AIAA 98-3880A NUMERICAL STUDY OF THE PULSE DETONATION WAVE ENGINE

WITH HYDROCARBON FUELS

Balu SekarAFRL/PRTC

Wright Patterson AFB, Ohio

andSampath Palaniswamy, Oshin Peroomian and Sukumar Chakravarthy

Metacomp Technologies, Inc.Westlake Village, California

ABSTRACT

This paper explores some issues that arise in theanalysis of pulse detonation wave engines withhydrocarbon fuels. One-dimensional andaxisyrnmetric/two-dimensional simulations areemployed along with reduced kinetic mechanisms toconfirm the ability of the numerical approach toaccurately compute relevant physical characteristicssuch as proper detonation wave speed, von-Neumannspike, aspiration, pressure time history and sequence ofcycle events. It is shown that qualitatively andquantitatively reasonable results can be obtained with acareful treatment of the finite-rate-chemistry sourceterms. Some of the numerical difficulties that arise indealing with unsteady detonation phenomena arediscussed and improvements demonstrated. One-dimensional test cases with simplified H2-O2 and C3Hg-Air kinetics are used to verify correct detonation wavespeed and testing boundary conditions. Anaxisymmetric case for the latter chemistry is studiedwith a generic inlet to illustrate the ability of themethodology to capture the relevant physics, namely,pressurization of thrust wall by the detonation wave andinteraction of the reflected wave with rarefaction wavesfrom the open end.

BACKGROUND

Pulse Detonation Wave Engines (PDWE) provide avery high specific impulse at operating frequencies of afew hundred Hz. They can be designed without the useof any rotating machinery or valves hi the flow path.PDWEs also operate at a higher thermal efficiency thanconventional constant pressure combustion engines(Ref. 1). The advantages of pulse detonation enginesover propulsion systems using rotating machinery canbe found in Refs. 2 and 3. But the design and operationof the PDWE is complicated by the unsteady flowwithin the engine. At the start of the power cycle, a

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spark-initiated detonation wave moves from the aft endof the engine towards the thrust producing walls onwhich they reflect, producing a very high pressure levelfor a short duration of time. The reflected shock wavethen moves through the burnt gases towards the aft endof the engine. The shock-induced motion naturallyaspirates the combustion chamber where fuel should bemixed with the incoming air to get prepared for the startof the next cycle. A number of issues that define theoperating envelope include:

1. Initiation of a sustainable detonation wave from theattend;

2. The ability to uniformly mix fuel and air in thedetonation tube within a short time while takingadvantage of the natural aspiration;

3. High heat transfer rates to the solid walls of theengine;

4. Efficient purging of the burnt products;5. The frequency of the valve closing and opening to

let the fuel-air mixture have a continued averagethrust level; and

6. The acoustic levels in the engine cycle.

The level of energy required to initiate and sustain adetonation from the aft end depends on the detonationlimits of the fuel mixture, on the ignition delay at theoperating conditions within the engine, and on theinteraction between the exhaust at the aft end and theflow within the detonation tube. Thus there is a tightcoupling between the mixing process, kinetics ofcombustion and the geometry of engine. At theoperating frequency of several hundred cycles persecond, time available to complete fuel injection andmixing is no more than a few milliseconds. One way toachieve proper mixing is to exploit the unsteadyturbulent flow within the engine. Ignition delay of thefuel air mixture determines the extent of the transitionregion of the detonation tube. Expansion from the aftend of the detonation tube also interacts with thedetonation wave as it traverses the transition region.

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Numerical simulation of a complete cycle of the PDWErequires the ability to model turbulent unsteady mixing,resolving the detonation wave at the right speed, theability to accurately reproduce ignition delay andresolve the acoustic waves traversing the detonationtube. While a number of turbulence models have theability to simulate turbulent mixing, they remain to betested for conditions that exist within the PDWE andimprovements may have to be made. Resolving theunsteady detonation wave at the right speed has been anissue which has seen many solutions including a"limiter" on the chemical source terms at the leadingedge of the wave. While it is straightforward toimplement this fix for planar detonation, it may have tobe modified to resolve distorted detonation fronts.Distorted fronts may occur within the PDWE due tonon-uniform mixture, interactions with bleed air, andviscous effects. In order to resolve the acoustic wavesand hence the interaction with the aft end exhaust, ahigher order algorithm with the ability to adapt gridsmay be required. Within this framework, theobjectives of the present study are: 1) 1-Dcharacterization of the wave speed, namely theChapman-Jouguet (CJ) velocity, and 2) modificationsneeded to address exit pressure boundary conditions in1-D and 2-D in limited computational domains.

DESCRIPTION OF THE NUMERICALPROCEDURE

The numerical and physical modeling issues mentionedabove can be studied with Computational FluidDynamics (CFD). CFD can also be used to properlyevaluate the parametric effects of the design parametersfor efficient design of these systems and to study thecritical problem areas in detail by analyzing time-dependent, chemically-reacting constant volume flows.Recently, a new tool (CFD-H-) was developed byChakravarthy et al., which utilizes a unified-grid ("grid-transparent") computational methodology (Refs. 4-7).The code allows for very easy treatment of complexgeometries by combining structured, unstructured, andmulti-block structured/unstructured grids. Its versatilityallows the use of various elements within the samemesh such as hexahedral, triangular prism, andtetrahedral elements hi 3-D. A multi-dimensionalsecond-order Total Variation Diminishing interpolationis used to avoid spurious numerical oscillations in thecomputed flow field, along with Riemann solvers toguarantee correct signal propagation. One-equation, twoequation and three- equation pointwise turbulencemodels are available to capture the turbulent flowfeatures. The aerothermodynamics capability includesmulti-species modeling with multi-range curve fits for

AIAA 98-3880thermodynamic properties, and finite-rate chemistry.The numerical procedure has been validated for a widerange of flow problems ranging from subsonic tohypersonic Mach numbers but nominally within thesteady-state regime.

hi the numerical experiments reported below to extendthe methodology to tackle unsteady detonationphenomena, the following options available hi CFD-H-were invoked: A form of Roe's Riemann Solvermodified for use with multi-species formulation wasemployed. The "MinMod" slope limiter was usedalong with characteristic variable interpolation. Secondorder spatial accuracy was used along with second ordertime accuracy achieved through a Runge-Kuttaformulation (RK2, explicit hi time). These werenecessary to capture the correct physics and trendsillustrated hi various review papers hi this technologyarea. Grid sensitivity studies were performed so thatthe results presented correctly represent the solutionqualitatively and quantitatively. The flow was assumedto be inviscid and the reaction mixture was premixed atstoichiometric proportions corresponding to H2-O2 orC3H8-Air, as applicable.

Constant-speed detonation wave.

The ability to resolve the detonation wave at the correctwave speed is tested in the following study where astoichiometric fuel-oxidizer (H2-O2) mixture hi a tubeclosed at one end is ignited from the closed end of thetube. The tube used hi the study is assumed to besufficiently long to establish a steadily propagatingdetonation wave. A schematic of the initiation ofdetonation from closed end is given hi Figure 1.

Figure 1. Schematic of initiation from closed end

A source of ignition such as a spark igniter or someother form of energy is required to initiate thecombustion process. Here we assume that the first sixcomputational cells adjacent to the closed end of thetube is fully burnt at constant volume conditions. Themixture hi these six cells is at higher temperature andpressure at the start of the computation. The flow hi thetube is modeled as one-dimensional, unsteady inviscidflow. Interaction between the tube wall and the

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detonation wave are neglected. Spatial and temporaldiscretization is second-order accurate. In addition tothe fluid dynamic conservation laws, the two speciesconservation equations complete the set of equationsused to simulate the flow in the tube. A simple one-step Arrhenius kinetics is used to model the combustionprocess. A shock propagates to the right compressingunburnt gases. This sets off the chemical reaction. Thereaction rate parameter chosen for this study is such thatthe combustion process is almost instantaneous oncethe shock compresses and heats up the fluid. At theseconditions there is little distinction between thereaction zone and the shock.

Figure 2 shows the distribution of the pressure along thelength of the detonation tube at several instances oftime. It can be seen that the pressure peak reaches asteady value soon after the initiation. At this wavevelocity the burnt gases leave the detonation front atsonic speed which was confirmed in this numericalexperiment. This stable, steady detonation wave iscalled the Chapman-Jouguet (CJ) detonation wave andthe corresponding wave speed is called CJ detonationvelocity..

0.15

Figure 2. Pressure-time history for initiation at theclosed end

Figure 3 shows as symbols on a p-v plot values takenfrom the numerical simulation across the detonationwave. It can be seen that the chemical reaction hasbegun to occur inside the numerical shock layer andcontinues to proceed after the shock front producing asmall von Neumann spike in pressure. The movingshock front is resolved over about 5 computationalcells. By the second point within the front, about 40%of the reactants have been consumed by the chemicalreaction. Figure 3 also shows Hugoniot curves from40% to 100% consumption of the reactants. At the CJ

AIAA 98-3880point, the reactants are completely consumed. Thisproves that the numerical method used is able to capturethe CJ wave speed very well. Since the extent of thechemical reaction within the numerical shock layer issignificant, the von Neumann spike observed hi Fig.2does not correspond to that of a frozen shock ahead ofthe reaction front. If the reaction rate parameter werelowered, the von Neumann spike will approach that ofthe frozen shock (p « 74) for this case.

0.25 0.5 0.75specific volume

Figure 3. Validation for CJ condition for simplified H2-O2 reaction kinetics

The initiation mechanism used in this study did not leadto overdriven detonation waves. No unstable orunsteady behavior (Ref. 8) of the wave was evident.Improvements to source term treatmentProper numerical treatment of the source terms isnecessary in order to reliably compute the wave speedsaccurately hi detonation wave simulations (Ref. 9). Anexample that illustrates the error possible with"standard" source term treatments is shown in Figure 4.This test was conducted as a solution to a Riemannproblem for left and right states which correspond to asingle right-moving detonation front. The spatialcoordinate x is a wave stationary system. At time t = 0,the conditions are stationary unburnt gases to the right(x > 0) and conditions corresponding to ChapmanJouget detonation to the left (jc < 0). The curvemarked as "Standard Source Term Evaluation" splitsthe total pressure rise across two waves. The correctsource term treatment involves evaluating the sourceterms using the minimum temperature hi theneighborhood of each computational grid cell. Thisresults hi the solution marked as "With Proper SourceTerm Treatment" in Figure 4. The detonation front is

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predicted with correct wave speed. A von Neumannspike in pressure is also evident. The lower peak inpressure at the von Neumann spike is again due tosignificant reaction within the shock layer.

40

30

20 Standard SourceTerm Evaluation

von NeumannSpike

With Proper SourceTerm Treatment

-0.05 0.05

Figure 4. Numerical Challenges - Wave Speed

Initiation from the open end.

Before PDWE cycle can be successfully simulated it isnecessary to quantify the amount of energy required toinitiate a detonation wave from the open end of thetube. The distance from the open end before thedetonation wave reaches steady propagating speed isalso of interest in the design of PDWE. To this end, thefollowing numerical experiment was conductedsimulating one-dimensional, inviscid, reacting flow.Figure 5 is a schematic of wave initiation from the openend of the tube. Initiation from the open end is similarto the previous test except that the task is complicatedby the presence of constant pressure boundary conditionat the open end. Compression or expansion wavesreflect off the open end as expansion or compressionwaves maintaining constant pressure while the flowaccelerates through the boundary. A constant pressureboundary condition based on the Riemann invariants,available in CFD++, was modified for use at the openend. When the back pressure to be imposed is smallerthan the local p* (pressure at local sonic conditions),the p* value was used as the back pressure.

The tube is initially filled with stoichiometric C3Hg-Airmixture at nominal pressure and temperature.Combustion kinetics is modeled by a single stepreaction:

AIAA 98-38805O2 -» 3 CO2 +4 H2O

where the rate of consumption of C3H8 is proportional

to |C3//g] \O2 I • The square brackets implymolar concentrations. This reduced mechanism wasproposed by Westbrook et. al.(Ref. 10).

A two-temperature-range curve fit was assumed tomodel the Cp variation of all the species, C3H8, O2,CO2, H2O, and N2 used in the simulation. Theproperties for C3H8 were obtained from NASA TM4513 (Ref. 11) and NASA SP-3001 (Ref. 12) was usedfor all the other species.

Expansion to atmosphericconditions

Figure 5. Schematic of initiation from open end

Pressure-time slices are shown in Figure 6a,b,c for a 6-inch tube, 8-inch tube, and 10-inch tube respectively.When the wave (moving from right to left, or from openend to closed end) reaches the closed end, thenondimensional pressure maximum of approximately60 occurs due to reflection as seen hi Figure 6c. Whenthe wave reaches the steady CJ velocity, it produces apeak nondimensional pressure of approximately 18 atthe detonation wave front. All the Figures 6a-c show aconstant peak value prior to reflection after an initialtransition period. In Figure 6a, the steady propagationspeed is reached just prior to reflection. Figure 7 showsthe pressure time history at the thrust wall. All the threetubes register the same peak pressure at the thrust wall.If the tube had been shorter than 6 niches, the peakpressure before reflection would predictably be lowerbecause the detonation front would not have reached CJvelocity yet. Thus, the length of the detonation tube formaximum thrust is determined only by the wavetransition distance.

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AIAA 98-3880

400 350 300 200

100 TimeSteps

1 2 3 4 5X (Inches)

Figure 6a. Pressure time slices for 6-inch tube

550 500 400300

200

X (Inches)

Figure 6b. Pressure time slices for 8-inch tube

60

50

40

lao

20

10

700 Time Steps

600 500 400 200

100

4 6X (Inches)

I- B'Tube50

40

30

20

10

2 3Nondimensional Time

Figure 7. Thrust wall pressure time history for 3 ducts

A cycle of PDWE with aspiration.

As a logical extension of the one-dimensional studiesdiscussed above, a numerical simulation of a simpleaxisymmetric PDWE configuration similar to thatanalyzed by Lynch et. al (Ref. 13) was carried out. Thedetonation tube is 8 cm in radius and about 8 cm long.It has an annular inlet 12 cm in length. The aft end ofthe detonation tube directly exhausts into the ambientah-. Fig. 8 shows a schematic of the cross section ofsuch an engine. Earlier studies have been performed (inRef. 13 and by the authors of this paper) for H2-Airmixtures.

shear layer

Inlet flow conditions

Fuel injection

I

unburnt gas

detonation waveBurnt

Ambient

Figure 6c. Pressure time slices for 10-inch tube

Figure 8. Schematic of axisymmetric/two-dimensionalPDWE

The incoming free stream Mach number is assumed tobe supersonic. The inlet is left open forming a shearlayer in the detonation tube which envelopes the spacefilled by the premixed C3Hg-Air. Detonation is initiatedat the aft end. For the initial conditions chosen, thetransmitted wave going into the annular inlet reversesdirection quickly. If the inlet pressure were lower, thetransmitted wave would proceed further upstream intothe inlet and would show a tendency to unstart the inlet

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before the incoming flow forces the flow back into thedetonation tube and aspirating the engine.Figures 9a-h show static pressure contours over one fullcycle of operation of duration about 0.4 milliseconds.The reference time of 3.16 milliseconds can be used toconvert the nondimensional tunes reported in the Figuretitles into dimensional time. In Figure 9a, thedetonation wave has traversed about half the length ofthe tube and reaches the thrust wall hi Figure lObcreating a high nondimensional pressure of about 32(reference pressure is 100,000 N/m2). Large variationsof pressure on the thrust wall in the radial direction (y)can be seen in Figure 9b. In Figure 9c, the transmittedwave is just entering the annular Met and reflectedwaves cause high pressure on the outer wall. Figures9d and 9e show the reversal in direction of thetransmitted wave from the inlet. Figures 9f, 9g, and 9hshow a sequence of pressure histories that return thedetonation tube to its starting condition approximately.

Figures lOa-h show the corresponding Oj mass fractioncontours. In Figure lOa, the detonation wave which hastraversed about half the length of the tube hasconsumed all oxygen in its wake. When the detonationwave reaches the thrust wall (Figure lOb), there isalmost no more Oxygen left in the combustionchamber. The high pressure at the thrust wall causes a"necking" in the oxygen feed supplied through the inlet(Figure lOc). As the pressure drops behind thetransmitted wave, the inflow is able to expand into thethrust chamber as is seen hi the remaining Figures lOd-h.

CONCLUSIONS

In this paper, important unsteady flow issues that arisein the numerical simulation of pulse detonation waveengines with hydrocarbon fuels have been investigatedusing simplified kinetics. The numerical featuresnecessary for accurate qualitative and quantitativesimulations have been tested and validated. Futurework will include more complex geometries, moreelaborate kinetic mechanisms and viscous effects.

REFERENCES

1. Bratkovich, T.E., and Bussing, T.R.A., "A PulseDetonation Engine Performance Model", AIAA 95-3155, 31st AIAA/ASME/SAE/ASEE JointPropulsion Conference, July 10-12,1995.

2. Bussing, T.R.A., and Pappas, G., "An introductionto Pulse Detonation Engines", AIAA 94-0263, 32nd

AIAA Aerospace Sciences Meeting and Exhibit,RenoNV, Jan 10-13,1994.

AIAA 98-38803. Eidelman, S., and Grossman, W., "Pulsed

Detonation Engine Experimental and TheoreticalReview", AIAA 92-3168, 28th

AIAA/ASME/SAE/ASEE Joint PropulsionConference and Exhibit, Nashville TN, July 6-8,1992.

4. Chakravarthy, S., and Peroomian, O., and Sekar,B., "Some Internal Flow Applications of a Unified-Grid CFD Methodology", AIAA Paper 96-2926,July 1996, Lake Buena Vista Florida.

5. Peroomian, O., and Chakravarthy, S., "A "Grid-Transparent" Methodology for CFD", AIAA Paper97-0724, January 1997, Reno Nevada.

6. Chakravarthy, S., Goldberg, U., Peroomian, O., andSekar, B., "Some Algorithmic Issues in ViscousFlows Explored using a Unified-Grid CFDMethodology", July 1997, Snowmass, Colorado.

7. Goldberg, U., Peroomian, 0., Chakravarthy, S., andSekar, B., "Validation of CFD-H- Code Capabilityfor Supersonic Combustor Flowfields", July 1997,Seattle Washington.

8. Bourlioux., A., Majda, A.J., and Roythurd, V.,"Theoretical and numerical structure for unstableone-dimensional detonations", SIAM J. Appl.Math., Vol 51, No.2, pp303-343, April 11.

9. Lynch, E.D., and Edelman, R.B., "Analysis of thePulse Detonation Wave Engine - Developments inHigh-Speed-Vehicle Propulsion Systems", S.N.B.Murthy & E.T. Curran editors, Progress hiAstronautics and Aeronautics, Vol 165, pages 473-516.

10. Westbrook, C. K., and Dryer, F. L., "SimplifiedReaction Mechanisms for Oxidation ofHydrocarbon Fuels," Comb. Sci. Tech., Vol. 27,1981.

11.McBride, B. J., Gordon, S. and Reno, M. A.,"Coefficients for Calculating Thermodynamic andTransport Properties for Individual Species,"NASATM-4513.

12. McBride, B. J., Heimel, S., Ehlers, J. G., andGordon, S., "Thermodynamic Properties to 6000Kfor 210 Substances Involving the First 18Elements," NASA SP-3001.

13. D. Lynch, R. Edelman, and S. Palaniswamy,"Computational Fluid Dynamic Analysis of thePulse Detonation Engine Concept," AIAA Paper94-0264.

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Static Pressure Contours

SWio Pressure Contours

Figure 9a. Static pressure contours at nondimensionaitime of 0.0093525

Figure 9e. Static pressure contours at nondimensionaitime of 0.0455043

Static Pressure ContoursStatic Pressure Contours

Figure 9b. Static pressure contours at nondimensionaltime of 0.0187072

Figure 9f. Static pressure contours at nondimensionaltime of 0.054538

Static Pressure ContoursStatic Pressure Contours

Figure 9c. Static pressure contours at nondimensionaltime of 0.027813

Figure 9g. Static pressure contours at nondimensionaltime of 0.06398

Static Pressure Contours

Statra Pressure Contours

Figure 9d. Static pressure contours at nondimensionaltime of 0.0362965

Figure 9h. Static pressure contours at nondimensionaltime of 0.112611

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Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc.

O2 P.te&e Fraction Contours

Figure lOa. O2 mass fraction contours atnondimensional time of 0.0093525

AIAA 98-388©02 Mass Fraction Contours

Figure lOe. O2 mass fraction contours atnondimensional time of 0.0455043

02 M«M Fraction Contour! O2 Mass Fraction Contours

Figure lOb. O2 mass fraction contours atnondimensional time of 0.0187072

Figure lOf. O2 mass fraction contours atnondimensional time of 0.054538

02 MIBS Fraction Contours O2 Mass Fraction Contours

Figure lOc. O2 mass fraction contours atnondimensional time of 0.027813

Figure lOg, O2 mass fraction contours atnondimensional time of 0.06398

02 Mm Fraction Contours O2 MMa Friction Contours

Figure lOd. O2 mass fraction contours atEondimensional time of 0.0362965

Figure lOh. O2 mass fraction contours atnondimensional time of 0.112611

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