =..

17 . rr v u Figure : Early-time rofiles of,, and reaction rate and of u, v and for a = ressure. ressure... time time rogress secific vol Figure : and v diagrams for a =. at early times. Profiles of,, u, v, and the reaction rate are shown in figure (a,b). Once again we see the coherence in the eaks of, and reaction rate. he eakressure rises more raidly with assage of time, increasing from the near-constant-volume elosion value of. at the second time ste to. at the last time ste shown. he corresonding rise in temerature is less dramatic, indicative of increased comression through the wave. he article velocity behind the wave is larger and the eansion in that region stronger, as is the rarefaction wave that reflects from the wall. Figures (a, b) confirm these features; the -lane now shows that while the reaction zone is still comressive, there is a hint of eansion within it near the very end. In the v -lane the Rayleigh-line segments corresonding to the quasisteady, weak-detonation ortion of the structure still dominate, but the less stee sloe of the rofile at the final time ste oints to a greater deceleration of the wave, as does the corresondingly higher value of the ressure eak... Phase III:transition ransition is characterized in figure by the crossing of the RH wave by the forward characteristic, an event that signals that the wave has slowed down to a sonic seed relative to the following flow. Imortantly, the figure also shows that the crossing occurs at, or very close to, the CJ seed, as foretold by the asymtotic analysis for a = O(). his stage is best viewed in figure, which shows the reaction-zone structure in 7

18 ressure time r r rogress Figure : diagram and rofiles of and reaction rate for a =. at transition shock locus reaction wave. t Figure : Profiles of, transition. and and loci of shockand maimum reaction rate for a =. at the -lane for si time stes in the transition interval. We see the hitherto monotonic ressure rofile showing sign of turning backon itself very close to the end of the reaction zone, and then steeening quickly to form a weakshock. he shockis born in the time interval.9 t., and the ressure at shock birth is in the range 7. <<7., which comares quite favorably with the CJ value of 7. at which the asymtotic analysis redicts the first aearance of the shock. he develoment of the shockat the rear of the reaction zone can also be seen in the reaction-rate rofiles of figure, drawn for the shorter time interval.999 t.. At the third time ste in this figure one sees the first sign of the shock, at the rear of the reaction zone and behind the eak in the reaction rate. Both figures and show the strengthening shockadvancing towards the front of the reaction zone. hat the advance is etremely raid is aarent from the etreme brevity of the transition interval; that it is also quasisteady is shown by the stationary nature of the subsonic and suersonic branches of the reaction-zone structure in figure. In the steady case of a CJ wave the shockconnecting the weakand strong detonation branches could be inserted at any arbitrary value of to give a hybrid detonation. Here the shocklocation is a raid function of time determined by the evolutionary rocess, while the branches remain essentially stationary. he rofiles of, and during transition are dislayed in figure, and show yet again how the structure of the wave evolves from that of a shockless, weak detonation towards that of a ZND detonation as

19 Figure 7: Post-transition rofiles of, and for a =.. the shockmoves forward through the reaction zone. Figure shows the shockcatching u to the locus of ositions of maimum reaction rate. We emhasize again that the shockseed is essentially constant; it finds itself rogressing towards less reacted regions because the suersonic ortion of the reaction zone structure ahead of it is slowing down. his is recisely the scenario redicted by the asymtotic analysis for the a = O() case... Phase IV:ost-transition Figure shows that subsequent to transition, the relative flow behind the wave is subsonic and D remains close to D CJ. he rofiles in figure 7 confirm that as the shockadvances, more and more of the reaction occurs in the subsonic region behind the shockand the structure aroaches the ZND structure. At the final time ste shown, the wave seed is. and the eakressure.7, the corresoinding ZND values being. and.7 resectively.. Large gradient, a u Figure : Very-early-time rofiles of u and of, and for a =. We now resent results for a =, a value that is on the large side but satisfies the criterion O / ɛ. For this case the small-disturbance asymtotic theory rovides only artial information, but redicts the 9

20 v u. r r v. r r u (c) Figure 9: Early-time rofiles of, and and of u, v and for a =. Snashots of all the variables at t =. aear in (c). ossibility of a route to detonation different from that observed for a of order unity. he starting value of U S is now., far less than the initial sound seed. he Zeldovich criterion suggests that the RH wave should roagate with the sontaneous wave seed with little ressure change across it, but comutations will reveal an altogether different scenario... Phase I:birth We begin by ehibiting the very-early-time snashots of the solution in figures (a,b). he u -rofiles shown therein are analogous to those of figure for a =., but now there is a larger velocity gradient at the wall at the last time ste, indicative of a stronger eansion for the larger a. he increased level of eansion is borne out by the substantially larger difference between the and rofiles of figure when comared to the corresonding situation in figure ; it also elains the longer induction delay for a =. When the reactant at the wall is nearly ehausted, the RH wave emerges from the wall. Its early evolution, shown in figures 9 and, is in starkcontrast to the way in which events unfolded for a =.. Gone is the strong similarity between the rofiles of ressure and temerature, and the near-erfect satial coincidence of eaks of ressure, temerature and reaction rate, so strikingly evident in figures (a,b). Also missing is the urely comressive reaction zone so rominently featured in figure. Instead, figure 9 shows that the wave starts out as a low-amlitude ressure ulse, generated by the elosive energy release and the associated eansion in the thin boundary layer at the wall. As the ulse advances it amlifies and broadens; the former because the ulse finds itself in a reactive environment, and the latter because the eak