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Direct numerical simulation of a realistic acoustic wave interacting with a premixed flame
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Combustion Institute
Proceedings of the Combustion Institute 32 (2009) 1473–1480
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Direct numerical simulation of a realistic acoustic wave interacting with a premixed flame H. Shalaby a, A. Laverdant b, D. The´venin a,* a
Laboratory of Fluid Dynamics and Technical Flows, University of Magdeburg ‘‘Otto von Guericke”, Universita¨tsplatz 2, D-39106 Magdeburg, Germany b Office National d’E´tudes et de Recherches Ae´rospatiales (ONERA), 29, avenue de la Division Leclerc, BP 72, 92322 Chaˆtillon-sous-Bagneux, France
Abstract The interaction of an acoustic wave with a turbulent flame and its consequences are still not well understood at present time. In the present paper the interaction of a syngas turbulent premixed flame with an acoustic wave is considered. At the difference of previous publications, a realistic, sinusoidal acoustic wave is considered for several periods. The interaction process is investigated by Direct Numerical Simulations (DNS) involving detailed physical models, so that an excellent accuracy is obtained. The DNS computations are systematically carried out twice, once with and once without adding an acoustic wave at the inlet, all other conditions being fixed. By a difference between both results, it is possible to quantify the interaction process. Local amplification, damping and structural modifications of the initially planar acoustic wave can be identified and analyzed. Moreover, a possible influence of the direction of propagation (acoustic wave reaching the flame from the fresh or from the burnt gas) has been investigated. Since a detailed reaction scheme is employed, the effect of each individual species on the wave amplification or damping can be quantified through the local Rayleigh’s criterion. It can be proved that the direction of propagation has no influence, confirming that amplification or damping is in the mean mainly controlled by the coupling process between pressure and heat release fluctuations through the chemical reactions. For the present case, species CO2, H and H2O mostly control wave amplification, while species O, OH and CO dominate damping. Two different reaction mechanisms have been employed and deliver almost identical results. The local version of the classical Rayleigh’s criterion presented in previous works is extended to take into account flame movement, which is now noticeable. Nevertheless, the results prove that the variation of flame position with time has a negligible influence on the coupling mechanism. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Flame; Acoustic; Rayleigh’s criterion; Combustion instabilities; DNS
1. Introduction The control of combustion instabilities is a fundamental and practical problem of great *
Corresponding author. Fax: +49 391 67 12 840. E-mail address: [email protected] (D. The´venin).
importance [1,2]. To understand the birth and development of such instabilities, the coupling between pressure and heat release fluctuations is essential. The interaction of a well-defined acoustic wave with a turbulent flame, leading to amplification or damping, is an important basic element of this coupling process. This phenomenon has first been explained qualitatively by Lord
1540-7489/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2008.06.202
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Rayleigh using a global analysis [3,4] and has been the subject of numerous subsequent investigations (e.g. [5–7]), relying mostly on theoretical considerations or experimental measurements. Alternative or extended forms of the stability criterion have also been proposed [8,9]. At the difference of all these global, system-based analysis, the present paper will consider local conditions associated with amplification or damping of the acoustic wave. As a consequence, it is not possible to analyze in this manner global stability conditions, since it is not even known if the geometry is open or closed. Nevertheless, essential physical information can be obtained and analyzed to understand the local coupling process between pressure and heat release fluctuations. As a complementary source of information, numerical simulations are very important, but require high-fidelity models and high-accuracy methods to reproduce quantitatively all important physical processes controlling this configuration. Direct Numerical Simulations (DNS) relying on detailed physical models are best suited for this purpose, but lead to considerable requirements in terms of computing time. As a consequence, most published studies use a simplified representation of chemical and transport processes. Such theoretical and numerical investigations are of high importance to delineate between different regimes and understand qualitatively the coupling process (see for example [10–14] and the included references), but cannot for example examine the effect of individual species or reactions. Previous studies relying on a realistic description of chemical processes have been restricted to laminar, one-dimensional configurations [15]. After deriving a specific version of the classical Rayleigh’s criterion, allowing to investigate local amplification or damping of a single Gaussian acoustic pulse interacting with a reaction front [16], extensive investigations of flame/acoustic interactions have been carried out in our group using DNS combined with accurate physical models. Premixed [16–18] as well as non-premixed [19] flames have been considered, using different fuels (hydrogen [16,17,19] and syngas [18]). All these studies have already delivered essential information in order to understand the local coupling between laminar and turbulent flames and acoustic waves. The corresponding results have been employed to check and consolidate Computational Aero-Acoustics models, particularly relying on the Acoustic Perturbation Equations (APE) [20]. Nevertheless, all these previous investigations suffer from a major limitation: they have considered a flame interacting with a single, isolated Gaussian acoustic pulse. It is therefore a priori questionable how general and applicable the obtained results might be for real acoustic waves.
In order to check this issue, the present study considers realistic, sinusoidal acoustic waves interacting with the flame over several periods. As in the previous studies the solution is obtained by Direct Numerical Simulations involving complete reaction schemes and multicomponent diffusion transport so that an excellent accuracy is ensured. The compressible DNS code parcomb is again employed for this purpose. In the previous studies the duration of the interaction were very small (single pulse). As a consequence the local version of the classical Rayleigh’s criterion developed in [16] was based on the assumption that most variables would vary spatially, but not in time: the flame was considered to be practically frozen in space during wave propagation, as indeed observed in the DNS results. For a quantitative analysis it is sufficient to introduce a few classical parameters averaged over the computational box: the speed of sound c, the flame thickness d and the flame speed S. For order of magnitude estimation, laminar quantities can be considered in the present case for flame thickness and flame speed, since DNS are only practicable for relatively low Reynolds numbers. In all previous investigations, the considered Gaussian pulse was much thinner than the corresponding flame thickness (high equivalent frequency). The interaction time is then simply d=c. During this time the relative flame displacement is D ¼ ðS d=cÞ= d ¼ S=c. The flame Mach number [21] appearing here is extremely small in all our previous studies so that the flame movement could indeed be considered as negligible during the interaction, as observed a posteriori in the DNS results. For the case considered now a realistic acoustic wave involving n periods at a frequency f will be considered. The spatial extension of the pressure wave is then much larger than the mean flame thickness. The interaction time becomes cn=ðf cÞ ¼ n=f and the corresponding relative flame displacement is D ¼ Sn=ðf dÞ. When considering several periods at a not exceedingly high frequency, D becomes non-negligible. In that case it becomes necessary to extend the previously developed, local Rayleigh’s criterion, in order to take into account the changes observed in time, and not only in space. This is the subject of the next section. Afterwards, DNS computations of turbulent premixed flames are carried out both with and without adding an acoustic wave, using the same code with exactly the same conditions. By a simple difference for each variable, the information characterizing wave/flame interaction can be obtained in a local, instantaneous manner. If needed, the local values can be averaged over the computational domain to deliver global information. This will be used in particular to investigate the influence of each individual chemical species on amplification or damping.
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2. Extended local Rayleigh’s criterion The developments in this section build on top of the analysis presented in [16] and the same notations are employed as far as possible. Nevertheless, all variables (for instance the speed of sound c) are now free to vary in space and time, (i.e. cðx; tÞ), and not only in space, as in [16]. Let us consider as a starting point the reduced non-conservative balance equations for momentum and energy, used classically for acoustic considerations at low Mach number M, neglecting all dissipative terms [6]: ov þ rp ¼ 0 ot ! N X 1 op þ r v ¼ ðc 1Þ hi w_ i cp ot i¼1
q
ð1Þ ð2Þ
Since c2 ¼ cp=q the second term in (1) can be rewritten after dividing by the density: 1 c2 c2 rp ¼ rp ¼ r log p q cp c
ð3Þ
In the following analysis each variable will be decomposed as a sum between the unperturbed value, written ðÞ0 , plus a fluctuation, written ðÞ0 . The pressure variation is classically considered through the variable P ¼ 1=c logðp=p0 Þ [16]. Since the pressure jump through the unperturbed premixed flame is negligibly small compared to the later acoustic perturbation, it may be safely assumed for the present developments that r log p ¼ r logðp=p0 Þ. Eq. (3) can then be rewritten: 1 c2 p rp ¼ r log ¼ c2 rP þ c2 Pr log c ð4Þ q p0 c A similar relation is found for the time derivative, so that we can write: oP o log c 1 o p þP ¼ log ð5Þ ot ot c ot p0 By substitution in Eqs. (1) and (2), one thus obtains for the momentum and energy equations: 1 ov þ rP ¼ Pr log c ð6Þ c2 ot ! N X oP o log c þ r v ¼ ðc 1Þ hi w_ i P ot ot i¼1 ð7Þ Multypling (6) by v, (7) by P and building the sum, one obtains: 1 o v2 o P2 þ þ v rP þ Pr v 2 c ot 2 ot 2 ! N X ¼ Pv rlog c þ ðc 1ÞP hi w_ i i¼1
P
2 olog c
ot
ð8Þ
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Defining in a classical manner [16] the acoustic energy E ¼ 1=2ðv2 =c2 þ P2 Þ and its flux F ¼ vP, it becomes possible to recognize in (8) a transport equation for the acoustic energy E, using: 1 o v2 o v2 v2 o log c ð9Þ ¼ þ 2 2 2 c ot 2 ot 2c ot c One obtains finally: I
zfflfflfflfflfflffl}|fflfflfflfflfflffl{ II oE olog c zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ þ r F ¼ M2 Pv rlog c ot ot XN ologc þ ðc 1ÞP i¼1 hi w_ i P2 ot |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} III
ð10Þ
IV
It is now sufficient to consider the right-hand side of this equation in order to determine stability conditions. Since the present developments are only valid at low Mach numbers, term (I) disappears automatically. Term (III) corresponds to the classical Rayleigh criterion. Splitting all variables as explained previously, (III) becomes: ðc 1ÞPQ ’ ðc0 1ÞP0 Q0 þ c0 P0 Q0 ð11Þ PN introducing Q ¼ i¼1 hi w_ i , the heat release induced by the chemical reactions (source term in the temperature equation) [6]. The last term in Eq. (11) is associated to fluctuations of c. Terms (II) and (IV) again involve c and become at leading order P0 v0 r log c0 P0 v0 r log c0 for (II) and P02 o log c0 =ot for (IV). The final stability condition thus reads: ðc0 1ÞP0 Q0 þ c0 P0 Q0 P0 v0 r log c0 o log c0 <0 P0 v0 r log c0 P02 ot
ð12Þ
Finally, the local stability analysis presented in [16] is still valid (the first term in Eq. (12) is the same as in [16], Eq.(16)), but involves now a supplementary correction (further terms in Eq. (12)). The importance of this correction will be quantified in what follows. In principle the sign of the correction term could be either positive or negative and thus directly influence the stability of the process. An asymptotic analysis (not included here due to lack of space) shows that all correction terms should be indeed negligible at low Mach numbers. In that case the classical stability condition [3,4] is found again: the acoustic wave is amplified (locally) for P0 Q0 > 0 and damped for P0 Q0 < 0. The term Q0 can be directly computed as a function of the fluctuations of the mass production rate of any species, w_ i (species number i from a total of N); or at the level of each chemical reaction, considering the reaction production rate x_ ik (reaction number k from a total of R).
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Q0 ¼
N X
hi w_ 0i
i¼1
¼
R X k¼1
N X ðmrik mfik ÞW i hi x_ 0ik
ð13Þ ! ð14Þ
i¼1
As a consequence, the presented stability criterion can be used to investigate the influence of any individual reaction (choosing a constant value for k in Eq. 14) or of any individual species (keeping a constant value for i in Eq. 13). 3. Numerical methods and configuration All simulations analyzed in this work have been carried out with the DNS code parcomb already employed in previous studies of flame/acoustic interaction [16–19]. It is a finite-difference DNS code solving the compressible Navier–Stokes equations for multicomponent reacting flows. Derivatives are computed using centered explicit schemes of order six except at boundaries where the order is progressively reduced to four. Temporal integration is realized with a Runge–Kutta algorithm of order four. Boundary conditions are treated with the help of the Navier–Stokes Characteristic Boundary Condition (NSCBC) technique [22], extended to take into account multicomponent thermodynamic properties. Transport coefficients and chemical kinetics are treated following methods similar to those used in the standard packages CHEMKIN and TRANSPORT [23]. This DNS code has been parallelized and widely used to investigate turbulent combustion (see [24–26] and the relevant references listed in these publications). DNS computations involving complete reaction schemes and multicomponent diffusion models remain extremely demanding in terms of computing time and memory. As a consequence, three-dimensional DNS relying on realistic physical models are extremely difficult, as further documented in [26]. Systematic studies requiring several computations are thus usually carried out in two dimensions. This will of course lead to a falsified evolution of the turbulent properties since turbulence is an intrinsically three-dimensional phenomenon. Considering that turbulence is not an essential feature of the present study, and considering the needed duration of the computation in physical time, two-dimensional DNS computations are requested and are expected to lead, at least qualitatively, to the same results as 3D DNS. Thus, only two-dimensional results employing detailed chemistry and transport models are considered in what follows. The present study is restricted to premixed flames. Previous investigations have revealed noticeable quantitative differences but no major
qualitative difference for acoustic interactions with premixed or non-premixed flames. Most published studies have considered hydrogen flames. Even if this is quite interesting to understand the basic coupling process, combustion applications usually rely on hydrocarbons as fuel. In order to keep acceptable computing times and memory requirements while introducing some amount of carbon chemistry, a syngas (CO/ H2/Air) flame is considered in the present work. For most computations the detailed reaction scheme of [27], involving 13 species (CO, HCO, CH2O, CO2, H2O, O2, O, H, OH, HO2, H2O2, H2, N2) and 67 individual reactions, is taken into account. In order to check the possible influence of the reaction scheme, the mechanism presented in [28], involving again 13 species (including Ar but neglecting CH2O) and 70 individual reactions is also considered for comparison. The corresponding laminar premixed flame is first computed for a one-dimensional flow, adapting dynamically the inlet velocity to keep the flame front in the middle of the domain. Conditions correspond to a stoichiometric CO/O2 ratio with slight hydrogen enrichment, leading to a mixture equivalence ratio / = 1.12. The obtained steady solution is transposed to two dimensions, with fresh gases on the left side and burnt gases on the right side. The final numerical domain under investigation is a box of length 1.2 cm on each side. A fixed mesh of 355 equidistant points is employed in each direction, leading to a spatial resolution of 34 lm, necessary to resolve correctly not only the smallest vortical structures but also stiff species like CH2O or H2O2. The left-hand boundary condition is a subsonic inlet with imposed values, while the right-hand boundary condition is a non-reflecting subsonic outlet. Top and bottom boundaries are periodic. A homogeneous isotropic turbulent velocity field is then superposed on top of the initially planar laminar flame when restarting the twodimensional computation (defined as t = 0). The distribution of turbulent kinetic energy follows a von Ka´rma´n spectrum coupled with Pao correction at near-dissipation scales (see [25] for more details). For the results shown afterwards an initial turbulence field with a velocity fluctuation u0 = 1.62 m/s and an integral length scale lt = 1.08 mm is finally generated. Considering a value of the kinematic viscosity of m ¼ 1:62 105 m2 =s, the Reynolds number based on the integral scale and corresponding to the turbulence actually resolved on the numerical grid is Ret ¼ 109. The characteristic time scale of this turbulence is st ¼ lt =u0 ¼ 0:67 ms. In a next step the interaction of the initially planar flame with the turbulent field is computed by DNS during a time equal to one turbulent time st , in order to obtain an appropriate turbulent flame structure. The resulting final time is then
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retained as the new origin of time, t0 = 0. DNS computations are continued by restarting from t0 = 0, once with and once without generating a realistic pressure wave through the inflow (left) boundary condition. The initiation of the acoustic wave at the inflow is easily realized thanks to the intrinsically time-dependent analysis underlying NSCBC [22]. Both calculations (with and without generating the acoustic wave) are fully identical except for the wave generation. These two computations are continued up to a physical time corresponding to five periods of the imposed sine wave. By simply taking the difference between both computations at exactly the same point in time and space, it becomes possible to quantify the interaction between the acoustic wave and the turbulent flame. The numerical cost associated with these computations grows proportionally to the number of periods and as the inverse of the wave frequency since the spatial extension of the wave considered in the DNS is directly given as nc=f . Five periods have been considered in the present study. An acceptable computing time on the employed Linux PC-cluster with 16 nodes is then obtained for frequencies of several kHz. In what follows, f = 5 kHz has been retained. This is clearly a high frequency but is still relevant for practical applications, in particular high-frequency oscillations in gas turbines or rocket engines (see for instance [29–31]). It is much lower than the equivalent frequency of the single pulse considered in [16–19]. For these conditions the relative displacement of the laminar flame is D ¼ Sn=ðf dÞ ’ 0:7 5=ð5000 0:0005Þ > 1. The movement of the laminar flame is thus already non-negligible, while turbulent flow structures will locally lead to an even larger displacement. Therefore, it is now impossible to consider a frozen flame in space. The variation of the variables with time at a given grid point should be considered, as done next.
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Fig. 1. Instantaneous value of the correction term appearing in Eq. (12) at t0 = 0.5 ms (third period). See also Figs. 2–4.
remains during all the interaction process of order 10. Since the order of magnitude of the first term within the local Rayleigh’s criterion presented in Eq. (12) is typically 1 000 (see for example later Fig. 4), there is roughly a factor 100 between the first (standard) term and the correction term appearing in Eq. (12). As a consequence the influence of flame movement can be safely neglected for such configurations. 4.2. Amplification and damping of the wave Considering now only the first term in Eq. (12), the local amplification and damping of the incoming acoustic wave can be quantified. For this purpose both the acoustically-induced pressure fluctuation P0 (Fig. 2) and the acoustically-induced
4. Results and discussion 4.1. Influence of the correction term For the present configuration, the maximum Mach number observed within the numerical domain is M max ¼ 0:022. As a consequence, compressibility effects will be small and the influence of the correction term in Eq. (12) is expected to be negligible as stated previously. This can be quantified by computing directly this correction term since all terms involved are known from the DNS. Corresponding results are shown in Fig. 1 for t0 = 0.5 ms (third period). As expected, the correction term is only active in the immediate vicinity of the instantaneous flame front since it describes the influence of flame movement. The magnitude of the correction term
Fig. 2. Acoustically-induced pressure fluctuation P0 at t0 = 0.5 ms (third period), shown with dashed lines. The instantaneous position of the flame front is shown with isolevels of heat release (black solid lines). See also Figs. 3 and 4.
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Fig. 3. Acoustically-induced heat release fluctuation Q0 at t0 = 0.5 ms (third period). See also Figs. 2 and 4.
Fig. 4. Rayleigh’s criterion at t0 = 0.5 ms (third period). See also Figs. 2 and 3.
heat release fluctuation Q0 (Fig. 3) are extracted from the DNS computations as explained before by a simple difference between both results corresponding to the same time. In Fig. 2 the instantaneous position of the flame front is represented by black solid lines. As in all previous studies, the amplitude of the acoustically-induced pressure fluctuation is small (Fig. 2), well below 0.1% of the baseline pressure. However, this still means a strong acoustic perturbation, with an order of magnitude of 100 Pa or 130 dB. In the present work, at the difference of the interaction considering an isolated Gaussian pulse [16–19], P0 is now distributed over the full computational domain, which seems much more realistic. Depending on physical time, conditions are found for which P0 is everywhere positive, everywhere negative (see for instance Fig. 2), or changes sign over the numerical domain. The acoustically-induced heat release fluctuation (Fig. 3) is of course only noticeable within the reaction zone and is thus limited to a very thin region, confirming previous studies involving premixed flames [16–18]. The local Rayleigh’s criterion can now directly be determined as ðc0 1ÞP0 Q0 (Fig. 4) and is thus proportional to the ‘‘product” of both previous figures. Confirming previous investigations using a Gaussian pulse, amplification and damping of the flame is limited to the active reaction zone since Q0 ’ 0 outside of this domain. Moreover, amplification and damping are very localized phenomena and occur in well-defined layers, while a large part of the flame front appears to play a negligible role. As already observed in previous studies, amplification and damping regions often appear as neighboring, sandwich layers. As a consequence, a possible influence of the direction of wave propagation must be checked, as presented next. Of course the presented Rayleigh’s criterion is time-dependent so that a different picture is
obtained at every time-step. Nevertheless, the qualitative features described previously are not modified during all the computation and even quantitative changes are of limited importance. This is illustrated in Fig. 5, corresponding to the fourth period. All previous observations still hold. 4.3. Influence of the direction of wave propagation As explained in the previous section regions leading to maximal amplification and damping often appear as neighboring, sandwich layers. It is therefore of interest to check the physical meaning of this observation in order to exclude the possibility of some numerical artifact. This has been already done in a previous publication for a laminar hydrogen non-premixed flame [19]. The case of a turbulent syngas premixed flame is now considered. An identical Gaussian wave is initiated once on the fresh gas side (left) and prop-
Fig. 5. Rayleigh’s criterion at t0 = 0.65 ms (fourth period). Compare to Fig. 4.
H. Shalaby et al. / Proceedings of the Combustion Institute 32 (2009) 1473–1480
agates to the right. In a second computation, the same wave (negative pulse) is initiated within the burnt gas and propagates towards the left. Thus, both identical waves (apart from their starting position) cross an initially identical turbulent syngas flame. Figure 6 shows the local Rayleigh’s criterion for a time corresponding to the same mean position of the wave. A direct comparison shows of course some quantitative differences, both in the localization and in the amplitude of the amplification and damping regions. This is unavoidable since the modification of the wave geometry depends on the gradients of acoustic impedance [19]; the corresponding field is of course not symmetrical. As a consequence, the instantaneous geometry of the wave during the interaction process is not identical in both cases. Nevertheless, both results are very similar, at least qualitatively. Furthermore, the double layers associated with amplification and damping are observed in both cases. It is therefore clear that these layers have a physical explanation and are not the result of a numerical problem, as already
Fig. 6. Instantaneous Rayleigh’s criterion for an identical wave coming from: top: the fresh gas, i.e. the left side or; bottom: the burnt gas, i.e. the right side of the numerical domain.
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demonstrated for a laminar non-premixed configuration [19]. These layers are always connected with the heat release fluctuation Q0 and correspond to opposite variations of heat release on both sides of the active reaction zone, leading to opposite effects regarding stability conditions. It is interesting to see that this effect is also found for the complex geometry associated with a turbulent flame, and not only when considering laminar, one-dimensional conditions, as in [19]. 4.4. Species leading to amplification or damping The local Rayleigh’s criterion is finally used to investigate the isolated influence of each species on wave modification. This can easily be done by considering Eq. (12) and keeping a constant value for index i (species index) in Eq. (13) in order to compute Q0 . The influence of the species is only contained in the first term of Eq. (12). In this manner, the influence of each species on amplification or damping is obtained as a function of space and time. In what follows the resulting influence is averaged over the complete numerical domain in order to get the global influence of each species (Fig. 7). Three different results are presented in this figure on which the influence of the most important species has always been normalized to 1 in order to facilitate comparisons. First, the direction of wave propagation is not important when determining if a species leads to amplification or damping, supporting the original, global analysis of Lord Rayleigh. Both results (wave originating from the fresh gas: squares; or from the burnt gas: circles) are almost identical, confirming also the results presented in the previous section. The only noticeable difference is observed for species CO2. Second, the employed kinetic scheme does not modify these findings. In order to check this issue, the same computation (wave starting in the fresh gas) has been repeated twice using two
Fig. 7. Normalized contribution of each species to Rayleigh’s criterion, averaged over the full numerical domain. Wave coming from the fresh gas, i.e. the left side (squares) or from the burnt gas, i.e. the right side of the numerical domain (circles), with both using the chemical scheme of [27]. Filled triangles: wave coming from the fresh gas using the chemical scheme of [28].
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different chemical schemes ([27]: squares; [28]: filled triangles). Once again, the two schemes lead qualitatively to the same findings and show only limited quantitative differences. Finally, the influence of the different species on amplification and damping is clear for all conditions: species CO2, H and H2O mostly control wave amplification, while species O, OH and CO dominate damping. As expected, short-lived radicals like HCO, CH2O, HO2, H2O2 and neutral species (N2, Ar) have a negligible influence on the process. 5. Conclusions and perspectives The interaction of a realistic acoustic wave with a turbulent premixed syngas flame has been investigated using Direct Numerical Simulations relying on detailed models for chemistry and diffusive transport. Amplification and damping occur only in well-defined layers within the active reaction zone, very often with a double-layer, sandwich structure. This observation is independent of the direction of wave propagation and has therefore a physical explanation. Such layers are always caused by the heat release fluctuation Q0 and correspond to opposite variations of heat release on both sides of the active reaction zone, leading to opposite effects regarding stability conditions. The global amplification or damping of the initial acoustic wave is therefore a complex average over contradictory effects. Two mechanisms control the interaction process: (1) the influence of the gradient of acoustic impedance, which is not symmetric with respect to the fresh or burnt gas but will globally play a minor role since both curvatures are found with a similar probability in highly turbulent flames, thus leading to large canceling effects; (2) coupling with the chemical reactions, as described originally by Lord Rayleigh. This last effect will play a dominant role in explaining wave amplification and damping, since it is almost independent of flame orientation or wave propagation direction. This is the most important coupling process with acoustics in laminar flames. For the present configuration species CO2, H and H2O mostly control wave amplification, while species O, OH and CO dominate damping. Short-lived radicals and neutral species have a negligible influence. This suggests new possibilities to enhance flame stability, either by modifying slightly the burning regime in order to shift the local composition towards more favourable conditions; or by injecting additional species, leading to damping.
Acknowledgments This work has been financially supported by the Deutsche Forschungsgemeinschaft in the frame of the Research Unit 486 ‘‘Combustion Noise”.
References [1] S. Candel, Proc. Combust. Inst. 29 (2002) 1–28. [2] S. Ducruix, T. Schuller, D. Durox, S. Candel, J. Propul. Power 19 (5) (2003) 722–734. [3] J.W.S. Rayleigh, Nature 18 (1878). [4] J.W.S. Rayleigh, The Theory of Sound, vol. II, Dover, New York, 1945. [5] G.H. Markstein (Ed.), Nonsteady Flame Propagation, Pergamon Press, Paris, 1964. [6] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, Edwards Publishing, Pennsylvania, 2001. [7] T. Lieuwen, J. Propul. Power 19 (2003) 765–781. [8] B.T. Chu, Acta Mech. (1965) 215–234. [9] F. Nicoud, T. Poinsot, Combust. Flame 142 (2005) 153–159. [10] A.C. McIntosh, Philos. Trans. R. Soc. Lond. A 357 (1999) 3523–3538. [11] T. Lieuwen, J. Fluid Mech. 435 (2001) 289–303. [12] S.H. Preetham, T.C. Lieuwen, Proc. Combust. Inst. 31 (2007) 1427–1434. [13] C. Pera, J. Re´veillon, Proc. Combust. Inst. 31 (2007) 2283–2290. [14] J. Teerling, A.C. McIntosh, J. Brindley, Combust. Theory Model. 11 (1) (2007) 147–164. [15] K. Prasad, Combust. Flame 97 (1994) 173–200. [16] A. Laverdant, D. The´venin, Combust. Flame 134 (1–2) (2003) 11–19. [17] A. Laverdant, D. The´venin, C.R. Me´canique 333 (2005) 29–37. [18] H. Shalaby, A. Laverdant, D. The´venin, in: Proceedings of the 12th International Congress on Sound and Vibration, Lisbon, Portugal, 2005, pp. 635/1–635/8. [19] A. Laverdant, L. Gouarin, D. The´venin, Combust. Theory Model. 11 (4) (2007) 585–602. [20] T. Phong Bui, M. Meinke, W. Schro¨der, F. Flemming, A. Sadiki, J. Janicka, in: Proceedings of the 11th AIAA Aeroacoustics Conference, AIAA Paper 2005-3014, 2005. [21] V. Smiljanovski, V. Moser, R. Klein, Combust. Theory Model. 1 (1997) 183–215. [22] T. Poinsot, S.K. Lele, J. Comput. Phys. 101 (1992) 104–129. [23] U. Maas, J. Warnatz, Impact Comput. Sci. Eng. 1 (1989) 394–420. [24] M. Lange, U. Riedel, J. Warnatz, in: Proceedings of the 29th AIAA Fluid Dynamics Conference, AIAA, Albuquerque, AIAA-98-2979, 1998, pp. 1–8. [25] R. Hilbert, D. The´venin, Combust. Flame 128 (1–2) (2002) 22–37. [26] R. Hilbert, F. Tap, H. El-Rabii, D. The´venin, Prog. Energy Combust. Sci. 30 (2004) 61–117. [27] U. Maas, S.B. Pope, Combust. Flame 88 (1992) 239– 264. [28] R.A. Yetter, F.L. Dryer, H. Rabitz, Combust. Sci. Technol. 79 (1991) 97–128. [29] R. Ebrahimi, K. Mazaheri, A. Ghafourian, J. Sound Vib. 229 (4) (2000) 973–991. [30] C.O. Paschereit, G. Mungal, ASME Turbo Expo, Vienna, ASME Paper 2004-GT-53587, 2004. [31] F. Richecoeur, S. Ducruix, P. Scouflaire, S. Candel, in: Proceedings of the 56th International Astronautical Congress, Fukuoka, Japan, 2005.
Proceedings of the
Combustion Institute
Proceedings of the Combustion Institute 32 (2009) 1473–1480
www.elsevier.com/locate/proci
Direct numerical simulation of a realistic acoustic wave interacting with a premixed flame H. Shalaby a, A. Laverdant b, D. The´venin a,* a
Laboratory of Fluid Dynamics and Technical Flows, University of Magdeburg ‘‘Otto von Guericke”, Universita¨tsplatz 2, D-39106 Magdeburg, Germany b Office National d’E´tudes et de Recherches Ae´rospatiales (ONERA), 29, avenue de la Division Leclerc, BP 72, 92322 Chaˆtillon-sous-Bagneux, France
Abstract The interaction of an acoustic wave with a turbulent flame and its consequences are still not well understood at present time. In the present paper the interaction of a syngas turbulent premixed flame with an acoustic wave is considered. At the difference of previous publications, a realistic, sinusoidal acoustic wave is considered for several periods. The interaction process is investigated by Direct Numerical Simulations (DNS) involving detailed physical models, so that an excellent accuracy is obtained. The DNS computations are systematically carried out twice, once with and once without adding an acoustic wave at the inlet, all other conditions being fixed. By a difference between both results, it is possible to quantify the interaction process. Local amplification, damping and structural modifications of the initially planar acoustic wave can be identified and analyzed. Moreover, a possible influence of the direction of propagation (acoustic wave reaching the flame from the fresh or from the burnt gas) has been investigated. Since a detailed reaction scheme is employed, the effect of each individual species on the wave amplification or damping can be quantified through the local Rayleigh’s criterion. It can be proved that the direction of propagation has no influence, confirming that amplification or damping is in the mean mainly controlled by the coupling process between pressure and heat release fluctuations through the chemical reactions. For the present case, species CO2, H and H2O mostly control wave amplification, while species O, OH and CO dominate damping. Two different reaction mechanisms have been employed and deliver almost identical results. The local version of the classical Rayleigh’s criterion presented in previous works is extended to take into account flame movement, which is now noticeable. Nevertheless, the results prove that the variation of flame position with time has a negligible influence on the coupling mechanism. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Flame; Acoustic; Rayleigh’s criterion; Combustion instabilities; DNS
1. Introduction The control of combustion instabilities is a fundamental and practical problem of great *
Corresponding author. Fax: +49 391 67 12 840. E-mail address: [email protected] (D. The´venin).
importance [1,2]. To understand the birth and development of such instabilities, the coupling between pressure and heat release fluctuations is essential. The interaction of a well-defined acoustic wave with a turbulent flame, leading to amplification or damping, is an important basic element of this coupling process. This phenomenon has first been explained qualitatively by Lord
1540-7489/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2008.06.202
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Rayleigh using a global analysis [3,4] and has been the subject of numerous subsequent investigations (e.g. [5–7]), relying mostly on theoretical considerations or experimental measurements. Alternative or extended forms of the stability criterion have also been proposed [8,9]. At the difference of all these global, system-based analysis, the present paper will consider local conditions associated with amplification or damping of the acoustic wave. As a consequence, it is not possible to analyze in this manner global stability conditions, since it is not even known if the geometry is open or closed. Nevertheless, essential physical information can be obtained and analyzed to understand the local coupling process between pressure and heat release fluctuations. As a complementary source of information, numerical simulations are very important, but require high-fidelity models and high-accuracy methods to reproduce quantitatively all important physical processes controlling this configuration. Direct Numerical Simulations (DNS) relying on detailed physical models are best suited for this purpose, but lead to considerable requirements in terms of computing time. As a consequence, most published studies use a simplified representation of chemical and transport processes. Such theoretical and numerical investigations are of high importance to delineate between different regimes and understand qualitatively the coupling process (see for example [10–14] and the included references), but cannot for example examine the effect of individual species or reactions. Previous studies relying on a realistic description of chemical processes have been restricted to laminar, one-dimensional configurations [15]. After deriving a specific version of the classical Rayleigh’s criterion, allowing to investigate local amplification or damping of a single Gaussian acoustic pulse interacting with a reaction front [16], extensive investigations of flame/acoustic interactions have been carried out in our group using DNS combined with accurate physical models. Premixed [16–18] as well as non-premixed [19] flames have been considered, using different fuels (hydrogen [16,17,19] and syngas [18]). All these studies have already delivered essential information in order to understand the local coupling between laminar and turbulent flames and acoustic waves. The corresponding results have been employed to check and consolidate Computational Aero-Acoustics models, particularly relying on the Acoustic Perturbation Equations (APE) [20]. Nevertheless, all these previous investigations suffer from a major limitation: they have considered a flame interacting with a single, isolated Gaussian acoustic pulse. It is therefore a priori questionable how general and applicable the obtained results might be for real acoustic waves.
In order to check this issue, the present study considers realistic, sinusoidal acoustic waves interacting with the flame over several periods. As in the previous studies the solution is obtained by Direct Numerical Simulations involving complete reaction schemes and multicomponent diffusion transport so that an excellent accuracy is ensured. The compressible DNS code parcomb is again employed for this purpose. In the previous studies the duration of the interaction were very small (single pulse). As a consequence the local version of the classical Rayleigh’s criterion developed in [16] was based on the assumption that most variables would vary spatially, but not in time: the flame was considered to be practically frozen in space during wave propagation, as indeed observed in the DNS results. For a quantitative analysis it is sufficient to introduce a few classical parameters averaged over the computational box: the speed of sound c, the flame thickness d and the flame speed S. For order of magnitude estimation, laminar quantities can be considered in the present case for flame thickness and flame speed, since DNS are only practicable for relatively low Reynolds numbers. In all previous investigations, the considered Gaussian pulse was much thinner than the corresponding flame thickness (high equivalent frequency). The interaction time is then simply d=c. During this time the relative flame displacement is D ¼ ðS d=cÞ= d ¼ S=c. The flame Mach number [21] appearing here is extremely small in all our previous studies so that the flame movement could indeed be considered as negligible during the interaction, as observed a posteriori in the DNS results. For the case considered now a realistic acoustic wave involving n periods at a frequency f will be considered. The spatial extension of the pressure wave is then much larger than the mean flame thickness. The interaction time becomes cn=ðf cÞ ¼ n=f and the corresponding relative flame displacement is D ¼ Sn=ðf dÞ. When considering several periods at a not exceedingly high frequency, D becomes non-negligible. In that case it becomes necessary to extend the previously developed, local Rayleigh’s criterion, in order to take into account the changes observed in time, and not only in space. This is the subject of the next section. Afterwards, DNS computations of turbulent premixed flames are carried out both with and without adding an acoustic wave, using the same code with exactly the same conditions. By a simple difference for each variable, the information characterizing wave/flame interaction can be obtained in a local, instantaneous manner. If needed, the local values can be averaged over the computational domain to deliver global information. This will be used in particular to investigate the influence of each individual chemical species on amplification or damping.
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2. Extended local Rayleigh’s criterion The developments in this section build on top of the analysis presented in [16] and the same notations are employed as far as possible. Nevertheless, all variables (for instance the speed of sound c) are now free to vary in space and time, (i.e. cðx; tÞ), and not only in space, as in [16]. Let us consider as a starting point the reduced non-conservative balance equations for momentum and energy, used classically for acoustic considerations at low Mach number M, neglecting all dissipative terms [6]: ov þ rp ¼ 0 ot ! N X 1 op þ r v ¼ ðc 1Þ hi w_ i cp ot i¼1
q
ð1Þ ð2Þ
Since c2 ¼ cp=q the second term in (1) can be rewritten after dividing by the density: 1 c2 c2 rp ¼ rp ¼ r log p q cp c
ð3Þ
In the following analysis each variable will be decomposed as a sum between the unperturbed value, written ðÞ0 , plus a fluctuation, written ðÞ0 . The pressure variation is classically considered through the variable P ¼ 1=c logðp=p0 Þ [16]. Since the pressure jump through the unperturbed premixed flame is negligibly small compared to the later acoustic perturbation, it may be safely assumed for the present developments that r log p ¼ r logðp=p0 Þ. Eq. (3) can then be rewritten: 1 c2 p rp ¼ r log ¼ c2 rP þ c2 Pr log c ð4Þ q p0 c A similar relation is found for the time derivative, so that we can write: oP o log c 1 o p þP ¼ log ð5Þ ot ot c ot p0 By substitution in Eqs. (1) and (2), one thus obtains for the momentum and energy equations: 1 ov þ rP ¼ Pr log c ð6Þ c2 ot ! N X oP o log c þ r v ¼ ðc 1Þ hi w_ i P ot ot i¼1 ð7Þ Multypling (6) by v, (7) by P and building the sum, one obtains: 1 o v2 o P2 þ þ v rP þ Pr v 2 c ot 2 ot 2 ! N X ¼ Pv rlog c þ ðc 1ÞP hi w_ i i¼1
P
2 olog c
ot
ð8Þ
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Defining in a classical manner [16] the acoustic energy E ¼ 1=2ðv2 =c2 þ P2 Þ and its flux F ¼ vP, it becomes possible to recognize in (8) a transport equation for the acoustic energy E, using: 1 o v2 o v2 v2 o log c ð9Þ ¼ þ 2 2 2 c ot 2 ot 2c ot c One obtains finally: I
zfflfflfflfflfflffl}|fflfflfflfflfflffl{ II oE olog c zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ þ r F ¼ M2 Pv rlog c ot ot XN ologc þ ðc 1ÞP i¼1 hi w_ i P2 ot |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} III
ð10Þ
IV
It is now sufficient to consider the right-hand side of this equation in order to determine stability conditions. Since the present developments are only valid at low Mach numbers, term (I) disappears automatically. Term (III) corresponds to the classical Rayleigh criterion. Splitting all variables as explained previously, (III) becomes: ðc 1ÞPQ ’ ðc0 1ÞP0 Q0 þ c0 P0 Q0 ð11Þ PN introducing Q ¼ i¼1 hi w_ i , the heat release induced by the chemical reactions (source term in the temperature equation) [6]. The last term in Eq. (11) is associated to fluctuations of c. Terms (II) and (IV) again involve c and become at leading order P0 v0 r log c0 P0 v0 r log c0 for (II) and P02 o log c0 =ot for (IV). The final stability condition thus reads: ðc0 1ÞP0 Q0 þ c0 P0 Q0 P0 v0 r log c0 o log c0 <0 P0 v0 r log c0 P02 ot
ð12Þ
Finally, the local stability analysis presented in [16] is still valid (the first term in Eq. (12) is the same as in [16], Eq.(16)), but involves now a supplementary correction (further terms in Eq. (12)). The importance of this correction will be quantified in what follows. In principle the sign of the correction term could be either positive or negative and thus directly influence the stability of the process. An asymptotic analysis (not included here due to lack of space) shows that all correction terms should be indeed negligible at low Mach numbers. In that case the classical stability condition [3,4] is found again: the acoustic wave is amplified (locally) for P0 Q0 > 0 and damped for P0 Q0 < 0. The term Q0 can be directly computed as a function of the fluctuations of the mass production rate of any species, w_ i (species number i from a total of N); or at the level of each chemical reaction, considering the reaction production rate x_ ik (reaction number k from a total of R).
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Q0 ¼
N X
hi w_ 0i
i¼1
¼
R X k¼1
N X ðmrik mfik ÞW i hi x_ 0ik
ð13Þ ! ð14Þ
i¼1
As a consequence, the presented stability criterion can be used to investigate the influence of any individual reaction (choosing a constant value for k in Eq. 14) or of any individual species (keeping a constant value for i in Eq. 13). 3. Numerical methods and configuration All simulations analyzed in this work have been carried out with the DNS code parcomb already employed in previous studies of flame/acoustic interaction [16–19]. It is a finite-difference DNS code solving the compressible Navier–Stokes equations for multicomponent reacting flows. Derivatives are computed using centered explicit schemes of order six except at boundaries where the order is progressively reduced to four. Temporal integration is realized with a Runge–Kutta algorithm of order four. Boundary conditions are treated with the help of the Navier–Stokes Characteristic Boundary Condition (NSCBC) technique [22], extended to take into account multicomponent thermodynamic properties. Transport coefficients and chemical kinetics are treated following methods similar to those used in the standard packages CHEMKIN and TRANSPORT [23]. This DNS code has been parallelized and widely used to investigate turbulent combustion (see [24–26] and the relevant references listed in these publications). DNS computations involving complete reaction schemes and multicomponent diffusion models remain extremely demanding in terms of computing time and memory. As a consequence, three-dimensional DNS relying on realistic physical models are extremely difficult, as further documented in [26]. Systematic studies requiring several computations are thus usually carried out in two dimensions. This will of course lead to a falsified evolution of the turbulent properties since turbulence is an intrinsically three-dimensional phenomenon. Considering that turbulence is not an essential feature of the present study, and considering the needed duration of the computation in physical time, two-dimensional DNS computations are requested and are expected to lead, at least qualitatively, to the same results as 3D DNS. Thus, only two-dimensional results employing detailed chemistry and transport models are considered in what follows. The present study is restricted to premixed flames. Previous investigations have revealed noticeable quantitative differences but no major
qualitative difference for acoustic interactions with premixed or non-premixed flames. Most published studies have considered hydrogen flames. Even if this is quite interesting to understand the basic coupling process, combustion applications usually rely on hydrocarbons as fuel. In order to keep acceptable computing times and memory requirements while introducing some amount of carbon chemistry, a syngas (CO/ H2/Air) flame is considered in the present work. For most computations the detailed reaction scheme of [27], involving 13 species (CO, HCO, CH2O, CO2, H2O, O2, O, H, OH, HO2, H2O2, H2, N2) and 67 individual reactions, is taken into account. In order to check the possible influence of the reaction scheme, the mechanism presented in [28], involving again 13 species (including Ar but neglecting CH2O) and 70 individual reactions is also considered for comparison. The corresponding laminar premixed flame is first computed for a one-dimensional flow, adapting dynamically the inlet velocity to keep the flame front in the middle of the domain. Conditions correspond to a stoichiometric CO/O2 ratio with slight hydrogen enrichment, leading to a mixture equivalence ratio / = 1.12. The obtained steady solution is transposed to two dimensions, with fresh gases on the left side and burnt gases on the right side. The final numerical domain under investigation is a box of length 1.2 cm on each side. A fixed mesh of 355 equidistant points is employed in each direction, leading to a spatial resolution of 34 lm, necessary to resolve correctly not only the smallest vortical structures but also stiff species like CH2O or H2O2. The left-hand boundary condition is a subsonic inlet with imposed values, while the right-hand boundary condition is a non-reflecting subsonic outlet. Top and bottom boundaries are periodic. A homogeneous isotropic turbulent velocity field is then superposed on top of the initially planar laminar flame when restarting the twodimensional computation (defined as t = 0). The distribution of turbulent kinetic energy follows a von Ka´rma´n spectrum coupled with Pao correction at near-dissipation scales (see [25] for more details). For the results shown afterwards an initial turbulence field with a velocity fluctuation u0 = 1.62 m/s and an integral length scale lt = 1.08 mm is finally generated. Considering a value of the kinematic viscosity of m ¼ 1:62 105 m2 =s, the Reynolds number based on the integral scale and corresponding to the turbulence actually resolved on the numerical grid is Ret ¼ 109. The characteristic time scale of this turbulence is st ¼ lt =u0 ¼ 0:67 ms. In a next step the interaction of the initially planar flame with the turbulent field is computed by DNS during a time equal to one turbulent time st , in order to obtain an appropriate turbulent flame structure. The resulting final time is then
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retained as the new origin of time, t0 = 0. DNS computations are continued by restarting from t0 = 0, once with and once without generating a realistic pressure wave through the inflow (left) boundary condition. The initiation of the acoustic wave at the inflow is easily realized thanks to the intrinsically time-dependent analysis underlying NSCBC [22]. Both calculations (with and without generating the acoustic wave) are fully identical except for the wave generation. These two computations are continued up to a physical time corresponding to five periods of the imposed sine wave. By simply taking the difference between both computations at exactly the same point in time and space, it becomes possible to quantify the interaction between the acoustic wave and the turbulent flame. The numerical cost associated with these computations grows proportionally to the number of periods and as the inverse of the wave frequency since the spatial extension of the wave considered in the DNS is directly given as nc=f . Five periods have been considered in the present study. An acceptable computing time on the employed Linux PC-cluster with 16 nodes is then obtained for frequencies of several kHz. In what follows, f = 5 kHz has been retained. This is clearly a high frequency but is still relevant for practical applications, in particular high-frequency oscillations in gas turbines or rocket engines (see for instance [29–31]). It is much lower than the equivalent frequency of the single pulse considered in [16–19]. For these conditions the relative displacement of the laminar flame is D ¼ Sn=ðf dÞ ’ 0:7 5=ð5000 0:0005Þ > 1. The movement of the laminar flame is thus already non-negligible, while turbulent flow structures will locally lead to an even larger displacement. Therefore, it is now impossible to consider a frozen flame in space. The variation of the variables with time at a given grid point should be considered, as done next.
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Fig. 1. Instantaneous value of the correction term appearing in Eq. (12) at t0 = 0.5 ms (third period). See also Figs. 2–4.
remains during all the interaction process of order 10. Since the order of magnitude of the first term within the local Rayleigh’s criterion presented in Eq. (12) is typically 1 000 (see for example later Fig. 4), there is roughly a factor 100 between the first (standard) term and the correction term appearing in Eq. (12). As a consequence the influence of flame movement can be safely neglected for such configurations. 4.2. Amplification and damping of the wave Considering now only the first term in Eq. (12), the local amplification and damping of the incoming acoustic wave can be quantified. For this purpose both the acoustically-induced pressure fluctuation P0 (Fig. 2) and the acoustically-induced
4. Results and discussion 4.1. Influence of the correction term For the present configuration, the maximum Mach number observed within the numerical domain is M max ¼ 0:022. As a consequence, compressibility effects will be small and the influence of the correction term in Eq. (12) is expected to be negligible as stated previously. This can be quantified by computing directly this correction term since all terms involved are known from the DNS. Corresponding results are shown in Fig. 1 for t0 = 0.5 ms (third period). As expected, the correction term is only active in the immediate vicinity of the instantaneous flame front since it describes the influence of flame movement. The magnitude of the correction term
Fig. 2. Acoustically-induced pressure fluctuation P0 at t0 = 0.5 ms (third period), shown with dashed lines. The instantaneous position of the flame front is shown with isolevels of heat release (black solid lines). See also Figs. 3 and 4.
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Fig. 3. Acoustically-induced heat release fluctuation Q0 at t0 = 0.5 ms (third period). See also Figs. 2 and 4.
Fig. 4. Rayleigh’s criterion at t0 = 0.5 ms (third period). See also Figs. 2 and 3.
heat release fluctuation Q0 (Fig. 3) are extracted from the DNS computations as explained before by a simple difference between both results corresponding to the same time. In Fig. 2 the instantaneous position of the flame front is represented by black solid lines. As in all previous studies, the amplitude of the acoustically-induced pressure fluctuation is small (Fig. 2), well below 0.1% of the baseline pressure. However, this still means a strong acoustic perturbation, with an order of magnitude of 100 Pa or 130 dB. In the present work, at the difference of the interaction considering an isolated Gaussian pulse [16–19], P0 is now distributed over the full computational domain, which seems much more realistic. Depending on physical time, conditions are found for which P0 is everywhere positive, everywhere negative (see for instance Fig. 2), or changes sign over the numerical domain. The acoustically-induced heat release fluctuation (Fig. 3) is of course only noticeable within the reaction zone and is thus limited to a very thin region, confirming previous studies involving premixed flames [16–18]. The local Rayleigh’s criterion can now directly be determined as ðc0 1ÞP0 Q0 (Fig. 4) and is thus proportional to the ‘‘product” of both previous figures. Confirming previous investigations using a Gaussian pulse, amplification and damping of the flame is limited to the active reaction zone since Q0 ’ 0 outside of this domain. Moreover, amplification and damping are very localized phenomena and occur in well-defined layers, while a large part of the flame front appears to play a negligible role. As already observed in previous studies, amplification and damping regions often appear as neighboring, sandwich layers. As a consequence, a possible influence of the direction of wave propagation must be checked, as presented next. Of course the presented Rayleigh’s criterion is time-dependent so that a different picture is
obtained at every time-step. Nevertheless, the qualitative features described previously are not modified during all the computation and even quantitative changes are of limited importance. This is illustrated in Fig. 5, corresponding to the fourth period. All previous observations still hold. 4.3. Influence of the direction of wave propagation As explained in the previous section regions leading to maximal amplification and damping often appear as neighboring, sandwich layers. It is therefore of interest to check the physical meaning of this observation in order to exclude the possibility of some numerical artifact. This has been already done in a previous publication for a laminar hydrogen non-premixed flame [19]. The case of a turbulent syngas premixed flame is now considered. An identical Gaussian wave is initiated once on the fresh gas side (left) and prop-
Fig. 5. Rayleigh’s criterion at t0 = 0.65 ms (fourth period). Compare to Fig. 4.
H. Shalaby et al. / Proceedings of the Combustion Institute 32 (2009) 1473–1480
agates to the right. In a second computation, the same wave (negative pulse) is initiated within the burnt gas and propagates towards the left. Thus, both identical waves (apart from their starting position) cross an initially identical turbulent syngas flame. Figure 6 shows the local Rayleigh’s criterion for a time corresponding to the same mean position of the wave. A direct comparison shows of course some quantitative differences, both in the localization and in the amplitude of the amplification and damping regions. This is unavoidable since the modification of the wave geometry depends on the gradients of acoustic impedance [19]; the corresponding field is of course not symmetrical. As a consequence, the instantaneous geometry of the wave during the interaction process is not identical in both cases. Nevertheless, both results are very similar, at least qualitatively. Furthermore, the double layers associated with amplification and damping are observed in both cases. It is therefore clear that these layers have a physical explanation and are not the result of a numerical problem, as already
Fig. 6. Instantaneous Rayleigh’s criterion for an identical wave coming from: top: the fresh gas, i.e. the left side or; bottom: the burnt gas, i.e. the right side of the numerical domain.
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demonstrated for a laminar non-premixed configuration [19]. These layers are always connected with the heat release fluctuation Q0 and correspond to opposite variations of heat release on both sides of the active reaction zone, leading to opposite effects regarding stability conditions. It is interesting to see that this effect is also found for the complex geometry associated with a turbulent flame, and not only when considering laminar, one-dimensional conditions, as in [19]. 4.4. Species leading to amplification or damping The local Rayleigh’s criterion is finally used to investigate the isolated influence of each species on wave modification. This can easily be done by considering Eq. (12) and keeping a constant value for index i (species index) in Eq. (13) in order to compute Q0 . The influence of the species is only contained in the first term of Eq. (12). In this manner, the influence of each species on amplification or damping is obtained as a function of space and time. In what follows the resulting influence is averaged over the complete numerical domain in order to get the global influence of each species (Fig. 7). Three different results are presented in this figure on which the influence of the most important species has always been normalized to 1 in order to facilitate comparisons. First, the direction of wave propagation is not important when determining if a species leads to amplification or damping, supporting the original, global analysis of Lord Rayleigh. Both results (wave originating from the fresh gas: squares; or from the burnt gas: circles) are almost identical, confirming also the results presented in the previous section. The only noticeable difference is observed for species CO2. Second, the employed kinetic scheme does not modify these findings. In order to check this issue, the same computation (wave starting in the fresh gas) has been repeated twice using two
Fig. 7. Normalized contribution of each species to Rayleigh’s criterion, averaged over the full numerical domain. Wave coming from the fresh gas, i.e. the left side (squares) or from the burnt gas, i.e. the right side of the numerical domain (circles), with both using the chemical scheme of [27]. Filled triangles: wave coming from the fresh gas using the chemical scheme of [28].
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different chemical schemes ([27]: squares; [28]: filled triangles). Once again, the two schemes lead qualitatively to the same findings and show only limited quantitative differences. Finally, the influence of the different species on amplification and damping is clear for all conditions: species CO2, H and H2O mostly control wave amplification, while species O, OH and CO dominate damping. As expected, short-lived radicals like HCO, CH2O, HO2, H2O2 and neutral species (N2, Ar) have a negligible influence on the process. 5. Conclusions and perspectives The interaction of a realistic acoustic wave with a turbulent premixed syngas flame has been investigated using Direct Numerical Simulations relying on detailed models for chemistry and diffusive transport. Amplification and damping occur only in well-defined layers within the active reaction zone, very often with a double-layer, sandwich structure. This observation is independent of the direction of wave propagation and has therefore a physical explanation. Such layers are always caused by the heat release fluctuation Q0 and correspond to opposite variations of heat release on both sides of the active reaction zone, leading to opposite effects regarding stability conditions. The global amplification or damping of the initial acoustic wave is therefore a complex average over contradictory effects. Two mechanisms control the interaction process: (1) the influence of the gradient of acoustic impedance, which is not symmetric with respect to the fresh or burnt gas but will globally play a minor role since both curvatures are found with a similar probability in highly turbulent flames, thus leading to large canceling effects; (2) coupling with the chemical reactions, as described originally by Lord Rayleigh. This last effect will play a dominant role in explaining wave amplification and damping, since it is almost independent of flame orientation or wave propagation direction. This is the most important coupling process with acoustics in laminar flames. For the present configuration species CO2, H and H2O mostly control wave amplification, while species O, OH and CO dominate damping. Short-lived radicals and neutral species have a negligible influence. This suggests new possibilities to enhance flame stability, either by modifying slightly the burning regime in order to shift the local composition towards more favourable conditions; or by injecting additional species, leading to damping.
Acknowledgments This work has been financially supported by the Deutsche Forschungsgemeinschaft in the frame of the Research Unit 486 ‘‘Combustion Noise”.
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