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Fine cellular structures produced by marginal detonations
Proceedings of the Combustion Institute, Volume 28, 2000/pp. 611–617
FINE CELLULAR STRUCTURES PRODUCED BY MARGINAL DETONATIONS V. N. GAMEZO,1 A. A. VASIL’EV,2 A. M. KHOKHLOV3 and E. S. ORAN3 1
Berkeley Research Associates, Inc. Springfield, VA 22150, USA 2 Lavrientyev Institute of Hydrodynamics SB Russian Academy of Sciences 630090 Novosibirsk, Russia 3 Laboratory for Computational Physics and Fluid Dynamics Naval Research Laboratory Washington, DC 20375, USA
A new regime of unstable detonation near the propagation limits was observed experimentally in a flat channel. Highly resolved, two-dimensional reactive Navier-Stokes numerical simulations with one-step Arrhenius kinetics were used to reproduce and analyze the detonation structures involved in this regime. The regime included three kinds of detonation instability: (1) the primary instability, which causes detonation waves moving in transverse directions (transverse detonations) to appear and disappear at regular intervals behind the leading shock; (2) the instability of the overdriven parts of the leading front, which is responsible for the formation of secondary detonation cells that coexist with transverse detonation waves; and (3) the instability of transverse detonations, which produces the fine structure called transverse detonation cells. Numerical simulations showed that transverse detonations appeared when a large induction zone formed behind the leading shock. The resulting reaction-zone structure was similar to that observed for spin detonations. The transverse detonation wave burned most of the material in the induction zone, thus preventing the formation of large unreacted pockets and leaving behind only thin tails of unreacted material. The material involved in transverse detonations was compressed by two shocks, which makes transverse detonation waves more unstable than the overdriven parts of the leading detonation front.
Introduction Experimental studies of gaseous detonations [1– 5] and two-dimensional numerical simulations [6] have shown that cellular patterns produced by an unstable detonation front may be complicated by small secondary cells that appear inside the main structure. This happens when the overdriven parts of the detonation front become unstable enough to develop secondary triple points during the time between two collisions of the primary triple-shock configurations [1]. According to experimental observations [2,5], and in agreement with numerical simulations [6], secondary detonation cells appear when the scaled activation energy exceeds a specific value—that is, h ⳱ Ea/RT ⬎ 6.5, where Ea is the dimensional acti* vation energy, R is the universal gas constant, and T * is the temperature behind the leading shock front moving with the average front velocity, D. This condition reflects the fact that h is the most important controlling parameter whose variation is the largest among explosive mixtures. Detonation stability is also controlled by the heat of chemical reaction, QM/RT , the overdrive, f ⳱ (D/DCJ)2, and the poly* tropic exponent, c (see, for example, Ref. [7]).
Secondary cells are usually observed near the limits of detonation propagation when the primary cell size is comparable to the size of the channel. The larger structures are much easier to visualize. Even for large main cells, experimental techniques are not always able to resolve the secondary cells, which can be from 10 to 1000 times smaller. In addition, marginal detonations (i.e., regimes near the detonation limit [8]) produce more regular structures that are easier to analyze. Secondary cells exist in systems with high activation energy that are characterized by very irregular cells [5,9]. When only one or a few primary triple-shock configurations fit into the channel, the system has fewer degrees of freedom, and the possibilities for irregular behavior are reduced. Finally, secondary cells are more likely to exist near the propagation limits, where the detonation velocity decreases due to energy losses. As a result, the postshock temperature, T , also decreases, h grows, * and the detonation front becomes more unstable. The transverse-wave structure associated with marginal detonations often includes transverse detonation waves propagating through the induction zone behind the leading shock [8,10–14]. The exact conditions under which transverse detonations form are still unclear. There are some indications
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Fig. 1. Open-camera record of the detonation in a flat channel for C2H2 Ⳮ 2.5O2 near the limits of propagation. P0 ⳱ 0.03 bar, channel width is 30 mm. 1, track of transverse detonation; 2, secondary detonation cells.
[8,10,13,14] that they can also exist far from the propagation limits, and the transitions between the transverse shocks and transverse detonations can occur both for marginal regimes [8] and far from the limits [13]. However, it seems that transverse detonations are more likely to appear when the induction zone becomes large, especially near the propagation limits when the average detonation velocity decreases. In particular, a transverse detonation provides the only mechanism of self-sustained propagation for the spin detonation, which is the ultimate marginal regime that occurs when only one primary triple-shock configuration exists in a round tube. Transverse detonation waves can also be unstable and produce another kind of the fine cellular structures observed for spin detonations [15–17]. We refer to these fine structures as the transverse detonation cells to distinguish them from the secondary detonation cells which are the result of instability of the overdriven parts of the leading detonation front. In this work, we combine the numerical simulations and experiments to investigate transverse detonations and fine detonation structures for the marginal regimes in two-dimensional systems. As an example, we consider a new marginal mode experimentally obtained in a flat channel. Highly resolved numerical solutions provide more detailed information about this regime, which involves transverse detonations and produces secondary and transverse detonation cells. Experimental Data The experiment was performed for a stoichiometric acetylene/oxygen mixture at the initial pressure P0 ⳱ 0.03 bar. The flat channel (1.5 mm thick) containing the mixture consisted of two sections. The first section, of constant 30 mm width, was connected to the second, slightly divergent (5⬚) section. The detonation initiated at the beginning of the first section, reached the quasi-steady-state regime, then entered the divergent section and died. The process was recorded using the open-camera technique. The ¯ ⳱ 0.8D0 ⳱ 1790 m/ average detonation velocity, D s, was determined from separate experiments in the same channel. Here, D0 ⳱ 2238 m/s was the ideal detonation velocity calculated for this system using the SAFETY code [18].
The open-camera photograph in Fig. 1 shows that the history of the detonation propagation looks very similar to the record expected for a spin detonation in a round tube. The difference is that the system considered here is practically two-dimensional. The tracks marked 1 were left by the transverse detonation wave which propagated from the top to the bottom, then disappeared, and appeared again at the top. Unlike a spin detonation, the track of the transverse detonation wave disappeared here because the transverse detonation actually died, not because it disappeared from view. After some time, the transverse detonation reignited again near the upper wall and moved down until it collided with the lower wall. These primary pulsations occurred at regular intervals of k1 ⳱ 77 Ⳳ 5 mm. The propagation of transverse detonation waves was accompanied by secondary pulsations produced by unstable overdriven parts of the leading detonation front. The smallest resolved secondary cells, marked 2, initially were about 2 mm wide. They grew as the overdrive decreased and essentially disappeared at the distance k1/3 from the transverse detonation track 1. The remnants of some of the secondary triple points survived until the next primary pulsation, and they might participate in the reignition of the transverse detonation. Transverse detonation cells that may result from the instability of the transverse detonation were not resolved in this experiment, though they were reported for the spin detonation in the stoichiometric acetylene/oxygen mixture diluted with argon [16]. The marginal detonation regime described here was observed only in flat channels and for a narrow range of parameters (P0 ⳱ 0.03 bar Ⳳ 10%) very close to the propagation limit. The detonation was easily destroyed in the slightly divergent section of the channel during only one primary pulsation. At a slightly higher P0, the detonation propagated in a different regime characterized by one primary triple-shock configuration reflecting from the walls and forming a regular half cell in the channel. A galloping mode was observed at a slightly lower P0. Similar to the spin detonation, the marginal regime described here provided a good opportunity to study different levels of detonation structures. It also had the advantage of being two-dimensional, an advantage which we exploited in numerical simulations of this regime.
FINE CELLULAR STRUCTURES TABLE 1 Physical parameters of the model system P0 (initial pressure) T0 (initial temperature) q0 (initial density) c (polytropic exponent) M (molecular weight) DCJ (CJ detonation velocity) PCJ (CJ pressure) TCJ (CJ temperature) qCJ (CJ density) PZND (ZND pressure) TZND (ZND temperature) qZND (ZND density) Q (heat of reaction) QM/RT0 QM/RTZND Ea (activation energy) Ea/RT0 Ea/RTZND A (pre-exponential factor) L05 (half-reaction length)
0.03 bar 298 K 0.03669 kg/m3 1.25 0.0303 kg/mol 1281 m/s 0.2807 bar 1627 K 0.06290 kg/m3 0.5315 bar 879 K 0.2203 kg/m3 1.282 MJ/kg 15.7 5.31 21.59 kcal/mol 36.5 12.4 6.146 ⳯ 109 m3/kg/s 0.00539 m
Numerical Model and Physical Parameters The time-dependent behavior of the reactive gaseous mixture is described by the reactive NavierStokes equations coupled with the polytropic equation of state and the one-step Arrhenius kinetics of energy release. The reaction rate is defined as d␣/dt ⳱ A(1 ⳮ ␣)q exp(ⳮEa/RT)
(1)
where A is the pre-exponential factor and ␣ is the mass fraction of reaction products. The density, q, in equation 1 reflects the experimental fact that the induction time for C2H2/O2 mixtures varies inversely with q [19]. The model takes into account the viscosity, heat conductivity, and diffusion defined in Ref. [20] as functions of density and temperature. The equations were solved on a structured mesh using an explicit, second-order, Eulerian, Godunovtype numerical scheme incorporating a Riemann solver [20,21]. The process-splitting and directionsplitting techniques [22] were employed to add the chemical sources and obtain the two-dimensional solution. An adaptive mesh refinement algorithm based on the fully threaded tree [21] was used to increase the resolution locally, as required by the changing physical conditions. The code was extensively tested in reactive fluid-dynamic simulations [20] including detonations [6]. We considered a rectangular two-dimensional computational domain, 30 mm wide and 240 mm long, bounded by free-slip solid walls on the top and bottom. The unreacted fluid entered the computational domain from the right with specified initial parameters q0, P0, and T0 and velocity ⳮDCJ. An
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extrapolated outflow boundary condition on the left allowed the pressure at the boundary to relax adiabatically to PCJ. The physical parameters of the reactive fluid were defined by taking into consideration that (1) the detonation regime that we were trying to model only existed for a narrow range of parameters and (2) the two-dimensional numerical model based on onestep kinetics could not quantitatively reproduce the experiment. Thus, we tried to obtain qualitatively the same regime, using the experimental conditions as a starting point and then adjusting the model parameters in a series of preliminary low-resolution runs. The resulting set of physical parameters used in the final simulations is summarized in Table 1. The unreacted mixture and the reaction products are described by the same polytropic equation of state, with c ⳱ 1.25, and the same transport properties that were selected in Ref. [20] to fit the laminar flame speed and thickness for acetylene/air mixtures. The heat of chemical reaction, Q, was one of the parameters that we varied to adjust the model. This parameter affects the detonation instability through Ea/RT and QM/RT , because it defines the con* * trolling temperature, T . The value of Q was chosen * so that the detonation was unstable enough to produce secondary triple points. As a result, the ideal thermodynamic detonation velocity, DCJ, for the model system equaled 0.57 D0. All the ChapmanJouguet (CJ) and Zeldovich-Neuman-Do¨ring (ZND) parameters in Table 1 correspond to that velocity. The activation energy, 18.1 kcal/mol, was obtained in Ref. [19] from the measurements of the induction times, s, for acetylene/oxygen mixtures, assuming that s ⬃ exp(Ea/RT). The asymptotic solution based on equation (1) gives for the adiabatic induction time s ⬃ T2 exp(Ea/RT). To take into account the term T2, we increased Ea by 2RTZND, taking TZND as the controlling temperature. The pre-exponential factor, A, in equation 1 was also an adjustable parameter. This parameter can be considered as a scale factor that controls the induction delays and the sizes of all the dynamic flow structures associated with them. It was selected to make the pattern of primary pulsations as close as possible to the experimental one. The calculations were performed for two numerical resolutions. For the higher resolution, the adaptive mesh contained six levels, with the minimum cell size dx ⳱ dy ⳱ 2.93 ⳯ 10ⳮ5 m, which corresponds to the effective resolution 184 cells per halfreaction length of CJ detonation. The minimum time step in this case was about 6 ⳯ 10ⳮ9 s. For the lower resolution, for which the mesh had five levels, the minimum cell size and time step were two times larger. Numerical Results The detonation in the low-resolution simulations was initiated by a strong shock placed near the left
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Fig. 2. Numerical smoke foils for the same primary pulsation calculated with low (a) and high (b) resolutions. The channel width is 30 mm. 1, transverse cells; 2, secondary cells.
boundary. This shock, with the initial pressure PZND, produced an overdriven detonation wave that moved to the right through the computational domain. Shortly after the initiation, the detonation wave developed several triple-shock configurations resulting from the specially introduced initial perturbations and the thermal instability of the front. Most of these triple points disappeared, but the remaining transverse shocks became stronger, sometimes generating transverse detonations in the large induction zone behind the leading shock. At the same time, small triple points appeared both in the overdriven parts of the leading front and in the transverse detonation waves and produced fine secondary and transverse cell structures, respectively. Eventually, the average detonation velocity became equal to DCJ, the average position of the detonation front stabilized in the middle of the computational domain (which had the inflow velocity ⳮDCJ), and the pattern of pulsations did not change. At this point, we increased the numerical resolution by introducing an additional level of mesh refinement and then continued the calculations. Fig.
2 compares the numerical smoke foils for one primary pulsation calculated with lower (Fig. 2a) and higher (Fig. 2b) resolutions. These figures recorded the maximum pressure reached at each location in space and show the trajectories of the triple points and transverse detonations. The maximum pressure was about 2 bar for the secondary triple points and 6 bar for the triple points inside the transverse detonation wave. Increasing the resolution did not noticeably affect the primary pulsations and had a little effect on the secondary cells. However, it substantially increased the number of transverse cells which are still not well resolved. The numerical smoke foil in Fig. 3 shows a typical part of the history of the detonation propagation in the quasi-steady-state regime over a longer period of time. Two tracks were left by transverse detonations, both containing traces of the fine triple points which form the transverse cells. The primary pulsations also produced multiple secondary cells, which grew and became more irregular until the next transverse detonation re-created the secondary structure. The transverse detonations periodically appeared near one of the walls and disappeared when they collided with the opposite wall. The distance between primary pulsations was about 15 cm. This regime existed only for a very narrow range of parameters. The detonation would die if we decreased the reaction rate by 5%. Increasing the reaction rate by 5% led to a regular half cell in the channel. A second primary triple point appeared when the reaction rate was increased by about 50%. The four snapshots of the temperature field behind the leading shock, shown in Fig. 4, provide direct information about the dynamic structure of the detonation front. In particular, we see that the transverse detonation formed when transverse shocks reflected from the wall interacted with the large induction zone behind the leading shock (frame 1). The transverse detonation propagated along the induction zone and burned almost all the material except for a thin tail in the vicinity of the primary triple point (frame 2). The fine structure of the transverse detonation included the small triple points moving along the front, which had relatively strong and weak parts with different induction-zone
Fig. 3. Numerical smoke foil for the marginal detonation regime calculated with high resolution. The arrows at the top correspond to the positions of the primary triple point in the frames shown in Fig. 4.
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Fig. 4. Temperature field behind the leading shock at different times. The position of the primary triple point for every frame is marked by an arrow at the top of Fig. 3. 1, transverse detonation; 2, strong part of the leading shock (overdriven detonation); 3, weak part of the leading shock (inert); 4, induction zone; 5, transverse shock; 6, unreacted tail; 7, primary unreacted pocket; 8, secondary unreacted pockets.
lengths. Due to insufficient numerical resolution, we did not see the unreacted pockets which could appear behind the transverse detonation. The pockets were well resolved for the secondary detonation structure behind the strong part of the leading shock. These pockets grew as the overdrive decreased, moved downstream, and burned at a distance of the order of the secondary cell size from the leading shock. Eventually, secondary triple points became so weak that they could no longer cause enough energy release to separate pockets from the induction zone. At this time, the transverse detonation wave collided with the wall and died. The reflected shock could not start a new transverse detonation because the induction zone was not large enough (frame 3). As the primary transverse shock propagated through the irregular reaction zone that contained remnants of the secondary structures, it induced some energy release and rapidly became attenuated. The induction zone behind the weak part of the leading shock grew. Because unreacted material did not burn in the transverse shock, it became a pocket the next time the primary triple point collided with the wall. Before the collision (frame 4), the induction zone behind the strong part of the leading shock was larger than it was in frame 3, but it was not yet large enough for transverse detonation to occur. Fig. 3
shows that the transverse detonation will reappear after one more reflection.
Discussion The marginal detonation regimes in the numerical simulations and those observed experimentally in a flat channel have similar detonation structures, even though the calculated detonation history looks more irregular. The difference is not surprising considering the approximate nature of the numerical model. It is important to note that the numerical smoke foil is a history of the maximum pressure and captures more pressure waves than the experimental opencamera technique that records light emitted by the detonation front. The calculations produce the same kind of primary pulsations as the experiments, and they show how the transverse detonations form and why they disappear and reappear. The numerical results also provide detailed information about fine detonation structures, including transverse cells that are not resolved in the experiment. Since we use an approximate model, and the numerical resolution is not high enough to produce transverse detonation cells of the correct size, we can only say that these cells are likely to exist in the real system, and their size, as follows from Fig. 2, should be well below 1
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mm. This is too small to be resolved by the experimental technique used. The existence of the transverse cells in this system is also confirmed by experimental observations [16] of spin detonations in a stoichiometric acetylene/oxygen mixture diluted with argon. Dilution with an inert monoatomic gas usually increases postshock temperatures, so the scaled activation energy, h, decreases, and the detonations become more stable. Nevertheless, transverse detonation cells were observed in Ref. [16] for the argon-diluted system. This means that these cells are even more likely to exist in the undiluted mixture. The smoke foil published in Ref. [16] also shows that a spin detonation in the mixture with argon does not produce secondary cells. Only a few short traces of secondary triple points are visible. This observation agrees well with the lower value of h for the diluted mixture, and it also means that the h that controls the stability of the overdriven part of the leading front is lower than the h that controls the stability of the transverse detonation. This may occur only if the controlling temperature, T , for the trans* verse detonation is lower than the T for the strong * leading shock. Indeed, analysis of the numerical results shows that the postshock temperature varies from 1050 to 1100 K for the transverse detonation wave and from 1120 to 1400 K for the strong part of the leading shock. At the same time, the postshock pressure is substantially higher for the transverse detonation (1.6–1.8 bar) than for the leading shock (0.75–1.0 bar). The higher pressure and lower temperature for the transverse detonation result from the double-shock compression that occurs when the shock-compressed material in the induction zone is compressed again by the transverse shock. A small part of the unreacted material also undergoes a triple-shock compression. This happens when the gas enters the induction zone through the weak leading shock in the vicinity of the primary triple point and then passes through two oblique shocks in a gap between the leading and the transverse detonation fronts. The details of similar complex Mach structures were described in Ref. [10] for spin detonations and investigated in Refs. [13,14] for twodimensional systems. As shown in frame 2 in Fig. 4, the temperature of the material passing through the gap is so low (about 950 K) that the mixture does not ignite but forms an unreacted tail. This tail burns slowly and separates the detonation products originating from the leading and the transverse detonation fronts. Similar unreacted tails were observed in experiments [12]. The transverse cells and the unreacted tails are characteristic features that can form only when the transverse shock moving through the induction zone is a detonation. Transverse shocks with small or no energy release are stable and do not produce any
cells. The material in the induction zone is still unreacted behind inert transverse shocks and forms unreacted pockets instead of tails. The secondary cellular structure is a more general feature. It appears because the leading detonation front is unstable and it does not require the presence of transverse detonations. For example, secondary cells without transverse detonations were obtained in numerical simulations [6] of detonations in systems with high activation energy. Conclusions Four major contributions result from this work: 1. We reported a new two-dimensional detonation regime observed in experiments near the propagation limits and confirmed by numerical simulations. The new feature of this regime is the periodic appearance and disappearance of transverse detonations that form when the induction zone becomes large enough behind the leading shock. 2. Fine cellular structures produced by unstable transverse detonations and unreacted tails associated with transverse detonations were shown in numerical simulations for the first time. 3. It was shown that transverse detonations are more unstable than the overdriven parts of the leading detonation front because of the doubleshock compression of the material involved in transverse detonations. 4. The numerical simulations showed that all of the complexity of the phenomena can be explained in the framework of the one-step Arrhenius kinetics. Acknowledgments We would like to thank Dr. Almadena Chtchelkanova for help with the parallel implementation of numerical smoke foils. REFERENCES 1. Shchelkin, K. I., Usp. Fiz. Nauk 87:273–302 (1965). 2. Manzhalei, V. I., Fizika Gorenija i Vzryva 13:470–472 (1977). 3. Libouton, J-C., Jacques, A., and Van Tiggelen, P. J., in Colloque International Berthelot-Vieille-Mallard-Le Chatelier, Vol. 2, The Combustion Institute, Pittsburgh, PA, 1981, pp. 437–442. 4. Bull, D. C., Elsworth, J. E., Shuff, P. J., and Metcalfe, E., Combust. Flame 45:7–22 (1982). 5. Vasil’ev, A. A., Mitrofanov, V. V., and Topchian, M. E., Fizika Gorenija i Vzryva 23:109–131 (1987). 6. Gamezo, V. N., Khokhlov, A. M., and Oran, E. S., “Secondary Detonation Cells in Systems with High Activation Energy,” in Proceedings of the Seventeenth International Colloquium on the Dynamics of Explosions and Reactive Systems, Universita¨t Heidelberg, Hei-
FINE CELLULAR STRUCTURES delberg, Germany, IWR, 1999. 7. Short, M., and Stewart, D. S., J. Fluid Mech. 382:109– 135 (1999). 8. Fickett, W., and Davis, W. C., Detonation, University of California Press, Berkeley, CA, 1979, p. 327. 9. Gamezo, V. N., Desbordes, D., and Oran, E. S., Combust. Flame 116:154–165 (1999). 10. Voitsekhovski, B. V., Mitrofanov, V. V., and Topchian, M. E., Proc. Combust. Inst. 12:829–837 (1969). 11. Strehlov, R. A., and Crooker, A. J., Acta Astronaut. 1:303–315 (1974). 12. Subbotin, V. A., Fizika Gorenija i Vzryva 11:96–102 (1975). 13. Lefebvre, M. H., and Oran, E. S., Shock Waves 4:277– 283 (1995). 14. Oran, E. S., Weber Jr., J. W., Stefaniw, E. I., Lefebvre, M. H., and Anderson Jr., J. D., Combust. Flame 113:147–163 (1998).
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15. Denisov, Yu. N., and Troshin, Ya. K., Doklady Akad. Nauk SSSR 125:110–113 (1959). 16. Lee, J. H., Soloukhin, R. I., and Oppenheim, A. K., Astronaut. Acta 14:565–584 (1969). 17. Manzhalei, V. I., and Mitrofanov, V. V., Fizika Gorenija i Vzryva 9:703–710 (1973). 18. Vasil’ev, A. A., in Proceedings of the Twenty-Eighth Annual ICT Conference, Fraunhofer Institute of Chemical Technologies, Karlsruhe, Germany, 1997, pp. V50.1–V50.14. 19. White, D. R., Proc. Combust. Inst. 11:147–154 (1967). 20. Khokhlov, A. M., Oran, E. S., Chtchelkanova, A. Yu., and Wheeler, J. C., Combust. Flame 117:99–116 (1999). 21. Khokhlov, A. M., J. Comput. Phys. 143:519–543 (1998). 22. Oran, E. S., and Boris, J. P., Numerical Simulations of Reactive Flows, Elsevier, New York, 1987.
COMMENTS Douglas Feikema, NASA Glen Research Center, USA. I would like further clarification of the initial conditions of your calculations. How does the unsteady behavior in the flow channel develop? In the movie you showed in the presentation, why did you choose the initial state you did? Author’s Reply. The detonation was initiated by a strongly perturbed shock. The initial perturbations made the shock non-planar and produced several triple-shock configurations. As the detonation propagated along the channel, most of the initial triple points disappeared. A few remaining transverse shocks became stronger, sometimes producing transverse detonations. The transverse detonations and the overdriven parts of the leading front developed small triple points. Eventually, the detonation became quasi–steady state and established a pattern of pulsations that did not depend on the initial conditions. One of these
established pulsations was shown in the movie. The first frame of the movie corresponds to the first frame of Fig. 4. ● Pierre Van Tiggelen, University of Louvain, Belgium. How sensitive are your results with respect to the kinetic parameters used for the computation? Would a two-step mechanism not be more accurate to describe the kinetics? Author’s Reply. The results are very sensitive to the kinetic parameters due to the fact that the marginal detonation regime which we are trying to model exists only for a very narrow range of parameters. For example, the detonation will die if we decrease the reaction rate by 5%. Increasing the reaction rate by 5% leads to a regular halfcell in the channel.
FINE CELLULAR STRUCTURES PRODUCED BY MARGINAL DETONATIONS V. N. GAMEZO,1 A. A. VASIL’EV,2 A. M. KHOKHLOV3 and E. S. ORAN3 1
Berkeley Research Associates, Inc. Springfield, VA 22150, USA 2 Lavrientyev Institute of Hydrodynamics SB Russian Academy of Sciences 630090 Novosibirsk, Russia 3 Laboratory for Computational Physics and Fluid Dynamics Naval Research Laboratory Washington, DC 20375, USA
A new regime of unstable detonation near the propagation limits was observed experimentally in a flat channel. Highly resolved, two-dimensional reactive Navier-Stokes numerical simulations with one-step Arrhenius kinetics were used to reproduce and analyze the detonation structures involved in this regime. The regime included three kinds of detonation instability: (1) the primary instability, which causes detonation waves moving in transverse directions (transverse detonations) to appear and disappear at regular intervals behind the leading shock; (2) the instability of the overdriven parts of the leading front, which is responsible for the formation of secondary detonation cells that coexist with transverse detonation waves; and (3) the instability of transverse detonations, which produces the fine structure called transverse detonation cells. Numerical simulations showed that transverse detonations appeared when a large induction zone formed behind the leading shock. The resulting reaction-zone structure was similar to that observed for spin detonations. The transverse detonation wave burned most of the material in the induction zone, thus preventing the formation of large unreacted pockets and leaving behind only thin tails of unreacted material. The material involved in transverse detonations was compressed by two shocks, which makes transverse detonation waves more unstable than the overdriven parts of the leading detonation front.
Introduction Experimental studies of gaseous detonations [1– 5] and two-dimensional numerical simulations [6] have shown that cellular patterns produced by an unstable detonation front may be complicated by small secondary cells that appear inside the main structure. This happens when the overdriven parts of the detonation front become unstable enough to develop secondary triple points during the time between two collisions of the primary triple-shock configurations [1]. According to experimental observations [2,5], and in agreement with numerical simulations [6], secondary detonation cells appear when the scaled activation energy exceeds a specific value—that is, h ⳱ Ea/RT ⬎ 6.5, where Ea is the dimensional acti* vation energy, R is the universal gas constant, and T * is the temperature behind the leading shock front moving with the average front velocity, D. This condition reflects the fact that h is the most important controlling parameter whose variation is the largest among explosive mixtures. Detonation stability is also controlled by the heat of chemical reaction, QM/RT , the overdrive, f ⳱ (D/DCJ)2, and the poly* tropic exponent, c (see, for example, Ref. [7]).
Secondary cells are usually observed near the limits of detonation propagation when the primary cell size is comparable to the size of the channel. The larger structures are much easier to visualize. Even for large main cells, experimental techniques are not always able to resolve the secondary cells, which can be from 10 to 1000 times smaller. In addition, marginal detonations (i.e., regimes near the detonation limit [8]) produce more regular structures that are easier to analyze. Secondary cells exist in systems with high activation energy that are characterized by very irregular cells [5,9]. When only one or a few primary triple-shock configurations fit into the channel, the system has fewer degrees of freedom, and the possibilities for irregular behavior are reduced. Finally, secondary cells are more likely to exist near the propagation limits, where the detonation velocity decreases due to energy losses. As a result, the postshock temperature, T , also decreases, h grows, * and the detonation front becomes more unstable. The transverse-wave structure associated with marginal detonations often includes transverse detonation waves propagating through the induction zone behind the leading shock [8,10–14]. The exact conditions under which transverse detonations form are still unclear. There are some indications
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Fig. 1. Open-camera record of the detonation in a flat channel for C2H2 Ⳮ 2.5O2 near the limits of propagation. P0 ⳱ 0.03 bar, channel width is 30 mm. 1, track of transverse detonation; 2, secondary detonation cells.
[8,10,13,14] that they can also exist far from the propagation limits, and the transitions between the transverse shocks and transverse detonations can occur both for marginal regimes [8] and far from the limits [13]. However, it seems that transverse detonations are more likely to appear when the induction zone becomes large, especially near the propagation limits when the average detonation velocity decreases. In particular, a transverse detonation provides the only mechanism of self-sustained propagation for the spin detonation, which is the ultimate marginal regime that occurs when only one primary triple-shock configuration exists in a round tube. Transverse detonation waves can also be unstable and produce another kind of the fine cellular structures observed for spin detonations [15–17]. We refer to these fine structures as the transverse detonation cells to distinguish them from the secondary detonation cells which are the result of instability of the overdriven parts of the leading detonation front. In this work, we combine the numerical simulations and experiments to investigate transverse detonations and fine detonation structures for the marginal regimes in two-dimensional systems. As an example, we consider a new marginal mode experimentally obtained in a flat channel. Highly resolved numerical solutions provide more detailed information about this regime, which involves transverse detonations and produces secondary and transverse detonation cells. Experimental Data The experiment was performed for a stoichiometric acetylene/oxygen mixture at the initial pressure P0 ⳱ 0.03 bar. The flat channel (1.5 mm thick) containing the mixture consisted of two sections. The first section, of constant 30 mm width, was connected to the second, slightly divergent (5⬚) section. The detonation initiated at the beginning of the first section, reached the quasi-steady-state regime, then entered the divergent section and died. The process was recorded using the open-camera technique. The ¯ ⳱ 0.8D0 ⳱ 1790 m/ average detonation velocity, D s, was determined from separate experiments in the same channel. Here, D0 ⳱ 2238 m/s was the ideal detonation velocity calculated for this system using the SAFETY code [18].
The open-camera photograph in Fig. 1 shows that the history of the detonation propagation looks very similar to the record expected for a spin detonation in a round tube. The difference is that the system considered here is practically two-dimensional. The tracks marked 1 were left by the transverse detonation wave which propagated from the top to the bottom, then disappeared, and appeared again at the top. Unlike a spin detonation, the track of the transverse detonation wave disappeared here because the transverse detonation actually died, not because it disappeared from view. After some time, the transverse detonation reignited again near the upper wall and moved down until it collided with the lower wall. These primary pulsations occurred at regular intervals of k1 ⳱ 77 Ⳳ 5 mm. The propagation of transverse detonation waves was accompanied by secondary pulsations produced by unstable overdriven parts of the leading detonation front. The smallest resolved secondary cells, marked 2, initially were about 2 mm wide. They grew as the overdrive decreased and essentially disappeared at the distance k1/3 from the transverse detonation track 1. The remnants of some of the secondary triple points survived until the next primary pulsation, and they might participate in the reignition of the transverse detonation. Transverse detonation cells that may result from the instability of the transverse detonation were not resolved in this experiment, though they were reported for the spin detonation in the stoichiometric acetylene/oxygen mixture diluted with argon [16]. The marginal detonation regime described here was observed only in flat channels and for a narrow range of parameters (P0 ⳱ 0.03 bar Ⳳ 10%) very close to the propagation limit. The detonation was easily destroyed in the slightly divergent section of the channel during only one primary pulsation. At a slightly higher P0, the detonation propagated in a different regime characterized by one primary triple-shock configuration reflecting from the walls and forming a regular half cell in the channel. A galloping mode was observed at a slightly lower P0. Similar to the spin detonation, the marginal regime described here provided a good opportunity to study different levels of detonation structures. It also had the advantage of being two-dimensional, an advantage which we exploited in numerical simulations of this regime.
FINE CELLULAR STRUCTURES TABLE 1 Physical parameters of the model system P0 (initial pressure) T0 (initial temperature) q0 (initial density) c (polytropic exponent) M (molecular weight) DCJ (CJ detonation velocity) PCJ (CJ pressure) TCJ (CJ temperature) qCJ (CJ density) PZND (ZND pressure) TZND (ZND temperature) qZND (ZND density) Q (heat of reaction) QM/RT0 QM/RTZND Ea (activation energy) Ea/RT0 Ea/RTZND A (pre-exponential factor) L05 (half-reaction length)
0.03 bar 298 K 0.03669 kg/m3 1.25 0.0303 kg/mol 1281 m/s 0.2807 bar 1627 K 0.06290 kg/m3 0.5315 bar 879 K 0.2203 kg/m3 1.282 MJ/kg 15.7 5.31 21.59 kcal/mol 36.5 12.4 6.146 ⳯ 109 m3/kg/s 0.00539 m
Numerical Model and Physical Parameters The time-dependent behavior of the reactive gaseous mixture is described by the reactive NavierStokes equations coupled with the polytropic equation of state and the one-step Arrhenius kinetics of energy release. The reaction rate is defined as d␣/dt ⳱ A(1 ⳮ ␣)q exp(ⳮEa/RT)
(1)
where A is the pre-exponential factor and ␣ is the mass fraction of reaction products. The density, q, in equation 1 reflects the experimental fact that the induction time for C2H2/O2 mixtures varies inversely with q [19]. The model takes into account the viscosity, heat conductivity, and diffusion defined in Ref. [20] as functions of density and temperature. The equations were solved on a structured mesh using an explicit, second-order, Eulerian, Godunovtype numerical scheme incorporating a Riemann solver [20,21]. The process-splitting and directionsplitting techniques [22] were employed to add the chemical sources and obtain the two-dimensional solution. An adaptive mesh refinement algorithm based on the fully threaded tree [21] was used to increase the resolution locally, as required by the changing physical conditions. The code was extensively tested in reactive fluid-dynamic simulations [20] including detonations [6]. We considered a rectangular two-dimensional computational domain, 30 mm wide and 240 mm long, bounded by free-slip solid walls on the top and bottom. The unreacted fluid entered the computational domain from the right with specified initial parameters q0, P0, and T0 and velocity ⳮDCJ. An
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extrapolated outflow boundary condition on the left allowed the pressure at the boundary to relax adiabatically to PCJ. The physical parameters of the reactive fluid were defined by taking into consideration that (1) the detonation regime that we were trying to model only existed for a narrow range of parameters and (2) the two-dimensional numerical model based on onestep kinetics could not quantitatively reproduce the experiment. Thus, we tried to obtain qualitatively the same regime, using the experimental conditions as a starting point and then adjusting the model parameters in a series of preliminary low-resolution runs. The resulting set of physical parameters used in the final simulations is summarized in Table 1. The unreacted mixture and the reaction products are described by the same polytropic equation of state, with c ⳱ 1.25, and the same transport properties that were selected in Ref. [20] to fit the laminar flame speed and thickness for acetylene/air mixtures. The heat of chemical reaction, Q, was one of the parameters that we varied to adjust the model. This parameter affects the detonation instability through Ea/RT and QM/RT , because it defines the con* * trolling temperature, T . The value of Q was chosen * so that the detonation was unstable enough to produce secondary triple points. As a result, the ideal thermodynamic detonation velocity, DCJ, for the model system equaled 0.57 D0. All the ChapmanJouguet (CJ) and Zeldovich-Neuman-Do¨ring (ZND) parameters in Table 1 correspond to that velocity. The activation energy, 18.1 kcal/mol, was obtained in Ref. [19] from the measurements of the induction times, s, for acetylene/oxygen mixtures, assuming that s ⬃ exp(Ea/RT). The asymptotic solution based on equation (1) gives for the adiabatic induction time s ⬃ T2 exp(Ea/RT). To take into account the term T2, we increased Ea by 2RTZND, taking TZND as the controlling temperature. The pre-exponential factor, A, in equation 1 was also an adjustable parameter. This parameter can be considered as a scale factor that controls the induction delays and the sizes of all the dynamic flow structures associated with them. It was selected to make the pattern of primary pulsations as close as possible to the experimental one. The calculations were performed for two numerical resolutions. For the higher resolution, the adaptive mesh contained six levels, with the minimum cell size dx ⳱ dy ⳱ 2.93 ⳯ 10ⳮ5 m, which corresponds to the effective resolution 184 cells per halfreaction length of CJ detonation. The minimum time step in this case was about 6 ⳯ 10ⳮ9 s. For the lower resolution, for which the mesh had five levels, the minimum cell size and time step were two times larger. Numerical Results The detonation in the low-resolution simulations was initiated by a strong shock placed near the left
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Fig. 2. Numerical smoke foils for the same primary pulsation calculated with low (a) and high (b) resolutions. The channel width is 30 mm. 1, transverse cells; 2, secondary cells.
boundary. This shock, with the initial pressure PZND, produced an overdriven detonation wave that moved to the right through the computational domain. Shortly after the initiation, the detonation wave developed several triple-shock configurations resulting from the specially introduced initial perturbations and the thermal instability of the front. Most of these triple points disappeared, but the remaining transverse shocks became stronger, sometimes generating transverse detonations in the large induction zone behind the leading shock. At the same time, small triple points appeared both in the overdriven parts of the leading front and in the transverse detonation waves and produced fine secondary and transverse cell structures, respectively. Eventually, the average detonation velocity became equal to DCJ, the average position of the detonation front stabilized in the middle of the computational domain (which had the inflow velocity ⳮDCJ), and the pattern of pulsations did not change. At this point, we increased the numerical resolution by introducing an additional level of mesh refinement and then continued the calculations. Fig.
2 compares the numerical smoke foils for one primary pulsation calculated with lower (Fig. 2a) and higher (Fig. 2b) resolutions. These figures recorded the maximum pressure reached at each location in space and show the trajectories of the triple points and transverse detonations. The maximum pressure was about 2 bar for the secondary triple points and 6 bar for the triple points inside the transverse detonation wave. Increasing the resolution did not noticeably affect the primary pulsations and had a little effect on the secondary cells. However, it substantially increased the number of transverse cells which are still not well resolved. The numerical smoke foil in Fig. 3 shows a typical part of the history of the detonation propagation in the quasi-steady-state regime over a longer period of time. Two tracks were left by transverse detonations, both containing traces of the fine triple points which form the transverse cells. The primary pulsations also produced multiple secondary cells, which grew and became more irregular until the next transverse detonation re-created the secondary structure. The transverse detonations periodically appeared near one of the walls and disappeared when they collided with the opposite wall. The distance between primary pulsations was about 15 cm. This regime existed only for a very narrow range of parameters. The detonation would die if we decreased the reaction rate by 5%. Increasing the reaction rate by 5% led to a regular half cell in the channel. A second primary triple point appeared when the reaction rate was increased by about 50%. The four snapshots of the temperature field behind the leading shock, shown in Fig. 4, provide direct information about the dynamic structure of the detonation front. In particular, we see that the transverse detonation formed when transverse shocks reflected from the wall interacted with the large induction zone behind the leading shock (frame 1). The transverse detonation propagated along the induction zone and burned almost all the material except for a thin tail in the vicinity of the primary triple point (frame 2). The fine structure of the transverse detonation included the small triple points moving along the front, which had relatively strong and weak parts with different induction-zone
Fig. 3. Numerical smoke foil for the marginal detonation regime calculated with high resolution. The arrows at the top correspond to the positions of the primary triple point in the frames shown in Fig. 4.
FINE CELLULAR STRUCTURES
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Fig. 4. Temperature field behind the leading shock at different times. The position of the primary triple point for every frame is marked by an arrow at the top of Fig. 3. 1, transverse detonation; 2, strong part of the leading shock (overdriven detonation); 3, weak part of the leading shock (inert); 4, induction zone; 5, transverse shock; 6, unreacted tail; 7, primary unreacted pocket; 8, secondary unreacted pockets.
lengths. Due to insufficient numerical resolution, we did not see the unreacted pockets which could appear behind the transverse detonation. The pockets were well resolved for the secondary detonation structure behind the strong part of the leading shock. These pockets grew as the overdrive decreased, moved downstream, and burned at a distance of the order of the secondary cell size from the leading shock. Eventually, secondary triple points became so weak that they could no longer cause enough energy release to separate pockets from the induction zone. At this time, the transverse detonation wave collided with the wall and died. The reflected shock could not start a new transverse detonation because the induction zone was not large enough (frame 3). As the primary transverse shock propagated through the irregular reaction zone that contained remnants of the secondary structures, it induced some energy release and rapidly became attenuated. The induction zone behind the weak part of the leading shock grew. Because unreacted material did not burn in the transverse shock, it became a pocket the next time the primary triple point collided with the wall. Before the collision (frame 4), the induction zone behind the strong part of the leading shock was larger than it was in frame 3, but it was not yet large enough for transverse detonation to occur. Fig. 3
shows that the transverse detonation will reappear after one more reflection.
Discussion The marginal detonation regimes in the numerical simulations and those observed experimentally in a flat channel have similar detonation structures, even though the calculated detonation history looks more irregular. The difference is not surprising considering the approximate nature of the numerical model. It is important to note that the numerical smoke foil is a history of the maximum pressure and captures more pressure waves than the experimental opencamera technique that records light emitted by the detonation front. The calculations produce the same kind of primary pulsations as the experiments, and they show how the transverse detonations form and why they disappear and reappear. The numerical results also provide detailed information about fine detonation structures, including transverse cells that are not resolved in the experiment. Since we use an approximate model, and the numerical resolution is not high enough to produce transverse detonation cells of the correct size, we can only say that these cells are likely to exist in the real system, and their size, as follows from Fig. 2, should be well below 1
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mm. This is too small to be resolved by the experimental technique used. The existence of the transverse cells in this system is also confirmed by experimental observations [16] of spin detonations in a stoichiometric acetylene/oxygen mixture diluted with argon. Dilution with an inert monoatomic gas usually increases postshock temperatures, so the scaled activation energy, h, decreases, and the detonations become more stable. Nevertheless, transverse detonation cells were observed in Ref. [16] for the argon-diluted system. This means that these cells are even more likely to exist in the undiluted mixture. The smoke foil published in Ref. [16] also shows that a spin detonation in the mixture with argon does not produce secondary cells. Only a few short traces of secondary triple points are visible. This observation agrees well with the lower value of h for the diluted mixture, and it also means that the h that controls the stability of the overdriven part of the leading front is lower than the h that controls the stability of the transverse detonation. This may occur only if the controlling temperature, T , for the trans* verse detonation is lower than the T for the strong * leading shock. Indeed, analysis of the numerical results shows that the postshock temperature varies from 1050 to 1100 K for the transverse detonation wave and from 1120 to 1400 K for the strong part of the leading shock. At the same time, the postshock pressure is substantially higher for the transverse detonation (1.6–1.8 bar) than for the leading shock (0.75–1.0 bar). The higher pressure and lower temperature for the transverse detonation result from the double-shock compression that occurs when the shock-compressed material in the induction zone is compressed again by the transverse shock. A small part of the unreacted material also undergoes a triple-shock compression. This happens when the gas enters the induction zone through the weak leading shock in the vicinity of the primary triple point and then passes through two oblique shocks in a gap between the leading and the transverse detonation fronts. The details of similar complex Mach structures were described in Ref. [10] for spin detonations and investigated in Refs. [13,14] for twodimensional systems. As shown in frame 2 in Fig. 4, the temperature of the material passing through the gap is so low (about 950 K) that the mixture does not ignite but forms an unreacted tail. This tail burns slowly and separates the detonation products originating from the leading and the transverse detonation fronts. Similar unreacted tails were observed in experiments [12]. The transverse cells and the unreacted tails are characteristic features that can form only when the transverse shock moving through the induction zone is a detonation. Transverse shocks with small or no energy release are stable and do not produce any
cells. The material in the induction zone is still unreacted behind inert transverse shocks and forms unreacted pockets instead of tails. The secondary cellular structure is a more general feature. It appears because the leading detonation front is unstable and it does not require the presence of transverse detonations. For example, secondary cells without transverse detonations were obtained in numerical simulations [6] of detonations in systems with high activation energy. Conclusions Four major contributions result from this work: 1. We reported a new two-dimensional detonation regime observed in experiments near the propagation limits and confirmed by numerical simulations. The new feature of this regime is the periodic appearance and disappearance of transverse detonations that form when the induction zone becomes large enough behind the leading shock. 2. Fine cellular structures produced by unstable transverse detonations and unreacted tails associated with transverse detonations were shown in numerical simulations for the first time. 3. It was shown that transverse detonations are more unstable than the overdriven parts of the leading detonation front because of the doubleshock compression of the material involved in transverse detonations. 4. The numerical simulations showed that all of the complexity of the phenomena can be explained in the framework of the one-step Arrhenius kinetics. Acknowledgments We would like to thank Dr. Almadena Chtchelkanova for help with the parallel implementation of numerical smoke foils. REFERENCES 1. Shchelkin, K. I., Usp. Fiz. Nauk 87:273–302 (1965). 2. Manzhalei, V. I., Fizika Gorenija i Vzryva 13:470–472 (1977). 3. Libouton, J-C., Jacques, A., and Van Tiggelen, P. J., in Colloque International Berthelot-Vieille-Mallard-Le Chatelier, Vol. 2, The Combustion Institute, Pittsburgh, PA, 1981, pp. 437–442. 4. Bull, D. C., Elsworth, J. E., Shuff, P. J., and Metcalfe, E., Combust. Flame 45:7–22 (1982). 5. Vasil’ev, A. A., Mitrofanov, V. V., and Topchian, M. E., Fizika Gorenija i Vzryva 23:109–131 (1987). 6. Gamezo, V. N., Khokhlov, A. M., and Oran, E. S., “Secondary Detonation Cells in Systems with High Activation Energy,” in Proceedings of the Seventeenth International Colloquium on the Dynamics of Explosions and Reactive Systems, Universita¨t Heidelberg, Hei-
FINE CELLULAR STRUCTURES delberg, Germany, IWR, 1999. 7. Short, M., and Stewart, D. S., J. Fluid Mech. 382:109– 135 (1999). 8. Fickett, W., and Davis, W. C., Detonation, University of California Press, Berkeley, CA, 1979, p. 327. 9. Gamezo, V. N., Desbordes, D., and Oran, E. S., Combust. Flame 116:154–165 (1999). 10. Voitsekhovski, B. V., Mitrofanov, V. V., and Topchian, M. E., Proc. Combust. Inst. 12:829–837 (1969). 11. Strehlov, R. A., and Crooker, A. J., Acta Astronaut. 1:303–315 (1974). 12. Subbotin, V. A., Fizika Gorenija i Vzryva 11:96–102 (1975). 13. Lefebvre, M. H., and Oran, E. S., Shock Waves 4:277– 283 (1995). 14. Oran, E. S., Weber Jr., J. W., Stefaniw, E. I., Lefebvre, M. H., and Anderson Jr., J. D., Combust. Flame 113:147–163 (1998).
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15. Denisov, Yu. N., and Troshin, Ya. K., Doklady Akad. Nauk SSSR 125:110–113 (1959). 16. Lee, J. H., Soloukhin, R. I., and Oppenheim, A. K., Astronaut. Acta 14:565–584 (1969). 17. Manzhalei, V. I., and Mitrofanov, V. V., Fizika Gorenija i Vzryva 9:703–710 (1973). 18. Vasil’ev, A. A., in Proceedings of the Twenty-Eighth Annual ICT Conference, Fraunhofer Institute of Chemical Technologies, Karlsruhe, Germany, 1997, pp. V50.1–V50.14. 19. White, D. R., Proc. Combust. Inst. 11:147–154 (1967). 20. Khokhlov, A. M., Oran, E. S., Chtchelkanova, A. Yu., and Wheeler, J. C., Combust. Flame 117:99–116 (1999). 21. Khokhlov, A. M., J. Comput. Phys. 143:519–543 (1998). 22. Oran, E. S., and Boris, J. P., Numerical Simulations of Reactive Flows, Elsevier, New York, 1987.
COMMENTS Douglas Feikema, NASA Glen Research Center, USA. I would like further clarification of the initial conditions of your calculations. How does the unsteady behavior in the flow channel develop? In the movie you showed in the presentation, why did you choose the initial state you did? Author’s Reply. The detonation was initiated by a strongly perturbed shock. The initial perturbations made the shock non-planar and produced several triple-shock configurations. As the detonation propagated along the channel, most of the initial triple points disappeared. A few remaining transverse shocks became stronger, sometimes producing transverse detonations. The transverse detonations and the overdriven parts of the leading front developed small triple points. Eventually, the detonation became quasi–steady state and established a pattern of pulsations that did not depend on the initial conditions. One of these
established pulsations was shown in the movie. The first frame of the movie corresponds to the first frame of Fig. 4. ● Pierre Van Tiggelen, University of Louvain, Belgium. How sensitive are your results with respect to the kinetic parameters used for the computation? Would a two-step mechanism not be more accurate to describe the kinetics? Author’s Reply. The results are very sensitive to the kinetic parameters due to the fact that the marginal detonation regime which we are trying to model exists only for a very narrow range of parameters. For example, the detonation will die if we decrease the reaction rate by 5%. Increasing the reaction rate by 5% leads to a regular halfcell in the channel.