- Email: [email protected]
Extinction processes during a non-premixed flame-vortex interaction
Twenty-Seventh Symposium (International) on Combustion/The Combustion Institute, 1998/pp. 719–726
EXTINCTION PROCESSES DURING A NON-PREMIXED FLAME–VORTEX INTERACTION D. THE´VENIN, P. H. RENARD, J. C. ROLON and S. CANDEL Laboratoire E.M2.C. Ecole Centrale Paris and C.N.R.S. Grande Voie des Vignes F-92295 Chaˆtenay-Malabry, France
Studies of flame–vortex interactions are quite valuable in the analysis of turbulent combustion. As turbulence may be viewed as a collection of vortices with different scales and intensities, the interaction of isolated vortical structures with flames defines the elementary process by which turbulence acts on flames. Experiments and interpretation are thus simplified because the unperturbed flame and the incoming vortex may be controlled with precision. We here investigate the influence of vortex velocity (directly related to its induced strain rate) and of global mixture ratio on the extinction limits. Three vortex types with different velocities interact with a non-premixed diluted hydrogen–air flame. The global mixture ratio of this flame has been varied between 0.5 and 1.2. Four different kinds of interaction are described, and the limits of the connected-flame regime, relevant for flamelet modeling, are identified. The growth of the flame surface during the interaction is also examined, showing very different effects depending on vortex velocity and global mixture ratio. The increase in flame surface area is maximum for slow vortices and intermediate values of the mixture ratio. The main features of the interaction and the relative importance of the increase in flame surface are then explained in the light of characteristic times and extinction strain rates obtained by asymptotic analysis. The extinction of the flame front is finally examined using direct numerical simulations of flame–vortex interactions, including complex chemistry, detailed thermodynamics, and multicomponent diffusion velocities. The relative importance of the strain rate acting on the flame front and of mixing effects is assessed, proving that unmixedness is not responsible for the extinction.
Introduction The interaction of flames with vortices is of theoretical and practical interest. Considering a field of turbulence as a collection of vortices with different sizes and strengths, it is possible to use the flame– vortex interaction as a model problem to investigate turbulent combustion. Flame–vortex interactions may also govern the combustion rate or generate combustion instabilities. Vortices are observed in free flames and in flames stabilized on bluff bodies or are used to enhance mixing in supersonic combustion applications. Many studies have concerned flame–vortex interactions. A more detailed list is proposed in Ref. [1], and we restrain ourselves here to some key publications and mainly to non-premixed flames. A first description of the flame structure and an estimation of the combustion enhancement for a flame rolled up in a vortex was proposed in Ref. [2]. Karagozian and Marble [3] showed that a two-dimensional analysis reveals the main features of more complex, three-dimensional configurations. Simplified numerical studies have also been carried out. Laverdant and Candel [4] solved the problem introduced by Marble [2] with infinitely fast
chemistry. They proved the accuracy of the theoretical predictions and illustrated some effects of the equivalence ratio. Ashurst [5] extended this work by introducing an evolution equation for the vorticity, hence proving the strong impact of the combustion on the flow field, as demonstrated also in Ref. [6]. A review paper on the flame–vortex interaction was given in Ref. [7]. All these studies concentrate mainly on flame–vortex interactions where the vortex grows in the plane of the flame surface. Numerical studies of the influence of vortices in jets for non-premixed flames with infinitely fast chemistry [8] or with detailed chemistry [9,10] may also be found. Considering a different configuration, Poinsot et al. [11] used one-step chemistry and direct numerical simulations to investigate the effect of a pair of counter-rotating vortices crossing a premixed flame. They deduced conditions for extinction and characteristics of the combustion regimes. In the same configuration, non-premixed calculations with one-step chemistry were reported in Ref. [12] to assess the validity of flamelet descriptions. The interaction of a premixed flame with a vortex has been investigated experimentally in several publications. These experiments consist of an upwardly
719
720
LAMINAR DIFFUSION FLAMES
propagating, premixed flame interacting with a vortex ring traveling downward [13–16] or of the interaction between an anchored premixed flame and a Ka´rma´n vortex street [17] or line-vortex pair [18]. Such configurations cannot be used to investigate diffusion flames. Experiments of a non-premixed type were only carried out in water using an acidbase reaction to simulate the flame [19]. Considering that experiments for diffusion flames are highly needed, a counterflow, non-premixed burner was built in our laboratory to investigate the effect of vortices in non-premixed cases [1,20]. The present paper provides detailed experimental and numerical results concerning such a non-premixed flame interacting with a vortical structure. The experimental configuration is based on a counterflow flame whereby a vortex generator is employed to inject a vortex ring into the flame front. Using planar laser-induced fluorescence (PLIF), the structure of the flame front and the evolution of the flame surface are studied. The associated numerical calculations are carried out using direct simulations with detailed chemistry and are used to examine the local extinction of the flame front.
our case, VG is a constant, but the value of sr was varied to obtain different vortex ring velocities. The vortex ring diameter is identical for all cases. Measurements are carried out using PLIF of the OH radical. All the details, as well as a discussion on the relevance of the OH radical as a flame tracer, are proposed in Ref. [23]. The fluorescence of OH radicals is induced by a frequency-doubled dye laser at 282.9 nm. Single laser-shot images of the region between both nozzles are made using an intensified charge-coupled device (CCD) camera. The fluorescence signal is collected between 305 and 320 nm [Q(0,0) and Q(1,1) vibrational bands]. The obtained spatial resolution is about 130 lm/pixel. Experiments were conducted as follows. A steady flame is first established and then at a selected instant t0, a toroidal vortex is impulsively injected by the action of the piston and rapidly accelerates toward the flame. At the same time t0, the shutter of the ICCD camera is opened. At a well-defined instant t1 $ t0, a single shot of the laser sheet is emitted. As the experiment is perfectly reproducible, we can get the whole interaction sequence by choosing different delays t1 1 t0.
Experimental Apparatus and Procedure
Experimental Results
A detailed description of the burner is given in Ref. [20], and a schematic illustration of the setup is found in Ref. [1]. Experiments are performed with hydrogen diluted with nitrogen on the upper side and air on the lower side, both at room temperature and pressure. A steady, non-premixed flame featuring a nearly one-dimensional structure in a small region around the stagnation point is initiated. This flame is subjected to a quasi-constant strain acting along its plane, which is kept constant at eb 4 20 s11, defined as the ratio between inflow velocity and burner diameter [21]. But we change the global mixture ratio fg, defined as the ratio between initial mass fraction of fuel on the fuel side Yfu,0 and initial mass fraction of oxidizer on the oxidizer side Yox,0, divided by corresponding values for stoichiometric combustion. When varying fg, the position of the flame does not change by more than 2 mm in this work, which can be neglected. Each injection nozzle has a diameter of 25 mm and is surrounded by another nozzle issuing nitrogen to isolate the reactive stream from outer perturbations. The distance between the nozzle exit planes is kept constant at 40 mm. The lower nozzle contains a cylindrical tube of internal diameter d 4 5 mm. This tube is connected to a cylindrical plenum chamber in which a piston is accelerated by an actuator. Theoretical developments [22] show that the rotational velocity of the vortex ring is proportional to V 2G /(sr d 5), where VG is the volume of fluid pushed through the tube and sr is the displacement time. In
Seven different global mixture ratios, ranging from 0.5 to 1.2, have been used. Three different types of vortex rings have been generated, employing piston rise time sr of 10, 20, or 30 ms. We will distinguish these three categories of vortices by their translational velocity, measured from the duration of their interaction with the flame. As explained in Ref. [23], it is possible to analyze the PLIF results to compute an instantaneous flame surface. The time at which the flame surface begins to vary is the starting time for the interaction. At some later time, the tip of the disturbed flame front reaches the level of the upper burner (Fig. 1). This is the final time for the useful part of the interaction, as the flame is not fully visible afterward. Resulting mean interaction times are ti 4 7.4 ms for sr 4 10 ms, ti 4 10.9 ms for sr 4 20 ms, and ti 4 15.3 ms for sr 4 30 ms. As the distance between the top of the initial flame front and the upper burner equals 2 cm, these times ti can be readily converted into a global translational vortex velocity vi that will be used to represent the relative position of the different vortex types in future diagrams. We find vi 4 2.70 m/s for sr 4 10 ms, vi 4 1.83 m/s for sr 4 20 ms, and vi 4 1.31 m/s for sr 4 30 ms. Regimes of Interaction We now describe the four different regimes observed during the flame–vortex interactions (Fig. 2). For a very low mixture ratio (fg 4 0.5) and a fast
EXTINCTION DURING FLAME / VORTEX INTERACTION
Fig. 1. Initial configuration (a) and a typical example of interaction type II (b), III (c), and IV (d) are schematically represented. For each regime, three different times during the interaction are displayed, for the beginning (solid line), the middle part (dashed line), and the end (dotted line) of the interaction.
Fig. 2. Interaction diagram for the non-premixed flame– vortex ring interaction. Four different interaction types are identified when varying the global mixture ratio fg and the vortex velocity vi. Approximate limits between these four regimes are displayed as lines. The solid line gives the limit between disrupted flames and cases without extinction.
vortex ring (vi 4 2.7 m/s), a global extinction of the flame is observed, which will be classified as type-I interaction. Increasing the global mixture ratio or using slower vortices leads to type-II interaction, where extinction is only observed locally at the place where the vortex ring impacts the flame front. For these two types of interaction, the flame front is disrupted by the vortex, and an isolated pocket of oxidizer travels afterward in the fuel stream but without any associated reaction zone. This consideration is of importance to define the range of validity of the flamelet
721
models, as such models imply that no path exists between fuel side and oxidizer side without crossing a flame zone. This is clearly not the case for the two interaction types previously described. For a global mixture ratio fg 4 0.7, and for fg 4 0.6 for the slowest vortex, a third type of interaction (type III) is found. In that case, no extinction is observed before the top of the flame reaches the level of the upper nozzle, but the flame is strongly elongated by the vortex ring. The left and right parts of the flame do not reconnect behind the vortex, but no rupture of the flame front is found. Finally, for mixture ratio fg $ 0.8, still no extinction is observed during the interaction with the vortex ring. But, for this type-IV interaction, the left and right sides of the flame reconnect very rapidly, and the final structure contains two different reaction zones: The flame rapidly goes back to its original structure, and the vortex cores are surrounded by another reacting zone. A faint reaction level is also observed in the vortex trail. For interaction types III and IV, the flame front always remains connected, meaning that the classical flamelet picture is acceptable. Interaction types II to IV are schematically pictured in Fig. 1. The physical analysis of the resulting interaction diagram may be carried out in terms of the two independent parameters (fg, vi). For low values of fg, the flame is very weak. In fact, for fg # 0.45, no stable flame can be established. On the other hand, high values of fg correspond to very strong flames, which are difficult to extinguish. Considering vi, fast vortices more often lead to a local extinction of the flame, as it has not enough time to adapt to the rapidly evolving conditions. From this first analysis, we expect a diagram where extinction prevails in the lower-right part, while no extinction and a fast return to an unperturbed flame would be found in the high fg region. This is confirmed in the experiment. Measurement of Global Stretch Rates In order to substantiate this view by a more accurate analysis, we now use results deduced from asymptotics. One may consider in this limit the evolution of the extinction strain rate for a plane counterflow flame, as given in Ref [24]. This extinction strain rate ee can be cast in the form [25] ee(fg) 4 K
1 [Te(fg)]6 d0E(fg) (1 ` fg)2
2 exp
311 `1 f 2
2
g
1
4
Ta Te(fg)
(1)
where K is a constant parameter for our investigation. In this expression, Ta is the activation temperature for the single-step reaction considered. Previous tests [26] have shown that Ta 4 8000 K leads to
722
LAMINAR DIFFUSION FLAMES
Fig. 3. Evolution of the flame surface Af in cm2 and of the global stretch rate S in s11 with time for three different operating points. All three cases correspond to the intermediate vortex type (vortex velocity vi 4 1.83 m/s), but for global mixture ratios fg 4 0.5 (2), 0.7 (`), and 1 (C).
a good agreement with experimental results. Te is the adiabatic temperature at equilibrium, which can be readily computed as [24] Te(fg) 4 Tox,0 `
Yox,0 Q fg s Cp 1 ` fg
(2)
Here, Tox,0 and Yox,0 are respectively the temperature and oxidizer mass fraction on the oxidizer side of the unperturbed counterflow flame. The parameter s is the stoichiometric factor, equal to 8 for a hydrogen–oxygen one-step reaction. The heat-release parameter Q/Cp can be estimated in our case as 7 2 104 K. To compute ee, we need to estimate K 4 8pYox,0B [sCp/(QYox,0Ta)]3. The only new parameter is the pre-exponential factor B. This factor is almost impossible to determine with sufficient validity, but it
may be adapted in order to get agreement at one experimental point. As seen in Fig. 2, for fg 4 0.7 and vi 4 2.7 m/s, one finds the limit between extinction and no-extinction regimes, as both cases can be observed when repeating the measurements. As no unsteady effect is observed for this point (see our analysis of the flame time), one may consider that, for these conditions, the exerted strain rate is at the limit of the extinction strain rate ee. Knowing the experimental value of the strain rate would therefore allow us to determine the pre-exponential factor B. It is worth remembering that, for the counterflow configuration considered here, the rate of stretch coincides with the strain rate. This will be used in what follows to estimate this last quantity. To do this, we postprocess the PLIF images to extract information concerning the flame surface, by working on a fixed central part of the experimental image around the symmetry axis. We showed in another publication [23] that the OH radical can be used as a tracer of the reaction zone in our configuration. In order to compute the evolution of the flame surface, the contour where the OH concentration is above 10% of the maximum in the unperturbed zone is extracted. It was proved that this value suitably separates reacting zones from extinguished ones. The total flame surface Af is deduced by rotation around the symmetry axis. We show in Fig. 3 the evolution of the flame surface for three different operating points, all corresponding to the intermediate vortex type, but for global mixture ratios fg 4 0.5, 0.7, and 1. The extinction for fg 4 0.5 is visible on the flame surface evolution by a drop of Af at t 4 8 ms. For fg 4 0.7 and fg 4 1, the flame surface increases significantly. It is now possible to compute the effective global stretch rate S encountered by the whole flame during the interaction, as S 4 1/Af (DAf /Dt). This quantity is also represented in Fig. 3. Important variations are observed, with peak values as high as 190 s11.
TABLE 1 Comparison of theoretical extinction strain rate with global stretch rate
fg
vi 4 2.7 m/s
0.5 0.6 0.7 0.8 0.9 1.0 1.2
75 (0) 215 (2.05) 250 (2.73) 260 (2.34) 260 (2.39) 320 (2.44) 290 (2.15)
vi 4 1.83 m/s 80 175 190 190 200 200 200
(1.13) (2.82) (2.94) (2.72) (2.33) (2.41) (2.22)
vi 4 1.31 m/s
ee (s11)
tf (s)
85 (3.42) 130 (3.00) n.a. n.a. n.a. n.a. n.a.
45.8 119.5 249.7 446.1 710.8 1038.3 1839.9
1.62 1012 3.97 1013 1.31 1013 5.30 1014 2.50 1014 1.32 1014 4.72 1015
The first three columns are experimentally measured global stretch rates S in s11 for the different global mixture ratios fg and vortex ring velocities vi. In parentheses are the mean growth factor for the flame surface gA during the interaction with the vortex rings. In the last two columns are extinction strain rates ee and characteristic flame times tf deduced from asymptotic analysis for the global mixtures ratios considered in the experiment.
EXTINCTION DURING FLAME / VORTEX INTERACTION
By repeating the experiment, we obtain mean values for S, listed in Table 1. This definition of the global stretch rate does not take into account the background counterflow, as we only wish to extract the influence of the vortex. We can then compute ee by adapting the pre-exponential factor to get ee 4 250 s11 for fg 4 0.7, in agreement with the experimental result. Corresponding values for the asymptotic extinction strain rates are also listed in Table 1. We observe a fast nonlinear increase of this extinction strain rate with fg. Going from a mixture ratio of 0.5 to fg 4 1.2 leads to a multiplication of ee by a factor of over 40. The strength of the flame is rapidly augmented when increasing fg, explaining why extinction is only observed in the lower part of the diagram. It is interesting to compare the theoretical extinction strain rate with the global stretch rate measured in the experiment. This comparison can be done readily using the values given in Table 1. The asymptotic value for the extinction strain rate is lower than the effectively measured stretch rate for all cases with fg 4 0.5 and fg 4 0.6, and for fg 4 0.7 with the fastest vortex. The agreement for this last point is clear, as it was used to adapt the preexponential factor. Nevertheless, the limit between extinction and no-extinction regimes appears to be adequately predicted by the asymptotic analysis. Moreover, the only point where the measured stretch rate falls very near to the predicted ee is for fg 4 0.6 and vi 4 1.31 m/s. Indeed, in the experiment, this point appears to lie on the boundary between extinction and no-extinction regimes. This shows again that asymptotic analysis provides suitable predictions for extinction limits. Flame Response Time The preceding analysis does not account for unsteady phenomena. The good agreement obtained tends to prove that dynamical effects are negligible for our operating points, but this must be checked. Again, asymptotic analysis [27] can be used to determine a characteristic response time for the flame: 1 fg3 CpT e2 4C 2 tf (ff) (fg ` 1) QTa
3
4
2
1 2
Ta exp 1 Te
(3)
The variables appearing in this expression have been described before. In order to determine tf, it is necessary to compute the constant C 4 16B/( q2f Wfu), where Wfu is the fuel molar mass (Wfu 4 2 g/mol for hydrogen) and qf is a typical value of the density that we choose as qf ; 1 kg/m3. Corresponding values for tf are displayed in Table 1. A fast decrease of tf occurs when increasing fg is observed. Between fg 4 0.5 and fg 4 1.2, tf is divided by a factor of 344. The variation of tf with fg is much faster than that of ee. Considering tf, one finds that for all global
723
mixture ratios higher than 0.6 and for all three types of vortices, the asymptotic flame time is much shorter than the duration of the interaction. This supports the previous analysis in terms of strain rate, where unsteady effects could not be found. Moreover, the computed value of tf should be regarded as an upper bound, considering the relatively high value used for qf. Thus, even for fg 4 0.5, dynamical effects should be regarded as unlikely, allowing us to understand why a steady-state analysis is appropriate to predict the extinction limits. Another approach can be followed to examine the time-dependent effects, based on the work presented in Ref. [28]. Using full-chemistry schemes in a numerical analysis of a counterflow configuration, a non-premixed hydrogen flame was submitted to step functions in strain rates, where the strain rate is abruptly increased from an initial, low value e0 to a high final level e1. It was found that the flame typically needs a time tr 4 1/(e1 1 e0) to respond. In our case, choosing e0 as the value associated with the background counterflow and e1 as the maximum measured stretch rate, we can easily compute a response time tr using Table 1, as 1/(S 1 20). Results are not presented here due to space limitations, but they confirm the previous analysis based on asymptotic results for tf. Again, comparing the response time tr with the interaction duration ti, only the points for fg 4 0.5 and the two fastest vortex rings could be associated with an incomplete response of the flame. Flame Surface Growth Finally, we wish to discuss the growth of the flame surface induced by the interaction with the vortex ring. Table 1 gives in parentheses the value of the flame surface Af at the end of the interaction divided by Af,0 for the initial flame (here 2.66 cm2). For flames without global extinction, the growth factor varies between 1.13 and 3.42. Considering, for example, the limits of the domain where this growth factor gA is higher than 2.6, one finds that this zone is located horizontally around a ratio fg 4 0.7 and widens when the vortex velocity is reduced. A physical explanation can be proposed, which is supported also by flame visualizations. For low values of fg, the flame is extremely sensitive to perturbations and is quickly extinguished. This extinction reduces the flame surface and limits the possibilities for flame growth during the interaction. On the other hand, for flames with high values of fg, extinction is not observed, but the characteristic time of the flame tf is very short as compared to the interaction duration. In this regime, the flame goes back quickly to its initial configuration and is less contorted by the vortex ring than for lower fg. The resistance of the flame is then too high to authorize a high level of wrinkling. We therefore expect a maximum value of
724
LAMINAR DIFFUSION FLAMES
Fig. 4. Time evolution of the degree of mixing (right side, solid line), of the strain rate exerted on the flame front (in s11, left side, dashed line), and of the maximal temperature (right side, long-dashed line) and OH mass fraction (right side, C symbols), both normalized by their starting value, along the symmetry line. Direct numerical simulation for fg 4 0.8 and vi 4 4.33 m/s.
gA for some intermediate value of fg, given as 0.7 by the experiment. Considering the velocity of the vortex ring, fast vortices are not favorable for flame roll-up and, therefore, for the growth of the flame surface, as the elongated flame remains close to the symmetry axis. Moreover, these fast vortices sometimes lead to extinction. Consequently, the domain for high values of gA should become larger when decreasing the vortex velocity, as is indeed found experimentally. Numerical Study Numerical Methods The numerical procedure was already described in Ref. [1] and is only briefly summarized here. The full, time-dependent, compressible Navier–Stokes equations are solved to investigate the flame–vortex interaction. Detailed models are implemented to compute the thermodynamic parameters, while accurate, fifth-order fits of experimental data are employed to determine the species characteristics. Standard multicomponent expressions are used to compute the diffusion velocities. Thermal diffusion is also taken into account because it is known to be important for hydrogen flames. The chemical scheme used to calculate the production terms consists of nine species and 19 reversible reactions [29]. One of the x boundaries is a line of symmetry, while the other is associated with a periodic condition to inhibit external perturbations from entering the numerical domain. Both y boundaries correspond to nonreflecting boundary conditions. The ignition of the non-premixed flame is first obtained by
depositing energy along the fuel–oxidizer interface. A vortex pair moving upward is later initialized inside the computational plane. The solution of the system is obtained by using a direct numerical simulations code [30]. The needed clock time is reduced by working on parallel supercomputers, using a domain-decomposition technique, whereby several processors cooperate simultaneously on the solution process. In the present case, we use eight processors of a Cray T3E supercomputer, yielding a total computational time of around 150 h per processor. A fixed, regular gridding is employed with a uniform grid spacing of 50 lm for a numerical domain size of 1 2 3 cm. Boundary and initial conditions for pressure, temperature, and compositions match the experimental configuration at global mixture ratio fg 4 0.8. The flame has an initial thermal thickness of 5 mm, as in the experiment. A pair of counter-rotating vortices characterized by a vortex diameter, dv, and a separation distance between vortex centers, D, both equal to 2 mm, is then initiated in the domain. The vortices propagate naturally toward the flame front with a translational velocity of 4.33 m/s. No direct comparison with the previously shown experimental results is possible, because the vortices are smaller and faster in the simulation in order to reduce computational costs, and the code is presently two dimensional. It is, however, possible to simulate the extinction process and investigate it in detail. An axisymmetric version of the code is under development. Description of the Local Extinction Process The computation directly yields the value of the strain rate acting on the flame front ef and of the degree of mixing M. The first one is computed as usual, while M is defined along the lines of Ref. [31] as M 4 YfuYox /(4Yfu,0Yox,0). A value of 1 is associated with a full mixture of the starting compositions on fuel and oxidizer sides, while M 4 0 in pure fuel or pure oxidizer. In order to determine where the flame front is located, we compute the Zeldovich variable Z 4 (fgYfu /Yfu,0 1 Yox /Yox,0 ` 1)/(1 ` fg). The flame front should theoretically lie along the stoichiometric line, corresponding to Zst 4 1/(1 ` fg). In the simulation considered here, Zst 4 0.556. Figure 4 shows the evolution with time, along the symmetry line, of the maximum degree of mixing, of the maximum temperature (normalized by its starting value Tmax,0 4 1968 K), and of the strain rate acting on the flame front, defined as the position where Z 4 Zst. The strain rate acting on the flame front increases very rapidly and goes over the value corresponding to the extinction strain rate predicted by the asymptotic analysis (446.1 s11 in this case) for t 4 2 2 1015 s. The maximum temperature along
EXTINCTION DURING FLAME / VORTEX INTERACTION
the symmetry line begins to decrease at an early time, this decrease is then rapidly accelerated after t 4 0.1 ms, when the strain rate acting on the flame front reaches high values exceeding 2000 s11. On the other hand, the degree of mixing is fairly constant, remaining at a low value for all these preliminary parts of the extinction process. This proves that extinction is a result of the intense straining and does not come from an insufficient mixing resulting from the presence of the vortex pair, which consists purely of oxidizer. During the later part of the interaction, the flame becomes extremely weak due to intense straining, and the intensity of the reaction is greatly reduced. This induces a strong increase of the degree of mixing, corresponding to this large reduction in the consumption rates of fuel and oxidizer. This better mixing does not prevent extinction, which is first seen on the OH mass fraction for t 4 0.173 ms and on the H mass fraction for t 4 0.181 ms. Conclusions Experimental and numerical results describing extinction processes for a non-premixed, hydrogen flame interacting with vortices are reported. Experiments are performed using a counterflow flame. Air and hydrogen diluted with nitrogen, both at room temperature, impinge on each other, and the global mixture ratio can be varied. Vortex rings with different velocities, exerting different strain rates, are then generated from a tube installed in the nozzle on the air side. A diagram defining the limits between four different interaction regimes is proposed. In particular, the domain where the flame always remains connected is identified, which is of particular importance for flamelet models. The influence of global mixture ratio and of vortex velocity on the global increase in flame surface is investigated. The trends observed are explained with results of asymptotic analysis. A numerical calculation of the flame structure and in particular of the extinction process is performed in a configuration similar to that used in the experiments. This simulation proves that straining effects are responsible for the extinction of the flame front and that the degree of mixing increases in fact at the end of the extinction process. A full coverage of the (mixture ratio, vortex strength) diagram is the object of our future experimental work. Numerical simulations with conditions identical to the experiment are also planned. Acknowledgments The numerical computations were carried out on a Cray T3E parallel computer of IDRIS, the CNRS computing center in Orsay. P.-H. Renard is financially supported by DRET.
725
REFERENCES 1. The´venin, D., Rolon, J. C., Renard, P. H., Kendrick, D. W., Veynante, D., and Candel, S., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1079–1086. 2. Marble, F. E., in Recent Advances in the Aerospace Sciences (C. Casci, ed.), Plenum, New York, 1985, pp. 395–413. 3. Karagozian, A. R. and Marble, F. E., Combust. Sci. Technol. 45:65–84 (1986). 4. Laverdant, A. M. and Candel, S. M., Combust. Sci. Technol. 60:79 (1988). 5. Ashurst, W. T., in Numerical Combustion (A. Dervieux and B. Larrouturou, eds.), Lecture Notes in Physics 351, Springer-Verlag, New York, 1989, p. 3. 6. Manda, B. V. S. and Karagozian, A. R., Combust. Sci. Technol. 61:101–119 (1988). 7. Sirignano, W. A., in Numerical Combustion (A. Dervieux and B. Larrouturou, eds.), Lecture Notes in Physics 351, Springer-Verlag, New York, 1989, p. 440. 8. Katta, V. R., Goss, L. P., and Roquemore, W. M., AIAA J. 32:84–94 (1994). 9. Hancock, R. D., Schauer, F. R., Lucht, R. P., Katta, V. R., and Hsu, K. Y., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1087–1093. 10. Takahashi, F. and Katta, V. R., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1151–1160. 11. Poinsot, T., Veynante, D., and Candel, S., J. Fluid Mech. 228:561–606 (1991). 12. Cue´not, B. and Poinsot, T., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, pp. 1383–1390. 13. Jarosinski, J., Lee, J. H. S., and Knystautas, R., in Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 505–514. 14. Roberts, W. L., Driscoll, J. F., Drake, M. C., and Ratcliffe, J. W., in Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 169–176. 15. Mueller, C. J., Driscoll, J. F., Sutkus, D. J., Roberts, W. L., Drake, M. C., and Smooke, M. D., Combust. Flame 100:323–331 (1995). 16. Mueller, C. J., Driscoll, J. F., Reuss, D. L., and Drake, M. C., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 347–355. 17. Nye, D. A., Lee, J. G., Lee, T. W., and Santavicca, D. A., Combust. Flame 105:167–179 (1996). 18. Nguyen, Q. V. and Paul, P. H., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 357–364. 19. Karagozian, A. R., Suganuma, Y., and Strom, B. D., Phys. Fluids 31:1862–1871 (1988). 20. Rolon, J. C., Aguerre, F., and Candel, S., Combust. Flame 100:422–429 (1995).
726
LAMINAR DIFFUSION FLAMES
21. Rolon, J. C., Veynante, D., Martin, J. P., and Durst, F., Exper. Fluids 11:313–324 (1991). 22. Roberts, W. L. and Driscoll, J. F., Combust. Flame 87:245–256 (1991). 23. Renard, P. H., Rolon, J. C., The´venin, D., and Candel, S., Combust. Flame, in press (1998). 24. Lin˜an, A., Acta Astron. 1:1007–1039 (1974). 25. The´venin, D., “Dynamique de l’Allumage de Flammes de Diffusion dans des E´coulements Cisaille´s,” Ph.D. thesis, Ecole Centrale Paris, ECP 1992-042, 1992. 26. The´venin, D. and Candel, S. M., Phys. Fluids A 7:434– 445 (1995).
27. Cue´not, B. and Poinsot, T., Combust. Flame 104:111– 137 (1996). 28. Darabiha, N., Combust. Sci. Technol. 86:163–181 (1992). 29. Maas, U. and Warnatz, J., in Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 1695–1704. 30. The´venin, D., Behrendt, F., Maas, U., Przywara, B., and Warnatz, J., Comput. Fluids 25:485–496 (1996). 31. Marble, F. E., in Chemical Reactivity in Liquids (A. Moreau and O. Turq, eds.), Plenum, New York, 1988, p. 581.
COMMENTS Alessandro Gomez, Yale University, USA. It has been established that vortex-induced extinction strain rates differ very significantly from steady-state extinction strain rates. It would appear then that using the asymptotics result from Linan’s theory (steady-state) for comparison purposes with your experimental results on vortex-induced extinction is not well justified. Comment on this point.
Author’s Reply. As detailed in our paper, the good agreement we observe between steady-state asymptotic predictions and experimental measurements is due to the fact that, in all cases presented here, the flame response time is very short. These flames can therefore adapt very rapidly to the perturbation induced by the vortex, in a similar way to what happens in a steady counterflow. But we expect that it is indeed possible to find regimes (e.g. fast vortices, lean flames) where this steady-state analysis is not sufficient to predict extinction limits.
EXTINCTION PROCESSES DURING A NON-PREMIXED FLAME–VORTEX INTERACTION D. THE´VENIN, P. H. RENARD, J. C. ROLON and S. CANDEL Laboratoire E.M2.C. Ecole Centrale Paris and C.N.R.S. Grande Voie des Vignes F-92295 Chaˆtenay-Malabry, France
Studies of flame–vortex interactions are quite valuable in the analysis of turbulent combustion. As turbulence may be viewed as a collection of vortices with different scales and intensities, the interaction of isolated vortical structures with flames defines the elementary process by which turbulence acts on flames. Experiments and interpretation are thus simplified because the unperturbed flame and the incoming vortex may be controlled with precision. We here investigate the influence of vortex velocity (directly related to its induced strain rate) and of global mixture ratio on the extinction limits. Three vortex types with different velocities interact with a non-premixed diluted hydrogen–air flame. The global mixture ratio of this flame has been varied between 0.5 and 1.2. Four different kinds of interaction are described, and the limits of the connected-flame regime, relevant for flamelet modeling, are identified. The growth of the flame surface during the interaction is also examined, showing very different effects depending on vortex velocity and global mixture ratio. The increase in flame surface area is maximum for slow vortices and intermediate values of the mixture ratio. The main features of the interaction and the relative importance of the increase in flame surface are then explained in the light of characteristic times and extinction strain rates obtained by asymptotic analysis. The extinction of the flame front is finally examined using direct numerical simulations of flame–vortex interactions, including complex chemistry, detailed thermodynamics, and multicomponent diffusion velocities. The relative importance of the strain rate acting on the flame front and of mixing effects is assessed, proving that unmixedness is not responsible for the extinction.
Introduction The interaction of flames with vortices is of theoretical and practical interest. Considering a field of turbulence as a collection of vortices with different sizes and strengths, it is possible to use the flame– vortex interaction as a model problem to investigate turbulent combustion. Flame–vortex interactions may also govern the combustion rate or generate combustion instabilities. Vortices are observed in free flames and in flames stabilized on bluff bodies or are used to enhance mixing in supersonic combustion applications. Many studies have concerned flame–vortex interactions. A more detailed list is proposed in Ref. [1], and we restrain ourselves here to some key publications and mainly to non-premixed flames. A first description of the flame structure and an estimation of the combustion enhancement for a flame rolled up in a vortex was proposed in Ref. [2]. Karagozian and Marble [3] showed that a two-dimensional analysis reveals the main features of more complex, three-dimensional configurations. Simplified numerical studies have also been carried out. Laverdant and Candel [4] solved the problem introduced by Marble [2] with infinitely fast
chemistry. They proved the accuracy of the theoretical predictions and illustrated some effects of the equivalence ratio. Ashurst [5] extended this work by introducing an evolution equation for the vorticity, hence proving the strong impact of the combustion on the flow field, as demonstrated also in Ref. [6]. A review paper on the flame–vortex interaction was given in Ref. [7]. All these studies concentrate mainly on flame–vortex interactions where the vortex grows in the plane of the flame surface. Numerical studies of the influence of vortices in jets for non-premixed flames with infinitely fast chemistry [8] or with detailed chemistry [9,10] may also be found. Considering a different configuration, Poinsot et al. [11] used one-step chemistry and direct numerical simulations to investigate the effect of a pair of counter-rotating vortices crossing a premixed flame. They deduced conditions for extinction and characteristics of the combustion regimes. In the same configuration, non-premixed calculations with one-step chemistry were reported in Ref. [12] to assess the validity of flamelet descriptions. The interaction of a premixed flame with a vortex has been investigated experimentally in several publications. These experiments consist of an upwardly
719
720
LAMINAR DIFFUSION FLAMES
propagating, premixed flame interacting with a vortex ring traveling downward [13–16] or of the interaction between an anchored premixed flame and a Ka´rma´n vortex street [17] or line-vortex pair [18]. Such configurations cannot be used to investigate diffusion flames. Experiments of a non-premixed type were only carried out in water using an acidbase reaction to simulate the flame [19]. Considering that experiments for diffusion flames are highly needed, a counterflow, non-premixed burner was built in our laboratory to investigate the effect of vortices in non-premixed cases [1,20]. The present paper provides detailed experimental and numerical results concerning such a non-premixed flame interacting with a vortical structure. The experimental configuration is based on a counterflow flame whereby a vortex generator is employed to inject a vortex ring into the flame front. Using planar laser-induced fluorescence (PLIF), the structure of the flame front and the evolution of the flame surface are studied. The associated numerical calculations are carried out using direct simulations with detailed chemistry and are used to examine the local extinction of the flame front.
our case, VG is a constant, but the value of sr was varied to obtain different vortex ring velocities. The vortex ring diameter is identical for all cases. Measurements are carried out using PLIF of the OH radical. All the details, as well as a discussion on the relevance of the OH radical as a flame tracer, are proposed in Ref. [23]. The fluorescence of OH radicals is induced by a frequency-doubled dye laser at 282.9 nm. Single laser-shot images of the region between both nozzles are made using an intensified charge-coupled device (CCD) camera. The fluorescence signal is collected between 305 and 320 nm [Q(0,0) and Q(1,1) vibrational bands]. The obtained spatial resolution is about 130 lm/pixel. Experiments were conducted as follows. A steady flame is first established and then at a selected instant t0, a toroidal vortex is impulsively injected by the action of the piston and rapidly accelerates toward the flame. At the same time t0, the shutter of the ICCD camera is opened. At a well-defined instant t1 $ t0, a single shot of the laser sheet is emitted. As the experiment is perfectly reproducible, we can get the whole interaction sequence by choosing different delays t1 1 t0.
Experimental Apparatus and Procedure
Experimental Results
A detailed description of the burner is given in Ref. [20], and a schematic illustration of the setup is found in Ref. [1]. Experiments are performed with hydrogen diluted with nitrogen on the upper side and air on the lower side, both at room temperature and pressure. A steady, non-premixed flame featuring a nearly one-dimensional structure in a small region around the stagnation point is initiated. This flame is subjected to a quasi-constant strain acting along its plane, which is kept constant at eb 4 20 s11, defined as the ratio between inflow velocity and burner diameter [21]. But we change the global mixture ratio fg, defined as the ratio between initial mass fraction of fuel on the fuel side Yfu,0 and initial mass fraction of oxidizer on the oxidizer side Yox,0, divided by corresponding values for stoichiometric combustion. When varying fg, the position of the flame does not change by more than 2 mm in this work, which can be neglected. Each injection nozzle has a diameter of 25 mm and is surrounded by another nozzle issuing nitrogen to isolate the reactive stream from outer perturbations. The distance between the nozzle exit planes is kept constant at 40 mm. The lower nozzle contains a cylindrical tube of internal diameter d 4 5 mm. This tube is connected to a cylindrical plenum chamber in which a piston is accelerated by an actuator. Theoretical developments [22] show that the rotational velocity of the vortex ring is proportional to V 2G /(sr d 5), where VG is the volume of fluid pushed through the tube and sr is the displacement time. In
Seven different global mixture ratios, ranging from 0.5 to 1.2, have been used. Three different types of vortex rings have been generated, employing piston rise time sr of 10, 20, or 30 ms. We will distinguish these three categories of vortices by their translational velocity, measured from the duration of their interaction with the flame. As explained in Ref. [23], it is possible to analyze the PLIF results to compute an instantaneous flame surface. The time at which the flame surface begins to vary is the starting time for the interaction. At some later time, the tip of the disturbed flame front reaches the level of the upper burner (Fig. 1). This is the final time for the useful part of the interaction, as the flame is not fully visible afterward. Resulting mean interaction times are ti 4 7.4 ms for sr 4 10 ms, ti 4 10.9 ms for sr 4 20 ms, and ti 4 15.3 ms for sr 4 30 ms. As the distance between the top of the initial flame front and the upper burner equals 2 cm, these times ti can be readily converted into a global translational vortex velocity vi that will be used to represent the relative position of the different vortex types in future diagrams. We find vi 4 2.70 m/s for sr 4 10 ms, vi 4 1.83 m/s for sr 4 20 ms, and vi 4 1.31 m/s for sr 4 30 ms. Regimes of Interaction We now describe the four different regimes observed during the flame–vortex interactions (Fig. 2). For a very low mixture ratio (fg 4 0.5) and a fast
EXTINCTION DURING FLAME / VORTEX INTERACTION
Fig. 1. Initial configuration (a) and a typical example of interaction type II (b), III (c), and IV (d) are schematically represented. For each regime, three different times during the interaction are displayed, for the beginning (solid line), the middle part (dashed line), and the end (dotted line) of the interaction.
Fig. 2. Interaction diagram for the non-premixed flame– vortex ring interaction. Four different interaction types are identified when varying the global mixture ratio fg and the vortex velocity vi. Approximate limits between these four regimes are displayed as lines. The solid line gives the limit between disrupted flames and cases without extinction.
vortex ring (vi 4 2.7 m/s), a global extinction of the flame is observed, which will be classified as type-I interaction. Increasing the global mixture ratio or using slower vortices leads to type-II interaction, where extinction is only observed locally at the place where the vortex ring impacts the flame front. For these two types of interaction, the flame front is disrupted by the vortex, and an isolated pocket of oxidizer travels afterward in the fuel stream but without any associated reaction zone. This consideration is of importance to define the range of validity of the flamelet
721
models, as such models imply that no path exists between fuel side and oxidizer side without crossing a flame zone. This is clearly not the case for the two interaction types previously described. For a global mixture ratio fg 4 0.7, and for fg 4 0.6 for the slowest vortex, a third type of interaction (type III) is found. In that case, no extinction is observed before the top of the flame reaches the level of the upper nozzle, but the flame is strongly elongated by the vortex ring. The left and right parts of the flame do not reconnect behind the vortex, but no rupture of the flame front is found. Finally, for mixture ratio fg $ 0.8, still no extinction is observed during the interaction with the vortex ring. But, for this type-IV interaction, the left and right sides of the flame reconnect very rapidly, and the final structure contains two different reaction zones: The flame rapidly goes back to its original structure, and the vortex cores are surrounded by another reacting zone. A faint reaction level is also observed in the vortex trail. For interaction types III and IV, the flame front always remains connected, meaning that the classical flamelet picture is acceptable. Interaction types II to IV are schematically pictured in Fig. 1. The physical analysis of the resulting interaction diagram may be carried out in terms of the two independent parameters (fg, vi). For low values of fg, the flame is very weak. In fact, for fg # 0.45, no stable flame can be established. On the other hand, high values of fg correspond to very strong flames, which are difficult to extinguish. Considering vi, fast vortices more often lead to a local extinction of the flame, as it has not enough time to adapt to the rapidly evolving conditions. From this first analysis, we expect a diagram where extinction prevails in the lower-right part, while no extinction and a fast return to an unperturbed flame would be found in the high fg region. This is confirmed in the experiment. Measurement of Global Stretch Rates In order to substantiate this view by a more accurate analysis, we now use results deduced from asymptotics. One may consider in this limit the evolution of the extinction strain rate for a plane counterflow flame, as given in Ref [24]. This extinction strain rate ee can be cast in the form [25] ee(fg) 4 K
1 [Te(fg)]6 d0E(fg) (1 ` fg)2
2 exp
311 `1 f 2
2
g
1
4
Ta Te(fg)
(1)
where K is a constant parameter for our investigation. In this expression, Ta is the activation temperature for the single-step reaction considered. Previous tests [26] have shown that Ta 4 8000 K leads to
722
LAMINAR DIFFUSION FLAMES
Fig. 3. Evolution of the flame surface Af in cm2 and of the global stretch rate S in s11 with time for three different operating points. All three cases correspond to the intermediate vortex type (vortex velocity vi 4 1.83 m/s), but for global mixture ratios fg 4 0.5 (2), 0.7 (`), and 1 (C).
a good agreement with experimental results. Te is the adiabatic temperature at equilibrium, which can be readily computed as [24] Te(fg) 4 Tox,0 `
Yox,0 Q fg s Cp 1 ` fg
(2)
Here, Tox,0 and Yox,0 are respectively the temperature and oxidizer mass fraction on the oxidizer side of the unperturbed counterflow flame. The parameter s is the stoichiometric factor, equal to 8 for a hydrogen–oxygen one-step reaction. The heat-release parameter Q/Cp can be estimated in our case as 7 2 104 K. To compute ee, we need to estimate K 4 8pYox,0B [sCp/(QYox,0Ta)]3. The only new parameter is the pre-exponential factor B. This factor is almost impossible to determine with sufficient validity, but it
may be adapted in order to get agreement at one experimental point. As seen in Fig. 2, for fg 4 0.7 and vi 4 2.7 m/s, one finds the limit between extinction and no-extinction regimes, as both cases can be observed when repeating the measurements. As no unsteady effect is observed for this point (see our analysis of the flame time), one may consider that, for these conditions, the exerted strain rate is at the limit of the extinction strain rate ee. Knowing the experimental value of the strain rate would therefore allow us to determine the pre-exponential factor B. It is worth remembering that, for the counterflow configuration considered here, the rate of stretch coincides with the strain rate. This will be used in what follows to estimate this last quantity. To do this, we postprocess the PLIF images to extract information concerning the flame surface, by working on a fixed central part of the experimental image around the symmetry axis. We showed in another publication [23] that the OH radical can be used as a tracer of the reaction zone in our configuration. In order to compute the evolution of the flame surface, the contour where the OH concentration is above 10% of the maximum in the unperturbed zone is extracted. It was proved that this value suitably separates reacting zones from extinguished ones. The total flame surface Af is deduced by rotation around the symmetry axis. We show in Fig. 3 the evolution of the flame surface for three different operating points, all corresponding to the intermediate vortex type, but for global mixture ratios fg 4 0.5, 0.7, and 1. The extinction for fg 4 0.5 is visible on the flame surface evolution by a drop of Af at t 4 8 ms. For fg 4 0.7 and fg 4 1, the flame surface increases significantly. It is now possible to compute the effective global stretch rate S encountered by the whole flame during the interaction, as S 4 1/Af (DAf /Dt). This quantity is also represented in Fig. 3. Important variations are observed, with peak values as high as 190 s11.
TABLE 1 Comparison of theoretical extinction strain rate with global stretch rate
fg
vi 4 2.7 m/s
0.5 0.6 0.7 0.8 0.9 1.0 1.2
75 (0) 215 (2.05) 250 (2.73) 260 (2.34) 260 (2.39) 320 (2.44) 290 (2.15)
vi 4 1.83 m/s 80 175 190 190 200 200 200
(1.13) (2.82) (2.94) (2.72) (2.33) (2.41) (2.22)
vi 4 1.31 m/s
ee (s11)
tf (s)
85 (3.42) 130 (3.00) n.a. n.a. n.a. n.a. n.a.
45.8 119.5 249.7 446.1 710.8 1038.3 1839.9
1.62 1012 3.97 1013 1.31 1013 5.30 1014 2.50 1014 1.32 1014 4.72 1015
The first three columns are experimentally measured global stretch rates S in s11 for the different global mixture ratios fg and vortex ring velocities vi. In parentheses are the mean growth factor for the flame surface gA during the interaction with the vortex rings. In the last two columns are extinction strain rates ee and characteristic flame times tf deduced from asymptotic analysis for the global mixtures ratios considered in the experiment.
EXTINCTION DURING FLAME / VORTEX INTERACTION
By repeating the experiment, we obtain mean values for S, listed in Table 1. This definition of the global stretch rate does not take into account the background counterflow, as we only wish to extract the influence of the vortex. We can then compute ee by adapting the pre-exponential factor to get ee 4 250 s11 for fg 4 0.7, in agreement with the experimental result. Corresponding values for the asymptotic extinction strain rates are also listed in Table 1. We observe a fast nonlinear increase of this extinction strain rate with fg. Going from a mixture ratio of 0.5 to fg 4 1.2 leads to a multiplication of ee by a factor of over 40. The strength of the flame is rapidly augmented when increasing fg, explaining why extinction is only observed in the lower part of the diagram. It is interesting to compare the theoretical extinction strain rate with the global stretch rate measured in the experiment. This comparison can be done readily using the values given in Table 1. The asymptotic value for the extinction strain rate is lower than the effectively measured stretch rate for all cases with fg 4 0.5 and fg 4 0.6, and for fg 4 0.7 with the fastest vortex. The agreement for this last point is clear, as it was used to adapt the preexponential factor. Nevertheless, the limit between extinction and no-extinction regimes appears to be adequately predicted by the asymptotic analysis. Moreover, the only point where the measured stretch rate falls very near to the predicted ee is for fg 4 0.6 and vi 4 1.31 m/s. Indeed, in the experiment, this point appears to lie on the boundary between extinction and no-extinction regimes. This shows again that asymptotic analysis provides suitable predictions for extinction limits. Flame Response Time The preceding analysis does not account for unsteady phenomena. The good agreement obtained tends to prove that dynamical effects are negligible for our operating points, but this must be checked. Again, asymptotic analysis [27] can be used to determine a characteristic response time for the flame: 1 fg3 CpT e2 4C 2 tf (ff) (fg ` 1) QTa
3
4
2
1 2
Ta exp 1 Te
(3)
The variables appearing in this expression have been described before. In order to determine tf, it is necessary to compute the constant C 4 16B/( q2f Wfu), where Wfu is the fuel molar mass (Wfu 4 2 g/mol for hydrogen) and qf is a typical value of the density that we choose as qf ; 1 kg/m3. Corresponding values for tf are displayed in Table 1. A fast decrease of tf occurs when increasing fg is observed. Between fg 4 0.5 and fg 4 1.2, tf is divided by a factor of 344. The variation of tf with fg is much faster than that of ee. Considering tf, one finds that for all global
723
mixture ratios higher than 0.6 and for all three types of vortices, the asymptotic flame time is much shorter than the duration of the interaction. This supports the previous analysis in terms of strain rate, where unsteady effects could not be found. Moreover, the computed value of tf should be regarded as an upper bound, considering the relatively high value used for qf. Thus, even for fg 4 0.5, dynamical effects should be regarded as unlikely, allowing us to understand why a steady-state analysis is appropriate to predict the extinction limits. Another approach can be followed to examine the time-dependent effects, based on the work presented in Ref. [28]. Using full-chemistry schemes in a numerical analysis of a counterflow configuration, a non-premixed hydrogen flame was submitted to step functions in strain rates, where the strain rate is abruptly increased from an initial, low value e0 to a high final level e1. It was found that the flame typically needs a time tr 4 1/(e1 1 e0) to respond. In our case, choosing e0 as the value associated with the background counterflow and e1 as the maximum measured stretch rate, we can easily compute a response time tr using Table 1, as 1/(S 1 20). Results are not presented here due to space limitations, but they confirm the previous analysis based on asymptotic results for tf. Again, comparing the response time tr with the interaction duration ti, only the points for fg 4 0.5 and the two fastest vortex rings could be associated with an incomplete response of the flame. Flame Surface Growth Finally, we wish to discuss the growth of the flame surface induced by the interaction with the vortex ring. Table 1 gives in parentheses the value of the flame surface Af at the end of the interaction divided by Af,0 for the initial flame (here 2.66 cm2). For flames without global extinction, the growth factor varies between 1.13 and 3.42. Considering, for example, the limits of the domain where this growth factor gA is higher than 2.6, one finds that this zone is located horizontally around a ratio fg 4 0.7 and widens when the vortex velocity is reduced. A physical explanation can be proposed, which is supported also by flame visualizations. For low values of fg, the flame is extremely sensitive to perturbations and is quickly extinguished. This extinction reduces the flame surface and limits the possibilities for flame growth during the interaction. On the other hand, for flames with high values of fg, extinction is not observed, but the characteristic time of the flame tf is very short as compared to the interaction duration. In this regime, the flame goes back quickly to its initial configuration and is less contorted by the vortex ring than for lower fg. The resistance of the flame is then too high to authorize a high level of wrinkling. We therefore expect a maximum value of
724
LAMINAR DIFFUSION FLAMES
Fig. 4. Time evolution of the degree of mixing (right side, solid line), of the strain rate exerted on the flame front (in s11, left side, dashed line), and of the maximal temperature (right side, long-dashed line) and OH mass fraction (right side, C symbols), both normalized by their starting value, along the symmetry line. Direct numerical simulation for fg 4 0.8 and vi 4 4.33 m/s.
gA for some intermediate value of fg, given as 0.7 by the experiment. Considering the velocity of the vortex ring, fast vortices are not favorable for flame roll-up and, therefore, for the growth of the flame surface, as the elongated flame remains close to the symmetry axis. Moreover, these fast vortices sometimes lead to extinction. Consequently, the domain for high values of gA should become larger when decreasing the vortex velocity, as is indeed found experimentally. Numerical Study Numerical Methods The numerical procedure was already described in Ref. [1] and is only briefly summarized here. The full, time-dependent, compressible Navier–Stokes equations are solved to investigate the flame–vortex interaction. Detailed models are implemented to compute the thermodynamic parameters, while accurate, fifth-order fits of experimental data are employed to determine the species characteristics. Standard multicomponent expressions are used to compute the diffusion velocities. Thermal diffusion is also taken into account because it is known to be important for hydrogen flames. The chemical scheme used to calculate the production terms consists of nine species and 19 reversible reactions [29]. One of the x boundaries is a line of symmetry, while the other is associated with a periodic condition to inhibit external perturbations from entering the numerical domain. Both y boundaries correspond to nonreflecting boundary conditions. The ignition of the non-premixed flame is first obtained by
depositing energy along the fuel–oxidizer interface. A vortex pair moving upward is later initialized inside the computational plane. The solution of the system is obtained by using a direct numerical simulations code [30]. The needed clock time is reduced by working on parallel supercomputers, using a domain-decomposition technique, whereby several processors cooperate simultaneously on the solution process. In the present case, we use eight processors of a Cray T3E supercomputer, yielding a total computational time of around 150 h per processor. A fixed, regular gridding is employed with a uniform grid spacing of 50 lm for a numerical domain size of 1 2 3 cm. Boundary and initial conditions for pressure, temperature, and compositions match the experimental configuration at global mixture ratio fg 4 0.8. The flame has an initial thermal thickness of 5 mm, as in the experiment. A pair of counter-rotating vortices characterized by a vortex diameter, dv, and a separation distance between vortex centers, D, both equal to 2 mm, is then initiated in the domain. The vortices propagate naturally toward the flame front with a translational velocity of 4.33 m/s. No direct comparison with the previously shown experimental results is possible, because the vortices are smaller and faster in the simulation in order to reduce computational costs, and the code is presently two dimensional. It is, however, possible to simulate the extinction process and investigate it in detail. An axisymmetric version of the code is under development. Description of the Local Extinction Process The computation directly yields the value of the strain rate acting on the flame front ef and of the degree of mixing M. The first one is computed as usual, while M is defined along the lines of Ref. [31] as M 4 YfuYox /(4Yfu,0Yox,0). A value of 1 is associated with a full mixture of the starting compositions on fuel and oxidizer sides, while M 4 0 in pure fuel or pure oxidizer. In order to determine where the flame front is located, we compute the Zeldovich variable Z 4 (fgYfu /Yfu,0 1 Yox /Yox,0 ` 1)/(1 ` fg). The flame front should theoretically lie along the stoichiometric line, corresponding to Zst 4 1/(1 ` fg). In the simulation considered here, Zst 4 0.556. Figure 4 shows the evolution with time, along the symmetry line, of the maximum degree of mixing, of the maximum temperature (normalized by its starting value Tmax,0 4 1968 K), and of the strain rate acting on the flame front, defined as the position where Z 4 Zst. The strain rate acting on the flame front increases very rapidly and goes over the value corresponding to the extinction strain rate predicted by the asymptotic analysis (446.1 s11 in this case) for t 4 2 2 1015 s. The maximum temperature along
EXTINCTION DURING FLAME / VORTEX INTERACTION
the symmetry line begins to decrease at an early time, this decrease is then rapidly accelerated after t 4 0.1 ms, when the strain rate acting on the flame front reaches high values exceeding 2000 s11. On the other hand, the degree of mixing is fairly constant, remaining at a low value for all these preliminary parts of the extinction process. This proves that extinction is a result of the intense straining and does not come from an insufficient mixing resulting from the presence of the vortex pair, which consists purely of oxidizer. During the later part of the interaction, the flame becomes extremely weak due to intense straining, and the intensity of the reaction is greatly reduced. This induces a strong increase of the degree of mixing, corresponding to this large reduction in the consumption rates of fuel and oxidizer. This better mixing does not prevent extinction, which is first seen on the OH mass fraction for t 4 0.173 ms and on the H mass fraction for t 4 0.181 ms. Conclusions Experimental and numerical results describing extinction processes for a non-premixed, hydrogen flame interacting with vortices are reported. Experiments are performed using a counterflow flame. Air and hydrogen diluted with nitrogen, both at room temperature, impinge on each other, and the global mixture ratio can be varied. Vortex rings with different velocities, exerting different strain rates, are then generated from a tube installed in the nozzle on the air side. A diagram defining the limits between four different interaction regimes is proposed. In particular, the domain where the flame always remains connected is identified, which is of particular importance for flamelet models. The influence of global mixture ratio and of vortex velocity on the global increase in flame surface is investigated. The trends observed are explained with results of asymptotic analysis. A numerical calculation of the flame structure and in particular of the extinction process is performed in a configuration similar to that used in the experiments. This simulation proves that straining effects are responsible for the extinction of the flame front and that the degree of mixing increases in fact at the end of the extinction process. A full coverage of the (mixture ratio, vortex strength) diagram is the object of our future experimental work. Numerical simulations with conditions identical to the experiment are also planned. Acknowledgments The numerical computations were carried out on a Cray T3E parallel computer of IDRIS, the CNRS computing center in Orsay. P.-H. Renard is financially supported by DRET.
725
REFERENCES 1. The´venin, D., Rolon, J. C., Renard, P. H., Kendrick, D. W., Veynante, D., and Candel, S., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1079–1086. 2. Marble, F. E., in Recent Advances in the Aerospace Sciences (C. Casci, ed.), Plenum, New York, 1985, pp. 395–413. 3. Karagozian, A. R. and Marble, F. E., Combust. Sci. Technol. 45:65–84 (1986). 4. Laverdant, A. M. and Candel, S. M., Combust. Sci. Technol. 60:79 (1988). 5. Ashurst, W. T., in Numerical Combustion (A. Dervieux and B. Larrouturou, eds.), Lecture Notes in Physics 351, Springer-Verlag, New York, 1989, p. 3. 6. Manda, B. V. S. and Karagozian, A. R., Combust. Sci. Technol. 61:101–119 (1988). 7. Sirignano, W. A., in Numerical Combustion (A. Dervieux and B. Larrouturou, eds.), Lecture Notes in Physics 351, Springer-Verlag, New York, 1989, p. 440. 8. Katta, V. R., Goss, L. P., and Roquemore, W. M., AIAA J. 32:84–94 (1994). 9. Hancock, R. D., Schauer, F. R., Lucht, R. P., Katta, V. R., and Hsu, K. Y., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1087–1093. 10. Takahashi, F. and Katta, V. R., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1151–1160. 11. Poinsot, T., Veynante, D., and Candel, S., J. Fluid Mech. 228:561–606 (1991). 12. Cue´not, B. and Poinsot, T., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, pp. 1383–1390. 13. Jarosinski, J., Lee, J. H. S., and Knystautas, R., in Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 505–514. 14. Roberts, W. L., Driscoll, J. F., Drake, M. C., and Ratcliffe, J. W., in Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 169–176. 15. Mueller, C. J., Driscoll, J. F., Sutkus, D. J., Roberts, W. L., Drake, M. C., and Smooke, M. D., Combust. Flame 100:323–331 (1995). 16. Mueller, C. J., Driscoll, J. F., Reuss, D. L., and Drake, M. C., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 347–355. 17. Nye, D. A., Lee, J. G., Lee, T. W., and Santavicca, D. A., Combust. Flame 105:167–179 (1996). 18. Nguyen, Q. V. and Paul, P. H., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 357–364. 19. Karagozian, A. R., Suganuma, Y., and Strom, B. D., Phys. Fluids 31:1862–1871 (1988). 20. Rolon, J. C., Aguerre, F., and Candel, S., Combust. Flame 100:422–429 (1995).
726
LAMINAR DIFFUSION FLAMES
21. Rolon, J. C., Veynante, D., Martin, J. P., and Durst, F., Exper. Fluids 11:313–324 (1991). 22. Roberts, W. L. and Driscoll, J. F., Combust. Flame 87:245–256 (1991). 23. Renard, P. H., Rolon, J. C., The´venin, D., and Candel, S., Combust. Flame, in press (1998). 24. Lin˜an, A., Acta Astron. 1:1007–1039 (1974). 25. The´venin, D., “Dynamique de l’Allumage de Flammes de Diffusion dans des E´coulements Cisaille´s,” Ph.D. thesis, Ecole Centrale Paris, ECP 1992-042, 1992. 26. The´venin, D. and Candel, S. M., Phys. Fluids A 7:434– 445 (1995).
27. Cue´not, B. and Poinsot, T., Combust. Flame 104:111– 137 (1996). 28. Darabiha, N., Combust. Sci. Technol. 86:163–181 (1992). 29. Maas, U. and Warnatz, J., in Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 1695–1704. 30. The´venin, D., Behrendt, F., Maas, U., Przywara, B., and Warnatz, J., Comput. Fluids 25:485–496 (1996). 31. Marble, F. E., in Chemical Reactivity in Liquids (A. Moreau and O. Turq, eds.), Plenum, New York, 1988, p. 581.
COMMENTS Alessandro Gomez, Yale University, USA. It has been established that vortex-induced extinction strain rates differ very significantly from steady-state extinction strain rates. It would appear then that using the asymptotics result from Linan’s theory (steady-state) for comparison purposes with your experimental results on vortex-induced extinction is not well justified. Comment on this point.
Author’s Reply. As detailed in our paper, the good agreement we observe between steady-state asymptotic predictions and experimental measurements is due to the fact that, in all cases presented here, the flame response time is very short. These flames can therefore adapt very rapidly to the perturbation induced by the vortex, in a similar way to what happens in a steady counterflow. But we expect that it is indeed possible to find regimes (e.g. fast vortices, lean flames) where this steady-state analysis is not sufficient to predict extinction limits.