d.m. davidenko, i. gökalp and a.n. kudryavtsev- numerical modeling of the rotating detonation in an...

8/3/2019 D.M. Davidenko, I. Gkalp and A.N. Kudryavtsev- Numerical Modeling of the Rotating Detonation in an Annular Co

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Numerical Modeling of the Rotating Detonation in an Annular Combustion

Chamber Fed with Hydrogen-Oxygen Mixture

D.M. Davidenko1*

, I. Gkalp1, A.N. Kudryavtsev

1,2

1ICARE Institut de Combustion, Arothermique, Ractivit et Environnement, CNRS, Orlans 45071, France2

Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk 630090, Russia

AbstractNumerical simulations of a transverse detonation wave continuously rotating in an annular cylindrical combustionchamber are performed. The investigation is aimed at a better understanding of the processes in the continuouslyoperating detonation wave rocket engine fed with stoichiometric hydrogen-oxygen mixture. The two-dimensionalEuler equations for chemically reacting flow are solved using a high-order weighted essentially non-oscillatoryscheme coupled with a semi-implicit Runge-Kutta time integration method. Finite-rate chemistry is described by akinetic model including 6 species and 7 reversible reactions. The flow structure in the combustion chamber isinvestigated and the dependence of main flow characteristics and combustion chamber performance on the injectiontotal pressure, chamber length and azimuthal period is studied.

* Corresponding author : [email protected] of the European Combustion Meeting 2007

Introduction

Theoretically, the rocket engine performance can beincreased by the use of the detonation regime ofcombustion instead of the conventional combustionmode. Several ways to obtain the detonation combustion

mode have been proposed. The well-known PulseDetonation Engine (PDE) concept is still problematic

for the practical implementation because it is difficult torun PDE at a high pulsation frequency which isnecessary for a satisfactory performance. An alternativedetonation engine concept, which is based on the spindetonation, was proposed in the early 1960s by Russianscientist B.V. Voitsekhovskii [1]. According to thisconcept, which is known as Continuous Detonation

Wave Engine (CDWE), one or more detonation wavefronts move across the flow of combustible mixturecontinuously injected into an annular combustion

chamber. The main advantages of CDWE are thecontinuous operation and a high combustion rate. Its

feasibility has been experimentally proven for the firsttime at the Lavrentyev Institute of Hydrodynamics(Siberian Branch of Russian Academy of Sciences, Novosibirsk). A detailed review of experimentalinvestigations of this concept and its comparison withthe conventional rocket engine can be found in therecent publications [2, 3].

Numerical simulations can contribute significantly

to the study of the CDWE internal process. However, tothe best authors knowledge, very few attempts to

simulate numerically the CDWE process wereundertaken [4, 5]. It can be explained by difficulties of

accurate modeling of detonation phenomena in aconcentrated combustible mixture at high pressure.

In the present work, the flowfield in an annularcylindrical combustion chamber of rocket-type CDWEis simulated numerically by solving the multispeciesEuler equations with a modern high-resolution shock-capturing scheme. The combustion of stoichiometrichydrogen-oxygen mixture is described by a detailed

thermochemical model. The investigation is aimed at

obtaining a clear picture of processes occurring in theCDWE combustion chamber, elucidating the underlying physical mechanisms, and studying the influence ofinjection and geometric parameters on the combustionchamber performance.

Problem Formulation

An annular cylindrical combustion chamber of

CDWE is schematically shown in Fig. 1. The mixture offuel and oxidizer (1) is injected from the closed end (2)of an annular cylindrical duct through a ring slit or alarge number of regularly arranged small holes.

Fig. 1. Schematic of a CDWE combustion chamber.

Transverse detonation waves (3) burn the layer ofcombustible gas (4), which is restored during the time period between successive fronts. Depending on theconditions of initiation, the detonation waves travelazimuthally in the clockwise or counter-clockwisedirection. For a stable detonation propagation, thethickness of the fresh mixture layerh must be about 10-20 mm and the spatial period l between the fronts is

typically (72) h depending on the injection conditions

THIRD EUROPEAN COMBUSTION MEETING ECM 2007

ack to Table of Contents


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[3]. The detonation wave propagation induces obliqueshock waves (5) in the combustion products. Thecombustion products flow towards the duct open end (6)and then discharge from the chamber through adivergent nozzle (not shown in the scheme). Due to thecombustion product expansion, a supersonic flow

velocity can be achieved though the device has nogeometrical throat.

The numerical simulation of CDWE in the full 3Dconfiguration is rather expensive as a first approach. Itis particularly true in the case of a parametric study. The problem can be simplified by assuming that the gap

between the duct walls is small in comparison with theirradii and the viscous effects on the walls can beneglected. Thus the flowfield variations in the radialdirection are not considered and the duct is represented by a 2D planar geometry with periodicity conditionsalong the vertical boundaries.

The computational domain is rectangular, its x-wise

size covers one spatial period l. Along y, the domainconsists of a constant-area and a divergent sectionswhose lengths are denoted by L and L1. In the present parametric study, the domain dimensions vary withinthe following ranges: l= 50-100 mm, L = 10-100 mm.Within the divergent section of the duct (L


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Fig. 2. A good agreement between the experiment andnumerical predictions is observed within thetemperature range corresponding to the detonationconditions (1000/Tini = 0.55-0.56).

Using a Zel'dovich-von Neumann-Dring (ZND)numerical model with the three kinetic mechanisms, 1D

solutions have been obtained for Chapman-Jouguet (CJ)detonations under the following conditions ahead of the

detonation front: H2:O2 = 66.7:33.3, T = 300 K, P =0.05-1 MPa. Figure 3 shows the dependence of theignition delay distance xign and the temperature Tsh just

behind the shock front onP. One can conclude that themost significant divergence ofxign is observed at high pressure when the present model underestimates theignition delay.

0.1 1P

, MPa

0.001

0.01

0.1

xign,

mm

1700

1800

1900

Tsh,

K

Fig. 3. Ignition distance and temperature behind theshock front in the CJ detonation from ZND model. Linecolors correspond to the same mechanisms as in Fig. 2.

Typically, the detonation wave engine operates athigh pressure when the ignition delay distance is smalland cannot be resolved in 2D numerical simulation offull-scale chamber without using prohibitively finegrids. Even if the internal structure of the detonationfront is unresolved, the numerical solution mustcorrectly predict the main detonation characteristicssuch as the propagation velocity and CJ conditions. Theinfluence of the spatial and temporal resolution on thedetonation velocity has been investigated by simulating

1D freely propagating detonations at P = 0.1 MPa,0.35 MPa, and 0.7 MPa. The mixture composition and

T are the same as for the ZND simulations. For the

case P = 0.35 MPa, Fig. 4 shows the variations of thedetonation wave velocity D with the time step t fordifferent spatial grid steps x. Results obtained on thecoarser grids (x = 0.05 mm and 0.1 mm) depend onboth t and x. The converged result, achieved withx 0.025 mm, corresponds exactly to the CJ velocityDCJ = 2904 m/s. ForP = 0.1 MPa and 0.7 MPa, theallowable grid step is respectively 0.1 mm and 0.01 mm.

It is worth of noting that development of 1D pulsating instability was observed in the computations

with much finer grids (x 1 m). 2D simulations forlower initial pressures, whenxign increases significantly,demonstrate also the development of transverse

disturbances and formation of cellular detonationwaves.

The 2D computational mesh is constructed ofparallel horizontal and vertical lines. Alongy, the meshhas a uniform step y = 0.05 mm near the lower boundary, within a layer slightly larger than thedetonation wave height. Then y is stretched graduallytowards the upper boundary. The number of cells in the

y-wise direction varies from 250 to 560 depending onL.The simulations can be performed both in the

laboratory frame of reference where the detonationwave rotates continuously and in the reference frame

moving with the detonation wave. The first caserequires either a uniform mesh in the x-wise direction,

which becomes too large when the pressure is high, or adynamically adapted mesh that is not yet available withthe present code. In the moving reference frame, thedetonation wave takes a fixed position with respect tothe computational mesh whosex-wise step can be easilyrefined in the vicinity of the detonation front. The twotechniques were utilized in the present study. At the

lowest injection mass flow rate, the simulation wasmade in the laboratory frame using a mesh with a

constant step x = 0.1 mm (1000 cells per horizontalrow). For the other mass flow rates, preliminarysolutions were obtained in the laboratory frame and thefinal ones were computed in the moving frame on

refined meshes. In the latter case, x has a minimumvalue of 0.025 mm or 0.01 mm and a maximum value of0.2 mm. The number of cells in the x-wise direction

varies from 500 to 900 depending on land xmin.A particular problem has been encountered with the

numerical treatment of the contact boundary betweenthe fresh mixture layer and combustion products. Even

though the flow is considered inviscid, the numericalscheme introduces some artificial diffusion that resultsin a pseudolaminar flame along that boundary. Theflame propagation velocity depends on the spatialresolution and local conditions. It is high enough to burnup to 10-20 % of mixture before the detonation wavearrives. Even if a similar effect is observed in theexperiment [3, 4], we prefer not to simulate it without a

suitable diffusion model. To avoid this problem, thechemical source terms on the contact boundary wereartificially suppressed. With this modification, thedifference between the injected mass and that burnt inthe detonation wave does not exceed 0.1%.

1x10-9

1x10-8

t, s

2900

2950

3000

3050D,m/s

x = 0.1 mm

0.05 mm

0.025 mm

0.0125 mm

Fig. 4. Dependence of the detonation wave velocity onthe time step and spatial resolution forP = 0.35 MPa.




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From the considerations presented in this section, wecan state that the present computational approach is notsuitable for predicting the limits of the detonation propagation because the internal detonation wavestructure is not accurately resolved. Hence our study isfocused on the regimes of stable propagation of

detonation. We do not analyze in detail the detonationinitiation and transition to a stable propagation. As a brief remark, it is assumed that at the initial timemoment the chamber is filled by the combustionproducts except a triangular region of fresh mixture nearthe injection boundary; a high-temperature and high-

pressure spot is placed within that triangular region toinitiate the detonation. After the detonation initiation,several tens of periods of detonation rotation are neededto reach a stable regime when the entire flowfield becomes steady in the moving frame of reference. Toreduce the computational time, we used whenever possible previous solutions to initialize a new

computational case.

Results of CDWE Simulations

Here we present numerical results on the CDWEcombustion chamber obtained for various injection

pressures Pt j and geometrical sizes L and l. Beforeanalyzing the effect of each parameter, we consider indetail a single case corresponding to the followingparameter set:Pt j = 3 MPa,L = l= 100 mm. Except fewdetails specially mentioned below, the other simulatedflowfields are very similar.

Fig. 5. Flowfield structure in the combustion chamber:Pt j = 3 MPa, L = l= 100 mm. The colored band

indicates the scale of Mach numberMD in the movingreference frame

The general flowfield structure within the constant-area section is shown in Fig. 5. The colored fielddisplays the variation of the Mach number MD in themoving reference frame along with superimposed

streamlines. Several characteristic zones are clearly

depicted. The fresh mixture layer (1) is continuouslygrowing from the lower boundary. In this layer, MD isvery high, from 5.5 up to 7. The detonation wave (2) is

situated close to the right boundary. In front of thedetonation wave, the mean static pressure andtemperature are respectively 0.34 MPa and 259 K. Thevelocity of detonation propagation in the laboratoryframe of reference, UD, is of 2889 m/s, and the meanvelocity of the detonation front with respect to the fresh

mixture, D, is of 2922 m/s. Under these particularconditions, the propagation velocity of the ideal CJdetonation is DCJ = 2915 m/s that is the simulateddetonation is well matched the CJ regime. Themaximum pressure and temperature obtained in thedetonation wave are respectively 9.5 MPa and 4090 K.

Behind the detonation wave, the flow is subsonicwithin a very narrow zone. It rapidly accelerates to

supersonic speeds due to a strong expansion ofcombustion products, which is clearly indicated by thedivergent streamlines. The injection is suppressed oversome distance behind the detonation wave because of ahigh pressure there. Two other waves are visible in

Fig. 5 that originate from the point where the detonationwave meets the boundary (3) of the fresh mixture layer:a shock wave (4), which recompresses the flow ofcombustion products produced by the previousdetonation, and a slip line or contact discontinuity (5),

which separates the two streams of combustionproducts. A white dashed line (6) delimits the flowfield

zones where the vertical velocity component is subsonic(My < 1) or supersonic (My > 1). One can notice that aty > 50 mm the flow is fully supersonic in the y-wisedirection. In this case the duct expansion is not

necessary to obtain a supersonic flow at the combustionchamber exit. The observed flow structure is

qualitatively similar to that described in [3, 4, 5].

Fig. 6. Flowfield structure around the detonation wave.Schlieren-like representation of density gradient in anarbitrary exponential scale.

A closer view of the region around the detonationwave is shown in Fig. 6. It gives a schlieren-like

representation of the computed field of density gradient.

The main flowfield features are highlighted by whitelines. The maximum height of the fresh mixture layerhis 8.9 mm. In this layer, the velocity vector angle with




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respect to the wall varies from 5 to 9. The detonationwave front (1) is slightly curved. It is practically normalto the mixture stream near the wall and becomes oblique

as the velocity vector angle increases. Immediately behind the detonation wave, the velocity vector angle

varies from 3 to 16. There are two expansion fans at

each end of the detonation front. A weak fan (2) isoriginated from the point where the detonation front

meets the wall. It turns the streamlines towards the wall.A strong fan (3) emanates from the triple point at thetop of the detonation front. This fan turns the streamupwards and aligns it with the deflected contactboundary (slip line) (4). The shock wave (5) originatingfrom the triple point (6) is oblique. When the flow passes the shock close to the triple point, its Machnumber falls from 2.95 to about 1.4-1.5. The shear between the two flows separated by the slip line isstrong as the velocity difference is about 1000 m/s.

Nearly the same velocity difference is found over the

contact boundary separating the fresh mixture andcombustion products in front of the detonation wave.

Main results characterizing the combustion chamber performance are summarized in Table 1 for all thesimulated cases. The quantities that are not yetintroduced have the following definitions: average specific mass flow rate

=l

xvl

g0

jjj d1

where j is the density and vj is the y-wise velocitycomponent of injected mixture;

frequency of the detonation rotationfD = UD /l

average wall pressure

=l

xPl

P0

ww d1

combustion chamber specific impulse

+=l

xPvlg

I0

2

jsp d)(

1

where , v, andPare taken aty =L although they-wisemomentum is globally conserved; relative variation of they-wise velocity component vwith respect to its mass-average value vm at thecombustion chamber exit (y =L)

)(

)],(min[)],(max[)(

Lyv

LyxvLyxvLyv

m =

=== ,

=ll

m xyxxyxvyxyv00

d),(d),(),()(

variation of the angle of velocity vector with respectto they-wise direction at the combustion chamber exit

)()()( minmax LyLyLy ==== ,

)/arctan( vu= where u is the x-wise velocity component in the

laboratory frame of reference. Vand are introducedhere to characterize the nonuniformity of the velocity profile at the combustion chamber exit that can affect

the nozzle performance.The effect of the injection pressure Pt j has been

studied for fixed dimensions L = l= 100 mm. The

choice ofPt j provides that the specific mass flow rate

jg corresponds to the experimental conditions [3] at

Pt j = 1 MPa, whereas it is within the range of CDWE

operating conditions [2] atPt j = 3-6 MPa. By analyzing

the data from Table 1, one can find that jg , wP ,Pw min,

and Pw max are linear increasing functions ofPt j. Theoverall growth in frequency fD and detonation velocity

UD is about 5 %. Even if jg is continuously growing,

the maximum height h of the mixture layer decreases

globally by 9 % because of the simultaneous rise ofPw

andfD. The overall increase in the specific impulse Isp is

about 3.5 %. The nonuniformity of the velocity profile

remains practically unchanged.

To investigate the influence of the dimensionsL andl, Pt j is fixed at 3 MPa. The value ofL is taken equal

40 mm, 20 mm, and 10 mm at constant l= 100 mm. Thedetonation propagation is insensible to the variation ofL

even ifL is close to h. WhenL 40 mm, the flow is notentirely supersonic at y = L. The subsonic-supersonictransition occurs at the very beginning of the divergentsection. At L = 10 mm, Pw slightly diminishes and thetop of the mixture layer appears in the divergent section(h >L). The detonation wave remains stable howeverIspglobally decreases by 4.9 %. At the same time, thevelocity profile nonuniformity strongly increases.

Finally, the azimuthal period l is reduced to 75 mmand 50 mm keeping the other dimension constant L =40 mm. From the data given in Table 1, one canconclude that the frequencyfD and mixture layer heighth are inversely proportional to l. The other quantities

related to the injection, detonation propagation, and wall pressure distribution have no any noticeable change.

Instantaneous distributions of wall pressure obtained forthe three different periods are plotted in Fig. 7. If we

normalize the x-coordinate by l, the three curves become coincident. We also verified that the entireflowfield is self-similar if both the coordinates arenormalized by l. Contrary to the reduction ofL, asmaller period result in more homogeneous velocityprofile at the combustion chamber exit.

0 0.025 0.05 0.075 0.1x, m

0

4

8

Pw,

MPal= 50 mm

75 mm

100 mm

Fig. 7. Instantaneous distributions of wall pressure fordifferent azimuthal periods.

As compared to the present results, the experiment[3] and simulations by Zhdan et al. [5] give lower

detonation speeds. The experimentally observed deficit




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in UD is certainly due to imperfect mixture (separateinjection of fuel and oxidizer, dilution with combustion products) and viscous losses. The models applied byZhdan et al. are probably tuned to match theexperimental velocity however their generalassumptions are the same as we adopted here.

Conclusions

Numerical simulations of continuous rotatingdetonation in the stoichiometric hydrogen/oxygen

mixture have been performed using high-resolutionnumerical schemes and a detailed thermochemical

model. Even if the present approach is rather idealized,it allowed us to investigate the flowfield structure andstudy the effects of the injection pressure andcombustion chamber geometry on the detonationpropagation and combustion chamber performance.

In future we plan to complete this study by aninvestigation of the internal detonation wave structure

and stability of detonation propagation in a layer ofcombustible mixture as well as by an evaluation of theCDWE performance accounting for the flow expansion

in the nozzle.

Acknowledgements

The present work was supported by MBDA France.Special thanks to Dr. E. Daniau for very helpfuldiscussions. The computations resources were provided by the EPEE Federation of CNRS and University ofOrlans.

References

1. B.V. Voitsekhovskii, J. Appl. Mech. Techn. Phys. 3(1960) 157-164.

2. E. Daniau, F. Falempin, S. Zhdan, Pulsed androtating detonation propulsion systems: first step

toward operational engines, Paper AIAA-2005-3233, 2005.

3. F.A. Bykovskii, S.A. Zhdan, E.F. Vedernikov, J.Propulsion Power 22 (2006) 1204-1216.

4. S.A. Zhdan, A.M. Mardashev, V.V. Mitrofanov,Combust. Explosion Shock Waves 26 (1990) 91-95.

5. S.A. Zhdan, F.A. Bykovskii, E.F. Vedernikov, in: G.Roy, S. Frolov, J. Sinibaldi (Eds.), Pulsed andContinuous Detonations, Torus Press Publ.,Moscow, Russia, 2006, p. 319.

6. G.-S. Jiang, C.-W. Shu, J. Comput. Phys. 126 (1996)202-228.

7. X. Zhong, J. Comput. Phys. 128 (1996) 19-31.8. A. Burcat, B. Ruscic, Third millennium ideal gas

and condensed phase thermochemical database forcombustion with updates from activethermochemical tables, Report No. ANL-05/20,

TAE 960, Argonne National Laboratory, Technion(2005).

9. D.M. Davidenko, I. Gkalp, E. Dufour, P. Magre,Systematic numerical study of the supersoniccombustion in an experimental combustion chamber,Paper AIAA-2006-7913, 2006.

10. A.A. Konnov, Khimicheskaya Fizika, 23 (2004) 5-

18.11. J. Li, Z. Zhao, A. Kazakov, F.L. Dryer, Int. J. Chem.

Kinet. 36 (2004) 566-575.12. E.L. Petersen, D.F. Davidson, M. Rohrig, H.K.

Hanson, Proc. 20th

Int. Symp. Shock Waves, 1996,pp. 941-946.

Table 1. Main characteristics of the detonation propagation and combustion chamber performance.

Pt jMPa

Lmm

lmm

jg

kg/(sm2)

jg /gj max

%

fDkHz

UDm/s

D,m/s

h/l%

1 100 100 173 86.2 27.9 2795 2829 9.45

3 100 100 513 85.5 28.9 2889 2922 8.88

6 100 100 1024 85.3 29.3 2933 2964 8.63

3 40 100 513 85.5 28.9 2889 2922 8.88

3 20 100 513 85.5 28.9 2889 2921 8.78

3 10 100 519 86.5 28.9 2889 2922 10.27

3 40 75 514 85.7 38.5 2886 2920 8.97

3 40 50 515 85.9 57.6 2881 2917 9.15

Pt jMPa

Lmm

lmm

wP

MPa

Pw min / wP Pw max / wP Ispm/s

v(y =L)%

(y =L)deg

1 100 100 0.434 0.320 6.79 2699 39.6 31.0

3 100 100 1.323 0.314 7.12 2765 40.2 31.9

6 100 100 2.668 0.320 6.91 2794 41.0 32.2

3 40 100 1.323 0.316 7.29 2760 49.5 56.3

3 20 100 1.323 0.314 7.11 2759 57.5 72.5

3 10 100 1.273 0.326 5.93 2631 65.2 99.6

3 40 75 1.323 0.314 7.20 2756 46.3 48.1

3 40 50 1.324 0.314 7.17 2749 40.4 36.9



d.m. davidenko, i. gökalp and a.n. kudryavtsev- numerical modeling of the rotating detonation in an...

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