Reactive Flow Phenomena in Pulse Detonation Engines

AIAA 2003-1171 Reactive Flow Phenomena in Pulse Detonation Engines X. He and A. R. Karagozian UCLA Los Angeles, CA 41st AIAA Aerospace Sciences Meeting and Exhibit 6–9 January 2003 Reno, Nevada For permission to copy or to republish, contact the American Institute of Aeronautics and Astronautics, 1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344. AIAA–2003–1171 REACTIVE FLOW PHENOMENA IN PULSE DETONATION ENGINES X. He ∗ and A. R. Karagozian † Department of Mechanical and Aerospace Engineering University of California, Los Angeles, CA 90095-1597 ‡ Abstract the thrust wall (Figures 1ef), allowing reactants to be drawn into the tube, and propagation of the expansion fan out of the tube (Figures 1gh), with simultaneous reflection of a compressive disturbance into the tube (Figures 1hij), which reflects from the thrust wall, ignites the fresh reactants, and reinitiates the cycle. Because the PDE concept holds promise for high thrust density in a constant volume device requiring little or no rotating machinery, a number of groups have been exploring PDEs for propulsion applications. This exploration has built 1–4 on fundamental PDE work over several decades , so that modern experimental diagnostic as well as computational methods may be used to bring about 2, 4–6 . significant advances in the state of the art Performance parameters commonly used to characterize the pulse detonation engine include the impulse, I, typically defined as
∞ I≡A ∆ptw (t)dt (1) This paper describes one- and two-dimensional numerical simulations, with simplified as well as full reaction kinetics, of a single cycle pulse detonation engine (PDE). Focus of the present studies is on 1) the presence of a nozzle extension at the end of the tube, and its effect on performance parameters as well as noise characteristics, 2) critical “spark ignition” energies associated with the initiation of a detonation in the PDE tube, and 3) quantification of performance parameters associated with full kinetics simulations of the PDE and comparison of these data sets with available experimental data. The present simulations demonstrate the ability to predict PDE reactive flow phenomena and associated performance and noise characteristics, and hence have promise as a predictive tool for the evolution of future PDE designs. 0 Introduction and Background where A is the area of the thrust wall and ∆ptw (t) is the time-dependent pressure differential at the thrust wall. The impulse is usually scaled to produce the engine’s specific impulse, Isp , The Pulse Detonation Wave Engine (often called the Pulse Detonation Engine or PDE) is a device which allows periodic ignition, propagation, and transmission of detonation waves within a detonation tube, with associated reflections of expansion and compression waves which can act in periodic 1, 2 fashion to produce thrust . A summary of the relevant gasdynamics within the PDE tube is shown in Figure 1. The figure indicates ignition and propagation of the detonation out of the PDE tube (Figures 1a-c), reflection of an expansion fan into the tube (Figures 1de), reflection of the expansion fan from Isp ≡ I ρV g (2) where ρ is the initial mass of the reactive gas mixture in the tube, V is the detonation tube volume (including the nozzle volume, if containing reactants), and g is the earth’s gravitational acceleration. As an alternative performance parameter, the fuel-based specific impulse, Isp,f , is often used: ∗ Graduate Researcher Associate Fellow, AIAA. Corresponding author (ark@seas.ucla.edu). ‡ Copyright (c) 2003 by X. He Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Isp,f ≡ † Professor; Isp Yf (3) where Yf is the fuel mass fraction present within the premixed reactants in the tube. Both Isp and Isp,f 1 American Institute of Aeronautics and Astronautics AIAA–2003–1171 are often used to compare performance among different PDE configurations and also to compare PDE 7 performance with that of alternative engine cycles . PDE tube is designed to operate in a cyclical manner, it is of interest to quantify the required energy input to be able to repetitively initiate a detonation front. Prior computational studies by our group pertain14 ing to detonation phenomena in general and the 15 pulse detonation engine in particular involve both one- and two-dimensional simulations, employing 16–18 the essentially non-oscillatory or ENO scheme for spatial integration. An examination of the one14 dimensional overdriven detonation suggests specific requirements for spatial resolution of the detonation front to be able to obtain accurate wave speeds, peak pressures, and frequencies of detonation oscillation. These ideas are incorporated into 15 1D and 2D simulations of the single cycle PDE with single step reaction kinetics. The studies suggest that useful performance and noise related estimates may be obtained even from one-dimensional computations of the pulse detonation wave engine with simplified reaction kinetics. The present study focused on using these high order numerical schemes to study the behavior of the pulse detonation engine, using simplified as well as complex reaction kinetics. Special attention was paid to the PDE’s geometrical, flow, and reaction characteristics and their influence on performance parameters as well as noise generation. NASA’s in19 terest in the PDE for advanced vehicle propulsion is incumbent upon the ability of the engine to efficiently generate thrust without having to pay a significant penalty in engine noise. While specific geometries for PDE nozzle extensions may have the effect of reducing the relative Isp , as suggested in 10 recent experiments , there may be benefits associated with noise reduction. Overviews of past and ongoing numerical simulations of PDEs are described in recent articles by 2, 8, 9 . Other recent simulations have foKailasanath cused on various flow and geometrical features of the PDE, including the effects of nozzles placed downstream of the detonation tube. Cambier and 4 Tegner , for example, find that the presence of the nozzle can have a significant effect on the impulse of a single cycle of the PDE. In 2D axisymmetric simulations, which employ a second order TVD (total variation diminishing) scheme, they find that increasing the ratio of the nozzle exit area to the tube area can produce monotonic increases in impulse I. Increasing the nozzle exit area causes a dropoff in Isp until Aexit /Atube reaches 4.0, since the nozzle volume increases; for area ratios larger than this value, Isp is seen to increase, suggesting that the impulse is increasing faster than the nozzle volume increases, per equation (2). In recent PDE experiments with nozzle exten10 sions, Johnson observes a decrease in the fuel specific impulse for converging-diverging nozzles as compared with straight nozzles or converging nozzles. Similarly, tests as well as modeling by Cooper 6 and Shepherd suggest that the relative impulse of a PDE tube with a flared nozzle is lower than that for a straight nozzle at a given fill fraction (or percentage of the PDE tube initially filled with reactants). It is of interest to understand the mechanisms whereby nozzles can increase or decrease PDE performance and what the associated changes in engine noise levels may be. Another issue of interest with respect to the successful performance of the pulse detonation engine is quantification of the required energy input needed to ignite and sustain a propagating detonation wave from the closed end of the tube. Thermally initiated detonations via the deflagration-to-detonation transition (DDT) have been examined over many 11–13 . When a mixture of reactants is ignited years by a bulk power deposition of limited duration, a sequence of events is initiated which eventually results in a sudden power burst or “explosion in the explosion”, accelerating the flame front and leading to formation of a propagating detonation. Since the Problem Formulation and Numerical Methodology The equations of mass, momentum, energy, and species conservation were solved in both one and two spatial dimensions, assuming inviscid flow. Single step reaction kinetics for CH4 − O2 and H2 − O2 , 15 as outlined in detail in He and Karagozian , as well as full reaction kinetics for mixtures of H2 − O2 , H2 −O2 −Ar, and H2 −O2 −N2 were employed. Both straight PDE tubes and tubes with nozzle extensions were explored. In the 1D simulations, the computational domain consisted primarily of the detonation tube or tube and nozzle (containing at least 2 American Institute of Aeronautics and Astronautics AIAA–2003–1171 600 grid points), with only a few grid points extending beyond the tube end in order to capture the external pressure. In the 2D simulations, the air external to the detonation tube was assumed to be uniformly at atmospheric pressure, and the computational domain extended well downstream of the end of the tube, in general at least one and one half tube lengths downstream and at least two tube diameters away from the detonation tube in the dimension perpendicular to the axial dimension. The effects of employing a 1D pressure relaxation length 9 (PRL), as done by Kailasanath and Patnaik , were 15 explored in our prior PDE study , but for most of the conditions examined, a relaxation length was not needed in order to obtain equivalent results between 1D and 2D simulations. In cases where alternative nozzle geometries were considered, a locally 1D flow approximation was employed to represent nozzle shapes of slowly varying cross-sectional areas A(x). In this quasi-onedimensional case, with a single step reaction, for example, the governing equations reduce to the following form: reaction-rate multiplier for the reaction source term. Through equation (4) it became possible, in an approximate way, to represent the effects of nozzle geometry in 1D PDE simulations. Four different nozzle extension shapes were explored here; these are shown in Figure 2. The nozzle shapes included a fifth order polynomial (configuration 1), a flared divergent section (configuration 2), a nozzle section with a constant conical divergence angle (configuration 3), and a straight tube (configuration 4). In the simulations of PDEs with nozzles, the straight portion of the PDE tube, of length L, was assumed to be initially filled with reactants, while the nozzle section, of length Ln , was filled with inert gas (for a single step reaction, effectively products). In the simulations of straight PDE tubes without a nozzle, the tube was assumed to be initially filled completely with reactants. Unless otherwise stated, the straight tube lengths L used in the present computations were 1 m, and the nozzle lengths Ln were also 1 m. For the simulations involving complex reaction kinetics, the equations (5) - (7) were replaced by relations for the straight PDE tube (with A(x) constant) but with N − 1 species equations for the N species involved in the reactions. Full kinetics simulations of the combustion reactions for H2 − O2 , H2 − O2 − Ar, and H2 − O2 − N2 (representing hydrogen-air) were considered here; the latter mechanism contained 23 elementary reactions and was part of the CHEMKIN 20 II library . ∂  ∂  1 dA   (H − F ) + S (4) U+ F = ∂t ∂x A dx where the vectors containing conserved variables, flux terms, and source terms are:     ρ ρu 2      =  ρu  F =  ρu + p  U  E   (E + p)u  ρY ρuY     0 0     0  =  = p  S H    0  0   TA  − 0 −KρY e T (5) 15 As in He and Karagozian , the present study used the Weighted Essentially Non-Oscillatory (WENO) 21 16–18 method , a derivative of the ENO method for spatial interpolation of the system of governing equations. The WENO scheme was fifth order accurate in smooth regions and third order accurate in the vicinity of discontinuities. The ENO/WENO schemes were tested on a variety of problems, including shock tubes with open ends, analogous to 15 the exit of the PDE , and that of the classical 14 one-dimensional, overdriven, pulsating detonation . For the single step kinetics simulations, the third order total variation diminishing (TVD) RungeKutta method was used for time discretization. For full kinetics simulations, the method of operator 22 splitting was used, whereby the system of governing equations (including N −1 species equations) was (6) Here E may be written  ρ u2 + v 2 p E= + + ρqY γ−1 2 (7) where ρ represents density, p is the static pressure, u is the x-component of the velocity vector, and γ is the ratio of specific heats. q is a heat release parameter which characterizes the amount of energy released during the reaction, and TA represents the activation temperature. Y is the reactant mass fraction, which varies from 0 to 1, while K is the 3 American Institute of Aeronautics and Astronautics AIAA–2003–1171 split into two separate equations, one which only included the advection-diffusion terms (solved via WENO) and one which only included the reaction rate source terms. A stiff ODE solver, DVODE (a 23 variation of VODE ) was employed for the solution of the rate equations; thermodynamic parameters and rate constants were obtained via the CHEMKIN 20 II subroutine . A computational “spark” adjacent to the thrust wall was used to initiate the detonation at the start of the PDE cycle. This narrow, high pressure, high temperature region (3 grid cells in width) was able to initiate a propagating shock and ignite the reactants; the flame front then caught up with the shock, form11–13 , ing a detonation. As suggested by prior studies however, such thermal initiation of detonation depends very strongly on the initial rate of deposition of energy in the reactants. This concept was explored in the present studies by altering the initial temperature and pressure in the computational “spark” to be able to determine minimum input energy densities leading to detonation initiation. In addition to the standard performance parameters used to characterize the PDE (I, Isp , and Isp,f ), the sound pressure level (SPL) at various locations within and external to the detonation tube was also 15 computed. As done previously , these noise levels were estimated by examining the Fourier transform of the time-dependent pressure measured at various locations within the computational domain. The SPL was then computed based on peak pressures in the Fourier spectrum. In most cases these peaks occurred at the PDE cycle frequency. 1D pressure relaxation length. 1D simulations of the PDE tube do not precisely replicate the pressure and Mach number evolution at the tube end, with or without a PRL. Yet the evolution of the tube’s interior pressure without a PRL is, in most 15 cases previously explored , sufficiently close to that obtained from the 2D simulations so as to produce similar PDE performance estimates. This is shown, for example, in Figure 4, which compares the specific impulse for 2D simulations with that for 1D simulations, with and without inclusion of a PRL. Time-series pressure data at specific locations were used to estimate the noise generated at various points in the flowfield over a single PDE cycle. Estimates of the sound pressure level were made using both 1D and 2D simulations of the straight PDE tube with a single step CH4 − O2 reaction. Since the 1D simulations only resolved the flow within the PDE tube, comparisons were made only for interior and tube exit noise levels. In all locations for this case we observed the peak in pressure to appear close to the frequency associated with the period of the PDE cycle, roughly 330 Hz. The noise levels at various tube locations are quantified in Table 1. Location 2D SPL 1D SPL Thrust Wall Mid-tube Tube end 212 dB 211 dB 202 dB 212 dB 211 dB 203 dB Table 1. Computed sound pressure level (SPL) at various locations within the tube (thrust wall, center of tube, and tube end). Results are computed from both 2D and 1D simulations of the CH4 − O2 reaction. Results Consistent with the evolution of the pressure field and the performance parameters (e.g., Figure 4), the noise levels were nearly the same for 1D as for 2D simulations. The magnitudes of the noise levels were 24, 25 close to those quantified in PDE experiments . The influence of the presence of the nozzle is shown in Figure 5, where the straight tube (nozzle configuration 4) had the same length as the tubes with the divergent nozzles. Again, a CH4 − O2 single step reaction was used in this set of simulations. Interestingly, the straight tube was observed to produce the highest values of I, Isp , and Isp,f , while the conical nozzle (configuration 3) produced the lowest values. These findings were generally consistent An example of the temporal evolution of the pressure distribution along the centerline of a straight PDE tube, over a single cycle, is shown in Figure 3 for a 2D axisymmetric configuration with a single step methane-oxygen reaction. Here the initiation and propagation of the detonation wave through the tube (Figures 3ab) and the exit of the shock from the tube and reflection of the expansion fan from the exhaust back into the tube (Figures 3cd) 15 are clear. Our prior studies demonstrate that a 1D simulation of this same PDE tube quantitatively yields a very similar pressure field evolution to that of the 2D simulation, even without inclusion of a 4 American Institute of Aeronautics and Astronautics AIAA–2003–1171 10 The full kinetics simulations of the reactant-filled, straight PDE tube without a nozzle allowed a more detailed examination of the detonation ignition and propagation process to be made, in addition to more quantitative comparisons with experimental data. Figure 7 displays the evolution of the 2D pressure field associated with the PDE tube and its surroundings, for a full H2 − O2 reaction. As seen in prior 2D simulations of the PDE but with simplified 15 kinetics , the propagation of the detonation out of the tube resulted in the propagation of a vortical structure coincident with the shock and simultaneous reflection of an expansion fan back into the tube. Increased complexity in the wave structures as compared with that for simplified reaction kinetics was observed, especially in the propagating shock/vortex structure downstream of the tube exit. The influence of the initial pressure and temperature (and resulting energy deposition) on initiation of a detonation wave was studied using these full kinetics simulations. Figure 8 shows the centerline pressure distribution for a 1D, full kinetics simulation of an H2 − O2 − Ar mixture, for different initial temperatures and pressures in the 3 grid cell-wide “spark” adjacent to the thrust wall. Critical combinations of temperature and pressure were observed to be necessary for the classical ZND detonation structure to evolve; if the initial energy deposition was too small, a weak shock front did not ignite the mixture and thus did not transition to a detonation, as seen by the solid lines in Figures 8ab. Tables 3 and 4, for H2 − O2 − Ar and H2 − O2 − N2 reactions, respectively, quantify the conditions that were required for ignition of a detonation. with the observations of Johnson , i.e., that Isp,f decreased with inclusion of a divergent exit nozzle. Our results were also consistent with those of Cam4 bier and Tegner , in that Isp decreased for nozzleto-tube area ratios of 4.0 (examined here). The fact that the straight tube produced the highest thrust (resulting from the highest sustained pressure at the thrust wall) has interesting implications for PDE noise estimates. Figure 6 shows the results of taking the Fourier transform of the time dependent pressure within (Figure 6ab) and at the end (Figure 6c) of the tube/nozzle, for the four different nozzle configurations explored here. For example, in the middle of the PDE tube, the straight nozzle case produced the smallest pressure perturbation at the PDE cycle frequency, yet at the end of the straight nozzle, the pressure and hence the SPL were both larger than those for the other nozzles, albeit at a higher harmonic (667 Hz) of the PDE cycle frequency (about 333 Hz). This behavior is reflected in Table 2 for SPL values at various locations. Location Thrust wall Mid-tube Tube end Nozzle end Config. 1 SPL 212 211 205 188 dB dB dB dB Config. 4 SPL 210 208 202 208 dB dB dB dB Table 2. Computed sound pressure level at various locations within the tube (thrust wall, center of tube, PDE tube end) and at the nozzle exit for two different nozzle configurations (see Figure 2). Results are computed from quasi 1D simulations of the CH4 − O2 reaction. The above behavior likely resulted from the weakened downstream-propagating shock that formed in the divergent nozzles as compared with that for the straight nozzle. The lower pressure and SPL in the upstream portions of the straight nozzle (as compared with the divergent nozzles) possibly could have resulted from stronger reflected expansion waves that occurred at the contact surface between reactants and air at the start of the nozzle. While there were clear tradeoffs between performance and nozzle exit noise conditions, it is unclear why the pressure perturbation of the higher harmonic (660 Hz) in the straight nozzle was so much larger than for the divergent nozzles. This and other noise related issues require further exploration in the future. Temp. 500K 1000K 1500K 2000K 1500K 1500K 1500K Press. Energy (erg/cm2 ) 3 atm 3 atm 3 atm 3 atm 5 atm 2.5 atm 2.0 atm 5 2.02 × 10 8.06 × 105 9.95 × 105 11.0 × 105 15.3 × 105 8.63 × 105 7.3 × 105 Deton.? No No Yes Yes Yes Yes No Table 3. Initial temperatures, pressures, and input energies for a computational “spark” used to ignite a H2 − O2 − Ar mixture, and determination of the possibility of detonation ignition. 5 American Institute of Aeronautics and Astronautics AIAA–2003–1171 Temp. Press. 800K 900K 1000K 1200K 1 1 1 1 atm atm atm atm Energy (erg/cm2 ) Deton.? 4.54 × 105 4.89 × 105 5.18 × 105 5.64 × 105 No No Yes Yes Conclusions High resolution numerical simulations of pulse detonation engine phenomena revealed useful information that may be used in future PDE designs. Simulations of the effects of the presence of a divergent nozzle downstream of the PDE tube suggested that, while performance parameters such as Isp may decrease with increasing nozzle exit area, the noise generation at the nozzle exit may actually be reduced, and hence these tradeoffs may be explored through simplified reaction kinetics studies. Simulations of PDE evolution with full chemical kinetics suggested that specific minimum energy densities were required to enable the initiation of a detonation, and hence to sustain the PDE cycle. Finally, it was observed that full kinetics simulations were able to capture quantitatively the physical phenomena and corresponding performance parameters for the PDE. Future studies will continue with this quantitative comparison as well as noise generation issues relevant to the PDE. Table 4. Initial temperatures, pressures, and input energies for a computational “spark” used to ignite a H2 − O2 − N2 mixture, and determination of the possibility of detonation ignition. As expected, the critical input energies for ignition of a detonation were found to be different for these different reactions. In the case of H2 − O2 − N2 , a critical energy deposition per unit area of about 5×105 erg/cm2 was required for detonation, whereas for the case of H2 − O2 − Ar, this critical value rose to about 8.5 × 105 erg/cm2 . The full kinetics simulations also allowed quantitative comparisons to be made between performance parameters from the present simulations and those obtained by experiment (for a single cycle PDE) or analysis. Table 5 below shows the current estimations of Isp for the PDE for H2 −O2 and H2 −O2 −N2 (hydrogen-air) reactions, as compared with the anal26 ysis and experiments described in Wintenberger 27 28 and the experiments of Zitoun and Schauer . Study Isp , H2 -air Present 26 Wintenberger 26 CIT expts. 27 Zitoun expts. 28 Schauer expts. 128.5 s 123.7 s – 149 s 113 s Acknowledgments This work has been supported at UCLA by NASA Dryden Flight Research Center under Grant NCC4153, with Dr. Trong Bui and Dave Lux as technical monitors, and by the Office of Naval Research under Grant ONR N00014-97-1-0027, with Dr. Wen Masters as technical monitor. Isp , H2 − O2 240 173 200 226 – s s s s References [1] Eidelman, S., Grossmann, W., and Lottati, I., “Review of Propulsion Applications and Numerical Simulations of the Pulsed Detonation Engine Concept”, J. Propulsion and Power, 7(6), pp. 857-865, 1991. Table 5. Comparison of specific impulse for single cycle PDE between the present simulations and corresponding experiments and modeling efforts, as noted. While the present simulations appeared quantitatively to replicate the experimentally observed performance parameters reasonably well, detailed comparisons of the pressure field evolution and noise estimates require further examination and are the subject of continued studies. [2] Kailasanath, K., “Recent Developments in the Research on Pulse Detonation Engines”, AIAA Paper 2002-0470 (Invited), AIAA 40th Aerospace Sciences Meeting, January, 2002. [3] Helman, D., Shreeve, R. P., and Eidelman, S., “Detonation Pulse Engine”, AIAA Paper 861683, June, 1986. 6 American Institute of Aeronautics and Astronautics AIAA–2003–1171 [4] Cambier, J.-L. and Tegner, J. K., “Strategies for Pulsed Detonation Engine Performance Optimization”, Journal of Propulsion and Power, 14(4), pp. 489-498, 1998. [14] Hwang, P., Fedkiw, R., Merriman, B., Karagozian, A. R., and Osher, S. J., “Numerical Resolution of Pulsating Detonation Waves”, Combustion Theory and Modeling, Vol. 4, No. 3, pp. 217-240, 2000. [5] L. Ma, S.T. Sanders, J.B. Jeffries, and R.K. Hanson, “Monitoring and Control of a Pulse Detonation Engine using a Diode-Laser Fuel Concentration and Temperature Sensor,” Proc. of the Comb. Inst., 29, 2002, to appear. [15] He, X. and Karagozian, A. R., “Numerical Simulation of Pulse Detonation Engine Phenomena” to appear in the SIAM Journal of Scientific Computing, 2003. [6] Cooper, M. and Shepherd, J. E., “The Effec of Nozzles and Extensions on Detonation Tube Performance”, AIAA Paper 02-3628, 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, July, 2002. [16] Harten, A., Osher S. J., Engquist, B. E., and Chakravarthy, S. R., “Some Results on Uniformly High-Order Accurate Essentially Nonoscillatory Schemes”, J. Appl. Numer. Math., Vol. 2, pp. 347-377, 1986. [7] Povinelli, L. A., “Pulse Detonation Engines for High Speed Flight”, Paper ID 17-5169, presented at the 11th AIAA/AAAF International Conference on Space Planes and Hypersonic Systems and Technologies, Orleans, France, Sept. 29 - Oct. 4, 2002. [17] Shu, C.W. and Osher, S., “Efficient Implementation of Essentially Non-Oscillatory Shock Capturing Schemes II”, Journal of Computational Physics, Vol. 83, pp. 32-78, 1989. [18] Fedkiw, R.P., Merriman, B., Osher, S., “High accuracy numerical methods for thermally perfect gas flows with chemistry”, Journal of Computational Physics, Vol. 132, No. 2, pp. 175-190, 1997. [8] Kailasanath, K., “A Review of PDE Research – Performance Estimates”, AIAA Paper 20010474, AIAA 39th Aerospace Sciences Meeting, January, 2001. [19] “Three Pillars for Success: NASA’s Response to Achieve the National Priorities in Aeronautics and Space Transportation”, NASA Office of Aeronautics and Space Transportation Technology, 1997. [9] Kailasanath, K. and Patnaik, G., “Performance Estimates of Pulsed Detonation Engines”, 28th Symposium (Intl.) on Combustion, 2000. [10] Johnson, C., “The Effects of Nozzle Geometry on the Specific Impulse of a Pulse Detonation Engine”, Final Report 16.622, MIT, December, 2001. [20] Kee, R. J., Miller, J. A., and Jefferson, T. H., “CHEMKIN: A general purpose, problem independent, transportable, Fortran chemical kinetics code package”, Sandia National Laboratories Report SAND80-8003, 1980. [11] Oppenheim, A. K., Manson, N., and Wagner, H. G., “Recent Progress in Detonation Research”, AIAA Journal, Vol. 1, pp. 2243-2252, 1963. [21] Jiang, G. S. and Shu, C. W., “Efficient Implementation of Weighted ENO Schemes”, Journal of Computational Physics, Vol. 126, pp. 202228, 1996. [12] Lee, J. H. S., “Initiation of Gaseous Detonation”, A. Rev. Phys. Chem., Vol. 28, pp. 74-104, 1977. [22] Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, Wadsworth and Brooks, 1989. [13] Sileem, A. A., Kassoy, D. R., and Hayashi, A. K., “Thermally Initiated Detonation through Deflagration to Detonation Transition”, Proc. Royal Soc. London A, Vo. 435, pp. 459-482, 1991. [23] Brown, P. N., Byrne, G. D., and Hindmarsh, A. C., “VODE: A variable coefficient ODE solver”, SIAM J. Scientific Statistical Computing 10, pp. 1038-1051, 1989. 7 American Institute of Aeronautics and Astronautics AIAA–2003–1171 [24] Schauer, F., private communication. detonation front [25] Perkins, H. D., private communication. reactants a) [26] Wintenberger, E., Austin, J. M., Cooper, M., Jackson, S., and Shepherd, J. E., “An Analytical Model for the Impulse of a Single Cycle Pulse Detonation Engine”, AIAA Paper 20013811, July, 2001. propagating detonation b) [27] Zitoun, R. and Desbordes, D., “Propulsive Performances of Pulsed Detonations”, Combustion Science and Technology, Vol. 144, pp. 93-114, 1999. products reactants detonation c) [28] Schauer, F., Stutrud, J., and Bradley, R., “Detonation Initiation Studies and Performance Results for Pulsed Detonation Engines”, AIAA Paper no. 2001-1129, 2001. products reflected expansion wave d) products expansion wave e) products reflected expansion wave reactants f) products enter expansion wave reactants g) reflected compression wave reactants h) compression wave i) reactants shock/detonation reflection j) reactants Fig. 1: The generic Pulse Detonation Engine (PDE) cycle. 8 American Institute of Aeronautics and Astronautics AIAA–2003–1171 30 Pressure (atm) 25 20 15 10 5 0 0 0.5 1 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 X (m) (a) 0.5 30 25 Pressure (atm) Radius (m) 0.4 nozzle 1 nozzle 2 nozzle 3 nozzle 4 0.3 20 15 10 5 0.2 0 0 0.5 1 X (m) (b) 0.1 30 0 0 0.5 1 1.5 25 20 15 10 5 0 0 0.5 1 X (m) (c) 30 25 Pressure (atm) Fig. 2: Different nozzle geometries explores in present computations. These include straight tubes (configuration 4), flared divergent sections (configuration 2), divergent sections with inflection (configuration 1), and a nozzle section with a constant divergence angle (configuration 3). In all case the reactants are assumed to lie initially upstream of nozzle, in the constant area tube, while the nozzle itself contains inert gas. Pressure (atm) X (m) 20 15 10 5 0 0 0.5 1 X (m) (d) Fig. 3: Evolution of the centerline pressure for a straight 2D axisymmetric PDE of 1 m length, at times (a) 0.06 ms, (b) 0.15 ms, (c) 0.49 ms, and (d) 9 2.89 ms. American Institute of Aeronautics and Astronautics AIAA–2003–1171 Impulse per unit area (ps.s) 2500 2000 1500 nozzle 1 nozzle 2 nozzle 3 nozzle 4 1000 500 350 0 0 0.001 Specific Impulse (sec.) 300 0.002 0.003 0.002 0.003 0.002 0.003 Time (sec.) 2D simulation 1D without pressure relaxation 1D with pressure relaxation 250 250 200 200 Isp (sec.) 150 100 150 100 50 50 0 0 0.001 0.002 0.003 0.004 0 Time (sec.) 0 0.001 Time (sec.) Fig. 4: Comparisons of time-dependent specific impulse for both 1D and 2D axisymmetric simulations of the PDE tube with a CH4 − O2 reaction, taken 15 from He and Karagozian . 1D simulations explored the use of a pressure relaxation length l = 0.5L. Here the 1D simulations incorporated a computational “spark” consisting of a pressure of 10 atm and a temperature of 3000K in order to match the initial conditions for the 2D simulation. 1300 1200 1100 1000 Ispf (sec.) 900 800 700 600 500 400 300 200 100 0 0 0.001 Time (sec.) Fig. 5: Comparisons of time-dependent performance parameters computed from 1D simulations using different nozzle geometries. Results shown are for impulse I, specific impulse Isp , and fuel specific impulse Isp,f . The reaction of methane and oxygen was simulated. 10 American Institute of Aeronautics and Astronautics AIAA–2003–1171 7 200 6 150 nozzle 1 nozzle 2 nozzle 3 nozzle 4 4 100 Y (cm) [P]/P0 5 50 3 0 2 -50 1 0 0 2500 5000 7500 50 10000 100 150 200 Frequency (HZ) X (cm) (a) (a) 250 300 350 4 200 nozzle 1 nozzle 2 nozzle 3 nozzle 4 3.5 3 150 100 Y (cm) [P]/P0 2.5 2 50 1.5 0 1 -50 0.5 0 0 2500 5000 7500 50 10000 100 150 200 Frequency (HZ) X (cm) (b) (b) 250 300 350 250 300 350 3 200 150 nozzle 1 nozzle 2 nozzle 3 nozzle 4 100 [P]/P0 Y (cm) 2 1 50 0 -50 0 0 2500 5000 7500 50 10000 100 150 200 Frequency (HZ) X (cm) (c) (c) Fig. 7: Temporal evolution of the 2D planar pressure field within and external to the PDE over one cycle, with pressure given in units of dyn/cm2 . Data shown are at times corresponding to (a) 0.15 ms, (b) 0.47 ms, and (c) 1.34 ms. A H2 − O2 reaction was simulated here with full chemical kinetics. Fig. 6: Comparisons of pressure spectra: (a) in the middle of the detonation tube, (b) at the end of the straight part of the detonation tube, and (c) at the end of the nozzle. Results are shown for different nozzle configurations (see Fig. 2). 11 American Institute of Aeronautics and Astronautics 9.66811E+06 9.09111E+06 8.51411E+06 7.93711E+06 7.36011E+06 6.78311E+06 6.20611E+06 5.62911E+06 5.05211E+06 4.47511E+06 3.89811E+06 3.32111E+06 2.74411E+06 2.16711E+06 1.59011E+06 AIAA–2003–1171 1.8E+07 Ps = 2.0 Ps = 2.5 Ps = 3.0 Ps = 5.0 1.6E+07 Atm Atm Atm Atm 2 Pressure (dyn/cm ) 1.4E+07 1.2E+07 1E+07 8E+06 6E+06 4E+06 2E+06 0 0 10 20 30 40 30 40 X (cm) (a) 1.8E+07 Ts = 500K Ts = 1000K Ts = 1500K Ts = 2000K 1.6E+07 2 Pressure (dyn/cm ) 1.4E+07 1.2E+07 1E+07 8E+06 6E+06 4E+06 2E+06 0 0 10 20 X (cm) (b) Fig. 8: Centerline pressure distribution for the H2 − O2 reaction with full kinetics, for different computational “spark” conditions: (a) fixed temperature 1500K and variable pressure, and (b) fixed pressure 3 atm and variable temperature, each at time 0.2 msec. 12 American Institute of Aeronautics and Astronautics