- Email: [email protected]
acoustic interactions
Proceedings of the
Proceedings of the Combustion Institute 31 (2007) 2283–2290
Combustion Institute www.elsevier.com/locate/proci
Direct numerical simulation of spray flame/acoustic interactions C. Pera, J. Reveillon
*
CORIA-University and INSA of Rouen, B.P. 12, 76800 Saint Etienne du Rouvray, France
Abstract Interactions between conical spray flames and sinusoidal velocity modulations due to the propagation of acoustic waves have been studied thanks to direct numerical simulations (DNS). A 2D axi-symmetric configuration has been used to capture the evolution of the pulsating laminar flames. The DNS solver has been coupled with a Lagrangian model to account for the dispersion and evaporation of the liquid fuel in the computational domain. Four main configurations, with a unitary global equivalence ratio, have been studied. Apart from a gaseous reference case, one polydispersed and two monodispersed Bunsen-type injections with various droplets density and inertia have been simulated. DNS results are in good agreement with experimental data. For significant acoustic Stokes numbers, results showed a double effect of the modulations on the flame: a direct disturbance of the flame front and a secondary impact through the local variation of the mixture fraction due to droplets preferential segregation. 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. PACS: 47.27Ek; 47.70Pq; 47.55Ca; 47.25Rs Keywords: DNS; Flame; Spray; Laminar; Acoustic; Modulation; Bunsen; Conical
1. Introduction Combustion instabilities are observed in numerous industrial systems and more particularly in aeronautical engines: turbojets, ramjets, rocket motors, etc. They create many undesirable effects as, for example, an increase of wall heat fluxes, flames extinction and flashback, or strong vibrations of the mechanical structure, which can lead to its destruction. In spite of many research tasks based on this topic, these instabilities are difficult, if not impossible, to predict.
*
Corresponding author. Fax: +33 232 959 780. E-mail address: [email protected] (J. Reveillon).
To begin with, it is necessary to identify the phenomena responsible for the presence of combustion instabilities and their consequences on various processes such as injection, atomization, spray evaporation, reactants mixing, chemical reactions, interactions between flames and walls, etc. For that reason, multiple theoretical, experimental and numerical works are dedicated to the understanding and the analysis of the fundamental physical mechanisms of the couplings between these various processes and acoustic phenomena in the chamber. To our knowledge, there is no numerical simulation dedicated to the analysis of the interactions between spray combustion and acoustic instabilities. The objective of this work is twofold: first to demonstrate the capability of
1540-7489/$ - see front matter 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2006.07.153
2284
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
a DNS solver to capture the complex spray/ flame/acoustic interactions and then, to focus on the impacts of velocity modulations on reaction rate through an analysis of the transfer function. A classical Bunsen configuration has been selected and experimental comparisons were made possible thanks to the data of the EM2C laboratory, Ecole Centrale Paris [1]. As pointed out previously, various works have been dedicated to acoustic instabilities in combustion chambers. We are focusing now on some configurations concerning low frequencies–low amplitude instabilities. Because of several difficulties, analysis are often simplified by considering some basic interactions: • Spray/acoustic (inert flow). Few studies [2,3] have been dedicated to the influence of acoustic disturbances on the liquid phase. However, it is of primary importance to know how atomized, dispersed and evaporated is a liquid jet undergoing acoustic modulations in aeronautical combustion chambers. • Combustion/acoustic. The most studied configurations are dedicated to the interactions between purely gaseous flames and acoustic waves. To understand all implications, basic configurations are generally considered [4]: stationary or propagating plane flames, V flames, M flames [5], cylindrical flames [6] and, of course conical or Bunsen flames that are studied in this paper. • Spray combustion/acoustic (experimental). Concerning two-phase flow combustion, it was shown by Saito et al. [7] that both evaporation and heat release rates might be increased by acoustic waves up to a ratio of three in some extremes. Although there are difficulties to resolve, it will be shown in this paper that DNS may be a useful tool to progress in the resolution of these problems. In the following section, definitions of several basic physical quantities are presented along with some methods used to characterize the response of the flame to the velocity modulations. The burner geometry is detailed in Section 3 with some indications concerning the numerical features of the simulations. In Section 4, reference stationary flames are presented and the description of the effects of the acoustic modulations is done in Section 5. 2. Flame modulation and combustion characterization 2.1. Dimensionless parameters Haile et al. [8] have suggested some physical parameters that could relate the characteristic
properties of the combustion phenomena and the acoustic modulations: • the time period Tac associated with the acoustic forcing frequency; • the flow characteristic time sfl = 2Rb/ufl, where Rb is the burner radius and ufl the flow mean velocity; • the kinetic time sp of the dispersed phase. Then, it is possible to define the acoustic Stokes number sp S tac ¼ ; ð1Þ T ac which characterizes the reactivity of the droplets to the carrier phase disturbances due to the acoustic wave. 2.2. Transfer function To characterize the response of the flame to the velocity modulation at the outlet of the injector, the flame transfer function Ftr has to be defined. A very thin flame is considered. If it undergoes the following modulation of the flow: uz ðx; tÞ ¼ uz0 ðxÞ þ u0z ðx; tÞ, where uz0 is the flow mean velocity and Au is the amplitude of the velocity modulation u0z whose frequency is given through x. The flame transfer function Ftr is defined by the ratio of the relative fluctuation of the heat release amplitude AQ/Q with the relative fluctuations of the velocity field amplitude Au/uz0 F tr ¼
AQ uz0 : Q Au
ð2Þ
More details may be found in [9]. 2.3. Reduced frequency A simple way to analyse flame interactions with a modulated acoustic field is to define a reduced frequency [6,9] whose main characteristics may be compared with the stationary (no acoustic modulations) conical flame properties x ¼
xRb ; S 0L cos a0
ð3Þ
where x is the effective pulsation, S 0L the reference stoichiometric planar flame velocity and a0, the half-angle at the top of the cone. 3. Burner geometry The selected geometry is based on the experiments carried out by Ducruix et al. from EM2C laboratory, Ecole Centrale Paris, in the framework of a European Project [1]. A Bunsen burner is considered. It leads to the formation of a
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
conical premixed flame. This configuration is well known and well documented in the case of purely gaseous combustion. Thus, a gaseous case (GAS) (experimental and numerical) is used as a reference when studying the interactions between the flame and the velocity modulations. A second experimental configuration (DRPE) involving spray combustion is selected from the experimental work of EM2C to validate the two-phase simulations. Two other two-phase simulations (DRP0 and DRP1) are carried out to analyse some extreme cases (small and large acoustic Stokes numbers).
2285
Table 1 Characteristics of the fuel injection of the various simulations Phase / /g /l S tac Min Max
GAS Gas
DRP0 Mono
DRPE Gas/poly
DRP1 Mono
1 1 0
1 0 1
1 0.75 0.25
1 0 1
0.001 0.03
0.004 0.1
0.01 0.3
In case: DRPE, the acoustic Stokes number is given for the heaviest droplets at the burner center.
3.1. Experimental setup N-Heptane is considered for both gaseous and two-phase configurations. The burner injection diameter is equal to 15 mm. The mean injection velocity is equal to 2 m/s. It is approximately 4 times faster than the reference stoichiometric laminar flame velocity (45 cm/s). Amplitude of the added velocity fluctuations reaches 10% of the mean flow velocity. Two experimental configurations have been selected: • The ‘‘GAS’’ configuration involves a purely gaseous flow and, thus, leads to a classical Bunsen flame with a unitary equivalence ratio. • The ‘‘DRPE’’ configuration is dedicated to two-phase flow combustion. N-heptane droplets are injected along with the main gaseous stream. The corresponding mean diameter profile may be seen in Fig. 1. However, to stabilize the conical flame near the burner lips, gaseous N-heptane is used as well. A mass ratio of 1/4 of liquid fuel and 3/4 of gaseous fuel has been used to reach a unitary global equivalence ratio.
Gaseous fuel has been obtained by preheating slightly a part of the liquid phase before the injection. 3.2. Numerical configurations From the DNS point of view, four different sets of simulations with a unitary global equivalence ratio have been carried out: the GAS and DRPE experimental configurations, the DRP0 case that corresponds to a monodispersed injection of small and light droplets ð0:001 < S tac < 0:03Þ and, eventually, the DRP1 case that involves heavier and sparser droplets 0:01 < S tac < 0:3. For both DRP0 and DRP1 cases, a purely liquid fuel injection is done (monodispersed) whereas, the other two-phase configuration (DRPE) is similar to the experiment (1/4 liquid, 3/4 gas) with polydispersed droplets 0:004 < S tac < 0:1. The liquid equivalence ratio is equal to 0.5 in the centre of the DRPE jet and diminishes toward 0.1 in the jet periphery. On the other hand, the gaseous equivalence ratio profile is constant and equal to 0.75. Thus, even if a unitary global equivalence ratio is used, the injection is rich close to the centre and lean in the jet periphery. A summary of these configurations may be seen in Table 1. 3.3. Numerical features
Fig. 1. Mean diameter a of the injected droplets with respect to the burner radius. Normalized by the maximum droplet size: a0 = 20 lm, case DRPE.
Because of the geometry of the configuration, a DNS code has been used with a 2D axi-symmetric system of coordinates. Resolved equations of the gaseous phase and their coupling with the Lagrangian solver for the dispersed phase may be found in [11] for a Cartesian system of coordinates. Chemical reactions are modeled by a onestep kinetic (F + sO fi (1 + s)P, with s = 15) through an Arrhenius law that has been adapted to spray combustion [11]. The axi-symmetric resolution imposed only some minor modifications. The axis of symmetry is at the centre of the burner and the first grid point is at r = Dr/2 (Dr being the radial grid mesh size) to avoid numerical inconsistencies. Along the
2286
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
radial direction, a symmetry boundary condition is used, when r = Dr/2 and a ‘cylindrically’ modified NSCBC is imposed on the external radius. Classical inlet and outlet NSCBC [12] are used along the streamwise direction. Note that droplets dispersion is characterized down to r = 0 exactly. An adaptation of the Lagrangian/Eulerian twoway coupling has been necessary as well. Characteristics of the coupling may be found in [11]. Cylindrical control volumes have to be defined for every Eulerian nodes. Thus, it is necessary to add to the simulation a new parameter describing the droplet density in each computational cell. Indeed, we have an expansion of the cell volume as the radius increases. Therefore, to obtain a constant density of droplets along the radial direction, it is necessary to consider more particles in the periphery than in the centre. A parcel method is used: one ‘numerical’ droplet may represent several ‘physical’ droplets. The following dimensions have been used: the radial direction Lr = 4Rb and Lh = 8Rb. The 2D grid uses 130,000 nodes. The burner lips are at a temperature high enough to evaporate and to ignite the mixture in the boundary layer of the injector at the inlet of the computational domain. It ensures the flame stabilization. The injected droplets are initially in dynamical equilibrium with the carrier phase (null slip velocity).
This fact is confirmed in Fig. 2 right, where comparison is done for the two-phase DRPE case. Again, the cone angle is well captured by DNS although it is different from the GAS case. Indeed, a smaller height of the cone has been found in this configuration. The cone height ratio of the DRPE flame with the GAS flame is HH dg jexp ¼ 0:88 as measured experimentally. The corresponding ratio obtained with the DNS is slightly different Hd j ¼ 0:85: a slight difference but a good catch H g DNS of the general decreasing behaviour. Note that the ‘stationary’ experimental flame (no acoustic modulation) shows various perturbations because some droplets are clustered and create disturbances of the flame front before burning outside the main cone. These disturbances are not reproduced by the DNS computations because of a controlled homogeneous injection of the droplets. Thus, it may be reasonable to state that a 3% difference is within the uncertainties of both experimental measures and numerical approximations. 5. Flame response to velocity modulations To study the flame response to acoustic modulations, a sinusoidal velocity signal has been
4. Stationary flames To begin with, comparisons of the experimental and numerical flame fronts are done for both GAS and DRPE configurations in Fig. 2. The stationary GAS flame obtained thanks to the DNS (Fig. 2, left) has the same normalized height and, thus, the same cone angle than the experimental results. Therefore, the simplified kinetic is able to capture the global behaviour of the real flame.
Fig. 2. Stationary experimental (cone left side) and numerical (cone right side) flames. Left: GAS, right: DRPE, dots: fuel droplets.
Fig. 3. Reacting rate of the various simulated flames, from left to right and top to bottom: cases GAS, DRP0, DRPE and DRP1. Dots: droplets. Heights normalized by the reference gaseous flame height Hg.
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
2287
Table 2 Equivalence between the reduced ‘flame’ frequencies (frequencies xDRP0 , xDRPE and xDRP1 , normalized by the various spray flame heights) and the reduced ‘real’ frequency (frequency xGAS normalized by the constant stationary GAS flame height) xGAS
xDRP0
xDRPE
xDRP1
0.9 1.5 2.2 3.5 4.5 5.6 6.5 7.1 7.8 8.7 17.4 25.1
0.9 1.6 2.4 3.7 4.8 6.0 6.8 7.5 8.3 9.2 18.4 26.6
0.8 1.3 2.0 3.0 3.9 4.9 5.6 6.1 6.7 7.5 15.0 21.6
1.0 1.7 2.6 4.0 5.1 6.4 7.4 8.1 8.9 9.9 19.9 28.8
These frequencies have been used to plot the transfer functions. Fig. 5. Pulsating experimental (cone left side) and numerical (cone right side) DRPE flame, experiment (xGAS ¼ 12) and simulation (xGAS ¼ 17). Phases, from left to right and top to bottom: p, p/4, 0 and p/4.
Fig. 4. Pulsating experimental (cone left side) and numerical (cone right side) GAS flame, experiment (xGAS ¼ 5:7) and simulation (xGAS ¼ 5:6). Phases, from left to right and top to bottom: p, 0, p/4 and p/2. Fig. 6. Pulsating experimental (cone left side) and numerical (cone right side) DRPE flame, experiment (xGAS ¼ 23) and simulation (x*jGAS = 25). Phases, from left to right and top to bottom: 0, p/4, p and 5p/4.
prescribed at the burner exit to mimic the loudspeakers effects. To compare experimental and numerical results, it is necessary to refine the reduced frequency definition. Indeed, the heights of the stationary flames are different depending on the injected spray properties (Fig. 3). Thus, for a unique ‘real’ frequency in the combustion chamber, it leads to different associated reduced frequencies. Two choices are possible:
(1) To study the flame shapes and their responses to the velocity modulations, we used a prescribed ‘real’ frequency similar in all the configurations to understand the influence of the dispersed phase and to differentiate it from the direct impact of the velocity modulations on the flame. Nevertheless, even if a ‘real’
2288
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
a
b
Fig. 7. Transfer function, gaseous (GAS) case.
frequency analysis is done, it may be reduced by a constant flame length for all the cases. In this paper, we are referring to the reduced frequency associated with the reference stationary GAS case xGAS . (2) To study the behaviour of a unique flame for several modulation frequencies, one prescribed a reduced ‘flame’ frequencies (that is, reduced with the cone heights of the corresponding stationary flame [6,9]). Table 2 offers an overall picture of the difference between reduced ‘real’ frequencies and reduced ‘flame’ frequencies used in our DNS.
Fig. 8. Transfer function, two-phase (DRPE) case.
a
5.1. Flame perturbations
b
Effects of low (xGAS 5), medium (xGAS 15) and high frequencies (xGAS 25) are shown in Figs. 4, 5 and 6, respectively. The first figure corresponds to the purely gaseous GAS case, the other two to the DRPE case. The inlet modulation generates a propagating signal along the flame front from the base up to the top of the cone. From a general point of view, the lower the frequency is, the more the flame lengthens. When frequency increases, the flame front is more and more perturbed. In some cases, it is possible to observe the formation of a pocket of fresh gases at the top of the cone and, when the flame retracts
Fig. 9. Transfer function, all DNS cases.
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
2289
itself, this pocket drifts away in the burnt gases and disappears very quickly. Similar burning pockets have been observed experimentally. Good agreements may be observed between simulations and experiments. 5.2. Transfer functions One of the aims of this study is to demonstrate the capability of DNS to characterize the interactions between a premixed spray flame and velocity modulations due to a sinusoidal acoustic wave. Encouraging comparisons have been made in the preceding section. However, a full agreement between experiments and simulations is needed. To ensure that the whole range of modulation frequencies may be captured by the DNS, transfer functions have to be determined. To begin with, we are focusing on the purely gaseous configuration, GAS. In that case, analytical models have been developed to predict the response of the flame to various low frequency modulations. Details concerning these models may be found in [9]. In Fig. 7, the analytical model and our simulations are compared with the experimental results of EM2C laboratory [1]. The amplitude of the experimental transfer function is well captured by DNS. The analytical amplitude shows a similar behaviour even if a less good agreement is observed. From the phase point of view, DNS is in very good agreement up to a reduced frequency equal to 30 with the experimental results. On the other hand, it is not the case for the analytical model that is unable to capture the phase when the reduced frequency is greater than seven. It is very interesting to note that the capability of DNS to capture both the amplitude and the phase of the transfer function is confirmed also in the two-phase EM2C case (Fig. 8).
Fig. 11. Sketch of the direct (top) and indirect (bottom) effects of acoustic waves on the flame.
These successful comparisons between DNS and experiments allowed us to validate the capability of the DNS to capture the physical phenomena embedded in the triple interactions: spray/ combustion/modulation. 5.3. DNS analysis In this section, DNS databases are used to plot the transfer functions of the four simulated configurations in Fig. 9. The gaseous analytical model prediction is plotted as well. As it was hinted by comparing some flame shapes, both GAS and DRP0 cases are giving similar results for all the studied modulation frequencies. In fact, ‘‘small acoustic Stokes number—short evaporation delay’’ droplets are not directly affected by the flow modulation and behave like a gas in a purely gaseous flow (DRP0). However, as soon as the acoustic Stokes number increases (DRPE, DRP1) the transfer function is strongly affected and the amplitude increases. It is explained by the modification of the local droplet density as it may be seen in Fig. 10, left. Consequently, the shape of the flame is affected by the resulting mixture fraction field that is not constant anymore. The flame front is more wrinkled, thus increasing the global reaction rate and the transfer function amplitude (Fig. 10, right). In conclusion, the acoustic wave affects directly the flame front (though combustion/acoustic interactions) but it has a secondary effect through the modification of the dispersed phase topology and, consequently, the vapour of fuel that will feed the flame. This double effect is summarized in Fig. 11. Note that a third effect has been neglected in this work: the direct influence of the acoustic wave on the evaporation rate of the droplets. On another hand, the phase of the transfer function is never really affected by the droplet presence (Fig. 9). This may be explained by the fact that, even if droplet density is affected by the acoustic wave, it is done following a phase similar to the one disturbing the flow velocity. 6. Conclusions
Fig. 10. DRP1 case, left: droplet density (normalized by the injected mean density), right: heat release (normalized by its stoichiometric value).
It is possible to predict correctly the transfer function of a pulsating conical flame undergoing acoustic instabilities thanks to a 2D axi-symmetric
2290
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
DNS. Both purely gaseous and spray combustion have been carried out and good agreements between experimental and numerical transfer functions have been found for the whole range of studied reduced frequency (up to 30). It may be a low-cost alternative to actual analytical models that are dedicated to gaseous combustion. Moreover, they seem to be limited to very low perturbation reduced frequency. The main perspectives of this work are to extend the validation to a higher frequency range and to other configurations like V-shape flames, for example.
Acknowledgments The authors thank deeply the experimental team of the EM2C laboratory, Ecole Centrale Paris: S. Ducruix, D. Durox and C. Pichard for their fruitful discussions and the quality of the experimental data. The authors acknowledge the support from IDRIS-CNRS (Institut de Developpement et de Ressources en Informatique Scientifique), where computations were performed. Support was also provided by the project ‘‘Modelling of UnSteady Combustion in Low-Emission Systems’’ (MUSCLES) under the EC Contract No G4RD-CT-2001-00644.
References [1] S. Ducruix, D. Durox, C. Pichard, Influence of acoustic waves on vaporization and combustion of sprays, Tech. rep., EEC, Modelling of UnSteady Combustion in Low-Emission Systems, Final report D.4.6, 2005. [2] R. Sujith, G. Waldherr, J. Jagoda, B. Zinn, J. Propul. Power 16 (1) (2000) 278–285. [3] B. Abramzon, W.A. Sirignano, Int. J. Heat Mass Transfer 32 (9) (1989) 1605–1618. [4] S. Candel, Proc. Combust. Inst. 1992, 1277– 1296. [5] T. Schuller, Me´canisme de Couplage dans les Interactions Acoustique-Combustion, Ph.D. thesis, Ecole Centrale Paris, 2003. [6] S. Ducruix, Dynamique des Interactions AcoustiqueCombustion, Ph.D. thesis, Ecole Centrale Paris, 1999. [7] M. Saito, M. Hoshikawa, M. Sato, Fuel 75 (1996) 669–674. [8] E. Haile, O. Delabroy, F. Lacas, D. Veynante, S. Candel, Proc. Combust. Inst. 26 (1996) 1663–1670. [9] S. Ducruix, D. Durox, S. Candel, Proc. Combust. Inst. 28 (2000) 768–773. [11] J. Reveillon, L. Vervisch, J. Fluid Mech. 537 (2005) 317–347. [12] T. Poinsot, S.K. Lele, J. Comput. Phys. 1 (101) (1992) 104–129.
Comments Jay Boris, U.S. Naval Research Laboratory, USA. How did you treat 3-D droplets in an axisymmetric geometry? How did you plot the particles to get a uniform density versus radius?
which the phase saturates. However, in a more recent unified model [1] we have been able to obtain an increasing phase like in your simulation and the match would then be much better.
Reply. In our axisymmetric geometry, ‘3-D volumes are considered’, i.e., we are simulating a portion (pie slice) of a cylinder. In this framework, fully spherical droplets may be considered. To deal with uniform droplet density with respect to the droplet radius, a parcel method is used (a ‘numerical’ droplet may represent various ‘physical’ droplets).
Reference
d
Sebastien Candel, EM2C, CNRS, ECP, France. In your transfer function plots for the gas phase case you are using one of our earlier models ([9] in paper) in
[1] Schuller et al., Combust Flame 134 (2003) 21–34. Reply. In this paper, we were focused on the achievement of direct numerical simulations of spray flames undergoing acoustic modulations; a configuration that no model can resolve yet. In this framework, concerning gaseous flows, we plotted the analytical model of Ducruix et al. However, it is true that the model of Schuller et al. should be used for any study considering purely gaseous flows.
Proceedings of the Combustion Institute 31 (2007) 2283–2290
Combustion Institute www.elsevier.com/locate/proci
Direct numerical simulation of spray flame/acoustic interactions C. Pera, J. Reveillon
*
CORIA-University and INSA of Rouen, B.P. 12, 76800 Saint Etienne du Rouvray, France
Abstract Interactions between conical spray flames and sinusoidal velocity modulations due to the propagation of acoustic waves have been studied thanks to direct numerical simulations (DNS). A 2D axi-symmetric configuration has been used to capture the evolution of the pulsating laminar flames. The DNS solver has been coupled with a Lagrangian model to account for the dispersion and evaporation of the liquid fuel in the computational domain. Four main configurations, with a unitary global equivalence ratio, have been studied. Apart from a gaseous reference case, one polydispersed and two monodispersed Bunsen-type injections with various droplets density and inertia have been simulated. DNS results are in good agreement with experimental data. For significant acoustic Stokes numbers, results showed a double effect of the modulations on the flame: a direct disturbance of the flame front and a secondary impact through the local variation of the mixture fraction due to droplets preferential segregation. 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. PACS: 47.27Ek; 47.70Pq; 47.55Ca; 47.25Rs Keywords: DNS; Flame; Spray; Laminar; Acoustic; Modulation; Bunsen; Conical
1. Introduction Combustion instabilities are observed in numerous industrial systems and more particularly in aeronautical engines: turbojets, ramjets, rocket motors, etc. They create many undesirable effects as, for example, an increase of wall heat fluxes, flames extinction and flashback, or strong vibrations of the mechanical structure, which can lead to its destruction. In spite of many research tasks based on this topic, these instabilities are difficult, if not impossible, to predict.
*
Corresponding author. Fax: +33 232 959 780. E-mail address: [email protected] (J. Reveillon).
To begin with, it is necessary to identify the phenomena responsible for the presence of combustion instabilities and their consequences on various processes such as injection, atomization, spray evaporation, reactants mixing, chemical reactions, interactions between flames and walls, etc. For that reason, multiple theoretical, experimental and numerical works are dedicated to the understanding and the analysis of the fundamental physical mechanisms of the couplings between these various processes and acoustic phenomena in the chamber. To our knowledge, there is no numerical simulation dedicated to the analysis of the interactions between spray combustion and acoustic instabilities. The objective of this work is twofold: first to demonstrate the capability of
1540-7489/$ - see front matter 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2006.07.153
2284
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
a DNS solver to capture the complex spray/ flame/acoustic interactions and then, to focus on the impacts of velocity modulations on reaction rate through an analysis of the transfer function. A classical Bunsen configuration has been selected and experimental comparisons were made possible thanks to the data of the EM2C laboratory, Ecole Centrale Paris [1]. As pointed out previously, various works have been dedicated to acoustic instabilities in combustion chambers. We are focusing now on some configurations concerning low frequencies–low amplitude instabilities. Because of several difficulties, analysis are often simplified by considering some basic interactions: • Spray/acoustic (inert flow). Few studies [2,3] have been dedicated to the influence of acoustic disturbances on the liquid phase. However, it is of primary importance to know how atomized, dispersed and evaporated is a liquid jet undergoing acoustic modulations in aeronautical combustion chambers. • Combustion/acoustic. The most studied configurations are dedicated to the interactions between purely gaseous flames and acoustic waves. To understand all implications, basic configurations are generally considered [4]: stationary or propagating plane flames, V flames, M flames [5], cylindrical flames [6] and, of course conical or Bunsen flames that are studied in this paper. • Spray combustion/acoustic (experimental). Concerning two-phase flow combustion, it was shown by Saito et al. [7] that both evaporation and heat release rates might be increased by acoustic waves up to a ratio of three in some extremes. Although there are difficulties to resolve, it will be shown in this paper that DNS may be a useful tool to progress in the resolution of these problems. In the following section, definitions of several basic physical quantities are presented along with some methods used to characterize the response of the flame to the velocity modulations. The burner geometry is detailed in Section 3 with some indications concerning the numerical features of the simulations. In Section 4, reference stationary flames are presented and the description of the effects of the acoustic modulations is done in Section 5. 2. Flame modulation and combustion characterization 2.1. Dimensionless parameters Haile et al. [8] have suggested some physical parameters that could relate the characteristic
properties of the combustion phenomena and the acoustic modulations: • the time period Tac associated with the acoustic forcing frequency; • the flow characteristic time sfl = 2Rb/ufl, where Rb is the burner radius and ufl the flow mean velocity; • the kinetic time sp of the dispersed phase. Then, it is possible to define the acoustic Stokes number sp S tac ¼ ; ð1Þ T ac which characterizes the reactivity of the droplets to the carrier phase disturbances due to the acoustic wave. 2.2. Transfer function To characterize the response of the flame to the velocity modulation at the outlet of the injector, the flame transfer function Ftr has to be defined. A very thin flame is considered. If it undergoes the following modulation of the flow: uz ðx; tÞ ¼ uz0 ðxÞ þ u0z ðx; tÞ, where uz0 is the flow mean velocity and Au is the amplitude of the velocity modulation u0z whose frequency is given through x. The flame transfer function Ftr is defined by the ratio of the relative fluctuation of the heat release amplitude AQ/Q with the relative fluctuations of the velocity field amplitude Au/uz0 F tr ¼
AQ uz0 : Q Au
ð2Þ
More details may be found in [9]. 2.3. Reduced frequency A simple way to analyse flame interactions with a modulated acoustic field is to define a reduced frequency [6,9] whose main characteristics may be compared with the stationary (no acoustic modulations) conical flame properties x ¼
xRb ; S 0L cos a0
ð3Þ
where x is the effective pulsation, S 0L the reference stoichiometric planar flame velocity and a0, the half-angle at the top of the cone. 3. Burner geometry The selected geometry is based on the experiments carried out by Ducruix et al. from EM2C laboratory, Ecole Centrale Paris, in the framework of a European Project [1]. A Bunsen burner is considered. It leads to the formation of a
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
conical premixed flame. This configuration is well known and well documented in the case of purely gaseous combustion. Thus, a gaseous case (GAS) (experimental and numerical) is used as a reference when studying the interactions between the flame and the velocity modulations. A second experimental configuration (DRPE) involving spray combustion is selected from the experimental work of EM2C to validate the two-phase simulations. Two other two-phase simulations (DRP0 and DRP1) are carried out to analyse some extreme cases (small and large acoustic Stokes numbers).
2285
Table 1 Characteristics of the fuel injection of the various simulations Phase / /g /l S tac Min Max
GAS Gas
DRP0 Mono
DRPE Gas/poly
DRP1 Mono
1 1 0
1 0 1
1 0.75 0.25
1 0 1
0.001 0.03
0.004 0.1
0.01 0.3
In case: DRPE, the acoustic Stokes number is given for the heaviest droplets at the burner center.
3.1. Experimental setup N-Heptane is considered for both gaseous and two-phase configurations. The burner injection diameter is equal to 15 mm. The mean injection velocity is equal to 2 m/s. It is approximately 4 times faster than the reference stoichiometric laminar flame velocity (45 cm/s). Amplitude of the added velocity fluctuations reaches 10% of the mean flow velocity. Two experimental configurations have been selected: • The ‘‘GAS’’ configuration involves a purely gaseous flow and, thus, leads to a classical Bunsen flame with a unitary equivalence ratio. • The ‘‘DRPE’’ configuration is dedicated to two-phase flow combustion. N-heptane droplets are injected along with the main gaseous stream. The corresponding mean diameter profile may be seen in Fig. 1. However, to stabilize the conical flame near the burner lips, gaseous N-heptane is used as well. A mass ratio of 1/4 of liquid fuel and 3/4 of gaseous fuel has been used to reach a unitary global equivalence ratio.
Gaseous fuel has been obtained by preheating slightly a part of the liquid phase before the injection. 3.2. Numerical configurations From the DNS point of view, four different sets of simulations with a unitary global equivalence ratio have been carried out: the GAS and DRPE experimental configurations, the DRP0 case that corresponds to a monodispersed injection of small and light droplets ð0:001 < S tac < 0:03Þ and, eventually, the DRP1 case that involves heavier and sparser droplets 0:01 < S tac < 0:3. For both DRP0 and DRP1 cases, a purely liquid fuel injection is done (monodispersed) whereas, the other two-phase configuration (DRPE) is similar to the experiment (1/4 liquid, 3/4 gas) with polydispersed droplets 0:004 < S tac < 0:1. The liquid equivalence ratio is equal to 0.5 in the centre of the DRPE jet and diminishes toward 0.1 in the jet periphery. On the other hand, the gaseous equivalence ratio profile is constant and equal to 0.75. Thus, even if a unitary global equivalence ratio is used, the injection is rich close to the centre and lean in the jet periphery. A summary of these configurations may be seen in Table 1. 3.3. Numerical features
Fig. 1. Mean diameter a of the injected droplets with respect to the burner radius. Normalized by the maximum droplet size: a0 = 20 lm, case DRPE.
Because of the geometry of the configuration, a DNS code has been used with a 2D axi-symmetric system of coordinates. Resolved equations of the gaseous phase and their coupling with the Lagrangian solver for the dispersed phase may be found in [11] for a Cartesian system of coordinates. Chemical reactions are modeled by a onestep kinetic (F + sO fi (1 + s)P, with s = 15) through an Arrhenius law that has been adapted to spray combustion [11]. The axi-symmetric resolution imposed only some minor modifications. The axis of symmetry is at the centre of the burner and the first grid point is at r = Dr/2 (Dr being the radial grid mesh size) to avoid numerical inconsistencies. Along the
2286
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
radial direction, a symmetry boundary condition is used, when r = Dr/2 and a ‘cylindrically’ modified NSCBC is imposed on the external radius. Classical inlet and outlet NSCBC [12] are used along the streamwise direction. Note that droplets dispersion is characterized down to r = 0 exactly. An adaptation of the Lagrangian/Eulerian twoway coupling has been necessary as well. Characteristics of the coupling may be found in [11]. Cylindrical control volumes have to be defined for every Eulerian nodes. Thus, it is necessary to add to the simulation a new parameter describing the droplet density in each computational cell. Indeed, we have an expansion of the cell volume as the radius increases. Therefore, to obtain a constant density of droplets along the radial direction, it is necessary to consider more particles in the periphery than in the centre. A parcel method is used: one ‘numerical’ droplet may represent several ‘physical’ droplets. The following dimensions have been used: the radial direction Lr = 4Rb and Lh = 8Rb. The 2D grid uses 130,000 nodes. The burner lips are at a temperature high enough to evaporate and to ignite the mixture in the boundary layer of the injector at the inlet of the computational domain. It ensures the flame stabilization. The injected droplets are initially in dynamical equilibrium with the carrier phase (null slip velocity).
This fact is confirmed in Fig. 2 right, where comparison is done for the two-phase DRPE case. Again, the cone angle is well captured by DNS although it is different from the GAS case. Indeed, a smaller height of the cone has been found in this configuration. The cone height ratio of the DRPE flame with the GAS flame is HH dg jexp ¼ 0:88 as measured experimentally. The corresponding ratio obtained with the DNS is slightly different Hd j ¼ 0:85: a slight difference but a good catch H g DNS of the general decreasing behaviour. Note that the ‘stationary’ experimental flame (no acoustic modulation) shows various perturbations because some droplets are clustered and create disturbances of the flame front before burning outside the main cone. These disturbances are not reproduced by the DNS computations because of a controlled homogeneous injection of the droplets. Thus, it may be reasonable to state that a 3% difference is within the uncertainties of both experimental measures and numerical approximations. 5. Flame response to velocity modulations To study the flame response to acoustic modulations, a sinusoidal velocity signal has been
4. Stationary flames To begin with, comparisons of the experimental and numerical flame fronts are done for both GAS and DRPE configurations in Fig. 2. The stationary GAS flame obtained thanks to the DNS (Fig. 2, left) has the same normalized height and, thus, the same cone angle than the experimental results. Therefore, the simplified kinetic is able to capture the global behaviour of the real flame.
Fig. 2. Stationary experimental (cone left side) and numerical (cone right side) flames. Left: GAS, right: DRPE, dots: fuel droplets.
Fig. 3. Reacting rate of the various simulated flames, from left to right and top to bottom: cases GAS, DRP0, DRPE and DRP1. Dots: droplets. Heights normalized by the reference gaseous flame height Hg.
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
2287
Table 2 Equivalence between the reduced ‘flame’ frequencies (frequencies xDRP0 , xDRPE and xDRP1 , normalized by the various spray flame heights) and the reduced ‘real’ frequency (frequency xGAS normalized by the constant stationary GAS flame height) xGAS
xDRP0
xDRPE
xDRP1
0.9 1.5 2.2 3.5 4.5 5.6 6.5 7.1 7.8 8.7 17.4 25.1
0.9 1.6 2.4 3.7 4.8 6.0 6.8 7.5 8.3 9.2 18.4 26.6
0.8 1.3 2.0 3.0 3.9 4.9 5.6 6.1 6.7 7.5 15.0 21.6
1.0 1.7 2.6 4.0 5.1 6.4 7.4 8.1 8.9 9.9 19.9 28.8
These frequencies have been used to plot the transfer functions. Fig. 5. Pulsating experimental (cone left side) and numerical (cone right side) DRPE flame, experiment (xGAS ¼ 12) and simulation (xGAS ¼ 17). Phases, from left to right and top to bottom: p, p/4, 0 and p/4.
Fig. 4. Pulsating experimental (cone left side) and numerical (cone right side) GAS flame, experiment (xGAS ¼ 5:7) and simulation (xGAS ¼ 5:6). Phases, from left to right and top to bottom: p, 0, p/4 and p/2. Fig. 6. Pulsating experimental (cone left side) and numerical (cone right side) DRPE flame, experiment (xGAS ¼ 23) and simulation (x*jGAS = 25). Phases, from left to right and top to bottom: 0, p/4, p and 5p/4.
prescribed at the burner exit to mimic the loudspeakers effects. To compare experimental and numerical results, it is necessary to refine the reduced frequency definition. Indeed, the heights of the stationary flames are different depending on the injected spray properties (Fig. 3). Thus, for a unique ‘real’ frequency in the combustion chamber, it leads to different associated reduced frequencies. Two choices are possible:
(1) To study the flame shapes and their responses to the velocity modulations, we used a prescribed ‘real’ frequency similar in all the configurations to understand the influence of the dispersed phase and to differentiate it from the direct impact of the velocity modulations on the flame. Nevertheless, even if a ‘real’
2288
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
a
b
Fig. 7. Transfer function, gaseous (GAS) case.
frequency analysis is done, it may be reduced by a constant flame length for all the cases. In this paper, we are referring to the reduced frequency associated with the reference stationary GAS case xGAS . (2) To study the behaviour of a unique flame for several modulation frequencies, one prescribed a reduced ‘flame’ frequencies (that is, reduced with the cone heights of the corresponding stationary flame [6,9]). Table 2 offers an overall picture of the difference between reduced ‘real’ frequencies and reduced ‘flame’ frequencies used in our DNS.
Fig. 8. Transfer function, two-phase (DRPE) case.
a
5.1. Flame perturbations
b
Effects of low (xGAS 5), medium (xGAS 15) and high frequencies (xGAS 25) are shown in Figs. 4, 5 and 6, respectively. The first figure corresponds to the purely gaseous GAS case, the other two to the DRPE case. The inlet modulation generates a propagating signal along the flame front from the base up to the top of the cone. From a general point of view, the lower the frequency is, the more the flame lengthens. When frequency increases, the flame front is more and more perturbed. In some cases, it is possible to observe the formation of a pocket of fresh gases at the top of the cone and, when the flame retracts
Fig. 9. Transfer function, all DNS cases.
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
2289
itself, this pocket drifts away in the burnt gases and disappears very quickly. Similar burning pockets have been observed experimentally. Good agreements may be observed between simulations and experiments. 5.2. Transfer functions One of the aims of this study is to demonstrate the capability of DNS to characterize the interactions between a premixed spray flame and velocity modulations due to a sinusoidal acoustic wave. Encouraging comparisons have been made in the preceding section. However, a full agreement between experiments and simulations is needed. To ensure that the whole range of modulation frequencies may be captured by the DNS, transfer functions have to be determined. To begin with, we are focusing on the purely gaseous configuration, GAS. In that case, analytical models have been developed to predict the response of the flame to various low frequency modulations. Details concerning these models may be found in [9]. In Fig. 7, the analytical model and our simulations are compared with the experimental results of EM2C laboratory [1]. The amplitude of the experimental transfer function is well captured by DNS. The analytical amplitude shows a similar behaviour even if a less good agreement is observed. From the phase point of view, DNS is in very good agreement up to a reduced frequency equal to 30 with the experimental results. On the other hand, it is not the case for the analytical model that is unable to capture the phase when the reduced frequency is greater than seven. It is very interesting to note that the capability of DNS to capture both the amplitude and the phase of the transfer function is confirmed also in the two-phase EM2C case (Fig. 8).
Fig. 11. Sketch of the direct (top) and indirect (bottom) effects of acoustic waves on the flame.
These successful comparisons between DNS and experiments allowed us to validate the capability of the DNS to capture the physical phenomena embedded in the triple interactions: spray/ combustion/modulation. 5.3. DNS analysis In this section, DNS databases are used to plot the transfer functions of the four simulated configurations in Fig. 9. The gaseous analytical model prediction is plotted as well. As it was hinted by comparing some flame shapes, both GAS and DRP0 cases are giving similar results for all the studied modulation frequencies. In fact, ‘‘small acoustic Stokes number—short evaporation delay’’ droplets are not directly affected by the flow modulation and behave like a gas in a purely gaseous flow (DRP0). However, as soon as the acoustic Stokes number increases (DRPE, DRP1) the transfer function is strongly affected and the amplitude increases. It is explained by the modification of the local droplet density as it may be seen in Fig. 10, left. Consequently, the shape of the flame is affected by the resulting mixture fraction field that is not constant anymore. The flame front is more wrinkled, thus increasing the global reaction rate and the transfer function amplitude (Fig. 10, right). In conclusion, the acoustic wave affects directly the flame front (though combustion/acoustic interactions) but it has a secondary effect through the modification of the dispersed phase topology and, consequently, the vapour of fuel that will feed the flame. This double effect is summarized in Fig. 11. Note that a third effect has been neglected in this work: the direct influence of the acoustic wave on the evaporation rate of the droplets. On another hand, the phase of the transfer function is never really affected by the droplet presence (Fig. 9). This may be explained by the fact that, even if droplet density is affected by the acoustic wave, it is done following a phase similar to the one disturbing the flow velocity. 6. Conclusions
Fig. 10. DRP1 case, left: droplet density (normalized by the injected mean density), right: heat release (normalized by its stoichiometric value).
It is possible to predict correctly the transfer function of a pulsating conical flame undergoing acoustic instabilities thanks to a 2D axi-symmetric
2290
C. Pera, J. Reveillon / Proceedings of the Combustion Institute 31 (2007) 2283–2290
DNS. Both purely gaseous and spray combustion have been carried out and good agreements between experimental and numerical transfer functions have been found for the whole range of studied reduced frequency (up to 30). It may be a low-cost alternative to actual analytical models that are dedicated to gaseous combustion. Moreover, they seem to be limited to very low perturbation reduced frequency. The main perspectives of this work are to extend the validation to a higher frequency range and to other configurations like V-shape flames, for example.
Acknowledgments The authors thank deeply the experimental team of the EM2C laboratory, Ecole Centrale Paris: S. Ducruix, D. Durox and C. Pichard for their fruitful discussions and the quality of the experimental data. The authors acknowledge the support from IDRIS-CNRS (Institut de Developpement et de Ressources en Informatique Scientifique), where computations were performed. Support was also provided by the project ‘‘Modelling of UnSteady Combustion in Low-Emission Systems’’ (MUSCLES) under the EC Contract No G4RD-CT-2001-00644.
References [1] S. Ducruix, D. Durox, C. Pichard, Influence of acoustic waves on vaporization and combustion of sprays, Tech. rep., EEC, Modelling of UnSteady Combustion in Low-Emission Systems, Final report D.4.6, 2005. [2] R. Sujith, G. Waldherr, J. Jagoda, B. Zinn, J. Propul. Power 16 (1) (2000) 278–285. [3] B. Abramzon, W.A. Sirignano, Int. J. Heat Mass Transfer 32 (9) (1989) 1605–1618. [4] S. Candel, Proc. Combust. Inst. 1992, 1277– 1296. [5] T. Schuller, Me´canisme de Couplage dans les Interactions Acoustique-Combustion, Ph.D. thesis, Ecole Centrale Paris, 2003. [6] S. Ducruix, Dynamique des Interactions AcoustiqueCombustion, Ph.D. thesis, Ecole Centrale Paris, 1999. [7] M. Saito, M. Hoshikawa, M. Sato, Fuel 75 (1996) 669–674. [8] E. Haile, O. Delabroy, F. Lacas, D. Veynante, S. Candel, Proc. Combust. Inst. 26 (1996) 1663–1670. [9] S. Ducruix, D. Durox, S. Candel, Proc. Combust. Inst. 28 (2000) 768–773. [11] J. Reveillon, L. Vervisch, J. Fluid Mech. 537 (2005) 317–347. [12] T. Poinsot, S.K. Lele, J. Comput. Phys. 1 (101) (1992) 104–129.
Comments Jay Boris, U.S. Naval Research Laboratory, USA. How did you treat 3-D droplets in an axisymmetric geometry? How did you plot the particles to get a uniform density versus radius?
which the phase saturates. However, in a more recent unified model [1] we have been able to obtain an increasing phase like in your simulation and the match would then be much better.
Reply. In our axisymmetric geometry, ‘3-D volumes are considered’, i.e., we are simulating a portion (pie slice) of a cylinder. In this framework, fully spherical droplets may be considered. To deal with uniform droplet density with respect to the droplet radius, a parcel method is used (a ‘numerical’ droplet may represent various ‘physical’ droplets).
Reference
d
Sebastien Candel, EM2C, CNRS, ECP, France. In your transfer function plots for the gas phase case you are using one of our earlier models ([9] in paper) in
[1] Schuller et al., Combust Flame 134 (2003) 21–34. Reply. In this paper, we were focused on the achievement of direct numerical simulations of spray flames undergoing acoustic modulations; a configuration that no model can resolve yet. In this framework, concerning gaseous flows, we plotted the analytical model of Ducruix et al. However, it is true that the model of Schuller et al. should be used for any study considering purely gaseous flows.