Lewis number and curvature effects on sound generation by premixed flame annihilation

Transcription

1 Center for Trblence Research Proceedings of the Smmer Program 2 28 Lewis nmber and crvatre effects on sond generation by premixed flame annihilation By M. Talei, M. J. Brear AND E. R. Hawkes A nmerical and theoretical stdy of sond generation by planar, axisymmetric and spherically symmetric premixed laminar flame annihilation is presented in this paper. The compressible Navier-Stokes, energy, and progress-variable eqations are solved sing Direct Nmerical Simlation DNS) with one-step chemistry. The sond prodced for these three configrations is compared for different Lewis nmbers. A theory is developed in the case of nity Lewis nmber sing the propagation velocity and consmption speed, which were identified as key parameters in the generation of sond in or previos stdy Talei et al. 2). The reslts show that by obtaining the propagation velocity of the flame from the simlations, the sond generation can be predicted accrately for nity Lewis nmber. Use of Markstein s linear theory leads to nder-prediction of the sond radiated. Cases involving non-nity Lewis nmber are then investigated. As the flames approach annihilation, either flame acceleration or deceleration is observed depending on the Lewis nmber, and this is shown to have a significant contribtion to the sond radiated. For Lewis nmbers greater than nity, the annihilation of the flame reslts in a local increase in consmption speed at annihilation, leading to more sond prodction compared with that of nity Lewis nmber. Lewis nmbers less than nity exhibit the opposite behavior.. Introdction Combstion is a significant sorce of noise polltion. Combstion-generated sond also plays a central role in the stability of many engineering devices sch as indstrial brners, gas trbines, and rockets e.g. Liewen 23; Candel et al. 29). The ongoing prsit of qieter and cleaner combstion in these devices provides a contined need for frther refinements in or nderstanding of combstion-generated sond. Strahle 97) constrcted a theory of combstion noise by extending Lighthill s acostic analogy Lighthill 95) to combsting flows. He arged that monopolar sond sorces can appear in trblent combsting flows. Under the assmptions of low Mach nmber and a constant, average-mixtre moleclar weight, Lighthill s acostic analogy can be simplified to an inhomogeneos wave eqation with a single monopolar sorce term Strahle 978; Dowling 992). This sorce term has been written in terms of time derivatives of the heat release rate or the flame volme e.g. Strahle 978). One mechanism affecting the flame volme, and hence sond generation, is flame annihilation Kidin et al. 984; Candel et al. 24). For example, when two flame srfaces interact the nbrnt gas trapped between these srfaces is consmed, reslting in a rapid redction in flame srface area and ths heat release. Candel et al. 24) stdied sond Department of Mechanical Engineering, University of Melborne School of Photovoltaic and Renewable Energy Engineering/School of Mechanical and Manfactring Engineering, University of New Soth Wales

2 282 Talei, Brear & Hawkes generation by mtal flame annihilation and conclded that the dominant mechanism of sond generation was flame srface destrction. Or previos work Talei et al. 2) examined sond generation by premixed, laminar flame annihilation for nity Lewis nmber. The main aim of this stdy is therefore to investigate Lewis nmber effects on the mechanism of sond generation in planar, axisymmetric and spherical flame annihilation. We also extend or earlier theoretical reslts to inclde the linear theory of Markstein 964) in an attempt to close the sond generation problem. These more general theoretical reslts are also compared with the simlations. 2. Nmerical Methods and Flow Parameters A modified version of NTMIX Cenot et al. 997) is sed. NTmix is a high-order accrate solver with single-step chemistry. It featres a 6 th order compact scheme for spatial derivatives, combined with a 3 rd order Rnge-Ktta time integrator. These modifications inclde the ability to simlate axisymmetric and spherically symmetric geometries. Flly three-dimensional characteristic bondary conditions Lodato et al. 28) were also implemented. Frther details of nmerical methods are presented in or previos stdy Talei et al. 2). The governing eqations were discretized into 3 nodes from ζ = to 5L ref, where L ref is the reference length and ζ is the spatial coordinate in the planar, axisymmetric and spherical cases. The laminar flame thickness δ is constant and eqal to.l ref. Three different Lewis nmbers Le =.5, and 2) have been stdied. The Prandtl nmber is.75. The temperatre ratio /T is 4 where T is the fresh gas temperatre and is the brned gas temperatre for an nstretched laminar flame with nity Lewis nmber. The Zel dovich nmber β is Reslts and discssion 3.. Nmerical reslts for nity Lewis nmber We first briefly review the process of annihilation and the associated sond prodction for Le =, which was reported in or previos stdy Talei et al. 2). The middle colmn in Fig. shows the reaction rate verss normalized distance x/δ at several instants. The flame propagates from right to left and the annihilation location is at the origin ζ/δ = ). It may be observed in this nity Lewis nmber case that the reaction-rate profile retains a similar spatial profile as the flame propagates. After the point of the peak reaction rate reaches the origin, the peak vale decreases ntil the flame is annihilated. Figre ii shows the redced temperatre θ = T T )/ T ) verss normalized distance for several instants. The redced temperatre increases as the two preheat zones start to merge. After the extinction event, the temperatre is niform and eqal to the brnt gas temperatre throghot the domain. Now consider the pressre field in Fig.. Before the annihilation event there is a small, hardly observable change in the pressre profile across the flame de to flame propagation. However, dring the extinction event, the pressre at the origin first increases then decreases, leading to a mch larger pressre plse that propagates away from the symmetry plane in the positive x-direction. The generated pressre wave after the annihilation event for the planar case has a steady-state vale that is less than the reference pressre.

3 Sond generation by premixed flame annihilation 283 a) Le =.5 b) Le = c) Le = 2 i) ii) iii) after annihilation after annihilation x/δ -.2 after annihilation 2 x/δ x/δ 2 Figre : i) non-dimensional reaction rate ω/ ω max), ii) redced temperatre θ, and iii) nondimensional pressre p p ref )/ρ ref c 2 ref verss distance from the origin non-dimensionalized by flame thickness δ at different instants before solid), dring dashed and dash-dot), and after long dash) planar annihilation Theory for nity Lewis nmber Or previos stdy Talei et al. 2) theorized that a constant propagation velocity cold adeqately predict the far-field steady state pressre for the planar configration. However, assming constant propagation velocity led to significant nder prediction of sond in the far-field in the axisymmetric and spherically symmetric cases. In this section we will relax one of the assmptions sed in or previos, infinitely thin flame approximation in an attempt to partially accont for finite flame-thickness effects. In or previos theory, we assmed that the flame displacement speed i.e. the speed at which the location of the maximm heat release moves) and the flame consmption speed i.e. the spatial integral of the reaction rate) were identical. Here we relax this assmption, which will be shown to lead to improved reslts. In or previos stdy the general soltion of Lighthill s eqation for flames of finite thickness, retaining only the heat release term as the sorce term of Lighthill s eqation, was p r, t) = T ) t r/cb ζ + ζ ω/ τ dζdτ t τ)2 r 2 /c 2 b 3.)

4 284 Talei, Brear & Hawkes for the axisymmetric annihilation and p R, t) = T ) Ht R/c b ) ζ 2 ω/ τdζ ζ + R τ=t R/cb 3.2) for spherically symmetric annihilation. The variable p is the pressre flctation in the far-field, r and R are the distances from the origin in the axisymmetric and spherically symmetric cases, respectively, ω is the reaction rate, t is the time and c is the sonic velocity. These expressions are not directly sefl becase of the appearance of the reaction rate and its time derivative in the integrands. Some approximation is therefore reqired to proceed. In order to evalate the integral terms in Eqs. 3. and 3.2, the reaction rate may be modelled by a delta fnction and assming a constant consmption speed, ω = ρ S L δζ f τ) ζ), 3.3) where ρ is the density, S L is the laminar flame speed, and ζ f is the flame position. Sbstitting Eq. 3.3 into Eqs. 3. and 3.2, the soltion of Lighthill s eqation for axisymmetric and spherically symmetric annihilations may be expressed as and p r, t) = ρ S L T ) t r/cb V f τ)dτ t τ)2 r 2 /c 2 b 3.4) p R, t) = 2ρ S L T ) ζ f t R/c b )V f t R/c b )Ht R/c b ), 3.5) respectively. The variable V f = dζ f /dτ is the propagation velocity of the flame. Note, we do not assme here that V f eqals the consmption speed. The remaining problem is to estimate the flame displacement velocity, which may depart from the laminar brning rate as annihilation is approached. A simple relation between laminar brning velocity and crvatre was proposed by Markstein 964), V f = S L + nl M /ζ f ), 3.6) where L M is the Markstein length and n = for axisymmetric and n = 2 for spherically symmetric cases respectively. Sbstitting Eq. 3.6 into Eqs. 3.4 and 3.5, the pressre in the far-field may be estimated for the axisymmetric annihilation as p r, t) = ρ S 2 L T ) t r/cb + L M /ζ f )dτ, 3.7) t τ)2 r 2 /c 2 b whereas for the spherically symmetric annihilation p R, t) = 2ρ SL 2 T ) ζ f + 2L M )Hζ f τ)). 3.8) τ=t R/cb The reslt for the spherically symmetric annihilation is similar to the theory proposed by Kidin et al. 984), which is repeated here, p R, t) = 2ρ T ) SL 2 + 2L ) ) M ζf + L M Hζ f τ)). 3.9) ζ f R τ=t R/cb

5 a) Sond generation by premixed flame annihilation 285 b) c) i) ii) E-5-2E-5-2E-5-4E-5 ζ/ L ref -4E-5 ζ/ L ref -4E-5 ζ/ L ref Figre 2: Comparison of DNS solid line) and soltion of Lighthill s eqation dashed line) a) sing the reaction rate sorce term, b) obtaining propagation velocity from the simlation reslts, and c) sing the Markstein length theory, dash-dot: analytical soltion of Lighthill s eqation assming constant propagation velocity, top row; axisymmetric, bottom row; spherically symmetric. We wold obtain the same reslt as Kidin et al. 984) if we had assmed that the displacement and consmption speeds were identical. These different theoretical approximations are now compared. The top row of Fig. 2, which is reprodced from data of or previos stdy, shows the DNS reslt, the reslt from the soltion of Lighthill s eqation retaining only the heat-release sorce term and the soltion assming an infinitely thin flame that propagates at the laminar brning speed. As can be seen, retaining only the heat release sorce terms reslts in good agreement with the simlation, while the infinitely thin, constant speed reslt is poor. The middle colmn of Fig. 2 shows a nmerical soltion of Lighthill s eqation sing Eqs. 3.4 and 3.5. Here, the propagation velocity is taken directly from the simlation reslt by measring the absolte displacement speed of the point of maximm reaction rate. As may be seen, for both axisymmetric and spherically symmetric annihilation events the theory can predict the sond radiated to the far field with a reasonable degree of agreement. The right colmn of Fig. 2 shows a comparison of simlation reslts with Eqs. 3.7 and 3.8 for axisymmetric and spherically symmetric cases, respectively. The Marsktein length is 3δ for this nity Lewis nmber case, which is obtained from the theory developed by Clavin & Jolin 983). As can be seen, the soltion of Lighthill s eqation sing the Markstein length for the displacement speed captres some of the sond generation de to flame acceleration. Becase the Markstein length is proportional to flame thickness, this shows that the effect of flame thickness needs to be considered Non-nity Lewis nmber Figre i shows the reaction rate at different instants before and dring the extinction event for the planar case for Le =.5,, and 2. It is observed that for low Lewis nmber, the reaction rate decreases when the flame approaches the origin. In this case, the thickness of the mass diffsion layer is larger than the thermal diffsion layer. Therefore, when the two flames approach each other the mass diffsion layers start to interact

6 286 Talei, Brear & Hawkes a) planar b) axisymmetric c) spherically symmetric i ) ii ) t c ref / L ref t c ref / L ref t c ref / L ref Figre 3: Relative pressre for flame annihilation from DNS i) at the symmetry axis and ii) in the far-field x/l ref = 7.5 for different Lewis nmbers: solid line, Le = ; dashed, Le = 2; dash-dot, Le =.5. earlier than the thermal diffsion layers, leading to the depletion of reactants before the region of greatest reaction rate reaches the symmetry plane. As a reslt, the extinction of the flame commences at a distance from the origin. This behavior is consistent with prior observations from planar flame extinction e.g. Sn & Law 998). As a reslt of the gradal decrease in the reaction rate the reactants brn slowly as the temperatre increases Fig. i and ii). The pressre shown in Fig. also changes gradally over a long period of time and significantly less sond is prodced compared with that in the other two cases. The case of Le = 2 is shown in the rightmost colmn of Fig.. The increase in the reaction rate as the flame approaches the symmetry plane is mch higher than for Le =. In this case, the reaction rate starts to increase at a greater distance from the origin, which also has a significant effect on the temperatre and pressre profile before and after flame annihilation. Indeed, in this case the peak temperatre exceeds the adiabatic flame temperatre at some instants. The observed far-field pressre is similar to the nity Lewis nmber case. Figre 3 shows a comparison of pressre verss time at two points in the domain for each of the different configrations and all three Lewis nmbers. The colmns from left to right correspond to the planar, axisymmetric, and spherical configrations. The Lewis nmbers of Le =.5,, and 2 are shown. As can be seen from Fig. 3, in the case of planar flame annihilation, the high and nity Lewis nmber cases prodce a similar steady-state far-field pressre, whereas the peak pressre is mch higher for the high Lewis nmber. The peak pressre is de to the peak reaction rate, which was mch higher in the case of the high Lewis nmber. In the low Lewis nmber case the pressre varies over a mch longer period compared with the other two cases Dynamics of flame annihilation In the theory developed in section 3.2, the propagation velocity and consmption speed were sed to predict the sond generation by the annihilation event. In this section

7 i) Sond generation by premixed flame annihilation 287 a) planar b) axisymmetric c) spherically symmetric 5 ii) 5 2 x/ δ 2 r/ δ 2 R/ δ Figre 4: i) normalized consmption speed R ωdζ/ρ S L, ii) magnitde of normalized propagation velocity V f /S L verss x/δ: solid line, Le = ; dashed, Le = 2; dash-dot, Le =.5. variation of these two parameters are investigated to assess the validity of the assmptions sed. Figre 4 shows non-dimensional consmption and propagation speeds verss nondimensional flame position. Flame position is defined as the position of maximm reaction rate. For Le >= the consmption speed starts to increase as the flame approaches the origin for all configrations. However, as the two flames get closer the consmption speed starts to drop and eventally tends to zero dring the extinction process. In the case of low Lewis nmber the consmption speed starts to decrease before the flame reaches the origin. The extinction commences at a point near the origin as the consmption speed tends to zero. The extinction at this point is not complete and there will be frther reaction as the point of maximm reaction rate approaches the origin. Figre 4ii shows the propagation velocity of the flame measred at the point of maximm reaction rate. In all cases an increase in the velocity is observed near the origin, providing jstification for the earlier attempt at closing the problem sing the Markstein length. As can be observed from Fig. 4i, constant consmption speed is a reasonable assmption to solve Lighthill s eqation for only nity Lewis nmber, ths demonstrating limitations in this approximation. 4. Conclsions This paper presents a nmerical and theoretical stdy of sond prodction by flame annihilation in planar, axisymmetric, and spherically symmetric configrations. A series of Direct Nmerical Simlations DNS) sing a higher order accrate solver appropriate for aeroacostic stdies was sed and the effects of flame crvatre and Lewis nmber were investigated. In order to predict the sond generation in axisymmetric and spherically symmetric cases, a previos work by the athors was revisited by modelling the reaction rate as a constant amplitde delta fnction moving toward the origin. The propagation velocity of the flame was measred by tracking the point of maximm reaction rate. There was a good agreement between the simlation reslts and soltion of Lighthill s eqation sing this method for nity Lewis nmber. In or attempt to find a more general theory,

8 288 Talei, Brear & Hawkes Markstein s linear theory was then sed to obtain the propagation velocity, bt this was shown to lead to an nder prediction of the sond radiated to the far-field. Simlations also showed that the assmption of constant consmption speed was not always valid. For high Lewis nmber, an increase is observed in both the propagation velocity and consmption speed, which leads to generation of higher amplitde sond compared with nity Lewis nmber. In the case of low Lewis nmber, a decrease in the consmption speed occrred, casing flame extinction to commence before it reached the origin. This case was shown to prodce significantly less sond compared with higher Lewis nmber cases. The athors acknowledge the generos spport of the Eropean Centre for Research and Advanced Training in Scientific Comptation CERFACS, in providing the athors with the sorce code for NTmix. Use of the facilities of the Center for Trblence Research CTR) at Stanford University dring the 2 smmer program is also grateflly acknowledged. REFERENCES Candel, S., Drox, D., Dcrix, S., Birbad, A. L., Noiray, N. & Schller, T. 29 Flame dynamics and combstion noise: progress and challenges. Int. J. Aeroacostics 8 &2), 56. Candel, S., Drox, D. & Schller, T. 24 Flame interactions as a sorce of noise and combstion instabilities. In Collection of Technical Papers-th AIAA/CEAS Aeroacostics Conference, pp Clavin, P. & Jolin, G. 983 Premixed flames in large scale and high intensity trblent flow. J. Phys. Lett. 44 ), 2. Cenot, B., Bedet, B. & Corjon, A. 997 NTMIX3D ser s gide manal, Preliminary Version.. Dowling, A. P. 992 Modern Methods in Analytical Acostics, chap. Thermoacostic sorces and instabilities, pp Springer. Kidin, N., Librovicht, V., Robertst, J. & Villermoz, M. 984 On sond sorces in trblent combstion. Dynamics of flames and reactive systems pp Liewen, T. 23 Modeling premixed combstion-acostic wave interactions: A review. J. Proplsion Power 9 5), Lighthill, M. J. 95 On sond generation aerodynamically I. General Theory. Proc. Roy. Soc. London 2, Lodato, G., Domingo, P. & Vervisch, L. 28 Three-dimensional bondary conditions for direct and large-eddy simlation of compressible viscos flows. J. Compt. Phys. 227 ), Markstein, G. H. 964 Nonsteady flame propagation. Pergamon. Strahle, W. C. 97 On combstion generated noise. J. Flid Mech. 49 2), Strahle, W. C. 978 Combstion noise. Prog. Energy Combst. Sci. 4, Sn, C. J. & Law, C. K. 998 On the consmption of fel pockets via inwardly propagating flames. Proceedings of the Combstion Institte pp Talei, M., Brear, M. J. & Hawkes, E. R. 2 Sond generation by premixed flame annihilation. Sbmitted to the J. Flid Mech.