Jonathan D. Regele, David R. Kassoy and Oleg V. Vasilyev- Detonation Initiation Modeling Using the...

8/3/2019 Jonathan D. Regele, David R. Kassoy and Oleg V. Vasilyev- Detonation Initiation Modeling Using the Adaptive Wavel…

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Detonation Initiation Modeling Using the Adaptive

Wavelet-Collocation Method

Jonathan D. Regele, David R. Kassoy∗

and Oleg V. Vasilyev†

Department of Mechanical Engineering, University of Colorado, Boulder CO, 80309-0427

Previous work has shown that thermal power deposition into a reactive gas mixture caninitiate a planar Deflagration-to-Detonation Transition (DDT) proces. Previous resultsusing a uniform grid MacCormack method are validated with new results obtained withthe Adaptive Wavelet-Collocation Method (AWCM). The previous MacCormack schemeresults used relatively unphysical non-dimensional inverse activation energies in the rangeof 0.0725− 0.08. Activation energies were limited to this range because proper resolution of the unsteady half-reaction lengths would be extremely difficult to achieve given the uniformgrid scheme. Due to the AWCM’s computational efficiency and scalability, it is possibleto use the AWCM to perform simulations with more realistic inverse activation energies.AWCM simulations were performed with non-dimensional inverse activation energies rang-ing from 0.04 − 0.075. The half-reaction lengths decreased dramatically over this relatively

small range. Care was taken to ensure there were100

or more points per half-reactionzone, due to the unsteady nature of the problem. Results show that there is a consistentsequence of events that lead to detonation initiation, regardless of the activation energy.

Nomenclature

∗ Inverse Activation Energys Non-dimensional timezi Center of heat deposition locationq Chemical heat release rate

I. Introduction

Spatially resolved, thermal power deposition of limited duration into a finite volume of reactive gas isthe initiator for planar deflagration to detonation transition (DDT) on the microsecond time scale. Sileemet al. (SKH8) and Kassoy et al. (KKNC2, 3), use an explicit MacCormack scheme to obtain solutionsto the one-dimensional reactive Euler equations with one-step Arrhenius kinetics. SKH8 conclude thatthe one dimensional unsteady mathematical model “appears to provide a physically plausible descriptionof detonation initiation through a transition from deflagration to detonation.” However, the high-speeddeflagration generated by the initial burst of power supplied by the external source does not itself evolveinto a shock-coupled reaction zone. Rather, numerous, localized reaction centers that appear spontaneouslyare found to be the sources of compression waves that strengthen an initially weak lead shock sufficiently tofacilitate closely coupled reaction zones. Localized reaction centers appear when the characteristic reactiontime is as short as the acoustic time defined by the ratio of the volume dimension to the local speed of sound.In that case momentary, partial inertial confinement is possible and local pressure rises with temperature.Subsequent expansion of the hot, high pressure spot driven by the locally large spatial pressure gradient drivescompression waves into the surrounding mixture. Combustion heat release and gasdynamic transients areessential to the formation of the detonation. Oppenheim5 observed the formation and evolution of reactioncenters described as “explosions in the explosion” in a system with transverse wave processes. Compression

∗Contact: [email protected]†Contact: [email protected]

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American Institute of Aeronautics and Astronautics



waves generated by these localized regions of very rapid chemical heat addition were deemed crucial tothe DDT8 process. Others have since observed similar phenomena in multidimensional numerical results.1

Further development of the SKH8 approach KKNC2, 3 has focused on solution reliability and a parametricstudy of solution sensitivity to the magnitude and location of the initial power burst as well as the activationenergy of the one step reaction. Results of this work indicate that the transient dynamics of planar DDTevolution are (a) dependent on the magnitude and location of the initial power deposition process, (b)sensitive to the activation energy of the one step Arrhenius reaction, (c) characterized by the spontaneousappearance of several localized hot spots (reaction centers) resulting from conditioning of isolated volumesof reactive mixture by the compression wave structure inherent in a reactive gasdynamic environment withthermal power addition from external and chemical sources and (d) likely to lead initially an overdrivendetonation, which relaxes smoothly to a Chapman-Jouget (CJ) wave. The amount of overdrive can bequantified by the global heat release rate maximum, which occurs when there are multiple reaction zones

behind the lead shock. This structure occurs during a period of shock strengthening, when the ignitiondelay time of adjacent fluid particles passing through the shock is significantly shortened as the post shocktemperature increases. Less conditioned particles explode farther behind the shock than more conditionedparticles that pass the shock subsequently. The global heat release maximum shown in the left plot of Fig.1 is nearly twice the value of the CJ heat release for the parameters of the problem, in spite of the fact thatthe shock Mach number is not nearly large enough to support that large heat release rate with a single postshock reaction zone.

II. AWCM Solution Results

The previous work in SKH8 and KKNC2, 3 used the MacCormack scheme, which is known to produceundesirable solution oscillations in the vicinity of large gradients (shocks). In principle, these oscillations

are sources of error in the local reaction rate modeled by an Arrhenius exponential term. The numericalscheme also used a uniform grid making it extremely difficult to solve problems with smaller reaction lengths.Sharpe and Falle7 showed that in unsteady problems, the half-reaction length may need in excess of 100 ormore points to obtain a convergent solution. In order to perform simulations with more realistic activationenergies, i.e. shorter reaction lengths, a more robust, scalable and numerically efficient algorithm is neededin order to properly resolve the transient processes involved in detonation formation.

An adaptive wavelet-collocation method (AWCM), developed originally by Vasilyev and co-workers9–11

for solving the Navier-Stokes equations in multiple dimensions, has been extended recently by Regele andVasilyev4 to hyperbolic systems like the reactive Euler equations that describe thermally initiated DDTprocesses. The efficiency of the adaptive wavelet-collocation solver is combined with the simplicity of aflux-limited type approach to explicitly add localized artificial viscosity near shocks. A series of inert andreacting compressible flow problems with known solutions are used as test cases to successfully validate theeffectiveness of the AWCM approach. Using this new algorithm the results generated in the original SKH 8

and KKNC2, 3

were reproduced with only minor deviations in the sequence of events leading to detonationformation.

The purpose of the current research effort is to demonstrate the similarity in the sequence of events lead-ing to thermal energy deposition Deflagration-to-Detonation Transition (DDT) regardless of the activationenergy. Due to the increased computational efficiency of the AWCM, the method is capable of handlingproblems with larger activation energies where the half-reaction length is substantially smaller. An effectiveresolution of over 100 points per half-reaction length was achieved since the method’s adaptation algorithmis controlled through the use of wavelets. Due to the fact that the reaction length decreases dramaticallywith increased activation energy, the AWCM is able to handle the more computationally challenging largeractivation energy problems.

The original inverse activation energy defined in the SKH paper and more recent work2 was limited to arange of 0.0725 to 0.08. Using the AWCM, inverse activation energies as small as 0.04 were used to modelinitiation of detonation waves. Figure 1 shows the time history of the nondimensional total heat release

rate associated with transient reactive gasdynamics of an evolving one-dimensional DDT 3 with an inverseactivation energy of 0.0725 using the MacCormack Scheme. The plateau shaped curve in the lower left cornerrepresents the time history of the initiating nondimensional thermal power source deposited in a slab shapedregion of width 1.4mm, located 0.35mm from the boundary surface, during a period of 10 microseconds. Thedimensional energy deposition is 1.53× 107J/m3.

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MacCormack Scheme

0 4 8 12 16 20 24 28 32 360

20

40

60

80

100

120

140

160

180

Time s

G l o b a l H e a t A d d i t i o n q

Global Heat Release

AWCM Scheme

Figure 1. Comparison of the original MacCormack Scheme results with the new AWCM results. These

plots show the evolution of the global heat release with time and that once the detonation is formed the

rate asymptotes to the CJ value for a steady detonation wave. Both simulations were run with zi = 3 and

∗ = 0.0725. The absolute maximum global heat release for the MacCormack scheme is q = 166.08 and occurs at

a non-dimensional time of s ≈ 21.6 while AWCM’s maximum global heat releaseis q ≈ 164 and occurs at s ≈ 20.4.

Both simulations exhibit the same sequence of events leading to detonation formation.

The right plot in Figure 1 shows the time history of the nondimensional total heat release rate forthe AWCM. There are a few minor differences between the two scheme’s solutions in regard to the overallmagnitude and exact timing of events, but the overall sequence of events remains the same. These differencescan be attributed to the different manners in which the shocks are stabilized and also due to the fact thatthe AWCM compresses the amount of information needed to be solved at each time step by approximatingthe solution at any given point in time. The error involved in this approximation is prescribed a priori bythe error threshold parameter. In the simulations performed for this study the error threshold parameterwas set to 1.0× 10−4.

The first local maximum, shown in both plots in Figure 1, is associated with the “explosion” of the reactivemixture in the vicinity of the external source. The second occurs when the first of the local reaction centersappears. The absolute maximum is attributed to a power burst from the appearance of a shock-coupledmultiple reaction zones (the overdriven detonationa.). The fourth local maximum occurs when previouslyunburned pockets of reactive mixture located far behind the wave front are finally consumed. In the AWCM

simulation this final local maximum is not present because the pockets burned earlier on in the simulation.The AWCM and its shock locator function can be used in these types of problems to construct an

informative (x, t)-diagram, used to describe the sequence of events leading to detonation formation. Thisplot helps to describe the sequence of events leading to detonation formation. Figure 2 shows the Fuel MassFraction on a (x, t)-diagram. Overlayed on top of the solution in black are regions of high activity. In theseregions the wavelet coefficients on the finest level of resolution are comparable if not larger than the errorthreshold parameter. This indicates that stabilization of the solution should take place in these regions. Inthis plot, each point whose shock locater function is larger than a prescribed value is plotted as a black point.These points signify shocks, contacts or sharp local maxima. The white arrows indicate the velocity fieldat given points in the solution. The velocity field helps one to visualize the path that a given fluid particletakes as it goes from rest into the post-shock induction heating zone and finally into the region where the“explosion within the explosion” occurs. Figure 2 used with additional plots such as Figure 3 help explainthe occurrence of localized reaction centers, which generate compression waves. These pressure waves reflect

off the walls and contacts, which then interact with localized reaction centers to create further explosionsand compression waves. The interaction of these compression waves with the localized reaction centers inthe induction zone eventually lead to a final explosion, which coalesces with the initial shock front and formsa detonation wave.

aSchauer et al.6 observe initially overdriven detonations in pulsed detonation engine spark initiated DDT events.

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x

t

Fuel Mass Fraction, X−t Diagram

5 10 15 20 25 30 35 40 45

2

4

6

8

10

12

14

16

18

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 2. A (x, t) diagram demonstrating the evolution of the fuel mass fraction and the induction zone where

the reactants are heated leading to a final explosion. The black dots indicate regions of high activity such as

shocks, compression waves, contacts and localized reaction centers. This simulation was run with zi = 3 and

∗ = 0.075.

(x, t)-diagrams like those in Figures 2 & 3 are generated for activation energies ranging from 0 .075 to0.04. Each plot shows a very similar sequence of events. The large activation energy simulations are moredifficult to achieve detonation initiation because they require a more complicated sequence of explosions, butthey still exhibit the same behavior. A detailed analysis of these processes for multiple activation energies

will be discussed in the final paper.

III. Conclusion

Previous results obtained using the MacCormack method are validated by the newly available AWCM.Since the earlier results used a uniform grid scheme and the half-reaction length decreases substantially asthe inverse activation energy is decreased, DDT simulations using the uniform grid scheme were limited tousing unphysically large inverse activation energies (∗ = 0.0725− 0.08). Due to the AWCM’s computationalefficiency and scalability, it is possible to use the AWCM to perform simulations with more realistic inverseactivation energies. Simulations were performed with inverse activation energies ranging from 0.04− 0.075.The half-reaction lengths decreased dramatically over this relatively small range. Care was taken to ensurethere were 100 or more points per half-reaction zone, due to the unsteady nature of the problem. There is aconsistent mechanism that drives the formation of the detonation waves, regardless of the activation energy.(x, t)-diagrams plotting the fuel, pressure, shock-wave fronts, localized reaction centers and velocity fieldhelp explain the mechanism that drives the detonation intiation. An in depth explanation of the sequenceof events leading to detonation initation will be provided in the final paper and presentation.

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x

t

Pressure X−t Diagram

5 10 15 20 25 30 35 40 45

2

4

6

8

10

12

14

16

18

2

4

6

8

10

12

14

16

18

20

Figure 3. Another (x, t) diagram demonstrating the evolution of the pressure. This simulation was run with

zi = 3 and ∗ = 0.075.

References

1V. N. Gamezo, D. Desbordes, and E. S. Oran. Formation and evolution of two-dimensional cellular detonations. Com-

bustion and Flame, 116:154–165, 1999.2D. R. Kassoy, J. A. Kuehn, M. W. Nabity, and J. F. Clarke. Modeling detonation initiation on the microsecond time

scale. 43rd AIAA Aerospace Sciences Meeting and Exhibit, Jan 10-13, Reno NV., 1169, 2005.3D. R. Kassoy, J. A. Kuehn, M. W. Nabity, and J. F. Clarke. Detonation initiation on the microsecond time scale: Ddt’s.

ms. submitted to Combustion Theory and Modeling , 2006.4D. R. Kassoy, J. D. Regele, and O. V. Vasilyev. Detonation initiation on the microsecond time scale: One and two

dimensional results obtained from adaptive wavelet-collocation numerical methods. 45th AIAA Aerospace Sciences Meeting

and Exhibit, January 8-11, 986, 2007.5Antoni K. Oppenheim. Dynamic features of combustion. Phil. Trans. R. Soc. Lond. A, 315:471–508, 1985.6F. R. Schauer, C. L. Miser, K. C. Tucker, R.P. Bradley, and J. L. Hoke. Detonation initiation of hydrocarbon-air mixtures

in pulsed detonation engine. 43rd AIAA Aerospace Sciences Meeting, Jan 10-13 , 1343, 2005.7G. J. Sharpe and S. A. E. G. Falle. Numerical simulations of pulsating detonations: I. nonlinear stability of steady

detonations. Combust. Theory Modeling , 4:557–574, 2000.8A. A. Sileem, D. R. Kassoy, and A. K. Hayashi. Thermally initiated detonation through deflagration to detonation

transition. Proc. R. Soc. Lond. A, 435:459–482, 1991.9O. V. Vasilyev and C. Bowman. Second generation wavelet collocation method for the solution of partial differential

equations. J. Comp. Phys., 165:660–693, 2000.10O. V. Vasilyev and N. K.-R. Kevlahan. An adaptive multilevel wavelet collocation method for elliptic problems. J. Comp.

Phys., 206:412–431, 2005.

11O.V. Vasilyev. Solving multidimensional evolution problems with localized structures using second generation wavelets.Int. J. Comp. Fulid Dyn., Special issue on High-resolution methods in CFD, 17:151–168, 2002.

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