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Numerical Study of Premixed Flames Parametric Acoustic InstabilityB. Deneta; A. Tomaa

a IRPHE Combustion, Université de Provence- Centre de Saint Jérome (S252), Marseille Cedex 20,France

To cite this Article Denet, B. and Toma, A.(1995) 'Numerical Study of Premixed Flames Parametric Acoustic Instability',Combustion Science and Technology, 109: 1, 23 — 33To link to this Article: DOI: 10.1080/00102209508951893URL: http://dx.doi.org/10.1080/00102209508951893

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Numerical Study of Premixed Flames ParametricAcoustic Instability

B. DENET* and A. TOMA IRPHE Combustion, Universite de Provence- Centrede Saint Jerome (S 252), 13397 Marseille Cedex 20, France

(Received June 30,1995; infinalform July 12, 1995)

ABSTRACT-Numerical simulations of flames unstable, due to a parametric acoustic forcing, are performed in this paper. Results concerning the threshold of this instability are compared to quasi-steadytheoretical predictions. Important disagreements are found at large frequencies. suggesting that theMarkstein lengths relative to curvature and stretch are frequency dependent. Results are discussed inconnection to experiments and to new theoretical ideas.

Key Words: Parametric instability, acoustic forcing, Markstein lengths

INTRODUCTION

Interaction of premixed flames with acoustics has been the subject of various experiments and interpretations. A major advance was realized when Markstein (1953) sawexperimentally a regime where the flame cellular structure oscillates at half the acousticfrequency observed in the tube. This situation reminded him of the parametricallyexcited oscillator problem, i.e. an oscillator whose natural.frequency oscillates in timedue to some external forcing. He was able to develop a theoretical analysis of this effectbased on a phenomenological description of the flame front, with an effective gravityterm oscillating sinusoidally in time due to the acoustic acceleration.

This analysis is valid only in the case where acoustics is imposed, and not for the fullproblem which occurs when a flame propagates in a tube and generates itselfthe sound.This motivated Searby and Rochwerger (1991) to design an experiment closer to thetheoretical analysis: a flame excited by a loudspeaker, i.e, in an imposed acoustic field.The main results of this experiment are:

- For low acoustic amplitudes, the flame shape consists of large cells caused by thehydrodynamic Darrieus Landau instability (Darrieus, 1938; Landau, 1944).However these cells oscillate a little at the acoustic frequency.

- For intermediate amplitudes, the flame becomes flat and is restabilized by theparametric forcing.

- For higher acoustic forcings, the flame becomes again cellular, but with smallercells oscillating at twice the acoustic period: this is the parametric instability zone.

These three phenomena correspond to three different stages of the freely propagatingexperiment, without imposed acoustics (Searby, 1992).

• Corresponding author.

23

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24 B. DENET AND A. TOMA

In the same article (Searby and Rochwerger, 1991), the analysis of Markstein isextended to include a more recent flame front model (Clavin and Garcia, 1983,see alsoPelce and Clavin, 1982). The essential prediction of this quasi-steady theory is thelocation of the parametric instability threshold in the plane (acoustic amplitude,pulsation). This threshold is shown to depend strongly on the diffusive properties of themixture, i.e, on the Markstein length.

However, experiments performed by G. Searby with gases of different diffusiveproperties, such as methane (Searby, 1991)and lean propane (Searby and Rochwerger,1991),show that the threshold is essentially the same in both cases at high frequencies.This observation remained unclear until G. Joulin (1994)suggested that the Marksteinlengths relative to curvature and stretch could be frequency dependent, so that theeffect of diffusive properties (Lewis number of deficient reactant) disappears at highfrequency. Actually, Joulin's analysis is limited to a thermo-diffusive model with no gasexpansion, but the qualitative results are expected to remain valid even when thecoupling with hydrodynamics is taken into account.

In this paper, we will test the previous ideas by numerically simulating the equationscorresponding to the quasi-steady theory (Searby and Rochwerger, 1991). We willcompare our measures of the parametric instability threshold to these theoreticalpredictions, showing large discrepancies at high pulsation and a very low dependenceon Lewis number, as qualitatively predicted by Joulin.

2 EQUATIONS AND NUMERICAL METHOD

We describe in this section the equations of premixed flames in the isobaric approximation (i.e. compressible effects on the flame structure are neglected) and with thesimplest chemistry: a single one-step chemical reaction is assumed. Non-dimensionalquantities are obtained using as unit of length the flame thickness obtained fromasymptotics, and as unit of velocity the asymptotic flame speed. The use of normalizedvariables allows us to write the model, in a frame moving with the flame front, asfollows:

with

ac 1p- +p(iJ-V)C=-~C - 0at Le

. o=Lc-p-ex ( P(T-l) )2 Le 1-)' P 1+ y(T - 1)

(11.1)

(11.2)

(11.3)


and

FLAME ACOUSTIC INSTABILITY

o(p) +v.(pv) = 0ot

O~v) + v.(pvv) = _ VP + p G+ lu:1v + (A. +143)V(V·v)

P=(L+_]'_T)-l1-]'

25

(11.4)

(11.5)

(11.6)

where T and C correspond to the reduced temperature of the gas mixture andconcentration of a reactant. (the other reactant being in excess). The boundaryconditions on T and Care

T(x = - co.y) = 0, T(x = + oo.y] = 1

C(X = - oo,y) = 1, C(x = + oo.y) = 0

all quantities being periodic in the y direction (the direction transverse to the flame).Le, fJ and]' are respectively the Lewis number of the reactant (ratio of the thermaldiffusivity to the molecular diffusivity of the deficient species), the reduced activationenergy (or Zeldovich number) (fJ = E(Tb- T.)/ RT;) and the heat release parameter(]' = (P. - Pb)/Pb)' the subscripts u and b referring to unburned and burned gases, p,v and P are the density, velocity and pressure; G is an external specific force per unitvolume

G = gd/(U 1)2 - wu cos(wt) (11.7)

where u, is the flame speed, d the flame thickness and g the gravity acceleration, wandu are respectively acoustic pulsation and amplitude. This alteration of the gravityterm to include the effect of acoustic acceleration has been used in various studies(Markstein, 1953;Searby and Rochwerger, 1991). See also for a derivation of this termthe interesting work of Pelceund Rochwerger (1992), particularly the equationsconcerned with the isobaric zone (Eqs. II and 12).~ = 0.7 and A. are respectively Prandtl numbers associated to shear and bulk

viscosities. As the second viscosity term can be incorporated into a new dynamicpressure gradient that serves only to satisfy the continuity equation, we completelyneglect this term.

We give now a very short summary of the numerical method used to solve theseequations (seeDenet and Haldenwang (1995)for more details). Fourier pseudo spectralmethods are used in the y direction and finite differences in the x direction. Thetemporal scheme is first order, implicit on the diffusivity terms but explicit on thereaction and convection terms. The equations on temperature and concentration arefirst solved, the density being deduced from the state equation. Then the isobarichydrodynamic equations are handled (there is of course coupling with the temperatureand species equations through the density): we solve the pressure field by transformingthe equations into a Poisson equation for pressure, the remaining equations on velocitybeing easy to solve when the pressure is known.



Note that in this study, we neglect completely the modification offlame structure bycompressible effects and consider that close to the flame, there is an isobaric zone.Theoretical studies of this possible modification or flame structure have been performed (McIntosh, 1991; Clavin Peke He, 1990). The amplification rate or acousticdisturbances by this mechanism in the case or flames propagating in tubes (withoutimposed acoustics) was obtained in the second paper. It was shown that this rate is fiftytimes lower (Clavin, 1994)than the rate observed in experiments (Searby, 1992).On thecontrary, the theory of Pelce and Rochwerger (1992),which neglects this compressibleeffect,gives the good order of magnitude for the amplification rate, and this is a strongargument in favor of neglecting this modification or flame structure and of concentrating on the effectof acoustic acceleration on isobaric hydrodynamics, as we will do here.

In the sequel, we will be concerned with the flame response on long time scales, butothers studies have been performed on pressure interaction with flames for short times(Liu et al., 1993;Edwards et al., 1994;Tsuruda and Hirano, 1991).Basically, they showthat the pressure wave can accelerate the flame and bend the flame front. Note that onshort time scales, intrinsic instabilities or the plane front can be neglected. For instance,the numerical work of Liu et al. (1993) considers the flame as a discontinuity wherejump conditions are prescribed and studies the compressible flow around the flame. Inour work, on the contrary, we will be concerned with flame behavior for long times,where the intrinsic Darrieus Landau instability is important, and we will solve theconcentration, temperature and isobaric hydrodynamic equations, excited by acousticacceleration.

3 QUALITATIVE RESULTS

We show in this section numerical solutions corresponding to the different qualitativebehaviors observed in experiments. For low acoustic forcings, the solution is relativelysimilar to a solution with no forcing at all and exhibits the characteristic shape ofa flame submitted to the hydrodynamic Darrieus Landau instability, i.e. a flame witha low number of cells oflarge amplitude. We will not show here such a solution, but seeDenet and Haldenwang (1995) and Denet and Bonino (1994) for figures of thisinstability.

When forcing is increased, plane flames restabilized due to parametric forcing, areobtained, as shown in Figure 1 (five temperature lines are plotted corresponding toT =0.1,0.3,0.5, O. 7and 0.9).This solution is plane although the width ofthe integrationdomain contains many Darrieus Landau unstable wavelengths. This effect is similar tothe experimental results and to the theoretical predictions.

Increasing the forcing further, a parametric instability is obtained. In Figure 2 (ab),a flame subject to this instability is studied during an acoustic period. In Figure 2a, the

FIGURE 1 Plane flame restabilized by parametric forcing with parameters f3 = 5 Le = I /l =0.7gd/(u,)' =O.Ot k =0.0275 «o =4 u =7.5.


FLAME ACOUSTIC INSTABILITY 27

forcing is plotted versus time. But now one acoustic period is not sufficient for thesystem to recover its initial state, and the true period of the oscillation is twice theacoustic period. This phenomenon is shown in Figure 2b, where the transversecoordinate ofthe point of'minimumxposition ofthe temperature line T = 0.9 is plotted:during the acoustic period, the minimum changes only once, at a time corresponding toa low amplitude flame, and a second period would be necessary to revert to the initialposition.

In Figure 3(a, b), the solutions during the period studied are presented (a largermagnification being used in the x direction). In Figure 3a, the solution at the initial timeis given. In Figure 3b, the solution is plotted one acoustic period after Figure 3a, andcan be seen to be exactly reversed.

In this section a good qualitative agreement between numerical results and theoryhas been obtained. In the next section, the threshold of parametric instability is studiedquantitatively and compared to the quasi-steady theoretical predictions.

(a) 16

14

12

10

8

6

4

2

0

-2

-4

~

-8

-10

-12

-14

-1615l.1

FIGURE 2(a,b) Parametric instability: evolution of the flame during one period of acoustic forcing(physical parameters (J = 5 Le = 1.51' =0.7 gd/(u,)' =om k =0.0275w = 2 u = 7.5)acoustic forcing (Fig. 2a)y coordinate of the flame minimum x position (Fig. 2b).



FIGURE 3(a, b) Parametric instability with the same parameters as in Figure 2: Flame at the beginning (Fig. 3a), and end (Fig. 3b) of the acoustic period.

4 THRESHOLD OF PARAMETRIC INSTABILITY

4.1 Quasi-Steady Theoretical Predictions

The theoretical results presented in the paper Searby and Rochwerger (1991), were obtained in a case where diffusion coefficients are temperature dependent. We have



used here the original computer programs of D. Rochwerger in a case of constant diffusion coefficients, because this case corresponds to the equations we solve numerically (see Section 3). These progra,ms use the theory of Mathieu functions to find out if the solution of the problem is linearly stable or unstable, and to obtain theoretical values for the parametric instability threshold.

It is thus possible to locate the position of the minimum of the parametric zone in the plane (wave vector, acoustic amplitude). A computer program is then used to obtain automatically these results for different pulsations.

4.2 Numerical Measurement of Threshold

We now explain how values of the amplitude and wavevector at threshold are numerically measured. In Section 3, results of simulations performed in a large domain, with a large number of cells, were shown. But in order to study the instability threshold we work now with a domain of small width, in order to have only one cell in the domain. Of course the width of the domain can be modified to vary the wavevector. As we are interested here only in the threshold value, we use a small number of Fourier modes in the transverse (y) direction, typically 10. These Fourier modes are however sufficient to describe the solution close to threshold, because the solution has only one cell, and on the other hand, the amplitude of the solution is small. For a given value of the amplitude and wavevector, the simulation is thus relatively inexpensive.

To locate precisely the threshold position, it is necessary to perform a lot of these calculations, with different values of wavevector and amplitude. In each calculation, we see if the solution is stable or not, and it is possible to get gradually closer to the minimum of the curve. The main errors in this numerical procedure seem to be of two types. First of all, the calculations are performed for a given value of the time step (this value is however reduced with increasing frequencies). There are temporal errors associated with the time discretization of the equations. To limit this effect, we have reduced the time step until the position of the minimum seemed to remain practically unchanged. Another problem is that, close to threshold, it is difficult to know whether the soiution is stable or unstable: as the threshold is approached, the integration should be performed for a longer time. Of course we have to limit the integration time to a reasonable value, but it seems that the transition to instability is rather sharp, s a that the error we have in locating the threshold is not very important.

4.3 Results

The results concerning the acoustic amplitude at threshold versus reduced acoustic pulsation are plotted in Figure 4. As both theoretical and numerical results are practically independant of g d / ( ~ , ) ~ , we will keep in the sequel this parameter to the constant value 0.01. The calculations are performed for a Zeldovich number of five, and different Lewis numbers: Le = 0.5, 1 and 1.5. The quasi-steady theoretical predictions are also plotted for the same values of the physical parameters. These predictions are very different at high frequency for the three Lewis numbers. On the contrary, in the



threshold vs pulsa tion

10.01.0

0.00 L-"--------'-'-'------'-------'---L-LJ-LJ-L__-'---------'-'-L-L..L--LJ-LJ

0.1

20.00

16.00

C'g 12.00Ql>u..,'"::s 8.000u

'"4.00

puisation

FIGURE 4 'Amplitudevs-pulsation at threshold: numerical and theoretical curves for p=5, Le=O.5,1 and 1.5.

high pulsation range, the numerical curves are very close. Actually the numerical curvesobtained for Le = 1and 1:5almost merge, the values obtained for Le = 0.5 being a littlehigher. The threshold values have been obtained 'Only for high.frequencies in this case;because for lower pulsations, the parametric and Darrieus Landau zones are very close,and it seems difficult to obtain a reliable estimate of the parametric threshold. Similarproblems were encountered by Sear by in. his experimental measurements' at lowpulsations.

When frequency is reduced-the numerical curves (for Le = 1 arid 1.5) continue to bepractically superposed.•The theoretical curve for Le = 1.5is a little higher, but it can beconsidered that at the minimum pulsation where calculations are performed, i.e.w = 0.3, the difference between the theoretical and numerical curves becomes of theorder of the numerical errors. Since the results would not be significant, we have notperformed calculations at pulsations lower than 0.3. Nevertheless, it is worth notingthat the theoretical curve for Le = 1 is closer to our numerical measurements for thethreshold' than the theoretical curves for Le = 0.5 and Le = 1.5.

In Figure 5, the wavenumber at threshold is plotted versus acoustic pulsation, for thesame values of the parameters as in Figure 4. The theoretical curves of the quasi-steadytheory are plotted as before: As in Figure 4, the theoretical curvesare quite different athigh frequency: On the-contrary, the numerical curves are very close, but here thesecurves are also relatively close to the theoretical curve for Le =; 1. When frequency isreduced, the numerical curves approach the theoretical curves, even for Le = 1.5.


FLAME ACOUSTIC INSTABILITY

wavevector at threshold vs pulsation

31

1.00

0.80

..0.60B

u

'">'">'" 0.40~

0.20

0.000.1

f i i

. Lee 0.5 (theory)~ - Le = 1.0 (theory)

Le =1.5 (theory)-e· Le =0.5~Le=1.0

-- -- Le > 1.5

, i

Ii:

: '

1.0

pulsation

10.0

FIGURE 5 Wavevector vs pulsation at threshold: numerical and theoretical curves for p= 5, Le = 0.5,1 and 1.5.

4.4 Discussion

Let us discuss the results obtained in Figures 4 and 5. First of all, the numerical curvesobtained are extremely weakly dependent on the different Lewis numbers at highfrequency. Although our numerical measurements correspond to a relatively low valueof the Zeldovich number and could include corrections in liP compared to thetheoretical results, this is in complete contradiction with the quasi-steady theoryresults. As in the experiments conducted with methane and lean propane, the numericalcurves almost merge at high frequency. It thus becomes obvious that the quasistationary hypothesis used in the constant Markstein length theory derivation is notvalid in the acoustic case, except at very low pulsations.

As suggested by Joulin, unstationary terms must be taken into account, and thisleads to the notion of frequency dependent Markstein lengths. The thermo diffusive,unstationary theory of Joulin predicts that the Markstein lengths become quicklyindependent of the Lewis number with increasing pulsations; this is compatible withour findings that the numerical thresholds for Le = I and 1.5are very close at relativelylow pulsations and also very close to the Le = I theoretical curve. The second effectpredicted is that the Markstein length relative to stretch decays slowly to zero at highpulsation: this may explain that the numerical curves move away from the Le = Itheoretical curve when frequency increases.

It would be interesting to extend the local study of frequency dependent Marksteinlengths to the full hydrodynamic problem in order to see if the qualitative behaviorobserved in the thermodiffusive case is preserved. This local study would be a first step



towards a theory of the parametric instability valid for pulsations of order unity.However, as in the frequency dependent case the Markstein lengths are complexnumbers, this theory would not lead again to Mathieu equations. According to Joulin,delay differential equations would be obtained and this would of course greatlycomplicate the resolution.

5 CONCLUSION

In this paper, the problem of a flame submitted to an acoustic forcing has been studiednumerically, in the approximation introduced by Markstein (1953) where acousticsinfluence an isobaric flame through a periodic acceleration, and where compressibleeffects on the flame structure are neglected. It was found that, in agreement withtheoretical and experimental works, a moderate acoustic forcing stabilize a planeflame, normally unstable because of the Darrieus Landau hydrodynamic instability.For higher acoustic forcings, the flame becomes cellular, with cells oscillating at twicethe acoustic period, this is the parametric acoustic instability zone. The threshold ofthis instability was predicted quantitatively by a quasi-steady, constant Marksteinlength theory (Searby and Rochwerger, 1991), we have compared this theory with ourmeasures for the instability threshold. It appears that there is as expected agreementwith theory at low frequencies, but that as far as the reduced pulsation (in units ofinverse of transit time) becomes of order unity, large discrepancies exist between theoryand numerics. Essentially, we have shown that as frequency increases, the instabilitythreshold becomes practically independent of the Lewis number, unlike the theoreticalprediction. This corroborates the experimental result (Searby and Rochwerger, 1991;Searby, 1991) that the thresholds of gases with very different diffusive properties arelocated approximately on the same curve. It suggests that the analysis of Joulin (1994),performed on the thermodiffusive model, showing that the Markstein lengths relativeto curvature and stretch are frequency dependent and different, might be relevant evenwhen density changes are considered. In this analysis, the Markstein lengths becomeprogressively independent of the Lewis number as frequency increases, and ourmeasures for the threshold values are in total qualitative agreement with this result.

ACKNOWLEDGEMENTS

We thank G. Joulin, P. Clavin and G. Searby for many helpful discussions. We also thank G. Searby forhaving given us access to the original computer programs of his paper with D. Rochwerger.

REFERENCES

Clavin C. (1994). Premixed Combustion and Gasdynamics, Annu. Rev. Fluid Meeh., 26, 321.Clavin P. and Garcia P. (1983). The Influence of the Temperature Dependence of Dilfusivities on the

Dynamics of Flame Fronts, J. Mec. Theor. Appl., 2(2),245.Clavin, C, Pelce, P. and He, L. (t990). One Dimensional Vibratory Instability of Planar Flames Propagat

ing in Tubes, J. Fluid Meeh., 216, 299.Darrieus, G. (1938). Propagation d'un Front De F1amme: Essai De Theorie Des Vitesses Anormales de

Deflagration par Developpement Spontane de La Turbulence, Communication Presented at LaTechnique Moderne.



Denet, B. and Bonino, J. L. (1994). Laminar Premixed Flame Dynamics: A Comparison of Model andComplete Equations, Combust. Sci. Tech., 99, 235.

Denet, B. and Haldenwang, P. (1995). A Numerical Study of Premixed Flames Darrieus Landau Instability,Combust. Sci. Tech., 104, 1-3, 143.

Edwards, N. R. McIntosh, A. C. and Brindley, J. (1994). The Development of Pressure Induced Instabilitiesin Premixed Flames Combust. Sci. Tech., 99, 179.

Joulin, G. (1994).On the Response of Premixed Flames to Time Dependent Stretch and Curvature, Combust.Sci. Tech., 97, 219.

Landau, L. (1944). On the Theory of Slow Combustion, Acta Physicochim. URSS, 19, 77.Liu, F., McIntosh, A. C. and Brindley, J. (1993). A Numerical Investigation of Rayleigh Taylor Effects in

Pressure wave Premixed Flame Interactions, Combust. Sci. Tech.; 91, 373.Mclntosh, A. C. (1991). Pressure Disturbances of Different Length Scales Interacting with Conventional

Flames, Combust. Sci. Tech.; 75, 287.Markstein, G. H. (1953). Instability Phenomena in Combustion Waves, Fourth Symp. on Combustion, 44-59,

Baltimore, Williams and Wilkins.Pelce, P. and Clavin, C. (1982). Influence of Hydrodynamics and Diffusion Upon the Stability Limits of

Laminar Premixed Flames, J. Fluid Mech., 124,219.Pelce, P. and Rochwerger, D. (1992). Vibratory Instability of Cellular Flames Propagating in Tubes, J. Fluid.

Mech., 239, 293.Searby, G. (1991). Experimental Investigations of Acoustic Instabilities in Laminar Premixed Flames,

Proceedings of the NATO ASI Summer School, "Nonlinear Phenomena Related to Growth and Form",Edited by Ben Amar M., Pelce P. and Tabeling P., Plenum Press.

Searby, G. (1992). Acoustic instability in premixed flames, Combust. Sci. Tech., 81, 221.Searby, G. and Rochwerger, D. (1991). A parametric Acoustic Instability in Premixed Flames, J. Fluid

Mech.;231, 529.Tsuruda, T. and Hirano, T. (1991). Local Flame Front Disturbance Development Under Acceleration,

Comb. and Flame, 84, 66.


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